Note: Descriptions are shown in the official language in which they were submitted.
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METHOD AND DATA SYSTEM FOR DETERMINING FINANCIAL INSTRU-
MENTS FOR USE IN FUNDING OF A LOAN.
INTRODUCTION
This invention relates to a method and a data processing
l0 system for calculating the type, the number, and the volume
of financial instruments for funding a loan with equivalent
proceeds to a debtor, the loan being designed to be at least
partially refinanced during the remaining term to maturity of
the loan. By the method according to the invention, the
15 remaining term to maturity of the loan is also determined at
the beginning of each period such that the debtor's payments
on the loan during the entire term to maturity of the loan
are within a band defined by a set of maximum and minimum
limits which may be determined for each period, and such that
20 the remaining term of the loan is within a band defined by a
maximum limit and a minimum limit. Furthermore, a special
financial instrument is determined which is designed to
ensure that given maximum limits for payments on the loan and
term to maturity are observed. The results of the method
25 according to the invention may be used by a lender, e.g. a
financing institution such as a mortgage credit institution,
to ensure that such a loan is~~funded such that both interest
rate risk and imbalances in the payment flows are prevented
or minimized. By applying the results of the method according
30 to the invention, the lender may thus create a hedge between
the lending and the funding.
When refinancing a loan, other financial instruments than the
' instruments forming the basis of the volume of the original
loan may be used, for which reason, in connection with the
35 refinancing, an adjustment of the interest rate on the loan
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may be necessary in relation to the interest level applicable
at the time of the refinancing. Loans which are fully or -
partially refinanced during the term to maturity of the loan
are thus termed Loan with Adjustable Interest Rates (LAIR).
One example of the financial instruments is non-callable
bullet bonds. In the following, the financial instruments are
also called funding instruments, just as funding volume is
also used as a term for the financial instruments consti-
tuting the volume.
BACKGROUND OF THE INVENTION AND INTRODUCTION TO THE
INVENTION.
In the Danish mortgage credit market callable loans have
historically been far the predominant type of loans and,
therefore, callable bonds in a pure "pass through" have been
equally predominant on the bond side. For a number of years,
up to the withdrawal in 1985 by the Danish Ministry of
Housing of the permission to grant cash loans, mortgage
credit institutions also offered the so-called loans with
adjustable interest rates. The previous loans with adjustable
interest rates were characterized by:
1) a long-term credit commitment.
2) funding every fifth year by the issue of bonds with a
term to maturity of 1 to 5 years.
3) the interest rate being fixed in successive periods of 5
years.
4) the underlying bonds with a term to maturity of 1 to 5
years being non-callable. The debtor is in a position to
terminate the loan at par prior to the next interest rate
adjustment.
The Danish loans with adjustable interest rates did not prove
very successful in that only per milles of the total lending
by mortgage credit institutions was granted as loans with
adjustable interest rates. The reasons were, presumably, that
the call premium was insignificant in those years due to a
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very large difference between the market rate and the coupon
rate and in addition, the investors were not as aware of the -
problem as they are today. Therefore, the difference in
interest rates between callable and non-callable bonds was
not sufficiently large in itself to make loans with adjust-
able interest rates attractive. Furthermore, the product was
not transparent seen from the point of view of the borrower.
An aspect which might have had some influence at times was
that a continued rise in the Danish interest level was
l0 expected so that the borrowers did not expect a loan with
adjustable interest rates to be advantageous in the long run.
Finally, the previous structure of loans with adjustable
interest rates involved an arbitrary and unpredictable
interest rate risk every fifth year. Most likely, these
conditions explain the poor supply in those years.
In June 1993 certain Danish tax laws were changed such that
the mortgage credit institutions were, in actual fact, once
again given the opportunity of offering loans with adjustable
interest rates.
This offers the prospect of changing the long-term mortgage
market such that in future the funding products will also be
attractive to foreign investors. One prerequisite is, in all
probability, that in future bonds are offered in conformity
with international practice, e.g. as non-callable bullet
bonds. It has therefore been of interest to examine whether
variants of loans with adjustable interest rates can be made
attractive to the borrowers.
The traditional loans with adjustable interest rates are
connected with a risk of, in principle, unlimited, intermit-
tent jumps in the interest rate on the loan. To many bor-
rowers, especially in the segment of private customers, this
risk must be assumed to be unacceptable, particularly in view
of the consequences as to the borrower's liquidity of a rise
in the interest rate on the loan to a very high level. It is
therefore of interest to consider whether the structure of
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loans with adjustable interest rates can be combined with an
adjustable term to maturity in which the interest rate on the
loan rising or falling, respectively; affects the debtor
payments only to a predetermined extent defined by a set of
maximum and minimum limits, whereas the remaining term to
maturity of the loan is varied in accordance with the
interest rate on the loan.
Typically, there is, however, both a maximum limit and a
minimum limit for the range of the remaining term to matur-
ity, which may be determined by both the borrower, the
lender, and by public authorities or legislation.
A characteristic feature of traditional loans with adjustable
interest rates was a match between the term to maturity of
the last maturing funding instrument and the period of time
between interest rate adjustments, viz. 5 years. If this
precondition is abolished, the Way is paved for an, in prin-
ciple, far wider range of opportunities as to funding and
interest rate adjustment.
Thus, the opportunity arises of securing a gradual adjustment
of the borrowing costs to the market rate with an adjustment
time depending on the maximum term to maturity of the under-
lying interest rate adjustment bond and on the weight with
which the individual interest rate adjustment bond is
included. This principle will, just as the above-mentioned
possibility of an adjustable term to maturity, reduce the
risk of large intermittent changes in the interest rate on
the Loan characterizing the traditional loans with adjustable
interest rates.
If the short-term interest rate is systematically minimum
than the long-term interest rate, it will be possible to
reduce the long-term borrowing costs for the borrowers.
Furthermore, the borrowing costs may, as mentioned above, be
reduced relative to callable bonds due to the absence of a
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call right and via increased liquidity and internationa-
lization of sales.
Whether it is possible to counter an interest rate adjustment
by adjusting the remaining term to maturity of the loan
5 depends on the determined maximum and minimum limits for
payments on the loan and term to maturity, as well as on the
extent to which the remaining debt of the loan is adjusted to
the market rate at the time of the adjustment of the interest
rate. The traditional loans with adjustable interest rates
were characterized by the remaining debt of the loan being
100 per cent adjusted to the market rate every fifth year.
Partly by allowing other frequencies with which the interest
rate adjustment is performed, and partly by allowing only a
partial adjustment of the interest rate of the remaining debt
of the loan, larger changes in the interest rate on the loan
than in the original structure will be compatible with the
maximum and minimum limits for the payments on the loan.
Therefore, it should be possible to combine a partial adjust-
ment of the interest rate of the remaining debt of the loan,
as well as other interest rate adjustment frequencies, with
an adjustable term to maturity. ,
In connection with loans with adjustable interest rates,
relations to the balance principle must be mentioned. It is a
leading principle in the legal regulation of the activities
of Danish mortgage credit institutions that the institutions
must not expose themselves to interest rate and funding
risks. On the face of it, the structure of loans with adjust-
able interest rates is contrary to these basic principles,
the funding side having a substantially shorter term to
maturity than has the lending side. Traditional loans with
adjustable interest rates are nevertheless regarded as lying
within the balance principle seen in the perspective that
borrowers accept to pay any interest rate that may occur in
connection with a future refinancing. In principle, there-
fore, this is a "pass through" which does not inflict any
risk upon the lender.
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In the funding of a new type of loans with adjustable inter-
est rates by a range of e.g. non-callable bullet bonds, four
conditions must be fulfilled according to present Danish
practice and legislation:
1. The volume of the individual volumes of each of the
financial instruments on the creditor side of the loan
must be determined such that the market price of the
financial instruments equals the volume of the loan on
the debtor side.
2. The debtor's interest rate on the loan must be determined
such that the interest rate on the loan is based on the
yield to maturity of the funding portfolio, said yield to
maturity being given by the interest rate at which the
present value of a future payment flow for funding
instruments equals the remaining debt on the debtor's
loan.
3. The requirement with respect to a balance between all
payments on the debtor side and payments on the creditor
side must be fulfilled.
4. Furthermore, the statutory requirement with respect to
terms to maturity and repayment profile must be ful-
filled, also for loans with adjustable interest rates
with adjustable terms to maturity.
In former calculations of volumes for traditional loans with
adjustable interest rates, no allowance was made for the
requirement as to the interest rate on the loan mentioned
herein.
In the funding of traditional loans with adjustable interest
rates. there was an unambiguous connection between the maxi-
mum terms to maturity of the funding instruments and the
interest rate adjustment period. This structure may briefly
be explained as follows: The funding principle was based on
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the assumption that there was a 5 year period in which the
debtor's interest rate was fixed. The traditional loans with
adjustable interest rates were funded by the debtor by the
issue of interest rate adjustment bonds with terms to matur-
ity to maturity of 1 to 5 years.
This funding principle is, however, not compatible with the
desire for issuing a range of e.g. 10 non-callable bullet
bonds with terms to maturity of 1 to l0 years, and at the
same time keeping the duration of the interest rate adjust-
ment period at e.g. 1-2 years.
Thus, in Denmark there is an interest in a general funding
principle comprising funding by the above range of non-
callable bonds or other financial instruments suitable for
the purpose. Presently, in international financial markets
there is no tradition of a close connection between the
lending and the funding of loans. Nevertheless, the broad
applicability of a principle linking the loan with a range of
financial instruments must be presumed to give rise also to
an international interest in a general funding principle of
the type described herein.
Thus, the funding principle may e.g. be used in mark-to
market pricing of loans and claims otherwise not traded. By
applying the principle, it will be possible to determine a
portfolio of traded financial instruments with an equivalent
payment flow on the basis of which the loan or the claim may
be priced in accordance with observed market prices.
Similarly, the funding principle may be applied to risk
management of loans and claims, the principle being appli-
cable to the determination of a hedge consisting of a port-
folio of financial instruments, as well as to the pricing of
such hedge. In recent years, the trend has been towards a
higher degree of attention being paid to financial risks,
including the possibility of hedging these risks, so it is
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within this area in particular that the international inter-
est in the funding principle is expected.
However, one technical problem in connection with such a
general funding principle has been that there was no know-
s ledge of an effective general calculation method for com-
puterized calculation of the volume of financial instruments
or funding volumes for funding a loan, the loan being at
least partially refinanced during the remaining term to
maturity of the loan, in which the calculation result must
fulfil both the requirement that lending institutions must
not expose themselves to interest rate and funding risks, or
at least they must not or will not expose themselves to such
risks above certain maximum limits, and be able to contribute
to minimizing the costs for the borrower such that the loan
with adjustable interest rates is as inexpensive as possible
within the given preconditions.
In Danish patent application no. 0165/96 and international
patent application no. PCT/DK97/00044, such convenient com-
puterized method for calculating the volume of financial
instruments or funding volumes for funding a loan of the type
described herein. In the following, this type of loan is
referred to as "Loans with Adjustable Interest Rates" (LAIR
I) .
A further development of the invention described in the above
patent applications has the following background:
In connection with Loans with Adjustable Interest Rates, it
is sometimes considered a problem that there is a risk of the
interest rate rising so much and for so long that it works
through to the borrower who may experience an increase in his
payments on the loan, at least during part of the term to
maturity of the loan, above the level that he can or wishes
to pay. It would be desirable to provide a possibility of
calculating the loans in such a way that instead of an
increase in payments on the loan, or in combination with a
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minor increase in payments on the loan, the term to maturity
was prolonged so that the borrower would be able to pay the
payments on the loan in view of his current financial posi-
tion.
Danish patent applications nos. 233/97, 308/97 and 770/97
concern this further development and relate to a method by
which not only the above parameters may be determined, but by
which requirements may also be laid down with respect to
maximum (or minimum) payments on the loan for the debtor in
one or more periods during the term to maturity of the loan,
the term to maturity of the loan optionally being calculated
as adjusted to these requirements. Conversely, it will be
possible by the method according to these patent applications
to lay down requirements with respect to the maximum (or
minimum) term to maturity of the loan, and then calculate an
adjusted payment on the loan. By the method according to the
mentioned patent applications, calculation results of a high
value may be achieved, which means, inter alia, that a high
degree of stability in the volumes of the calculated payments
on the loan is achieved despite relatively large interest
rate fluctuations in the different funding periods being
input. In the following, the type of loan calculated by means
of a method as described in Danish patent applications nos.
233/97, 308/97 and 770/97 is referred to as "Loans with
Adjustable Interest Rates II" (LAIR II).
In connection with Loans with Adjustable Interest Rates
(LAIR), it is sometimes considered a problem that the pay-
ments on the loan may rise to a level above that which the
debtor can pay. There are alternative types of loans which
have some characteristics in common with a LAIR and which, as
a facility, contain a maximum limit on the payments on the
loan.
Turning towards the American bond market which is tradi-
tionally seen as the most developed bond market in the world,
there are types of loans in which a short-term interest rate
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is combined with a ceiling on the possible interest rate to a
varying degree.
A first example is "step up" bonds. "Step up" bonds are long-
term bonds for which the coupon rate changes periodically
5 according to a predetermined pattern. Typically, the pattern
is based on the structure of the forward rates. If the for-
ward rate structure is rising, the coupon rate will typically
also rise over time.
The adjustment to the structure of the forward rate means
10 that, in principle, "step up" bonds will carry the short-term
interest rate initially. In periods with a rising yield
curve, the debtor may thus gain an interest rate advantage
comparable to the interest rate advantage of a LAIR.
In certain cases, the changes in the coupon rate are combined
with a call right. The debtor thus gets the possibility of
prepaying the remaining debt at par in connection with the
change in the coupon rate. The loan is thereby in the nature
of a loan with a short-term interest rate combined with an
option on the future interest rate, and the comparison to a
LAIR combined with an option on the interest rate adjustments
springs to mind.
The use of "step up" bonds for funding mortgage loans is, as
far as it is known, limited. "Step up"-bonds have mainly been
used in the high-risk bond market, where the lower coupon is
initially to secure the debtor's financial survival in the
short run.
Another example of a similar type of loan is "adjustable rate
mortgages" which are found in the American mortgage market.
"Adjustable rate mortgages" are loans in which the interest
rate is pegged to an interest rate index optionally added an
interest differential as a reflection of a credit risk or the
like.
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The interest rate index may be e.g. a "treasury°-based index
with a term to maturity of 1/2 year, 1 year, or 5 years. The
interest rate on the loan is adjusted at fixed intervals
typically of the same length as the interest rate index.
Thus, the loan has characteristics in common with a LAIR I.
A variant of the "adjustable rate mortgages" has as a faci-
lity a band in the interest rate index. The interest rate is
bound upwards by a "cap", whereas a "floor" sets a minimum
limit for the interest rate. The interest rate on the loan
will thus float within a band during the entire term to
maturity of the loan.
Both "step up" bonds and "adjustable rate mortgages" are
characterized by the hedging of the debtor's risk of rising
payments on the loan being built into the underlying bonds.
This structure has certain advantages. Once the "step up"
bonds or the "adjustable rate mortgages" are accepted by the
investors, the entire funding side of the loan is in confor-
mity with the market.
On the other hand, the structure is considered to have
several disadvantages.
Firstly, the tight linking of the debtor and bond sides
inhibits the flexibility on the debtor side. For each loan
with individual characteristics on the debtor side, a bond
with individual characteristics must exist on the funding
side. An adjustment of the debtor side to individual debtor
preferences will thus quickly lead to a situation in which a
wealth of different "step up" bonds or "adjustable rate
mortgages" must be opened.
Secondly, it is subject to some uncertainty whether the "step
up" bonds or the "adjustable rate mortgages" can attain the
necessary market conformity within a foreseeable number of
years.
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In connection with the above loans with adjustable interest
rates with adjustable terms to maturity, situations may dccur-
in which the interest rate has increased such that the pay-
ments on the loan cannot be kept within an established maxi-
s mum limit even if the term to maturity is prolonged, e.g.
because of statutory limits for the duration of the term to
maturity. Against this background, it may be considered to
supplement the applied financial instruments with a special
financial instrument, hereinafter termed a "payment guarantee
instrument", from which payments are made in situations in
which the maximum limits for payments on the loan and term to
maturity would otherwise have been exceeded. It may be con-
venient also to consider making payments to the instrument in
situations in which payments on the loan and term to maturity
would otherwise have fallen below their minimum limits.
The present invention permits an appropriate and realisti-
cally practicable computerized calculation of the above
parameters which are calculated in accordance with the above
patent applications, as well as further calculation of pay-
ments from (or to) a "payment guarantee instrument" of the
above type. As will appear from the following, it is possible
by use of available calculation methods to determine such a
price for such instrument that a professional market might
buy the instrument or a volume of instruments corresponding
to a volume of loans.
The payment guarantee instrument is considered particularly
convenient when apart from granting payments to the debtor in
situations in which agreed maximum limits for payments on the
loan and term to maturity are exceeded, it also receives
payments from the debtor in situations in which payments on
the loan and term to maturity would otherwise have fallen
below their minimum limits. Therefore, this type of payment
guarantee instrument is in particular the basis of the fol-
lowing explanation of the method according to the invention,
even if it is understood that a payment guarantee instrument
not designed to receive payments from the debtor could also
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be included and treated in the same way by the method accord-
ing to the invention.
Thus, the invention relates to a method for determining, by
means of a first computer system, the type, the number, and
the volume of financial instruments for funding a loan,
determining the term to maturity and payment profile of the
loan, and further determining the payments on a payment
guarantee instrument designed to ensure that the payments on
the loan and the term to maturity of the loan do not exceed
predetermined limits, and from which instrument payments are
made to the debtor in situations in which the maximum limits
for payments on the loan and term to maturity would otherwise
have been exceeded, the loan being designed to be at least
partially refinanced during the remaining term to maturity of
the loan,
- requirements having been laid down stipulating that
- the term to maturity of the loan is not longer than a
predetermined maximum limit nor less than a predeter-
mined minimum limit,
- debtor's payments on the loan are within predeter-
mined limits,
requirements having been laid down stipulating a maximum
permissible difference in balance between, on the one
hand, payments on the loan and refinancing amounts and,
on the other hand, net payments to the owner of the
financial instruments applied for the funding, and pay-
ments to and from the payment guarantee instrument,
- requirements having been laid down stipulating a maximum
permissible difference in proceeds between, on the one
hand, the sum of the market price of the volume of the
financial instruments applied for the funding of the
loan, and payments to and from the payment guarantee
instrument and, on the other hand, the volume of the
loan,
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- and requirements optionally having been laid down stipu-
lating a maximum permissible difference between the ' -
interest rate on the loan and the yield to maturity of
the financial instruments applied for the funding,
said method comprising
(a) inputting and storing, in a memory or a storage
medium of the computer system, a first set of data specifying
the parameters: the volume and the repayment profile of the
loan,
(b) inputting and storing, in a memory or a storage
medium of the computer system, a second set of data specify-
ing
(i) a maximum and a minimum limit far the
debtor's payments on the loan in each of a number of
periods collectively covering the term to maturity of
the loan,
(ii) a maximum and a minimum limit for the term to
maturity of the loan, and
(iii) optionally, a desired/intended payment on the
loan or a desired/intended term to maturity when the
maximum and the minimum limits for the payment in the
first period are not equivalent (i), or when the
maximum and the minimum limits for the term to matur-
ity are not equivalent (ii),
(c) inputting and storing, in a memory or a storage
medium of the computer system, a third set of data specifying
a desired/intended refinancing profile, such as one or more
points) in time at which refinancing is to take place, and
the amount of the remaining debt to be refinanced at said
points) in time,
and/or said third set of data specifying a
desired/intended funding profile, such as a desired/intended
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number of financial instruments applied for the funding
together with their type and volumes,
(d) inputting and storing, in a memory or a storage
medium of the computer system, a fourth set of data com-
5 prising a maximum permissible difference in balance within a
predetermined period, a maximum permissible difference in
proceeds and, optionally, a maximum permissible difference in
interest rates equivalent to the difference between the
interest rate on the loan and the yield to maturity of the
10 financial instruments applied for the funding and, optional-
ly, the payment guarantee instrument,
(e) determining and storing, in a memory or a storage
medium of the computer system, a fifth set of data specifying
a selected number of financial instruments with inherent
15 characteristics such as the type, the price/market price, and
the date of the price/market price,
(f) determining and storing, in a memory or a storage
medium of the computer system, a sixth set of data repre-
senting a first profile of the interest rate on the loan and
either.a first term to maturity profile or a first payment
profile of the loan,
(g) calculating and storing, in a memory or a storage
medium of the computer system, a seventh set of data repre-
senting
- a first term to maturity profile or a first payment
profile (depending on what was determined under (f))
corresponding to interest and repayments on the part of
the debtor,
- and a first remaining debt profile,
said term to maturity profile or payment profile, as well as
the remaining debt profile, being calculated on the basis of
- the volume and repayment profile of the loan as input
under (a),
- the set of data input under (b),
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the refinancing profile and/or the funding profile input
under (c)
- and the profile of the interest rate on the loan and
either the payment profile or the term to maturity pro-
s file determined under (f),
(gl) if necessary/if desired, calculating and storing, in
a memory or a storage medium of the computer system, an
eighth set of data representing payments (positive, zero or
negative) on the payment guarantee instrument, the require-
ments with respect to a maximum permissible difference in
balance and a maximum permissible difference in proceeds, as
well as the limits for payments on the loan and term to
maturity, always being fulfilled.
(h) selecting a number of financial instruments among the
financial instruments stored under (e), and calculating and
storing a ninth set of data specifying these selected finan-
cial instruments with their volumes, for use in the funding
of the loan, said ninth set of data being calculated on the
basis of
- the payment profile determined under (f) or calculated
under (g) and
- the remaining debt profile calculated under (g),
- the payments on the payment guarantee instrument option-
ally calculated under (gl),
- the refinancing profile input under (c) and/or the fund-
ing profile input under (c),
- the set of data input under (b),
- the requirements input under (d), and
- in the case of a refinancing where financial instruments
from a previous funding have not yet matured, the type,
the number, and the volume of these instruments,
one or more recalculations being made if necessary, including
if necessary, selection of a new number of the financial
instruments stored under (e),
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storing, in a memory or a storage medium of the computer
system, after each recalculation
- the recalculated profile of the interest rate on the
loan,
- the recalculated term to maturity profile,
- the recalculated payment profile,
- the recalculated remaining debt profile, and
- the selected financial instruments with their calculated
volumes,
until all the conditions stated under (b) and (d) have been
fulffilled,
and the payments on the payment guarantee instrument option-
ally being calculated in accordance with (gl), and the
recalculated payments being stored in a memory or a storage
medium of the computer system after each recalculation,
after which, if desired, the thus determined combination of
the type, the number and the volume of the financial instru-
ments for funding the loan,
- together with the calculated term to maturity,
- together with the calculated payment profile,
- optionally, together with the payments on the payment
guarantee instrument,
- preferably, together with the calculated interest rate on
the loan, and
- preferably, together with the calculated remaining debt
profile,
is output, transferred to a storage medium or sent to another
computer system.
In the following, the type of loan calculated by means of a
method according to the invention is referred to as ~~Loans
with Adjustable Interest Rates III~~ (LAIR III).
Apart from the input, determined and/or calculated data being
stored in a memory or on a storage medium, the data may be
output to a display or a printer. The memories applied may be
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e.g. electronic memories such as ROM, PROM, EEPROM or RAM,
and the storage media may be e.g. tapes, discs or CD-ROM.
It will also be possible to input data for use in or result-
ing from the data processing according to the invention in
one set of memories or storage media which may form part of
the first or a second computer system, and to transfer these
data to a second set of memories or storage media which may
form part of the second or the first computer system, these
data being optionally transferred e.g. via a data trans-
mission line or a net combining the first and the second
computer systems, or in a wireless manner, e.g. electro-
magnetically or optically.
The method according to the invention calculates, inter alia,
the volume of the individual financial instruments to be sold
to fund the loan. These volumes are normally not whole or
round volumes, and in certain cases they may be fairly small.
Usually, the institution issuing the loan solves the
divisibility problem by pooling many small loans when finan-
cial instruments are sold in the market. The lending institu-
tion makes an exact registration of the individual financial
instruments applied for the funding of each individual loan
with their volumes.
When in the present description and claims it is stated that
by the method according to the invention, the volume, the
type and the number of financial instruments for funding a
loan, the term to maturity and payment profile of the loan
and optionally the payments on the payment guarantee instru-
ment are determined, this indicates that the information
resulting from the method according to the invention may be
used e.g. as the basis of the actual physical action of the
lender (e. g. a mortgage credit institution) issuing/selling
the said financial instruments. The information resulting
from the method according to the invention may of course also
be applied for pricing a loan in connection with a loan
offer, and/or for calculating the lender's risk profile, e.g.
CA 02297990 2000-02-O1
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19
with a view to countering the risk by means of a hedge, or
for other analysis purposes etc., without the said financial -
instruments actually being issued.
Tt will be understood that the sequence of the
inputs/determinations/storage operations (a)-(e) stated above
is arbitrary and, therefore, the sequence of letters do not
specify an equivalent compulsory sequence of the steps. Step
(f) may also be carried out at an arbitrary stage in the
sequence unless it is chosen, as is often preferred, to have
the computer calculate a first guess at a profile of the
interest rate on the loan, and either a first term to matur-
ity profile or a first payment profile, in which case step
(f) will definitely follow step (e). Instead of expressing
that data are input/determined and stored in the individual
steps, it may simply be expressed (and should be considered
equivalent to the first expression form) that by the method
according to the invention, calculations are made by means of
the computer system on the basis of stored inputs of sets of
data (a)-(f). It will also be understood that these and other
inputs to be used in order to start the individual calcu-
lations, e.g. the first profile of the interest rate on the
loan, and either a term to maturity profile or a payment
profile (f), may be, as mentioned above, a guess or an
initial value which is also performed/determined by means of
the computer system according the predetermined rules, and
stored/applied as an initial value. Another example of data
being either input or guessed/calculated is the
desired/intended payment or the desired/intended term to
maturity under (b) (iv); if no initial value thereof has been
input/stored, the computer system is conveniently designed to
"guess" or calculate a value according to an established
rule, e.g. as an average of the values stored under (b) (i)
and (b) (ii).
A number of the inputs mentioned above are inputs applying to
a corresponding period. This is the case e.g. for the maximum
permissible difference in balance and the payment limits. In
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these cases, either the corresponding period is input, said
input applying, or the period has already been generally'
input to the computer system. For annuity loans, the period
mentioned under (b)(i) is preferably a refinancing period,
5 which will therefore normally be a default in the computer
system but, in principle, this period may be any period
desired by the debtor, said period normally being input
together with the mentioned limits.
The requirement with respect to the maximum permissible
10 difference in balance is linked to a period which, depending
on the legislation or the practice which is to form the basis
in connection with the calculations, may be a calendar year,
a year not following the calendar year but comprising the
time of a payment to the creditor, or another period either
15 comprising or not comprising the time of a payment to the
creditor. In Denmark a strict balance requirement must be
fulfilled per calendar year.
In the calculation of data corresponding to the volume of the
financial instruments applied for the funding, the require-
20 ment with respect to maximum permissible difference in
balance is, according to the current Danish rules of mortgage
loans, given by a strict balance, i.e. no appreciable differ-
ence in balance occurs or, to put it differently, the dif-
ference is practically zero. However, the method according to
the invention may also be used where a certain difference in
balance is tolerated or perhaps even desired, this tolerance
or this positive difference in balance then being stored as
part of the data set in (d) .
In the calculation according to the invention, both the
requirement with respect to the difference in proceeds, the
requirement with respect to the difference in interest rates
as well as the requirement with respect to the difference in
balance may be specified in different ways. Data may e.g. be
input, specifying a direct maximum permissible difference in
balance between, on the one hand, the sum of the market price
CA 02297990 2000-02-O1
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21
of the volume of the financial instruments applied for the
funding of the loan and payments to and from the payment -
guarantee instrument and, on the other hand, the volume of
the loan, as well as data may be input, specifying a direct
maximum permissible difference between the interest rate on
the loan and the yield to maturity of the financial instru-
ments applied for the funding, and data directly specifying a
maximum permissible difference in balance. The requirement
with respect to maximum permissible difference in proceeds
may also be input as data specifying a convergence condition
for the difference in proceeds, and/or the requirement with
respect to maximum permissible difference between the inter-
est rate on the loan and the yield to maturity may be given
by input data specifying a convergence condition for the
difference in interest rates, and/or the requirement with
respect to maximum permissible difference in balance may be
given by a convergence condition for the difference in
balance.
Usually, when obtaining the loan there is no match between,
on the one hand, the disbursement date of the loan and/or the
prepayment date of the loan and, on the other hand, the
settlement date of the financial instruments, for which
reason correction is preferably performed in the. calculation
by the method according to the invention of a possible dif-
ference between, on the one hand, the disbursement date of
the loan and/or the prepayment date of the loan and, on the
other hand, the settlement date of the financial instruments
by proportionally correcting for the already elapsed part or
the remaining part of the disbursement period and the prepay-
ment period, respectively. Here, data specifying a correction
factor for use in the calculation may e.g. be input or calcu-
lated.
In practice, one may select to have the maturity of the loan
coincide with a creditor payment date, which would normally
require that the last element of the term to maturity profile
is calculated as adjusted to attain this coincidence. Here,
CA 02297990 2000-02-O1
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22
it is most appropriate that the term to maturity is calcu-
lated prolonged, which in turn implies that the payment on
the loan falls, for which reason the minimum limit for the
payments on the loan on the loan in the last funding period
is suspended in the calculation.
The calculation method according to the present invention is
also applicable in situations in which the input data spe-
cifies that more than one debtor payment on the loan will be
made within one creditor payment period.
Under (c) information is to be input or be available, con-
cerning the point or points in time at which refinancing is
desired to take place, and concerning the amount to be
refinanced at said points in time. In one instance, which is
important in practice, the input data specify that full
refinancing of the remaining debt is performed at the end of
a predetermined period which is shorter than the term to
maturity of the loan, and in a second important instance, the
input data specify that refinancing of the remaining debt is
performed with a fixed annual fraction.
The method according to the invention may be applied for
determining the number and the volume of the financial
instruments, the term to maturity and the payment profile in
the situation in which the loan is to be calculated for the
first time, i.e. in the first funding situation, as well as
in the situation in which a refinancing is to be calculated.
The expression funding thus covers both "new funding" and
"refinancing". Apart from the parameters mentioned under (a)-
(f), information concerning the type, the number, and the
volume of the financial instruments which have not yet
matured at the time of refinancing is included in the calcu-
lations in the refinancing situation. This information is
often stored in the computer system from the previous calcu-
lation, but inputting this information is evidently within
the scope of the invention. It will be understood that the
parameters under (a)-(f) are parameters which are related to
*rB
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23
the funding situation in question, so that for the case in
which a refinancing is calculated, they are naturally rezated-
to the remaining debt of the loan as the volume of the loan
and to the remaining term to maturity of the loan as the term
to maturity of the loan.
Reference in the claims to the "remaining term to maturity"
or "term to maturity", means - depending on the context - the
remaining term to maturity or the term to maturity which is
the basis of the first calculation in the funding period for
which the calculation is performed.
The result of the method according to the invention as
defined above is usually at least one set of data which may
be applied in the next funding situation, whether this situ-
ation is the first funding period of the loan, or a later
refinancing situation.
The expression "term to maturity profile" is related to a
term to maturity being calculated by the method according to
the invention, as mentioned above, usually for each funding
or refinancing period. Hence, the expression term to maturity
profile refers to the series of terms to maturity which being
assigned to the refinancing period at each calculation in
connection with a refinancing.
The expression "profile of the interest rate on the loan" is
similarly related to a calculation of the interest rate on
the loan being performed by the method according to the
invention, usually for each funding or refinancing period.
Hence, the expression profile of the interest rate on the
loan refers to the series of interest rates on the loan being
assigned to funding periods at each calculation in connection
with the refinancing.
In the present description and claims, the term "financial
instruments" has the meaning normally used and covers thus
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24
e.g. all types of interest rate related claims, i.e. all
types of bonds, including zero-coupon bonds. ~ -
When by means of the method according to the invention, the
calculations are performed with financial instruments which
are not directly interest-bearing, first, a calculation is
conveniently performed of the expected payment flows such
that a calculation of an internal interest rate may be per-
formed, causing the payment flow or flows or the likely
payment flow or flows to be expressed in parameters corre-
sponding to the above-mentioned parameters for interest-
bearing claims, primarily a yield to maturity. Thus, e.g. for
an option which has a price of DKK 100 and which has a pro-
bability of 50 per cent of resulting in proceeds amounting to
DKK 210 at the end of a term to maturity of one year and a
probability of 50 per cent of resulting in proceeds amounting
to DKK 0, this can be done by a purely statistical calcu-
lation of the average proceeds of DKK 105 and by formulation
of the relevant parameters as a price of DKK 100, a quotation
of 100 and an interest rate of 5 per cent per annum which -
together with the interest rate on the other financial
instruments applied - is to constitute the basis on which it
is checked whether the requirement with respect to maximum
permissible interest difference has been fulfilled. These
parameters may then be input to the computer system. Alterna-
tively, and often preferably, the data being stored as cha-
racteristics of the instruments in section (a) above may be
data directly defining the financial instruments in question,
and the computer system may be adapted to perform a conver-
sion into parameters characterizing an interest-bearing claim
according to predetermined principles. In the case of CAPS or
FLOORS, the procedure is similar as the same payment flows
may be expressed by corresponding interest-bearing instru-
ments, the characteristics of which may then be stored as,
stated in section (e), or the computer system may preferably
be adapted to perform a conversion into parameters charac-
terizing an interest-bearing claim according to predetermined
principles. It will be understood that in each individual
CA 02297990 2000-02-O1
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case, the calculation may also be performed on the basis of
the prerequisite that in the individual funding or
refinancing situation, a combination of different types of
financial instruments is used, the characteristics to be
5 applied to the calculation then being specified for each type
of instrument. In this situation, the fulfilment of the
requirement with respect to the maximum permissible differ-
ence in interest rates is preferably checked on the basis of
an overall calculation based on the total payment flows from
10 all the financial instruments applied. Alternatively, a
weighted average of the interest rates of the individual
instruments may be applied.
By the method according to the invention, calculations may
thus be performed on the basis of different types of finan-
15 cial instruments or funding volumes, but in one instance,
which is important in practice, the calculation is performed
on the basis of bonds with a maximum term to maturity corre-
sponding to the refinancing period. The bonds are usually
non-callable bullet bonds, also including zero-coupon bonds.
20 The method according to the invention may, as explained
above, also be used advantageously in the calculation in
connection with other types of financial instruments, such as
e.g. serial bonds or annuity bonds.
The type of a financial instrument refers conventionally to
25 the combination of all the basic information or basic data
collectively and unambiguously defining the said financial
instrument, for bullet bonds thus the nominal volume, the
nominal interest rate, the date of maturity, all interest
rate payment dates, and the ex-coupon date, i.e. the deadline
determining who will receive the next interest rate payment
on the bond, as well as optionally the day count convention,
i.e. the formula applied to the conversion of the payment
f lows of the bond into an annual yield to maturity. The
number of financial instruments specify how many different
financial instruments to be stated as applied. The volume
specifies how many entities of the individual financial
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26
instruments, or the volume of nominal amount of the indivi-
dual financial instrument to be indicated as applied.
In accordance with common practice, the expressions "repay-
ment profile", "remaining debt profile" and "payment profile"
specify the development over time in repayments, remaining
debt and payments on the loan, respectively.
The repayment profile may follow the annuity loan principle
as well as the serial loan principle. In addition, any arbi-
trary placing in time of the repayments is naturally pos-
sible. For types of loans with an repayment profile depending
on the interest rate on the loan, the repayment profile may
be determined either on the basis of the interest rate on the
loan applying at the time in question, or on the basis of the
original interest rate on the loan, or on the basis of an
arbitrarily determined interest rate.
The expressions "financing profile" and "funding profile"
respectively specify the type, the number, and the volume of
the financial instruments applied for the funding. In the
present description and claims, the expression may be used
about the desired or intended funding profile which is input
and stored under (c), and which might not be fulfilled, as
well as about the accurate funding profile which is the
result of the calculations following application of the
method.
Here, the expression "refinancing profile" specifies at which
points in time and with which amounts the loan is to be
refinanced.
It should be noted that in some cases, the desired/intended
refinancing profile stored as a second set of data under (c)
above may be rewritten as a funding profile, viz. as a number
of financial instruments with their type and volumes. An
indication of a desired annual interest rate adjustment
percentage of 100 may e.g. be rewritten into the loan being
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27
desired to be funded exclusively through sales of bullet
bonds with a term to maturity of 1 year. It will be under-
stood that the invention also extends to the situation in
which such a rewriting has taken place in the data stored
under ( c ) .
The method according to the invention may be implemented in
different ways, as simultaneous calculation of all parameters
to be calculated may be performed, except that payments to
and from the payment guarantee instrument are always to be
calculated following the calculation of the payment on the
loan (and normally the term), or a division of the calcula-
tions according to different criteria may be performed. In a
presently preferred embodiment which is described in detail
in the example section below, the method comprises
obligatorily an "inner loop" determining volumes of the
underlying financial instruments and the interest rate on the
loan in consideration of the balance condition and the pro-
ceeds condition, and an "outer loop" determining the term to
maturity in consideration of the maximum and minimum limits
for the payment on the loan. If payments on the loan and term
to maturity are within the established limits, the calcula-
tion may be seen to be concluded, but if the limits for
payments on the loan and term to maturity are not observed,
particularly if the term to maturity, in consideration of the
limits for the payments on the loan, is not within the
accepted interval, a model is activated which calculates
payments to or from the payment guarantee instrument such
that the conditions with respect to payments on the loan and
term to maturity are fulfilled.
It will be understood that the invention also extends to an
embodiment in which the term to maturity is calculated at the
same time as the other parameters in the "inner loop". This
may e.g. be carried out by increasing the dimension of the
iteration procedure by one. (In a preferred embodiment, a
Gauss-Newton iteration algorithm is applied).
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In connection with the maturity of a loan, an immediate
result of the calculations may indicate that the date of
maturity does not coincide with the date of maturity of the
last maturing financial instrument considered. It is natur-
ally possible to apply such a result but in a preferred
embodiment, the date of maturity of the loan is corrected
such that it corresponds to the date of maturity of the last
maturing financial instrument. The correction comprises
determining whether the term to maturity is to be round up to
a creditor payment date (a date of maturity of a financial
instrument) or be round down to the preceding creditor pay-
ment date (a date of maturity of a financial instrument one
period earlier). In this case, the adjustment of the date of
maturity may preferably be performed as follows:
When the set of data under (c) specifies that calculations
are to be performed for the case in which full refinancing of
the remaining debt is to be performed periodically with a
predetermined period which is shorter than the term to matur-
ity of the loan, and the remaining term to maturity of the
loan is shorter than the period which according to (c)
elapses between two successive interest rate adjustments, and
the remaining term to maturity does not correspond to the
maturity of the last maturing financial instrument selected
under (h), but it is desired that the loan matures at the
same time as the maturity of the last maturing financial
instrument selected under (h), the term to maturity may
conveniently be determined by the method according to the
invention as
(i) the term to maturity prolonged as little as
possible to a date of maturity of one or more of
the selected financial instruments, provided the
payment profile does not thereby exceed the mini-
mum limit for the payments on the loans as spe-
cified under (b) (i), or
(ii) the term to maturity shortened as little as
possible to a date of maturity of one or more of
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29
the selected financial instruments, provided the
payment profile does not thereby exceed the m~xi-
mum limit for the payments on the loans as spe-
cified under (b) (i), and provided the condition
under (i) is not fulfilled, or
(iii) the term to maturity prolonged as little as
possible to a date of maturity of one or more of
the selected financial instruments, if none of
the conditions specified under (i) and (ii) is
fulfilled.
When on the other hand the set of data (c) specifies that
calculations are to be performed for the case in which par-
tial refinancing of the remaining debt is to be performed
', periodically with a predetermined period which is shorter
than the term to maturity of the loan, e.g. in such a way
that the refinancing equals a fixed fraction of the remaining
debt of the loan, and the remaining term to maturity of the
loan is less than or equal to a set value and it is desired
that the loan matures no later than the date of maturity
stated under (e) of one or more of the funding instruments
applied for the refinancing of the loan, the term to maturity
is conveniently determined by the method according to the
invention as the term to maturity prolonged as little as
possible to a date of maturity of one or more funding instru-
ments.
As indicated above, it is preferred at present to implement
the method according to the invention by use of an "outer
loop" and an "inner loop", the specified recalculations being
performed in the "outer loop" on the basis of a first term to
maturity profile, changing the term to maturity at substan-
tially each recalculation until the payment on the loan in
each funding period is within the limits established in (b)
(i) ,
the determination of the type, the number and the volume
of the financial instruments for funding the loan being
calculated and recalculated at each iteration over the
CA 02297990 2000-02-O1
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term to maturity until the relevant variables with
respect to the type, the number and the volume of the-
financial instruments are established in observance of
the other requirements/conditions/desires,
5 after which, if the term to maturity for which the payment
profile is within the limits established therefor is not
within the limits specified in (b) (ii) for the term to
maturity, the payments on the payment guarantee instrument
are calculated such that the limit for the term to maturity
10 as well as the limits for the payments on the loan are
observed.
The calculation of the payments on the payment guarantee
instrument is conveniently performed on the basis of an
interest rate on the loan which is recalculated such that the
15 limits for payments on the loan as well as term to maturity
are observed, and wherein either resulting differences in the
payments on the debtor side and the payments on the financial
instruments or resulting differences in the market price of
sold financial instruments and the funding demands correspond
20 to the payments on the payment guarantee instrument.
The funding demand is defined at the disbursement of the loan
as the volume of the loan and at the adjustment of the inter-
est rate of the loan as the amount at which the requirement
with respect to maximum permissible difference in balance is
25 fulfilled in the year immediately preceding.
Conveniently, the payments on the payment guarantee instru-
ment correspond to the differences in the market price of
sold financial instruments and the funding demand resulting
from the recalculation, the volume of the financial instru-
30 ments being determined such that the requirement with respect
to maximum permissible difference in balance is fulfilled.
It will be understood that this embodiment of the method
extends to a series of recalculations in the outer loop, each
of these recalculations normally occasioning a series of
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31
recalculations in the inner loop. The recalculations in the
inner loop are performed each time until the conditions of
the inner loop are fulfilled.
A distinguishing feature of the inner loop in the embodiments
preferred at present is that its calculations are based on an
interest rate on the loan. However, it will be understood
that any mathematical expression which represents the inter-
est rate on the loan may be applied, e.g. the remaining debt
profile, the payment profile or the repayment profile of an
annuity loan, or the payment profile of a bullet loan or a
serial loan, the other calculation parameters being adjusted
thereto in accordance with common and obvious mathematical
principles. It will also be understood that, in principle, a
first calculation of the number, the type and the volume of
the financial instruments may be performed prior to the
determination of the first value of the interest rate on the
loan, but even if this is the case, calculations and option-
ally recalculations are subsequently to be performed, e.g. of
the interest rate on the loan according to the principles
stated above. Thus, it would be possible under (f) to replace
the profile of the interest rate on the loan by the volume of
the financial instruments and under (g) either to calculate
the profile of the interest rate on the loan or recalculate
the volume of the financial instruments.
A possible equivalent way of expressing steps (f) and (g)
could thus be to replace, in those two steps, the profile of
the interest rate on the loan by the volume of the financial
instruments, in such a way, however, that the volume of the
financial instruments fulfils the proceeds criterion, and
then calculate the interest rate on the loan in step (h).
If no instrument is selected in (e) with payment in the
period up to the next point in time at which a refinancing is
to be performed according to the refinancing profile input
under (c), the calculations in the inner model in the pre-
ferred embodiment relate to a situation in which the result
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32
for at least one of the financial instruments applied for the
funding will be negative, i.e. corresponding to the borrower -
having to buy one or more financial instruments in the next
period in order to fulfil the balance requirement. As will
appear from the following, it is preferred at present that
steps are taken to change the calculations, so that they do
not result in negative volumes of the financial instruments.
In the cases in which it is stated in the input refinancing
profile that full refinancing is to be performed, the finan-
cial instruments applied for the refinancing may e.g. be
calculated in the inner model in the same way as the finan-
cial instruments applied for the initial funding, in other
words, it would be possible to perform a new calculation
according to the method of the volume of financial instru-
menu for funding a new loan, the volume of the new loan
corresponding to the amount to be refinanced.
In another embodiment of the inner loop, it may be specified
in the input data corresponding to the refinancing profile
that a partial refinancing of the remaining debt is to be
performed. In the inner model a solution may be found to the
volume of the financial instruments constituting the volume,
if it has been input e.g. that refinancing is to be performed
periodically with a predetermined period which is shorter
than the term to maturity of the loan. A solution may also be
calculated if it is specified that periodic refinancing of a
fraction of the remaining debt of the loan is to be per-
formed, the denominator of the fraction corresponding to the
whole number of years of the financial instrument having the
longest term to maturity when the loan was obtained. Here,
the selected period may be e.g. 1 year, but other periods
such as 2, 4, 5, 6 or 10 years may be selected. Furthermore,
periods corresponding to a whole number of months, e.g. 2, 3,
4 and 6 months may be selected.
In connection with a full or a partial refinancing, it is
normally necessary in the inner loop to calculate with one or
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more new refinancing instruments not included in the range of
initial financial instruments constituting the range of
funding volumes which were applied according to the given
data when the loan was obtained or in a previous refinancing
of the loan. Normally, these new refinancing instruments have
such a term to maturity that they mature at a later point in
time than the points in time at which the initial financial
instruments mature.
l0 In the partial refinancing, the refinancing in the inner loop
may further comprise a funding by use of additional funding
for the financial instruments or funding volumes remaining at
the time of the refinancing. In the following, the volume of
such additional funding and new refinancing instruments are
also designated as the addition to the volume of the finan-
vial instruments.
The calculation method according to the present invention
will also provide a solution to the volumes of the additions
to the financial instruments applied for the refinancing.
When calculating the volume of these additions, data compris-
ing possible new refinancing instruments within the range of
selected financial instruments must be input. In the calcula-
tion in the case of a refinancing, the proceeds criterion may
e.g. be given as a requirement with respect to the difference
between, on the one hand, a funding demand given by the
balance requirement and, on the other hand, the sum of the
market price of the addition to the financial instruments.
As mentioned above, the issue of new financial instruments,
as well as additional issue of financial instruments already
applied may be made in connection with a refinancing. How-
ever, in theory, it will also be possible to repurchase the
financial instruments already applied, but this involves a
number of inconveniences, inter alia, an additional depreci-
ation risk on the part of the borrower and problems pertain-
ing to the mortgages, for which reason repurchase is not
effected in practice.
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Therefore, according to a preferred embodiment of the method,
the volume of the additions to the financial instruments'will-
be calculated in consideration of the volumes of the previ-
ously applied financial instruments remaining at the time of
refinancing.
When in the present description and claims reference is made
to payments from the payment guarantee instrument and pay-
ments to the payment guarantee instrument, this is not neces-
sarily to be taken to represent a direct "communication"
between the debtor and the instrument. In most cases, it is
convenient that the payments from the payment guarantee
instrument is experienced by the debtor as a reduction in the
interest rate payable, possibly via an increase in the market
price of the financial instruments to be applied in the
refinancing, and that the payments to the payment guarantee
instrument is experienced by the debtor as an increase in the
interest rate.
For a number of reasons further discussed in the example
section, it is convenient that the payment guarantee instru-
ment has a price or a value of zero. This may be achieved by
the desired/intended term to maturity of the loan being input
under (b) (iii) and/or the limits for the payments on the
loan and/or the limits for the term to maturity are estab-
lished such that the present value of the payments on the
payment guarantee instrument is zero.
The calculation of the present value of the payments on the
payment guarantee instrument may conveniently be performed by
use of a stochastic yield curve model. The stochastic yield
curve model is preferably calibrated to a yield curve which
is determined at the time of calculation.
The stochastic yield curve model is conveniently formulated
in discrete time and implemented in a yield curve lattice,
appropriately in e.g. a trinomial lattice according to Hull &
White (references to Hull & White in the present text com-
prise: "On derivatives. A compilation of articles by John
CA 02297990 2000-02-O1
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Hull and Alan White", Risk Publications 1996) in which the
yield curve model in a preferred embodiment is the extended
Vasicek model. A more detailed discussion of these aspects
are found in the example section.
5 In an embodiment of the method according to the invention,
also called Type F in the subsequent detailed description, it
is specified by the set of data under (c) that calculations
are to be performed for the case in which full refinancing of
the remaining debt is to be performed periodically with a
10 predetermined period which is shorter than the term to matur-
ity of the loan. If the limits for payments on the loan as
well as term to maturity are fulfilled, and the payment on
the payment guarantee instrument is at the same time zero,
the method for determining the volumes of financial instru-
15 ments specified in step (h) comprises calculating the differ-
ence in proceeds of the calculated volumes of the financial
instruments applied for the funding, and/or calculating an
interest rate adjustment of the interest rate on the loan,
said interest rate adjustment preferably being calculated in
20 consideration of the calculated difference in proceeds,
calculations being performed as to whether the interest rate
adjustment is so small that the requirement with respect to
maximum permissible difference in proceeds or a convergence
condition for the difference in proceeds is fulfilled. This
25 situation which will manifest itself in practice in many, and
perhaps moat, of the calculations for which the method accor-
ding to the invention will be applied, is described in more
detail in the following.
It should be noted that when a requirement with respect to a
30 difference in proceeds of 0 or very close to 0 is fulfilled,
the difference in interest rates is automatically 0 or very
close to zero, meaning that the requirement with respect to
difference in interest rates may conveniently be omitted from
the input starting conditions in this situation.
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36
This is the reason why the input concerning maximum permiss-
ible difference in interest rates is stated as being '
optional, whereas the input concerning maximum permissible
difference in proceeds is stated as being compulsory in all
situations. It should be noted, however, that a compulsory
input concerning maximum permissible difference in proceeds
may be fulfilled by inputting information which is completely
equivalent thereto, e.g. an interest rate input, and that the
present invention naturally extends to such substitutions.
i
If the requirements or conditions laid down with respect to
the difference in proceeds or the difference in interest
rates are not fulfilled, the recalculations in this part of
Type F comprise one or more interest rate iterations, each
interest rate iteration comprising
calculating and storing, in a memory or a storage medium
of the computer, data specifying a new interest rate on the
loan which is preferably based on the previous interest rate
on the loan and the calculated interest rate adjustment,
', calculating and storing, in a memory or a storage medium
of the computer, data specifying a new payment profile and
remaining debt profile for the debtor, said payment profile
and remaining debt profile being calculated in consideration
of the new interest rate on the loan, the volume, the term to
maturity, and the repayment profile of the loan as input
under (a), and the refinancing profile and/or the funding
profile input under (b), and
calculating and storing, in a memory or a storage medium
of the computer system, data specifying a new set of volumes
of the financial instruments applied for the funding.
Here, the interest rate iteration is preferably performed
applying a numerical optimization algorithm or by "grid
search".
Examples of numerical optimization algorithms are a Gauss-
Newton algorithm, a Gauss algorithm, a Newton-Ramphson
algorithm, a quadratic hill climbing algorithm, a quasi-
CA 02297990 2000-02-O1
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37
Newton algorithm, a maximum likelihood algorithm, a method of
scoring algorithm and a BHHH algorithm. A Gauss-Newton '
algorithm has proved very appropriate.
When the relevant requirement or requirements with respect to
maximum permissible difference in proceeds and/or maximum
permissible difference in interest rates is/are fulfilled in
the Type F embodiment, it is convenient to determine whether
all the calculated volumes of financial instruments are
positive. If the calculated set of volumes comprises at least
one negative volume, the result as such may optionally be
applied, implying that the calculation specifies that the
borrower is to buy one or more financial instruments in order
for the balance requirement to be fulfilled. As mentioned
I above, this is normally not preferred, for which reason in
this case the method according to the invention normally
further comprise either
i) selecting a new number of financial instruments among
the financial instruments stored under (e), one or more of
the instruments in the new number of instruments being deter-
mined such that the payments on this/these instruments) are
due relatively later compared to the original number of
financial instruments, after which a recalculation is per-
formed as stated in connection with the description herein
and in the following of the Type F embodiment, or
I 25 ii) the negative volume or volumes is/are assigned the
value zero after which a recalculation is performed as stated
in connection with the description herein and in the follow-
ing of the Type F embodiment.
Where the input data specify that partial refinancing is to
be performed, the volume of the financial instruments applied
for the funding or the refinancing is calculated in a pre-
ferred embodiment of the inner loop such that they follow the
development in the remaining debt given by the remaining debt
profile. This calculation may comprise the use of a first
function which is adjusted to the remaining debt profile as
explained in the following. If e.g. the input data indicate a
CA 02297990 2000-02-O1
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38
difference between, on the one hand, the disbursement date
and/or prepayment date of the loan and, on the other hand,
the settlement date of the financial instruments, the
volumes) of one or more financial instruments may be deter-
mined in the calculation according to a preferred embodiment
of the invention such that this instrument or these instru-
ments does/do not follow the polynomial function, but con-
tributes/contribute to a solution to the above marginal
conditions.
In a preferred embodiment of the invention, it is determined
,in the inner loop whether the thus calculated volume of
financial instruments fulfils one or more predetermined
convergence conditions. If such condition or conditions are
not fulfilled, one or more iterations may be calculated until
the new set of financial instruments fulfils one or more of
the convergence conditions.
In a preferred embodiment of the inner loop, the function
coefficients are calculated on the basis of a calculated
difference in proceeds, and/or a calculated refinancing
difference preferably corresponding to the difference
between, on the one hand, a funding demand given by the
balance requirement and, on the other hand, a desired
refinancing. The function coefficients may be found either
analytically or by iteration.
If it is determined in the calculation in this embodiment of
the inner loop that the volume of financial instruments
calculated in connection with the funding or the refinancing
do not fulfil the requirements laid down by the input data
with respect to the difference in interest rates, one or more
recalculations in the form of interest rate iterations will
be performed in a preferred embodiment, a new interest rate
on the loan being determined or calculated, after which a new
set of financial instruments is calculated. Interest rate
iteration is performed until the requirements with respect to
difference in interest rates have been fulfilled. In the
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39
description below, examples are given with a detailed expla-
nation of this embodiment. The-situation in which the furic- -
tion coefficients to the function expression, which is
adjusted to the development in the remaining debt, are found
by iteration, as well as the situation in which the function
coefficients are found analytically are described.
A more detailed description follows of the situation in which
the set of data (c) specifies that calculations are to be
performed for the case in which partial refinancing of the
remaining debt is to be performed periodically with a prede-
termined period which is shorter than the term to maturity of
the loan, e.g. in such a way that the refinancing corresponds
to fixed fraction of the remaining debt of the loan. In this
embodiment generally termed Type P in the following detailed
description, some or all of the financial instruments applied
for the funding in the first calculation in step (g) is/are
calculated in the inner loop in the case in which the limits
for payments on the loan as well as term to maturity are
fulfilled, and the payment on the payment guarantee instru-
ment is at the same time zero such that they substantially
follow a shifted level remaining debt profile, after which
recalculations are made, if necessary, until all the condi-
tion mentioned under (d) have been fulfilled. This situation
which will often apply to one or more periods when applying
the method according to the invention is explained in more
detail in the following.
The adjustment to a shifted level remaining debt profile are
conveniently performed by the volume of some or all the
financial instruments being calculated in the calculation in
step (h) and optionally in one or more recalculations in step
(h) by use of a function which is adjusted to a shifted level
remaining debt profile. This function is conveniently a
polynomial function with a maximum degree which is one (1)
less than the number of financial instruments applied.
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The polynomial function is conveniently calculated by use of
a statistical curve-fitting method. The least squares' method -
has been found to be an appropriate statistical curve-fitting
method, but other statistical curve-fitting methods, such as
5 other maximum likelihood methods or the cubic splines method
would also be useful for this application.
In a preferred embodiment of the invention, a calculation (as
opposed to an iteration) is performed of one or more coeffi-
cients in the polynomial function. The calculation is per-
10 formed such that the subsequent determination of the adjust-
rnents to the volumes of the financial instruments fulfils the
requirements with respect to maximum permissible difference
in proceeds, said adjustments corresponding to the difference
between the polynomial function and the volumes of the finan-
15 cial instruments already issued and, if possible, the
requirement with respect to maximum permissible difference in
balance, the actual interest rate adjustment fraction at the
same time corresponding to the desired interest rate adjust-
ment fraction.
20 If coefficients are calculated in the polynomial function
such that one or more of the resulting adjustments to the
volumes of the financial instruments is/are negative, said
adjustments will not be used, which is indicated by an
adjustment of an indicator function. In this embodiment, the
25 indicator function is an M-dimensional vector in which the
elements have either the value one or the value zero, and the
value zero indicates that the said financial instrument is
not applied for the funding. On the basis of the adjusted
indicator function, a new calculation of one or more coeffi-
30 cients is performed in the polynomial function, the resulting
adjustments to the volumes of the financial instruments are
checked and, it necessary, the indicator function is read-
justed.
*rg
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The calculation of coefficients in the polynomial function
and the adjustment of the indicator function continue until
either
all adjustments to the financial instruments are non-negative
(viz. positive or zero)
either the first element of the indicator function has the
value zero, or the sum of the elements in the indicator
function is strictly less than 2, in each of which cases only
one coefficient is calculated in the polynomial function such
that the resulting range of adjustments to the volumes of the
financial instruments fulfil the requirement with respect to
maximum difference in proceeds; the resulting adjustment of
the interest rate will be determined by a residual calcula-
tion in accordance with the requirement with respect to
maximum permissible difference in balance.
It is also possible to adjust only one element at a time in
the indicator function.
The above-mentioned analytical method for determining the
function coefficients in the polynomial function is a method
which is easy to calculate and hence time-saving.
On the other hand, the function coefficients may also be
calculated by iteration as discussed in the now immediately
subsequent sections.
In the embodiment termed Type P, the recalculations of all or
some of the data mentioned in (g) and (h), and/or one or more
function coefficients for the function representing the
shifted level remaining debt profile, and/or the interest
rate on the loan in the inner loop are performed by iteration
carried out by applying numerical optimization algorithms or
by grid search. One of the optimization algorithms mentioned
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42
above in connection with Type F may be used as optimization
algorithm here as well, and here the optimization algorithm
is also conveniently a Gauss-Newton algorithm.
If the requirements laid down with respect to the difference
in proceeds and/or the difference in interest rates and/or
the difference in balance calculated in consideration of the
refinancing profile input under (c) are not fulfilled, the
recalculations in the inner loop may comprise one or more
iterations in connection with the Type P embodiment, each
iteration comprising
calculating and storing data specifying a new interest
rate on the loan, and/or
calculating and storing data specifying a new payment
profile and a new remaining debt profile for the debtor, said
payment profile and remaining debt profile being calculated
in consideration of the new interest rate on the loan, the
volume, the term to maturity and the repayment profile of the
loan as input under (a), and the refinancing profile and/or
the funding profile input under (c), and/or
calculating and storing data specifying a new set of
coefficients for the function which is adjusted to the
shifted level remaining debt profile, and/or
calculating and storing data specifying a new set of
volumes of the financial instruments applied for the funding,
said new set of volumes being calculated on the basis of the
financial instruments already issued for the funding and the
new payment profile and remaining debt profile, as well as
the requirement with respect to maximum permissible differ-
ence in balance.
One may choose to iterate over the proceeds requirement and
the difference in balance widened in consideration of the
refinancing profile input under (c), and iteration over the
interest rate on the loan is not performed until the two
requirements have been fulfilled. It will be understood that
the iteration may be performed in an arbitrary order, and
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43
that the iteration of the function applied, the so-called
trend function, may also be included herein.
In this case; the method in step th) in the inner loop may
comprise determining whether the calculated volumes of finan-
cial instruments fulfil at least two of two or more predeter-
mined convergence conditions which are preferably calculated
in consideration of a calculated difference in proceeds and a
difference in balance calculated in consideration of the
refinancing profile input under (c), and in cases in which
the calculated volumes of financial instruments do not fulfil
these conditions, the recalculations comprise one or more
iterations of the coefficients for the function which is
adjusted to a shifted level remaining debt profile, each
iteration comprising
calculating and storing data specifying two or more new
function coefficients for the function representing the
shifted level remaining debt profile,
calculating and storing data specifying a new set of
volumes of the financial instruments applied for the funding,
said new set of volumes being calculated in consideration of
the new function representing the shifted level remaining
debt profile.
determining whether the new set of calculated volumes of
financial instruments fulfils the at least two or more prede-
termined convergence conditions until the new set of calcu-
lated volumes of financial instruments fulfils these condi-
tions. The new function coefficient or function coefficients
may conveniently be calculated in consideration of the calcu-
lated difference in proceeds and a difference in balance
calculated in consideration of the refinancing profile input
under ( c ) .
A calculation of the difference between the interest rate on
the loan and the yield to maturity of the calculated volumes
of the financial instruments may be performed in the inner
loop, calculations being performed as to whether the differ-
ence in interest rates is so small that that is fulfils the
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44
requirement with respect to maximum permissible difference in
interest rates or a convergence condition for the difference
in interest rates, and in cases in which the requirements
laid down are not fulfilled, the recalculations may comprise
one or more interest rate iterations, each interest rate
iteration comprising
calculating and storing an interest rate adjustment, said
interest rate adjustment preferably being calculated in
consideration of the difference between the interest rate on
the loan and the yield to maturity of the calculated volumes
of the financial instruments, e.g. by use of a Gauss-Newton
algorithm,
calculating and storing data specifying a new interest
rate on the loan which is preferably based on the previous
interest rate on the loan and the calculated adjustment of
the interest rate on the loan,
calculating and storing data specifying a new payment
profile and a new remaining debt profile for the debtor, said
payment profile and remaining debt profile being calculated
in consideration of the new interest rate on the loan, the
volume, the term to maturity and the repayment profile of the
loan as input under (a), and the refinancing profile and/or
the funding profile input under (c),
calculating and storing data specifying a new set of
I 25 coefficients for the function which is adjusted to the
shifted level remaining debt profile, and
calculating and storing data specifying a new set of
volumes of the financial instruments applied for the funding.
In connection with the calculations in the inner loop of Type
P, it is also within the ambit of the invention at a time to
determine whether the calculated volumes of financial instru-
ments fulfil at least three of three or more predetermined
convergence conditions which are preferably calculated in
consideration of a calculated difference in proceeds, a
difference in balance calculated in consideration of the
refinancing profile input under (c), and a maximum permis-
sible difference in interest rates, and, in cases in which
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the calculated volumes of financial instruments do not fulfil
these conditions, have said recalculations comprise one or
more iterations, each iteration comprising
calculating and storing an interest rate adjustment, said
5 interest rate adjustment preferably being calculated in
consideration of the difference between the interest rate on
the loan and the. yield to maturity of the calculated volumes
of the financial instruments,
calculating and storing data specifying a new interest
10 rate on the loan which is preferably based on the previous
interest rate on the loan and the calculated adjustment of
the interest rate on the loan,
calculating and storing data specifying a new payment
profile and a new remaining debt profile for the debtor, said
15 payment profile and remaining debt profile being calculated
in consideration of the new interest rate on the loan, the
volume, the term to maturity and the repayment profile of the
loan as input under (a) and the refinancing profile and/or
the funding profile input under (c),
20 calculating and storing data specifying a new set of
coefficients for the function which is adjusted to the
shifted level remaining debt profile, and
calculating and storing data specifying a new set of
volumes of the financial instruments applied for the funding,
25 said new set of volumes being calculated in consideration of
the new function representing the shifted level remaining
debt profile.
determining whether the new set of calculated volumes of
financial instruments fulfils said at least three or more
30 predetermined convergence conditions,
In this instance, the iterations may also be performed apply-
ing a numerical optimization algorithm, preferably e.g. a
three-dimensional Gauss-Newton algorithm.
In cases in which the calculated set of volumes in the calcu-
35 lations in the inner model of Type P comprises at least one
negative volume, the negative volume or volumes may conveni-
CA 02297990 2000-02-O1
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46
ently be assigned the value 0 so as to avoid negative volumes
in the result, cf. above remarks on the normally undesired -
situation in which it is stated that the borrower is to buy
financial instruments, after which the calculations are
continued on the basis of the thus determined volumes of the
financial instruments.
According to one embodiment of the invention, it will also be
possible to determine the volume of the financial instruments
applied for the loan in the cases in which data corresponding
to a funding profile or a financing profile desired by the
debtor have been input, said funding or financing profile
comprising the desired financial instruments. In this case
the calculations may also comprise calculation as to whether.
the volume of the financial instruments in the specified
funding profile fulfils the requirement with respect to
maximum permissible difference in proceeds, and in the case
in which the specified volumes do not fulfil this require-
ment, one or more adjustments to the previously specified
volumes are made according to a preferred embodiment of the
method, adjustments being made until the new set of financial
instruments fulfils the requirement with respect to maximum
permissible difference in proceeds.
Apart from the calculation of the proceeds requirement, here
it is also preferred that a calculation is performed as to
whether the requirement with respect to maximum permissible
difference in balance is fulfilled, and in the case in which
the found volumes do not fulfil this requirement, one or more
calculations of new financial instruments for at least one of
the financial instruments not fulfilling the requirement with
respect to maximum permissible difference in balance.
Here, the calculation of new financial instruments is prefer-
ably performed for one or more financial instruments to which
repayments are payable in a period in which the requirement
with respect to maximum permissible difference in balance is
not fulfilled. In a preferred embodiment, the calculation
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47
will be performed for one or more financial instruments to
which repayments are payable in the last period in which the
requirement with respect to maximum permissible difference in
balance is not fulfilled. Here, the calculation of the new
ffinancial instruments may preferably be based on the differ-
ence in balance for the periods in which the corresponding,
previously found financial instruments do not fulfil the
requirement with respect to maximum permissible difference in
balance.
Generally, it applies to the method according to the inven-
tion that in many situations it will be possible, after a
result has been obtained, fio perform a new calculation on the
basis of other instruments to establish whether a less expen-
sive loan may thereby be achieved.
The range of financial instruments determined under (e) is
selected among a number of available financial instruments.
It will be understood that this number of instruments may, if
desired, be input to a data base in the computer system or be
available via a network, and that the determination may, if
desired, be performed automatically or semi-automatically by
means of the computer system according to predetermined
criteria or functions.
The invention also relates to a data processing system, such
as a computer system for determining the type, the number and
the volume of financial instruments for funding a loan,
determining the term to maturity and payment profile of the
loan, and further determining the payments on a payment
guarantee instrument designed to ensure that the payments on
the loan and the term to maturity of the loan do not exceed
predetermined limits, and from which instrument payments are
made to the debtor in situations in which the maximum limits
for payments on the loan and term to maturity would otherwise
have been exceeded, the loan being designed to be at least
partially refinanced during the remaining term to maturity of
the loan,
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- requirements having been laid down stipulating that
- the term to maturity of the loan is not longer than a
predetermined maximum limit nor less than a predeter-
mined minimum limit,
- debtor's payments on the loan are within predeter-
mined limits,
- requirements having been laid down stipulating a maximum
permissible difference in balance between, on the one
hand, payments on the loan and refinancing amounts and,
20 on the other hand, net payments to the owner of the
financial instruments applied for the funding, and pay-
ments to and from the payment guarantee instrument,
- requirements having been laid down stipulating a maximum
permissible difference in proceeds between, on the one
hand, the sum of the market price of the volumes of the
financial instruments applied for the funding of the
loan, and payments to and from the payment guarantee
instrument, and, on the other hand, the volume of the
loan,
- and requirements optionally having been laid down stipu-
lating a maximum permissible difference between the
interest rate on the loan and the yield to maturity of
the financial instruments applied for the funding,
said data processing system comprising
(a) means, typically input means and a memory or a stor-
age medium, for inputting and storing a first set of data
specifying the parameters: the volume and the repayment
profile of the loan,
(b) means, typically input means and a memory or a stor-
age medium, for inputting and storing a second set of data
specifying
(i) a maximum and a minimum limit for the debtor's
payments on the loan in each of a number of periods
CA 02297990 2000-02-O1
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collectively covering the term to maturity of the
loan,
(ii) a maximum and a minimum limit for the term to
maturity of the loan, and
(iii) optionally, a desired/intended payment on the
loan or a desired/intended term to maturity when the
maximum and the minimum limits for the payments in
the first period are not equivalent (i) or when the
maximum and the minimum limits for the term to matur-
ity are not equivalent (ii),
(c) means, typically input means and a memory or a stor-
age medium, for inputting and storing a third set of data
specifying a desired/intended refinancing profile, such as
one or more points) in time at which refinancing is to take
place, and specifying the amount of the remaining debt to be
refinanced at said points) in time,
and/or said third set of data specifying a
desired/intended funding profile, such as a desired/intended
number of financial instruments applied for the funding
together with their type and volumes,
(d) means, typically input means and a memory or a stor-
age medium, for inputting and storing a fourth set of data
comprising a maximum permissible difference in balance within
a predetermined period, a maximum permissible difference in
proceeds and, optionally, a maximum permissible difference in
interest rates equivalent to the difference between the
interest rate on the loan and the yield to maturity of the
financial instruments applied for the funding and, optional-
ly, the payment guarantee instrument,
(e) means, typically input means and a memory or a stor-
age medium, for determining and storing a fifth set of data
specifying a selected number of financial instruments with
inherent characteristics such as the type, the price/market
price, and the date of the price/market price,
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(f) means, typically input means and a memory or a stor-
age medium, for determining and storing a sixth set of data
representing a first profile of the interest rate on the loan
and either a first term to maturity profile or a first pay-
5 ment profile of the loan,
(g) means, typically calculating means and a memory or a
storage medium, for calculating and storing a seventh set of
data representing
- a first term to maturity profile or a first payment
10 profile (depending on what was determined under (f))
corresponding to interest and repayments for the debtor
- and a first remaining debt profile,
said term to maturity profile or payment profile, as well as
the~remaining debt profile, being calculated on the basis of
15 - the volume and repayment profile of the loan as input
under (a),
- the set of data input under (b),
- the refinancing profile and/or the funding profile input
under (c)
20 - and the profile of the interest rate on the loan and
either the payment profile or the term to maturity pro-
file established under (f),
(gl) means, typically calculating means and a memory or a
storage medium for, if necessary/desired, calculating and
25 storing an eighth set of data representing payments (posi-
tive, zero or negative) on the payment guarantee instrument,
the requirements with respect to maximum permissible differ-
ence in balance and maximum permissible difference in pro-
ceeds, as well as the limits for payments on the loan and
30 term to maturity, always being fulfilled,
(h) means, typically calculating means and a memory or a
storage medium, for selecting a number of financial instru-
ments among the financial instruments stored under (e), and
calculating and storing a ninth set of data specifying these
35 selected financial instruments with their volumes for use in
*rg
~.~--
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the funding of the loan, said ninth set of data being calcu-
lated on the basis of
- the payment profile established under (f) or calculated
' under (g) and
- the remaining debt profile calculated under (g),
I - the payments on the payment guarantee instrument option-
ally calculated under (gl),
- the refinancing profile input under (c) and/or the fund-
ing profile input under (c),
- the set of data input under (b),
- the requirements input under (d), and
- in the case of a refinancing where financial instruments
from a previous funding have not yet matured, the type,
the number and the volume of these instruments,
said means being adapted to perform, if necessary, one or
more recalculations, including, if necessary, selecting a new
number of the financial instruments stored under (e), said
means further being adapted to store, after each
recalculation,
- the recalculated profile of the interest rate on the
loan,
- the recalculated term to maturity profile,
- the recalculated payment profile,
- the recalculated remaining debt profile, and
- the selected financial instruments with their calculated
volumes,
until all the conditions stated under (b) and (d) have been
fulfilled,
and the means further being adapted to optionally recalculate
the payments on the payment guarantee instrument in accord-
ance with (gl), and store, after each recalculation, the
recalculated payments in the memory or the storage medium,
means for outputting the hereby determined combination of the
type, the number, and the volume of financial instruments for
funding the loan,
- together with the calculated term to maturity,
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- together with the calculated payment profile,
- optionally, together with the payments on the payment
guarantee instrument,
- preferably, together with the calculated interest rate on
the loan, and
- preferably, together with the calculated remaining debt
profile,
or means for transferring the combination, if desired, to a
storage medium or sending it to another computer system.
A computer system which may be applied for the method accord-
ing to the present invention may comprise means for inputting
and storing the data necessary for the calculations. The
input means may comprise a keyboard or a mouse, a scanner, a
microphone or the like but may also comprise means for carry-
ing out electronic inputting via a storage medium or via a
network. As mentioned above, the storage media may be elec-
tronic memories such as ROM, PROM, EEPROM or RAM, or storage
media such as tapes, discs or CD-ROM.
Furthermore, the system comprises calculating means adapted
to perform the calculations necessary for the implementation
of the method. Here, the calculating means may typically
comprise one or more microprocessors.
The system may be a computer system programmed such that the
system is capable of performing the calculations necessary
for the implementation of the method according to the inven-
tion. In this connection it will be understood that there may
be different embodiments of the system meaning that these
different embodiments are adapted to perform the calculations
specified in the different embodiments of the method accord-
ing to the invention mentioned above and in the claims.
Embodiments and details of the method and the system accord-
ing to the present invention further appear from the claims
and the detailed description in connection with the drawing
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53
and the example section. The example section contains - apart
from preferred examples of the method according to the in'ven-
tion - a description of a number of preconditions for the
invention, and of a number of preferred applications of the
method according to the invention, and of the results
obtained by the method.
In the example section, the expression "bonds" is used about
a financial instrument in the ordinary meaning of the word.
Thus, the expression covers all types of interest-bearing and
non-interest-bearing claims, including financial instruments
and bonds.
In the example section, the expression "financial instrument"
is used about a payments guarantee instrument as previously
defined.
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Brief description of the drawings .
giQure 1 shows an example of a lattice structure and tree
structure of a binomial model. A tighter structure is achieved
in the trinomial models.
Figure 2 shows the connection between the continuous structure
and the lattice structure. The average value for the interest
rate is determined by the initial yield curve to which the
model is calibrated. This appears from the figure by the graph
(1). In a deterministic model, pricing is performed solely on
the basis of this graph. The connection is illustrated in
principle. No calculations form the basis of the figure.
Figure 3 shows an example of the dynamic adjustment of the
lattice structure.
Figure 4 shows an example of the yield curve for r* prior to
the calibration to the initial, observed yield curve.
Figure 5 shows the possible impact of the recalculation of the
interest rate on the loan on the payment profile and remaining
debt profile of the loan.
Figure 6 shows the calculation of probabilities in the lattice
by means of Bayes' rule. In the figure the shown lattice is
calibrated to a flat yield curve.
Figure 7 shows the flow diagram of the model for a LAIR III
' type F.
Figure 8 shows the determination of the next value of the
interest rate on the loan in the iteration according to the
Gauss-Newton algorithm.
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Figure 9 shows the flow diagram of the model for a LAIR III
' type P.
Figure 10 shows an example of the adjustment of the trend
function. The volumes shown are not calculated. The loan could
5 be a LAIR type P20,0 in which extreme yield curve has resulted
in the volume of the bond with a term to maturity of four
years being disproportionate, for which reason the trend
function "breaks".
Figure 11 shows the possible pattern of the payments on the
10 financial instrument. In the upper part of the lattice, the
payments are positive due to the high interest rate. In the
lower part of the lattice, the low interest rate implies
negative payments on the instrument.
Figure 12 shows the pricing of the financial instrument in
15 each node according to the backward induction principle.
Figure 13 shows the flow diagram of the model for quoting the
limits for payments on the loan and term to maturity.
Figure 14 shows the flow diagram of the model for Type F' in
the case in which the limits for payments on the loan and term
20 to maturity are compatible, payments on the financial
instrument thus not being necessary.
Figure 15 shows the flow diagram of the model for Type F' in
the case in which the limits for payments on the loan and term
to maturity are incompatible, payments on the financial
25 instrument thus being necessary.
Figure 1S shows the alternative modelling of a LAIR III type
P.
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Figure 17 shows an example of the initial adjustment of the
trend function. The trend function is parallel-shifted upwards
when determining G, as G1=1.25. In this way the model obtains
better information on the marginal issue in the individual
years.
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1. Modelling the yield curve
1. 0 Iatroductioaa~
The underlying yield curve is important for the pricing of
financial claims (claims are to be taken in a broad sense as
securities, debt and financial instruments). The yield curve
is an expression of the interest rate of different claims as a
function of a selected characteristic feature. Usually, the
selected characteristic feature is the remaining term to
maturity or duration of the claim, and thus it is the
horizontal yield curve. By contrast, the vertical yield curve
is an expression of the interest rate of claims with identical
terms to maturity but with different credit risks, liquidity,
or the like.
In order for claims with different cash flows to be priced,
the horizontal yield curve is most often formulated as a
zero-coupon yield curve. Zero-coupon rates express the
interest rate of a claim with only one payment in the entire
term to maturity of the claim. Claims with different cash
flows may thus be seen as different portfolios of
zero-coupons, and once the zero-coupon yield curve has been
determined, a pricing of any known cash flow is possible.
The pricing of the bonds and the financial instrument
underlying the LAIR III is based on the horizontal zero-coupon
I' 25 yield curve. Therefore, a model thereof will be set up in the
following.
The finance~theory has many different suggestions as to the
modelling of the yield curve. The different suggestions
deviate by including, in different ways, factors, such as
volatility, observed market prices etc. At the same time, the
models are widely different in their degree of
operationnbility, which should be considered fairly important
for this purpose. Prior to the presentation and description of
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the selected model, aspects of the selection of yield curve
model are explained without this developing into a review of
recent yield curve theory, however.
1.1 Aspect~ of the selection of yield curve
The selection of yield curve rests on a number of criteria,
which are reviewed in the following.
Firstly, the question of the modelling of the stochastics
in the model is to be considered.
The possibilities in the finance theory range from completely
disregarding the stochastics in the deterministic models to a
modelling of several stochastic variables in the so-called
', multiple factor models.
~ Secondly, it has to be clarified whether the modelling is
to be based on an equilibrium model or a no arbitrage
model.
~ Thirdly, the question of the handling of the volatility
in the interest rates is to be clarified.
~ Fourthly, the pricing of the specific financial
instrument lays down a number of requirements as to the
modelling of the yield curve.
1.1.1 Stochastic mo~elliaQ of the yield curve
The use of stochastic processes in the economic theory is
generally justifiable in that no reliable models with perfect
predictability may be set up.
It is a prerequisite for perfect predictability, firstly, that
all economic connections are known and, secondly, that new
information does not become available to the agents of the
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economy on a current basis. For obvious reasons, perfect .
predictability is therefore considered unattainable.
In the financial markets in particular, there is a long way
from perfect predictability to the facts of the world. The
financial markets react currently to a very large quantity of
information, and the reaction patterns change on a current
basis. Thus. a deterministic modelling of the future
development in the interest rate will be very insufficient
indeed, and will not provide the basis of a reliable pricing.
By introducing stochastics in the description of the future
interest rate, a random principle is actually left to rule.
Therefore, it is far from i~nnaterial how the stochastics are
introduced.
In the main part of the finance theory, the stochastics are
introduced in such a Way that the interest rate will approach
an (equilibrium) level in the long term, whereas the
short-term movement may fluctuate quite significantly around
the long-term trend. This modelling of the stochastics seems
plausible assessed on the basis of economic principles. In the
short term, the disclosure of more or less irrelevant
information may influence the formation of interest rates
solely because economic agents predict the reaction of other
agents ete., whereas in the long term, the interest rate will
cormerge towards an equilibrium level which is not affected by
the irrelevant information. Realization of the equilibrium
level requires, however, that no new information is revealed
for a period of time, and will thus not necessarily occur.
One category of stochastic processes fulfilling the
above-mentioned properties is the so-called Ito processes. In
its general form, the Ito process is formulated by (1.1).
(1.1) dx=~(t, x)dt+a(t, x)dZ(t)
* rE~'
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where x is the state variable and t is time. In the Ito
process the drift is given by ~.(t,x)dt, whereas the diffusion
of the process - i.e. the fluctuations around the long-term
trend - is given by a(t,x)dZ(t).
5 In (l.l) dZ-(t) is a so-called Wiener process. (The process is
also termed a generalized Brownian motion). The Wiener process
is to be seen as the counterpart to a random walk in
continuous time, and is thus a random walk in continuous time.
In differential form, the Wiener process complies with the
10 following equation.
(1.2) dZ(t)=a ~E ~t -E dt
This connection is interesting in relation to the value of the
state variable of the Ito process at a future point in time t.
An expression thereof is obtained by writing (1.1) in integral
15 form.
r r
( 1. 3 ) xr =xa + j~1(z, x)dt+ Ja(~, x)d2(~)
0 0
It appears from (1.2) and (1.3) that the drift term will
dominate the diffusion in the long term as a result of the
higher order of dt in the drift term (dt) as compared with the
20 diffusion (~).
The Wiener process, and thus also the Ito process, are also
characterized by being Markovian. A stochastic process X1 is
said to be Markovian if
( 1. 4 ) P(xe~ ~xri , . . . , xrm ) = P(xr~ (xe~-~ )
25 A Markovian process has no memory. Only the immediately
preceding value xt_1 is crucial for the value of the process in
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the current period xt, whereas all other preceding values of
the process are immaterial.
When setting up a stochastic yield curve model, it is
exceptionally important that the process is Markovian. This
may be interpreted as path independence. For Markovian
processes, it is sufficient at any future point in time to
know only the current interest rate to determine the further
developments. The development in the movement up to the
current period is thus immaterial. If a discrete modelling
method is followed, it is thus possible to set up a lattice
instead of a tree causing the number of nodes to be reduced
drastically, even though the lattice spans a corresponding
interval of possible interest rates.
Examples of the lattice and tree structures appear from figure
1.
The Ito process given by (l.l) defines a one-factor model, the
only variable in the process being x. In the light of the
interaction of the interest rate with a wide range of other
economic variables, it would be desirable, from a theoretical
point of view, to have other factors influence the process for
the interest rate.
The finance theory has accepted this challenge in the
so-called ma~ltiple factor models which include factors such as
inflation, the interest rate in other countries, or similar
factors in the stochastic process for the interest rate. In
the nature of the case, also the included other factors are
described by a stochastic process. A prominent example is
Heath, Jarrow and Morton (1991) "bond Pricing and the
structure of interest rates: A new methodology for contingent
claims valuation" Working paper Cornell University.
*rB
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The multiple factor models have obvious theoretical
advantages, but suffer from the weakness of not being
Markovian. This means that in practice, the models may be
operationalized only to a small extent, which is a central
property for this purpose. Therefore, the modelling is limited
to a one-factor model in the following.
1.1.2 8quilibriumn models versus no arbitrage models
Stochastic yield curve models are generally divided into two
categories.
The first category consists. of equilibrium models. The basis
of these models is of a microeconomic nature. The yield curve
is determined in accordance with the preferences of the agents
so as to provide a balance in the capital markets.
However, it may be difficult to determine the preference
structure for the agents of the economy. First, the preference
structure should reflect the degree of risk aversion of the
agents, said risk aversion traditionally causing opinions to
differ.
The advantage of the equilibrium models is that as soon as the
preference structure is described, all claims may be priced.
The use of parameters in the models is thus limited. An
example of the equilibrium model is the CIR model (Cox,
Ingersoll and Ross 1985) "A theory of the term structure of
interest rates", Econometrics 53) and the Vasicek model
tVasicek (1977) "An equilibrium characterization of the term
structure", Journal of Financial Economics 5).
The problems related to modelling the preference structure of
the agents have led to the development of a new category of
models, the so-called no arbitrage models.
*rB
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The no arbitrage models are characterized in that the ,
modelling of the future interest rate is calibrated to an
observed initial yield curve and optionally to a volatility
structure. The modelling of the future yield curve is thus no
arbitrage, as no possibilities of arbitrage occur between the
observed prices and the claim prices fixed in the model.
However, the theoretical basis of the no arbitrage models is
not significantly different from the balance models, as no
arbitrage must be considered a prerequisite for an economic
equilibrium.
with the no arbitrage models, the modelling of the yield curve
is subject to a more narrow interpretation. The future yield
curves, which may be calculated in the model, are not to be
interpreted as an actual prediction of the future development
in interest rates, but as a calculation of what the observed
prices reveal about the expectations of the financial markets
for one to agree with or not. With this interpretation it
becomes legitimate not to include macroeconomic conditions in
the modelling of the yield curve.
The no arbitrage models have gained ground to an increasing
extent in recent years, especially in practical applications.
It may be an obvious advantage that the yield curve model is
calibrated to observed yield curves, causing an actual
estimate of the preference structure to be rendered
superfluous. At the same time, the risk of obtaining a pricing
which is distorted in relation to the market is reduced.
Thus, in the following, the focus is solely on no arbitrage
modellings.
1.1.3 Volatility
In the Ito process, the volatility in the future interest
rates is given by the diffusion term.
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cs'(t, x)dZ(t)
where the diffusion coefficient a may depend on t and x.
Consequently, both the yield curve and the volatility
structure change over time according to the Ito process.
In the no arbitrage models, the formulation of the diffusion
coefficient as time-dependent opens up the prospect of the
modelling of the yield curve being calibrated to the initial
yield curve as well as the volatility structure. At first
glance, this also seems like a natural element in the setup of
a reliable yield curve model.
However, Hull and White (1996)/(1994a) (Hull and White (1996) ,
is a collection of previously published articles.
(1996)/(1994a) refers to Hull and White's article from 1994
that is included in the collection of articles. This reference
is used henceforth) adduce an argument against the diffusion
coefficient being time-dependent.
In simulations of the extended Vasicek model, which will be
dealt with later on, the volatility structure proves to
develop very differently from the traditional perception of
the volatility. The future volatility structure is
particularly sensitive to the initial estimate of the
volatility of claims with a long term to maturity.
Hull & White compare the time-dependent diffusion coefficient
with an excessive parameterization of the model and conclude,
on this basis, that the most reliable results are obtained
with a value of Q(t,x) which is not time-dependent. It is
preferred that the following recommendation is followed in a
method according to the invention.
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1.1.4. Requiremaents with respect to the modelling of the yield
curve derived from the f3.naacial i.nstrnmant
The pricing of the financial instrument lays down requirements
with respect to the modelling of the yield curve.
5 The most important requirement is that the modelling is to be
performed in discrete time. The financial instrument is
characterized in that the payments on the instrument are
dependent on the other variables on the debtor and funding
sides of the loan, said variables being dependent of the yield
10 curve. This implies that the payments are determined at each
adjustment of the interest rate for the period up to the next
adjustment of the interest rate on the basis of the yield
curve. This pattern cannot immediately be described in a model
in continuous time.
15 For the modelling of the stochastics, the transition to
discrete time means that the continuous process for the
interest rate must be approximated by the discrete expression
(1.5) t1x=~.(t, x)Ot+a{t, x)OZ(t)
Even though the majority of the stochastic yield curve models
20 are based on the Ito process, far from all stochastic interest
rate models may be adjusted to discrete time.
In a series of modellings, Hull and White have developed a
general frame in which different, originally continuous yield
curve models may be made discrete and be implemented in a
25 trinomial lattice. Further, the model frame distinguishes
itself by being more operational than other discrete yield
curve models. Thus, there are good arguments in favour of
following Hull and White s (1996) approach to a stochastic
modelling of the yield curve in discrete time
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1.2 The currently preferred model - Hull ~ W'hite's extended
Vasicek model
In the following the selected stochastic modelling of the
yield curve is presented.
First, the general Ito process is specified, and the issue of
negative interest rates is discussed. Then the structure of
the lattice in the trinomial model is explained, and the
adjustment of the process to the lattice is deduced.
Eventually, the analytical results of the model are deduced,
the most important result being the deduction of the yield
curve in each individual point in the lattice.
1.2.1 The stochastic process for the interest rate
In order to be able to claim a stochastic process for the
interest rate, the Ito process must be concretized, the term
and diffusion coefficients having to be specified.
Hull and White's very general model frame permits an
implementation of a number of yield curve models. In its
general form the process for the interest rate may be set up
as in (1.6)
(1.6) dr=a(b-r)dt+ar~dZ(t)
The formulation of the term coefficient in (1.6) corresponds
to a so-called Ornstein Uhlenbeck process characterized by
being mean reverting. It follows from the formulation that the
interest rate will be drawn towards an equilibrium level
expressed by the parameter b at a velocity a.
Further, it follows from the specification that a must be less
than 1 in order for the interest rate to converge. The value
of a, together with the diffusion term, will determine the
volatility of the interest rate. A high value of a (close to
1) implies that the interest rate quickly returns to the
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equilibrium level. Thus, the volatility of the interest rate
is limited.
The value of ~ is crucial to the properties of the model.
If ~=0. Vasicek's (1977) model appears. This model is
characterized in that the diffusion is not dependent in the
level of the interest rate. The relatively simple model has a
number of favourable analytical properties (see e.g. Hull and
White (1996)/(1990)), primarily in relation to the pricing of
European options.
One disadvantage of Vasicek's model is that the negative
interest rates are not excluded, but will occur in the model
with a positive predictability. The occurrence of negative
interest rates constitutes a technical problem, as an
argumentation based on arbitrage arguments can hardly be
extended to a situation with negative interest rates.
In practice, the problem is manageable. Firstly, the
likelihood of negative interest rates will be limited by a
realistic determination of the parameters of the model.
Secondly, a situation has arisen in practice, in which the
interest rates have been 0 (zero) ar negative as a consequence
of imperfections of the market. At the same time, the model -
like most other models - paves the way for very high interest
rates. The focus on the possibility of negative interest rates
with merely a minor probability may therefore seem
exaggerated.
It appears from the following that Bull and W'hite's method for
implementing the model contains a facility which reduces the
possibility of negative interest rates.
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The probability of negative interest rates may be reduced to 0
(zero) at a second specification of (3. If a value within the
interval ]0.1[ is assigned to Vii, the effect of the diffusion
of the interest rate process will grow drastically if the
interest rate approaches 0 (zero). With the probability of 1,
the interest rate will thus be increased by the diffusion
before it assumes a negative value. r=0 will thus for ~iE]0.1[
constitute a reflective barrier to the interest rate. Instead
of a reflective barrier, an absorbent barrier could be
modelled, in which the interest rate remains 0 (zero)
following the observation of r=0.
In the CIR model, the value 'f~ is assigned to (3 , and thus the
CIR model do not allow negative interest rates with a positive
probability. Thus, in comparison with the Vasicek model, the
CIR model has an obvious theoretical advantage. In practice,
the formulation of the diffusion results in the model being
difficult to implement. To Hull and White this is a crucial
argument in favour. of applying Vasicek's model as a basis,
said argument being the one currently preferred to follow in a
method according to the invention.
As formulated in (1.6) the model cannot immediately be
calibrated to an initial yield curve. Hull and White
(1996)/(1994a) makes the model no arbitrage by introducing a
further time-dependent parameter in the drift of the model.
Thus, the model appears as follows:
( 1. 7 ) dr = (9( t) - ar)dt + o'dZ( t)
In this formulation the equilibrium level is thus seen to be
decided by A(t) which is determined on the basis of the
initial yield curve. (1.6) is termed the extended Vasicek
model, and in the following, this model will be implemented in
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a discrete trinomial model. That is to say that (1.7) in Hull
and White is approximated 'by
( l . 8 ) 0r = (9( t) - ar)~ t + aAZ( t)
1.2.2 The lattice structure in the trinoanial model
Hull and White (1996)/(1994a) implement the extended Vasicek
model in a trinomial lattice.
The idea underlying the implementation of the model in a
trinomial lattice is that the lattice is to reflect the
development in the underlying continuous interest rate
process. In the underlying continuous process for the interest
rate, a continuous distribution of the adjustment to the
interest rate will exist for every t, the distribution, the
average, and the variance being determined by the continuous
process for the interest rate.
The continuous distribution is approximated in the lattice by
a discrete distribution consisting of an increasing number of
nodes. From each node there are three branching possibilities:
up, middle, and down. The probabilities of each of these
results are determined in each node such that the process in
the discrete lattice develops (approximately) in the same way
as the underlying continuous process. However, a difference
will always occur as a consequence of the transition from a
continuous to a discrete distribution.
The difference becomes evident in the Vasicek model in which
the adjustment to the interest rate at any point in time will
follow a continuous normal distribution. This follows from the
specification of the drift and diffusion coefficients. Thus,
in the continuous distribution of the interest rate, no
maximum and minimum limits for the adjustments will exist, cf.
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the discussion of the positive probability of negative ,
interest rates in section 1.2.1.
In the lattice there will be a maximum and a minimum limit for
the adjustment to the interest rate for a given value of t.
5 This follows from the distribution being discrete. Hence, the
lattice will not span an interval of adjustments as wide as
will the continuous distribution.
The connection between the underlying, continuous process and
10 the lattice structure is illustrated in figure 2.
Hull and White do not only adjust the probabilities to the
drift and volatility of the process, but also to the branching
structure. In the model the adjustment of the branching
structure to the drift of the process is introduced by the
15 parameter
k={-1,0,1)
which expresses whether the branching is to be pushed up or
down. The adjustment of the branching in the lattice is
performed for extreme values of the interest rate. For a very
20 high interest rate in the lattice, the drift downwards of the
process - which is mean reverting cf. section 1.2.1 - may be
so strong that the discrete distribution will be able to match
only the underlying continuous distribution by one or two of
the probabilities being negative in the said node. Thus, the
25 probabilities will not be probabilities in a theoretical
sense. This problem is solved by the branching being pushed
downwards. Similarly, for extremely low interest rates, a
situation might occur in which the branching is to be pushed
upwards.
30 Depending on the parameters of the model and the yield curve
to which the model is calibrated, the minimum limit will
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fluctuate. It is not a foregone conclusion that the minimum
limit is determined at a positive interest rate (>_0). Thus,
the branching procedure does not preclude for certain the
occurrence of negative interest rates in the lattice.
In figure 2 it is presumed that k assumes the value 0 in all
nodes, so that the branching structure is fixed in the entire
lattice. As an example of the adjustment of the lattice
structure, it is presumed in figure 3 that k has the value -1
in the upper node (2) and 1 in the lower node (3) for t=2.
At the same time, the adjustment of the lattice structure has
the favourable feature that the range of the lattice is
limited. The model thus obtains a higher degree of
operationability, as it is not necessary to operate with
extreme interest rates, which is also uninteresting in
practice.
By contrast, the adjustment constitutes a (minor) theoretical
problem for the pricing. The adjustments of the branching
structure and the probabilities are performed such that the
interest rate process still corresponds to the underlying
continuous distribution. In so far as the payments on the
financial instruments do not follow the same distribution,
i.e. they are not linear in the interest rates, the adjustment
will result in an imbalance in the determination of the
payments. Whether this imbalance affects the pricing is
difficult to assess. Hull and White's model frame is applied
in many situations, and e.g. in the pricing of options that
must be considered very sensitive, indeed, to these
imbalances. Thus, the consensus is that these imbalances may
be ignored.
1.2.3 Deduction of the probabilities and the branching
structure
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In Hull and White (1996)/(1994x) and (1996)/(1996), the
deduction of the probabilities and the branching structure is
performed in two stages. First, the lattice structure of the
model that is not calibrated to the initial yield curve is
determined. In this model, the probabilities of the different
results and the branching are determined. Then the calibration
is introduced as a displacement of the lattice.
Hull and White begin by considering the continuous interest
rate process for dr*, which appears by setting A(t) and the
l0 initial value of r to 0 (zero) in (1.7).
( 1. 9 ) dr' =-ar'dt+adZ( t)
The adjustment to the interest rate during a time interval of
At is distributed in a normal manner. The following
assumptions are made about the distribution of the discrete
adjustments
(1.10) Ar'=r"(t+~t)-r'(t)-1N(r'(t)E, ~ where
rs(t)E~ (e_~ne _ 1)r.(t) =_8r.(t)~t
(1.11)
VeQ21 2a~t =aaAt
are the average and the variance, respectively, in the
distribution of the adjustments.
A(t) is the step size in the lattice. The step size may be
determined arbitrarily in consideration of the claim that is
being priced. The determination of the step size is discussed
in more detail under the pricing of the financial instrument
in section 3.
The adjustments of the process to the interest rate are kept
fixed in the entire lattice. Since the lattice is based on an
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initial value of 0 (zero) far the interest rate, the lattice
is symmetrical at r*(t)=0.
A value for Dr' is to be determined. On the basis of numerical
analysis, Hull and White recommend to set
Ar' = 3V
which is here followed without further argumentation.
In the lattice each node may be described at the point in time
and at the interest rate, i.e. (t,r). The fixed values for
both At and Dr' open the prospect of a more appropriate
notation based on the adjustments. Each node may thus be
described by tg.h). where g denotes the number of periods
elapsed and h denotes the number of up results. This gives the
following relations
(1.12) t=gOt~g=of and r'=hAr'~h=~,.
for g=0,1,2,... and h=O,tlt2... The probabilities of up,
middle, or down branching are denoted P~, Pm and P". Three
requirements may be laid down with respect to the
probabilities. Formally, it is the fulfilment of these three
requirements which makes Po, Pm and P" eligible for being
perceived as probabilities.
Firstly, the average of the adjustment in each node in the
discrete lattice is to correspond to the average of the
underlying continuous process r*(t)E. This may be formalized
to
( 1. I3 ) Po(k+ 1)Or' + PmlsOr' + Pn(k-1)~r' = hllr'E
It should be noted that the expression applies to the average
of the adjustment to the interest rate and not of the interest
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rate as in e.g. Hull and White (1996)/(1993). Thus, h is not
included on the left-hand side of the relation. h appearing on
the left-hand side of the relation is due to the average of
the adjustment being level-dependent.
Secondly, the expressions of the variance must be in
accordance.
( 1.14 ) Po(k+ 1)24x''2 + Pmk2Ar'a + Pn(k-1)2~r'2 = V+(hAr'E)2
The expression follows from the relation E[X']-E[X)~=VAR[X].
Finally and thirdly, the probabilities are to sum to one.
(1.15) Po+P",+P"=1
Thus, the probabilities may be found as the solution
to three equations with three unknown quantities.
The probabilities may thus be found as the solution
to the matrix equation
(1.16) AB=C where
k+1 k k-1
A= {k+ 1)ZOr'Z k2~r'Z (k-1)ZAr'a
1 1 1
Po
( 1.17 ) B = Pm and
Pn
hE
C= V+(hAx''E)2
1
In (1.13) it applies that fir' may be reduced out in (1.13). The
probabilities are found by inverting the A matrix.
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a apr.~2 a
(1.18) Ai= 2k -~.f (1-k)(1+k)
ax+i ~, xcxm>
a anrw
The probabilities may then be found by A-1C.
(1.19) Po=lixhE+a~.z~V+(hAr*E)a ~+~zl
( 1.20 ) P~, = 2khE- ~., ( V+(hAr*E)a ,+(1-k)(1+k
5 ( 1. 21 ) P" _ -laakhE+ a~.~ ( V+(hOr*E)a , + ~ 2 i
As mentioned previously, k may assume the values (-1,0,1} in
the lattice. If the possible values for k are inserted in
(1.19) to (1.21), the actual probabilities for the branching
are deduced, at the same time applying that
10 V= 3Ar*a
In nodes in which the branching is normal, i.e. k=0, the
following probabilities apply
(1.22) Po=6+a(haEa+hE)
(1.23) Pm=3-haEa
15 (1.24) Pn=6+a(haEa-~)
It appears from (1.22) and (1.24) that the probabilities of an
up and a down branching are symmetrical functions of E, as
could be expected when the branching is normal.
In nodes in which the branching is increasing, i.e. for k=1,
20 the following probabilities may be determined.
(1.25) Po= 6+Z{haEa-hE)
(1.26) Pm= 3-haEa+2hE
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(1.27) Pn=6+a(hZEZ-3hf~
Finally, the probabilities at a decreasing branching for k=-1
are given by
(1.28) Po=6+2(h2E2+3hE)
(1.29) Pm= 3-h2Ez-2hE
(1.30) P"=6+i(haEa+hE)
The only factor lacking in the determination of the lattice
structure is the determination of a value for k. For a
sufficiently large value of h - and thus for a high interest
rate - the drift downwards of the process is so strong that
one or more of the probabilities is/are immediately to be
assigned a negative value in order to fulfil the requirement
given in (1.13). For this value of h, k=-1.
Similarly, a low value of h forms a floor below the interest
rates in the lattice. Thus, the lattice structure will appear
as shown in figure 4.
For which values of h the branching structure is to be changed
is determined by the probabilities on the basis of a
requirement that these must not be negative. The probabilities
for all values of k are to be tested, not only a maximum value
of h having to be determined prior to the branching being
pushed downwards, but also a lower value of h, and vice versa
Notwithstanding the value of h, the probabilities for both Po
and P" will be positive. Therefore, the focus is now on Pm. It
has proved convenient to define the variable x = E. For k=0 we
have
(1.31) x2+3 ~0 ~-0, 81655xS0, 8165
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and for k=1
(1.32) --.xa-2x+6~0~0, 18355xS1, 8165
and for k=-1
(1.33) -xa+2x-3 ~0~-0, 1835Sx5-1, 8165
The definition of E means that it will always assume negative
values for a>0. This means that the maximum limit for h (hmdx)
is to be found among the negative values of x. The maximum
limit for h is given by a integer value fulfilling
(1.34) -0,1835 shy ~-o.aiss
E E
Similarly, the minimum limit for h is given by
(1.35) °, fiss Shun 5 °' s3s
Once again the limits are sym;netrical, as expected.
1.2.4 The adjustment to the initial yield curve
The calibration to an initial yield curve is performed by a
new lattice being formed as a displacement in the vertical
plane of the old lattice. The displacement is determined such
that the lattice prices zero-coupon bonds in accordance with
the observed zero-coupon rates constituting the initial
zero-coupon yield curve. Thus, the underlying process for the
interest rate is once again given by
( 1. 8 ) dr = (8( t) - ar)dt + adZ( t)
The displacement of the lattice is introduced by the parameter
a*9 which is time-dependent but not level-dependent. For a
given g. the displacement of all nodes will thus be identical,
making it possible to determine the interest rate in each node
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as the interest rate in the previous lattice plus a9. The .
lattice thus maintains its symmetrical structure at h=0.
w
The determination of a9 is carried out on the basis of 9g(~t)
8(gOt) which is an estimate for 8(gAt). In the previous
lattice, t=0 is in the vertical centre of the lattice.
Further, r*=0 for t=0. This means that the drift (for h=0) in
the new lattice is given by
w
9(gAt)-aa~ .
From time (g-1)Ot to get, an adjustment of the interest rate
in the new lattice is given by
(r'(t)+a~)-(r'(t)+aø1) =a~-aø.l
The adjustment of the interest rate is to correspond to the
drift of the process. Hence, the following relation between
w
9g(At) and a9.
(1.36) (9(gAt)-aa~)fit=ag-a~la8(gAt)=°''o°~'+aa~,
where
w
( 1.37 ) a ..~.og(g~t) =9(t)
It follows from (1.36) and (1.37) that the calibration of the
interest rate process is in place when an expression for a9 has
been found. This expression is found by following a forward
induction method.
First, a new set of probabilities is to be introduced in the
lattice. Let q (h',h") denote the probability of a movement
from the node (g',h') to (g'+1,h"). The q-probabilities are
merely rewritings of P~, P~,. and P" which serve to facilitate
the notation, the branching being built into qt.).
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An Arrow Debreu asset with the price (Q(g,h) is then to be
introduced. This asset is characterized by having solely the
payment 1 in the node (g,h), whereas no payments from the
asset are found in any other node. Thus, the price of the
asset will reflect the discounting up to gOt as well as the
probability with which (g, h) is expected to occur. It follows
from the definition of the Arrow Debreu asset that Q(0,01=1 as
the payments of the asset are secure
The advantage of applying Arrow Debreu assets may be
illustrated by an arbitrage argument showing the determination
of Q(g'+1,h') given Q(g',h')for all h.
The state (g'+1,h') is obtainable in three ways from the nodes
( g' , h'-1 ) , ( g' , h' ) and ( g' , h'+1 ) . The argument is based on the
normal branching but may be immediately extended to situations
in which the node is obtainable in less or more ways. In the
node (g', h'-1 ) , the probability of (g'+1, h' ) being realized is
given by c~(h'-1,h'). In the node the interest rate is given by
r(g',h'-1). The expected value of the payment 1 in (g'+1,h') is
thus ( g' , h'-1 ) in the node .
(1.38) q(h'-1, h')e-~~'',w-We
and in the node (0,0)
(1.39) ~9r', h'-1)g(h'-1, h')e'rc~r',h'-We
Similarly, the values in the other nodes are discounted to
(0,0) given by
(1.40) fig', h')9~h', h')e-~°'.n')oc i (g',h') and
(1.41) Q(g', h*+1)q(h'+1, h')e ri~'~h'+lo' i (g',h'+1)
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In a state of arbitrage equilibrium, the sum of these values
is to correspond exactly to the Arrow Debreu asset with the
payment 1 in (g'+1,h'), as the expected payment is the same.
Consequently, in a state of no arbitrage, it must apply that
5 the sum of (1.39) to (1.41) equals Q(g'+1,h'). This may be
generalized to
( 1. 42 ) Q(g' + 1, h') _ ~'r ~9~' , h)~h, h')e n~'',hnc
h~-1
If the notation is adjusted to the specific problem and to all
10 possible branching structures, (1.42) appears as follows
( 1. 43 ) ~9~~' + 1, h') _ ~ Q(g' . h)9~h. h')e'~a~'+hnr~et
h
The deduction of (1.43) follows a forward induction argument,
as it is possible on the basis of Q(0,0) to determine Q(1,-1),
Q(1,0) and Q(l,l), etc. A corresponding forward induction
15 argument is applied in the determination of ag such that the
a~s are determined successively for g=0,1,2,...
First, afl is to be determined on the basis of the zero-coupon
bond with a maturity of lAt. The zero-coupon bond is assumed
to have the observed price
20 ( 1. 44 ) P(0, lit) = el~o,otiae
where R(.) is the observed zero-coupon rate. In the interest
rate lattice, the bond is to have the price
( 1. 45 ) Q(0, 0)e-°~~ec = e-aone
as Q(0,0)=1. The q probabilities are not included in the
25 expression. The zero-coupon has payments in all the nodes at
time lit. Therefore, it is of secondary importance for the
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payment which node is realized after (0,0). Tt follows from
(1.45) that
( 1. 46 ) e-te~o.°e~°c = e-ao°c p ao =R(0. At)
This Was immediately apparent since the old lattice is
constructed in such a way that r=0 in (0,0). The same method
is applied to the determination of a,
(1.47) P(0, 20t)=Q(1,-1)e'~°'~'°ro°c+Q(1, 0)e-
m°c+Q(1, 1)e~«i+~r'~t
which, reduced for al, provides the expression
r i
( 1. 48 ) al = QtClogl ~1 Q(1, h)e'h°t'~°r ~ _ log(p(0 ~ 20t))
The expressions in (1.47) and (1.48) may be generalized into
an expression for a9.. On the assumption that Q(g,h) is
determined for all g g<-g' , a~, is to fulfil
( 1. 4 9 ) P( 0 , g' D t) = F.r Q( g' . h) e~a°..~her~ )°t
h
which solved for a.~. produces the expression
(1.50) a~~=o ~log~Q(g", h)e~~~°t-logP(O, g'~t)~
h
where Q(.) is given by (1.43) and P(.) is the observed price
of the zero-coupon bond.
The lattice has thus been calibrated to the initial yield
curve. One example of the possible appearance of the lattice
is shown in figure 3.
1.2.5 The results of the arodel
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The interest rate modelled above is defined over a period of
fit. It is thus a short-term interest rate, depending on the
exogenous selection of the step size in the lattice, but not
as short as in continuous models in which the instant interest
rate is modelled.
The modelled interest rate may be sufficient for the pricing
of claims in which the payments are independent of the
prevailing yield curve. In such instances, the modelled
interest rate is applied to the discounting of the payments
via the lattice, causing the present value, and hence the
price, to be determined.
However, it may apply to the specific problem that the
payments may be dependent on the yield curve. The payments are
determined as the loss or gain in proceeds in connection with
the bond funding of the loan, provided the payments on the
loan and the term to maturity are within the allowed limits,
cf. section 2. Since the loan may be funded by bonds with a
maximum term to maturity of 11 years, the payments on the
instrument will consequently to a considerable extent depend
on the interest rate curve, and not just on the 0t interest
rate. In each node in the lattice, therefore, a yield curve,
and not just an interest rate, is to be determined.
Hull and White (1996)/(1996) deduce an expression for the
yield curve, which may be calculated in each individual node.
The deduction goes via Ito's lemma which is too comprehensive
to be reviewed herein. Therefore, the focus is solely on Hull
and White's deduction of the expression.
For this purpose, the price of a zero-coupon bond at time t
with maturity at time T>_t is defined as P(t,T). P(t,T) will
then fulfil
(1.51) P(t. T7=A(t, T)e &t.T~P where
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(1.52) logA(t, T)=logpto.~'~'&t~ ~~~~ t)'4a{1'e Zat)8(t~ ~2
(1.53) E(t, T)=;{1-ea«t))
It follows from the expressions that P(T,T)=1, as B(T,T)=0 and
A(T,T)=1. This is an obvious result, the price of the secure
payment of 1 being 1 in a state of no arbitrage. P(T,T)=1 is
also the marginal condition for the solution to the partial
differential equation under Ito's lemma, and is therefore
fulfilled by definition.
In the expression of A(t,T) F(O,t) is included, which is the
instant forward rate at time t seen from time 0. F(O,t) is
given by
( 1. 54 ) F(0, t) = aioQrto, t)
Thus, the deduction of the future yield curve presupposes that
the estimated initial yield curve (P(O,t)) is given by an
expression which may be differentiated.
Further, the future zero-coupon rates are seen to depend on p
which is the instant interest rate, whereas r in the lattice
is a 0t period interest rate. However, a conversion from r to
p is possible by means of (1.51). (1.51) provides the
possibility of calculating a longer interest rate on the basis
of a short-term interest rate. Let (t, T) be given by (t,t+0t).
( 1. 55 ) P(t, t+~t) =A(t, t+~t)e ~t~ t'''°t~
As P(t, t+At) =e~°' it follows from ( 1. 55 ) that
( 1 . 56 ) p = ret+A(t, c.r~t)
s(t, t+ot)
Subsequently, this expression may be inserted in (1.51)
*rB
.._
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ree,Ma.c.a~1
(1.57) p{t,'I'~=A(t,'!'~e e~c,z7 e,c.c,ec~
According to (1.51), the zero-coupon rate from time t to T may
be calculated on the basis of the relationship between the
interest rate at time 0 of zero-coupon bonds with maturity at
time t and T, respectively. This implies that it is impossible
to calculate zero-coupon rates stretching further forward than
the initial yield curve which typically corresponds to the
length of the lattice.
1.2.6 Estimation of the ina.tial yield curve and the parameters
of the model.
A few comments are to be made concerning the estimation of the
initial yield curve.
Firstly, it has to be clarified what bonds are applied for the
estimation. Government bonds are often applied, optionally
together with secured money market investments with a very low
risk The resulting yield curve is hence essentially risk-free.
At the same time, liquidity premiums do usually not form part
of the estimated yield curve, as the government bonds applied
are typically liquid securities.
It should be expected that it applies to the specific problem
that the financial instruments have a credit premium as well
as a liquidity premium, as the instruments are most often not
in conformity with the market. The yield curve is also used in
the calculation of future prices of the underlying bonds which
are also priced with an interest differential in relation to
similar government bonds.
This is not specifically allowed for in a preferred modelling
of the yield curve. However, credit risk and liquidity
premiums may be introduced in the pricing by the initial yield
curve being estimated on the basis of similar securities.
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However, it would be of great importance that correction for
the term to maturity effect is performed, so that a credit
premium, which was high at the time of the estimation, of a
bond with a long term to maturity, does not make itself felt
5 in the future short-term interest rates. Consequently, the
built-it credit and liquidity premiums are to be fairly
constant for~~the different terms to maturity.
In the determination of the parameters of the model. Hull and
White (1996)/(1994a) recommend a very general, but not very
10 operational principle. According to Hull and White, the
parameters of the model are to be determined such that the
volume of
~(pi -vita
i
is minimized, where P1 is the market price of the ith financial
15 claim, and Vj is the pricing in the model of the said asset.
On an operational level, the parameterization of the model is
a question of determining values for a and Q which are both
crucial for the volatility of the interest rates.
a determines the volatility of the short interest rate. In the
20 long term, the drift will, as already mentioned, dominate the
diffusion and hence the part of the volatility originating
from a. a should therefore be estimated on the basis of the
observation of the volatility in the short end of the interest
rate spectrum.
25 a determines how fast the interest rate approaches the
equilibrium level which is determined in the model by the
initial yield curve. Thus, the relationship between the
volatility of the short interest rate and the volatility of
the long interest rate is decisive for the determination of a.
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Therefore, a should be determined on the basis of observations
of this relationship.
1.3 Summmsy
The basis of the pricing of the financial instrument in a LAIR
III is a modelling of the future interest rate.
Firstly, the future interest rate will be decisive for the
volume of the payments on the financial instrument. Secondly,
the future interest rate will be decisive for the present
value of the future payments, and hence for the price of the
instrument. The modelling of the future interest rate is thus
an important part of the method according to the invention.
The applied model is Hull and White's extended Vasicek model
which has a number of theoretical as well as practical
advantages.
Firstly, the interest rate in the model is considered
stochastic. Perfect predictability should be precluded for
obvious reasons. Therefore, a reliable modelling of the
interest rate must involve stochastics. The stochastics is
introduced in the model via the general Ito process which has
a number of favourable characteristics. It follows from the
Ito process that the interest rate i the short term will
fluctuate about an equilibrium level which is approached by
the interest rate in the long term. And the Ito process is
Markovian, which permits the implementation of the model in a
discrete interest rate lattice.
Secondly, Hull and White's extended Vasicek model belongs to
the category of no arbitrage models. This means that the model
may be calibrated to an initial, observed yield curve. Thus,
provision is made for the theoretical prices of the model
being in accordance with the observed market prices. The
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calibration also implies that an estimation of the risk .
aversion of the agent is unnecessary.
Thirdly, Hull and White's model frame is characterized by a
high degree of operationability. Firstly, the model frame
permits the setting up of continuous stochastic processes for
the interest rate in discrete time. Secondly, the model may be
implemented in a discrete trinomial lattice. The very
possibility of implementation in a discrete lattice is
essential in the determination of the payments on the
financial instrument.
The construction of the interest rate lattice constitutes the
most important part of the model. The lattice structure is
adjusted to the drift in the stochastic process on a current
basis such that the lattice structure reflects the initial
yield curve. The lattice structure may also be made dynamic so
that a modelling of extremely low or extremely high interest
rates is avoided.
An essential result of the model is the deduction of the yield
curve in each node in the lattice. The payments on the
financial instrument depend not only on the short-term
interest rate but on the entire yield curve. Therefore, the
result is a prereguisite for the determination of the payments
of the instrument.
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2. Modelling the debtor and funding sides of a
LAIR III
2.0 Iatroductioa
In this section, all variables on the debtor side as well as
the funding side of the loan are modelled.
A LAIR III comprises a financial instrument combined with a
LAIR II or a LAIR I. A LAIR I may be perceived as a special
case of a LAIR II. In the situation in which identical maximum
and minimum limits are determined for the term to maturity,
l0.these limits will be fixed during the entire term to maturity
of the loan. Thus, said LAIR II degenerates to a LAIR I.
When setting up a model, this implies that it is sufficient to
model the debtor and funding sides of a LAIR II. Via the
determination of input to the model, a LAIR I emerges by
itself. If it is concluded, however, that a LAIR III is to
function solely as an extension of LAIR I, it is possible to
reduce the scope of the model.
The model for a LAIR III consists of a model for a LAIR II, as
well as an extension managing the financial instrument. Prior
to the review of the model, the section contains a more verbal
review of central aspects of the model. In section 2.1 aspects
regarding an adjustable term to maturity, including the
determination of the term to maturity, the term to maturity
concept, etc. are explained. In addition, it is discussed in
the section how the limits for the term to maturity and the
payments on the loan are determined in consideration of the
debtor's costs pertaining to the financial instrument.
Section 2.2 contains a general description of how to hedge the
limits of the loan in combination with a LAIR II. This is may
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be done in different ways, each of which is discussed before
the established method is described.
In section 2.3 the implementation of the model in the lattice
is explained. The model set up calculates the debtor and the
funding sides of one interest rate adjustment period at a
time. Therefore, the calculations in the model are to be
performed in each node which coincides with an interest rate
adjustment, which requires e.g. a determination of the input
to the model in each node. A method for determining input is
l0 thus deduced in the section.
Section 2.4 contains a review of the adjustment of the
interest rate on a LAIR. The adjustment of the interest rate
on a LAIR (for both LAIR I, II and III) divides the product
into two types of loans, LAIR type f arid type p, respectively,
with very different characteristics with regard to the
determination of the bond volumes, etc. Consequently,~a
distinction must be made in the modelling between these two
types of loans.
Subsequently, the models are set up for type F and then type
P. Furthermore, it should be noted that appendix A contains
the modelling of a variant of type f which occurs with certain
structures of input. Moreover, apgendix B contains an
alternative method for modelling type P.
2.1 Adjustable term to maturity - the general problem
If, for one moment, the payments on the financial instrument
are not taken into account, the interest rate on the loan will
rise and fall in line with the interest rate level at the time
of the interest rate adjustment. A falling interest rate is
not a problem for the debtor (the interest rate risk of the
remaining debt of the loan not being taken into account), and,
therefore, a falling interest rate does not give rise to
considerations regarding the product. However, the adjustment
to a higher interest rate level constitutes a potential
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problem for the debtor. A rising interest rate may influence
the loan in two ways.
Firstly, the payments on the loan may increase.
~ Secondly, the term to maturity of the loan may increase.
5 The optimum method seen from debtor's point of view cannot be
generally established but depends on the individual, and
possibly level-dependent, preferences of the debtor. Thus, it
is imaginable that the debtor prefers that minor increases in
the interest rate affect the payments on the loan, the
10 adjustment to large increases in the interest rate being
performed via the term to maturity of the loan.
A LAIR with an adjustable term to maturity is characterized in
that the payments on the loan float within a band defined by a
set of maximum and minimum limits for the payments on the
15 loan. The payments fluctuate as the loan is interest rate
adjusted to the prevailing market rate. The limits are denoted
YD~' og YD jin
respectively, YD generally denoting the debtor's payments on
the loan, and J indexing the interest rate adjustment periods.
20 J=0 specifies the disburse~nt date of the loan, and J=M
specifies the most recent interest rate adjustment such that
O SJSM
The fluctuation within the band is ensured by a correction of
the term to maturity of the loan when the payrc~ents on the loan
25 would otherwise have fluctuated outside the bead. The possible
corrections are defined on an interval defined by a maximum
and a minimum limit for the term to maturity. Similarly, the
limits for the term to maturity are denoted L'"°" and L"''°, L
generally denoting the term to maturity.
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No requirements in the model stipulate that the term to
maturity is in integer years or payment periods. The possible
corrections of the term to maturity are defined on a
continuous interval limited by L'"a" and L""°. This is necessary
if it is to be possible at each adjustment of the interest
rate on the loan to calculate a payment which is within a
relatively narrow band. At the maturity of the loan, however,
the term to maturity is corrected such that the loan matures
on 1 January at the same time as the underlying bonds.
The fact that the possible corrections of the term to maturity
are defined on a continuous interval also opens up the
possibility of offering the borrower fixed payments on the
loan. In the model fixed payments on the loan correspond to
the maximum and minimum limits for the payments on the loan
being equivalent such that
max min
~J -~J
In the model, there is no general need of distinguishing
between fixed payments on the loan and payments on the loan
within a maximum and a minimum limit. For a LAIR III identical
determination of the maximum and the minimum limits is
conditional on the initial term to maturity being very far
from the maximum limit if the financial instrument is not to
be too cost-intensive.
2.1.1 The limits for payments on the loan and term to maturity
The limits for payments on the loan and term to maturity are,
basically, selected by the debtor, and are thus considered
exogenous in the model. However, in practice, a need will
arise of the limits being determined with a view to the
resulting pricing of the financial instrument, which will be
elaborated on in section 2.1.2.
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The notation permits the definition of different limits for
each interest rate adjustment period during the term to
maturity of the loan. In the case of annuity loans, it would
be sensible to have a fixed maximum limit and a fixed minimum
limit corresponding to
m i i M
2 .1 ) Y~p~" _ 1 =~ ~ . _ and YDp n = Yt~i" ' = yD'Qin
The subsigns are maintained in the notation in order not to
lose generality. In principle, the notation also opens up the
prospect of having the serial loans being covered by LAIR III.
The payment profile of a serial loan is decreasing, for which
reason fixed limits for the payments on the loan would produce
inconvenient results. By defining a decreasing band over time
such that
YD~"' >_ YD~°" >_ Y~ . . . ~ YD~ " and
Y~n Z YD~in ~ Yt~in ~ . . . Z Yt~in
sensible results may be obtained.
The limits for the term to maturity are typically determined
by legislative or credit policy considerations, and are thus
given exogenously. However, the possibility for the debtor of
defining more narrow limits than stipulated by the exogenous
conditions is kept open.
But, the term to maturity concept of a LAIR with an adjustable
term to maturity paves the way for several possible
interpretations of limits for the term to maturity. The
adjustable term to maturity means that for a loan there will
be a sequence of terms to maturity given by
(2.2) Lo ~Ll ~ . . . ~LJ* . . . ELM where
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L,, specifies the term to maturity at the Jth
adjustment of the interest rate
On the face of it, it would be most obvious to demand that the
term to maturity at each interest rate adjustment fulfils
( 2 . 3 ) dJ : Lmin ~ LJ S L'°a"'
The actual term to maturity of the loan defined as the period
from the disbursement to the maturity is, however, given by L".
Hence, another possible interpretation of the limits for the
term to maturity is that only LM is to be imposed on these
limits, whereas it is allowed that either L,,>L"~" or L_<L'"'° for
J<M. The possibility of adjusting the term to maturity is
hereby enhanced.
If this interpretation is followed, a method must be
established for determining LM for each J<M. If LM is
calculated on the assumption that the interest rate is
falling, this would pave the way for values of L~ for J<M which
is high above L'"°", the assumed fall in interest rates
subsequently causing the term to maturity to be at the level
allowed. This would clearly be contrary to the intentions of
the provisions of the Danish Mortgage Credit Act.
A more reasonable assumption in the calculation of Lu would be
a future unchanged yield curve. The fact that a future
unchanged yield curve opens up the prospect Of L,,>L"'°" for J<M,
while LM 5 L"'v' is due to a shift in the composition of the bond
portfolio underlying the loan. The funding of a LAIR in
several bonds means that the future interest rate on the loan
will change not only as a result of the adjustments of the
yield to maturity of the bonds, but also as a result of the
current changes in the distribution of the bond volumes
provided the yield curve is not horizontal. If the yield curve
is rising, the effect thereof is a falling interest rate on
the loan at the end of the term to maturity of the loan, when
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the funding is effected in shorter and shorter bonds. Thus,
the illicit fall in the interest rate opens up the prospect
that L,,>L'"°" for J<M at the same time as L,~ SL'n°" .
However, a calculation of LH on the assumption that the yield
curve is unchanged will complicate the implementation of the
model in the trinomial lattice. Thus, in each node not only
calculations for the forthcoming adjustment of the interest
rate on the loan are to be performed, but for the entire
remaining term to maturity of the loan, if solely the limits
for the payments on the loan are applied to LM. At the same
time, numerical tests show that the variation between L~ and LM
is limited. The possibility of applying only the limits for
the term to maturity to LH is therefore not a part of the
currently preferred embodiment according to the invention.
If the limits for the term to maturity are determined
identically, the possibility of adjusting the term to maturity
would be exhausted in advance. The term to maturity is thus
fixed, and said LAIR II will degenerate to a LAIR I. The same
result could be obtained by determining the limits for the
payments on the loan at ~ (infinite) and 0 (zero)
respectively. However, the loan is thereby unable to function
in combination with the financial instrument, as there are no
limits to be hedged.
2.1.2 The determination of the limits for payments on the loan
and term to maturity
The very purpose of the introduction of the financial
instrument is to ensure that the debtor's payments on the loan
does not exceed the band. Payments outside the band are so to
speak covered by the instrument. This implies that payments
from the instrument are conditional on
~ firstly, the interest rate rising so that YDJ°" becomes
binding and the term to maturity is prolonged, and
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secondly, the term to maturity having reached the maximum
limit, excluding the possibility of further prolonging
said term to maturity
whereas payments to the financial instrument, by contrast, are
S conditional on
firstly, the interest rate falling so that YD~° becomes
binding and the term to maturity is shortened, and
~ secondly, the term to maturity being shortened to the
minimum limit, meaning that said term to maturity cannot
10 be further shortened
Apart from exogenous conditions such as the prevailing yield
curve, the pricing of the financial instrument will depend on
the initial term to maturity of the loan, the maximum and
minimum limits for the term to maturity, and the maximum and
15 minimum limits for the payments on the loan.
The model must be so flexible that an arbitrary determination
of all variables within the legislative and credit policy
framework is possible. A favourable solution would be that the
financial instrument has the price 0 (zero) at the
20 disbursement of the loan; causing the variables of the loan to
be interdependent.
The initial term to maturity of the loan may be applied for
the determination of a level for the payment initially. In the
case of a rising yield curve, the positive price of hedging
25 the maximum limit for the payments on the loan will be
significantly higher (numerically) than the negative price of
hedging the minimum limit. In order to achieve a negative
price of the hedging of the minimum limit which is as high as
possible, this price may be conveniently fixed as the initial
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payment on the loan. At the same time, the minimum limit for
the term to maturity is fixed conveniently as the initial term
to maturity.
Even a marginal fall in interest rates will thereby trigger
payments to the financial instrument. Thus, the minimum limits
provide, to the greatest extent possible, the possibility of
determining a maximum limit for the payments on the loan at a
relatively low level. This possibility is supported by a
determination of the maximum limit for the term to maturity at
a level as high as possible within the legislative and credit
policy framework.
If the variables of the loan are determined according to the
mentioned directions, the maximum limit for the payments on
the loan is determined unambiguously on the secondary
condition that the price of the instrument is 0 (zero). The
directions imply that the maximum limit will be determined at
the lowest possible level.
However, the model is to allow that the limits are determined
according to other directions determined by the debtor.
Therefore, the limits will be considered exogenous. S~Jhen
pricing the financial instrument in section 6, the above
method for determining the limits is modelled.
2.2 T'he financial instrument
As described above, the financial instrument is to prevent the
payments on the loan from fluctuating outside the band defined
by the maximum and minimum limits.
There are, however, several methods which may be applied for
the hedging of the maximum and minimum limits. Depending on
the method selected, the instrument will be more or less in
conformity with the market, which must be considered very
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important for the possibilities of the product. By contrast,
the debtor side of the loan must not suffer from the hedging
of the limits for the payments on the loan being designed with
a view to market conformity.
In the following, two methods are identified which may be
followed in the introduction of the financial instrument for
hedging the limits. The emphasis is on advantages and
disadvantages of the method.
2.2.1 A cap/floor approach
Firstly, the hedging of the limits for the payments on the
loan may be performed by a cap/floor approach.
According to this approach, the financial instrument is
defined directly by the payments which is outside the band in
the model for an adjustable term to maturity. Thus, this would
i5 be a complete hedging of the risk of fluctuations outside the
band.
The model may be used with adjustable as well as fixed terms
to maturity - i.e. as a further development of both a LAIR I
and a LAIR II. In principle, the model will be a pure further
development of either an underlying LAIR I or LAIR II, which
renders implementation of the model relatively easy, as the
underlying LAIR I or LAIR II may, in principle, be calculated
in an existing model.
The legislative and fiscal conditions mean that most
conveniently, the financial instrument exists only on the
debtor side of the loan. On the basis of the payments from the
underlying LAIR corrected by the payments on the financial
instrument, a new debtor side is thus to be calculated.
In the situation in which the payments from the financial
instrument are positive (the maximum limit for the term to
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maturity is thus binding), the recalculated debtor payment-is
to be smaller than on the underlying loan. This is achieved by
reducing the interest rate. The new interest rate on the loan
may be determined in a relatively simple way as the interest
rate which in the expression for the calculation of an annuity
produces a payment corresponding to the maximum limit given
the term to maturity.
However, the recalculation of the interest rate on the loan do
not only affect the volume of the payment but also the
distribution thereof on interest and repayments. It follows
from the expression for the, calculation of an annuity that a
lowering of the interest rate increases the volume of the
repayment, both relatively and absolutely. Consequently, the
remaining debt of the loan at the end of the interest rate
adjustment period will be smaller on the recalculated debtor
side than on the underlying loan. The consequences of the
recalculation for the payments and term to maturity profiles
in principle are illustrated in figure 5.
The reduced remaining debt at the end of the interest rate
adjustment period expressed by the difference between (4) and
(5) in the figure gives rise to imbalances. Bonds
corresponding to the remaining debt of the underlying loan
mature on the bond side, whereas the interest rate adjustment
amount on the debtor side is calculated on the basis of the
reduced remaining debt.
The imbalance may be equalized by the reduction in the
remaining debt also being covered by the payments of the
financial instruiaent. However, this produces a very distorted
development in the payments, the fixed payments for each
payment date in the interest rate adjustment period thus being
supplemented with a large payment at the end of the period.
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Alternatively, the imbalance may be equalized by the repayment
profile being fixed. The fixing may be performed at the
disbursement of the loan, which is a model that has previously
been applied for floating rate mortgage credit loans. However,
the model does not allow an adjustable term to maturity and
must, on this basis, be precluded. Secondly, the fixing may be
performed at the beginning of each interest rate adjustment
period so that e.g. the repayment profile of the underlying
loan is maintained. This model is difficult to comprehend for
the debtor and will result in either the payments or the
interest on the loan not being constant during the interest
rate adjustment period.
Apart from the consequences outlined, the approach implies
that the funding side of the loan is made artificially large
in certain cases. The approach implies that bonds are issued,
the payments of which are covered by the financial instrument.
This does not represent a problem but must be considered a
less than optimum solution from a theoretical point of view.
2.2.2 An approach based on put or call options
The other method for hedging the limits for the payments on
the loan follows an approach comparable to a collar, i.e. a
combination of a long position in put options for hedging the
maximum limit and a short position in call options for hedging
the minimum limit.
The approach implies that the payments on the financial
instrument are not defined directly by he payment profile of
the underlying loan. but instead as the necessary reduction of
the volume of underlying bonds ensuring that the payments on
the loan are within the band defined by the maximum and
minimmm limits.
The approach is based on the development of the adjustment of
the interest rate on a LAIR.
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At the beginning of each interest rate adjustment period, a
funding demand arises as a result of the adjustment of the
interest rate on the loan at the end of the preceding interest
rate adjustment period. At the interest rate adjustment, the
underlying volume of bonds matures fully or partially
(depending on the interest rate adjustment fraction) on the
bond side. On the debtor side, these payments correspond to a
new volume of underlying bonds being issued. The market price
thereof equals the nominal value of the bonds which have only
just matured corrected for bond repayments on account, so that
a balance between the payments is achieved.
The market price of the sold bonds influences the interest
rate on the loan in the interest rate adjustment period. At a
low market price, the extent of the necessary sale of bonds
increases. The larger volume of bonds causes larger payments
on the bond side and thereby also on the debtor side, the
balance principle having to be respected. The interest rate on
the loan is thus increased.
The mechanics of the adjustment of the interest rate on a LAIR
means that a reduction in the volume of the payments, so that
a maximum limit for the payments on the loan is observed, is
obtainable by means of a smaller sale of bonds at the
beginning of the interest rate adjustment period. A loss in
proceeds will thereby occur at the beginning of the period in
the relationship between, on the one hand, the nominal value
of the mature bonds and, on the other hand, the market price
of the sold bonds. According to the approach, this loss in
proceeds defines the payments from the financial instrument.
Correspondingly, a gain in proceeds may occur if an increased
volume of bonds is sold to enhance the volume of the payments
such that a minimum limit for the payments on the loan is
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observed. The gain in proceeds constitutes the payments to, the
financial instrument.
Firstly, the approach implies that payments on the financial
instrument will be concentrated at the beginning of the
interest rate adjustment periods. Consequently, there will be
no current payments on the instrument, which will facilitate
the pricing.
Secondly, it follows from the approach that the bond side and
the debtor side of the loan will be in agreement on a current
basis. The hedging of the limits is not in the nature of a
further development equalizing differences between the bond
side and the debtor side of the loan, but is, however, an
integrated part of the funding side. This also speaks in
favour of the approach.
The financial instrument formed by losses and gains in
proceeds is comparable to a collar.
The hedging of the maximum limit by means of covering the loss
in proceeds corresponds to the debtor having a long position
in put options with an exercise price corresponding to the
interest rate on the loan when the maximum limit has been
observed. Similarly, the hedging of the minimum limit
corresponds to a short position in a call option with an
exercise price corresponding to the interest rate on the loan
when the minimum limit is binding.
In order for the financial instrument to be formed as a
portfolio of put options in conformity with the market, a
number of preconditions must be met.
Firstly, the composition of the bond portfolio underlying the
loan must be known at all future interest rate adjustments. If
the distribution of bonds in the individual years is known, it
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naturally follows that it will be impossible to take positions
in options on the individual bonds.
However, if the distribution of bonds on the individual years
is to be determined ex ante, this requires that the
amortization to maturity is known ex ante. This precludes
adjustable terms to maturity as well as annuities calculated
with a floating interest rate. Consequently, conformity with
the market presupposes that the loan degenerates from a LAIR
II to a LAIR I with a fixed repayment profile.
Secondly, characteristics of the underlying bonds up to the
maturity must be known. Among these, the coupon rates of the
bonds must be known, which constitutes a problem in practice
as a floating minimum interest rate over time may result in
floating coupon rates over time.
In comparison with the financial instrument defined directly
by the payments outside the band, it must be assessed that the
financial instrument described above should obtain a far
higher degree of conformity with the market, primarily as a
consequence of the concentration of the payments on the
instrument at one point in time in each interest rate
adjustment period.
On the basis of the obvious advantages, the focus will, in the
following, be on an implementation of the last-mentioned
approach for hedging the limits for the payments of the loan.
2.3 RemsininQ clebt~ tezm to maturity and voluare lattice
The model for calculating the debtor and funding sides of the
loan must be implemented in the trinomial lattice set up in
section 1.
At each interest rate adjustment, the payments of the debtor
and bond sides are calculated in each node on the basis of the
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yield curve in the node in question. However, the debtor side
as well as the bond side depend not only on the yield curve. A
number of variables determined at the preceding interest rate
adjustment will also affect the current interest rate
adjustment.
Firstly, the volume of the remaining debt and the interest
rate adjustment amount at the end of the preceding interest
rate adjustment will affect the current interest rate
adjustment. Subsequently, the term to maturity in the
preceding interest rate adjustment is also to be used in the
calculation of the current interest rate adjustment.
Basically, the term to maturity i~ not changed provided the
maximum and minimum limits for the payment are not exceeded.
Further, for type P the bond volumes from the preceding
interest rate adjustment and the associated coupon rates are
to be input to the model because of the rolling movement in
the bond funding, cf. section 2.4.
In the lattice, it is usually impossible to identify
unambiguously the node for the preceding interest rate
adjustment. Typically, a node may be reached in several ways
for which reason it is not unambiguous in which node - and
thus under which yield curve - the preceding interest rate
adjustment was calculated. Only in the nodes in the lattice
which may be reached by an unambiguous path, will the input to
the calculation of the current interest rate adjustment be
unambiguously given.
In the other nodes, the said inputs must be determined as an
expected value of the nodes from which the current node may be
reached with a positive probability. In other words, input are
determined by a projection through the lattice in which the
probabilities of the different branching structures are
weighted.
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The projection is complicated by the fact that there is not
necessarily a one-to-one correlation between the step size in
the lattice (At) and the length of the interest rate
adjustment periods. Typically, there are four annual payment
dates on the debtor side for which reason a step size greater
than '~ may produce inaccurate results. However, the interest
rate adjustment periods may last up to 10 years for which
reason there may be many nodes between each interest rate
adjustment.
In order not to complicate the projection unduly, it is
preferable to project inputs in each node in the lattice,
though an adjustment of the interest rate on the loan is not
performed in the node in question. The projection may thus
follow a forward induction method comparable to that applied
in section 1.
First, the vector x(g,h) is defined. The elements in the
vector are constituted by the above input variables, i.e.
x=(remaining debt and interest rate adjustment amount
at end of period; term to maturity of the loan; bond
volumes at end of period, coupon rates associated to
the volumes?
For type F the bond volumes at end of period will all assume
the value 0 (zero), and there will be a match between the
remaining debt and the interest rate adjustment amounts.
In the projection, Bayes' rule is applied for the
determination of the probabilities. Bayes' rule expresses the
probability of an event having occurred by a specific path as
probabilities of said path seen from time 0 (zero) divided by
the cumulative probability of the event, also seen from time 0
(zero).
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The cumulative probabilities q(g,h) are determined according to
the principle illustrated in figure 6.
The method in the figure leads to the following general
expression
( 2 . 4 ) Wig, h) _ ~ X9-1, k)9(k, h)
x
It follows from (2.4) that the cumulative probabilities are to
be found successively for g=0,1,2... corresponding to the
forward induction method. With the definition of the
cumulative probabilities, Bayes' rule may be formulated. Let
( 2 . 5 ) peU9-1, k)~(g, h))
denote the probability of the immediately preceding node being
(g -1,k) provided one is in (g, h). Then it follows from Bayes'
rule that
( 2 . 6 ) Pa(~9-1, k)~~9', h)) = 99-1, k)4'~k, h)
4~9r, h)
Consequently, the expected value of x in each node is found by
the following expression
( 2 . 7 ) x'(p, h) _ ~ PB(~9'-1, k)~(g, h))x(9-1 ~ k)
x
This expression is also based an forward induction. x (0,0)
will be exogenously determined. On the basis of x (0,0) it
will be possible to determine successively the value of x for
g=1,2,3... until the projection has been performed through the
entire lattice.
Up to the first interest rate adjustment, x (g, h) will have
the same value in all nodes, as only the debtor and bond sides
of the loan have been calculated, viz. in (0,0). In connection
with the first interest rate adjustment, new values are
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assigned to x (g. h) on the basis of the interest rate
adjustment period, etc. Apart from the changes resulting from
the weighting over the different probabilities, x will thus
change values for the values of g which coincide with an
interest rate adjustment.
2.4 The adjustment of the interest rate on a LAIR
The adjustment of the interest rate on a LAIR may follow two
patterns, for which reason it is necessary to distinguish
I between two types of LAIR.
~ Firstly, the interest rate adjustment may take place at a
fixed frequency as 100 per cent of the remaining debt of
the loan at the interest rate adjustment. This type of
loan is termed LAIR type F
~ Secondly, the interest rate adjustment may take place
annually as a fixed fraction of the remaining debt at
beginning of period in the year in question. This type
of loan is termed LAIR type P.
It is preferred to perform the interest rate adjustment in
connection with a payment date on the bond side of the loan
regardless of the fraction and frequency of the interest rate
adjustment.
Especially the bond side is different for the two types of
loans. For LAIR type F, the full adjustment of the interest
rate on the loan means that all underlying bonds mature at the
interest rate adjustment. Thus, bonds with a term to maturity
longer than the period up to the next interest rate adjustment
period will not be issued.
For a LAIR type P, the partial adjustment of the interest rate
on the loan means that the bond issue follows a rolling
movement. The majority of the bonds are issued with a term to
maturity longer than the period up to the next interest rate
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adjustment. The rolling movement complicates the calculation
of the bond side. as e.g. bonds already issued are to be taken
into account currently at each interest rate adjustment.
The differences in the organization of the bond side are so
extensive that it is necessary to distinguish between the two
types of loans in the modelling a LAIR III~ In the following,
the models for type F and then type P are set up.
The necessity of operating with both a LAIR type F and type P
stems from the preferences of the debtors. Where a debtor with
a LAIR type f is exposed to a relatively large risk at few
points in time, a debtor with a LAIR type P is exposed to risk
more often, said risk being limited in scope, however. With
the introduction of a maximum limit for the payments on the
loan, one may argue for a limitation of the number of types of
loans to either type F or type P. Which type of loan is most
advantageous in combination with a hedging of the risk of a
fluctuating payment on the loan depends, however, on several
factors, e.g. the yield curve. A limitation of the types of
loans may thus have inconvenient consequences.
2.5 The model for a LAIR III type F
A preferred model for a LAIR III type F is built on the
conditions laid down for a solution to the model. Thus, the
description of the model begins by a formulation of these
conditions. Then the problem is formulated, which problem is
solved by the model, and the general model structure is
described in section 2.5.2. The specific model is set up in
section 2.5.3. A variant of the LAIR type F - type F' - is
described in the appendix.
2.5.1 The cor~itioas in the model
In each interest rate adjustment period, the debtor and
funding sides are subject to four conditions.
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Firstly, the term to maturity of the loan must be within
the interval from L"'° to L'"°"
Secondly. the debtor's payments on the loan must be
within the band defined by the maximum and minimum
limits.
These conditions are discussed above. However, the conditions
need to be concretized and formalized.
( 2 . 3 ) ~lJ : Lmin <- L.7 S L'nau'
(2.8) dJ: YD~'SYDa(~)sY~
Neither (2.3) nor (2.8) are operational and these conditions
will be insufficient to determine the term to maturity of the
loan generally. The conditions are rendered operational by
also requiring that corrections of the term to maturity are
minimized in scope. If the payments on the loan are obviously
larger than the maximum limit, the term to maturity is
corrected such that the payments on the loan correspond
exactly to the maximum limit, and vice versa. L,, must thus
further fulfil the condition
( 2 . 9 ) Ls a argmin( ~L~ - L~ i I ~ Y~i" ~ ~J( ~ ) 5 Y~a" , L"'in 5 Ls 5
L'""')
for 1 <-JSM
Thirdly. the payments on the funding side of the loan are
to balance the total payments on the debtor side.
On the funding side of the loan, the current payments comprise
coupon rates and mature bonds. In order to fulfil the
requirement, bonds in all years with maturity before the next
interest rate adjustment are sold (to begin with, the type F'
variant of type F is ignored). Consequently, bonds will
mature, not just in connection with the adjustments of the
interest rate on the loan.
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On the debtor side of the loan, the payments are constituted
by the debtor's payments of interest and repayment. In
addition, the remaining debt on the debtor side is included in
the payments at the end of the interest rate adjustment
period. The remaining debt is interest rate adjusted by bonds
being sold once again at the beginning of the next interest
rate adjustment period. In the current interest rate
adjustment period, the interest rate adjustment amount
balances the volume of mature bonds.
The balance condition may be formalized. Thus, it applies for
every year (j') until the interest rate adjustment that
m
(2.10) ~a(7~)=H~~7~)+
~ Ri(J)HJ~J),
j~<m where
j (not to be confused with J) denotes
the year
within the interest rate adjustment
period and
numbers the funding volumes. Therefore,
j is
assigned the value 0 following each
adjustment
of the interest rate on the loan.
In the nota-
tion, there is thus a direct connection
betwe-
en the year and the bond maturing
in this ye-
ar.
' YD,,(j) is the debtor's payments on the loan
in year j
following the Jth adjustment of the
interest
rate on the loan.
', 1~,(j) is the jth bond volume at a given
point in ti-
' me.
', RJ N is the nominal interest rate on the
jth bond
volume.
m (not to be confused with M) is the
number of
bonds at the beginning of the interest
rate
', adjustment period and at the same
time in the
next interest rate adjustment period.
That j may denote as well as funding volumes is due
year
solely to the t the bonds have only one annual
fact tha
creditor payment date on l January. If the number of annual
creditor payment dates is adjusted, the notation is to be
changed.
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On the debtor side it is possible, however, to have both 1 and
4 annual payment dates. Consequently, j cannot denote the
debtor payment dates as well. The notation is facilitated by
the debtor's payments on the loan being specified by YDJ(j)
where a summing over the payment dates within the year are
performed. Let n denote the number of debtor payment dates per
year
(2.11) YD~(7)_~~'Da(i)+~ n'RG,i(i-1) where
t=1 t=1
is the interest rate on the loan
RG,,(j) is the remaining debt of the loan at time j
AFD,,(i) is the debtor's repayment in payment period i
i denotes the payment dates within the year, i.e.
i=1,2,...,n
(2.10) does not include the year of the disbursement of the
loan in which the, payments on the underlying bonds will be
reduced by the accrued interest. Therefore, a regulation
factor is introduced in (2.10). At the disbursement of the
loan, (2.10) is formulated as
( 2 .12 ) YD~( 1) = Hs( 1) + REG~ ~ R~( j )H~( j ) where
j=i
', ~J is a regulation factor determining the amount
of the interest payment the creditor is to
receive from the debtor on the next creditor
payment date. REG,, is determined as that part
of the year the loan has existed measured from
30 November when the bonds turn ex-coupon.
Therefore, REG,, may assume values between 1/12
(if the loan is obtained on 30 November) and
13 / 12 ( if the loan is obtained on 1 December,
so that yeax 1 lasts 13 months). For J>0
REG,,=1, the adjustment of the interest rate on
the loan being performed on the creditor
payment date.
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In addition, (2.10) is to be modified for the years in which
the loan is interest rate adjusted, cf. j'<m, the interest
rate adjustment amount being included in the payments on the
debtor side of the loan.
( 2 .13 ) ~s(m) + RGa(m) = HJ(m) + RJ(m)HJ(m)
by means of which the total balance conditions may be
formulated
( 2 .12 ) Year 1 YD,r(1) =HJ(1)+REGJ~ RJ(j)HJ(j)
j~1
~"
( 2 .10 ) Year 2 YD,~(2) = H~(2)+ ~ RJ(a)HJ(,7)
j=2
( 2 .13 ) Year m YD,r(m) + RGa(m) = HJ(m) + R J(m)Ha{m)
~ Forthly, the market price of the sold bonds and the
payments from the financial instrument must cover the
funding demand of the debtor in connection with the
adjustment of the interest rate on the loan at the end
of the preceding interest rate adjustment period.
The requirement is termed the proceeds condition. Payments on
the financial instrument are conditional on either the maximum
limit for payments on the loan and term to maturity being
binding, or the corresponding minimum limits being binding. In
these situations, the payments on the financial instrument are
calculated as the residue between the funding demand and the
market price of the sold bonds. The proceeds condition may
then be formulated as
(2.14) F~=RG~(0)-~K,t(j)H,r(j) where
F,, is the payments on the financial instrument
IC,,(j~ is the market price of the jth funding instru-
went
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Thus, the proceeds condition will be fulfilled by definition,
corresponding to the criterion being suspended.
The market prices of the underlying bonds are observable at
the disbursement of the loan, but at all other paints in time
the calculation is performed on the basis of the modelled
yield curve. Consequently, the model is dependent on the
interest rate via the market prices.
In all other situations, there will not be payments on the
financial instrument, making it possible to formulate the
proceeds condition as a balance between the market price of
I sold bonds and the funding demand. This may be formulated as
m
( 2 .15 ) RGa~O) _ ~r KJ~~)H,~(J)
W
RG,,(0) is the remaining debt at beginning of period,
corresponding, at the disbursement of the loan, to the volume
of the loan, and at the adjustments of the interest rate on
the loan to the refinancing amount, the entire remaining debt
being interest rate adjusted.
The coupon rates of the underlying bonds are included in the
balance conditions as well as in the proceeds condition. At
the disbursement of the loan, the coupon rates of each bond
are known. At the future adjustments of the interest rate,
however, the coupon rate will depend on the policy of the
lending institute with regard to the opening of bonds and,
equally important, on the development in the minimum interest
rate, fiscal factors in practice precluding coupon rates under
the minimum interest rate.
The fixing of a minimum interest rate is governed by
subsection 3 of section 7 of the Danish Gains on Securities
and Foreign Currency Act. The minimum interest rate is fixed
every six months on the basis of a 20 trading days' average of
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the bond yield of all fixed-interest bonds denominated in
kroner listed on the Copenhagen Stock Exchange. The bond
volume is weighted in the calculation of the average. Callable
bonds above price 100 are not included in the calculation of
the minimum interest rate. The calculated average bond yield
is regulated by the factor g and is rounded down to the next
integer percentage. The rounded down interest rate constitutes
the mininnun interest rate
Consequently, the minimum interest rate will not only vary
with the develo~nent in the interest rates but also with the
composition of the open bonds. One example thereof could be
observed when in October 1996 the minimum interest rate was
reduced extraordinarily to 4 per cent following the closing in
August 1996 of the very large mortgage series with maturity in
2026.
An accurate prediction of the minimum interest rate on the
basis of the yield curve is thus difficult. Going further
back, a reasonable approximation of the minimum interest rate
seems to be the 10-year interest rate less 1 percentage point
and round down to the next integer percentage. This
approximation will not be very accurate in the current
situation which must be considered extraordinary due to the
very steep yield curve.
The approximation of the minimum interest rate constitutes a
floor below the future coupon rates. However, it is not given
in advance that the coupon rate is determined such that it
corresponds to this floor. One relevant consideration in the
policy for opening new bonds could be that the bonds must be
applicable for as long as possible, the volume and liquidity
of the bonds being maximized. This consideration speaks in
favour of the bonds being opened with a coupon above the
minimum interest rate so that an increase in the minimum
interest rate does not close the bonds. Product considerations
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also speak in favour of the bond being opened with a price
close to 100.
In the model the future coupon rates are determined such that
they fulfil the requirement with respect to a minimum interest
rate and such that the price is as close to 100 as possible.
2.5.2 The general structure of the model
Collectively, the conditions define the problem that the model
solves. 'Fhe problem may be formulated as follows:
T$e problem for a LAIR III type F
Determine the term to maturity of the loan. the interes t
rate on the Loan, the volumes of the underlying bonds
and the volume of the payments on the financial
instrument such that
1. the term to maturity is within the maximum and the
minimum Iuni ts, c . f . (2 . 3 )
2, the payments on the loan are within the maximum and
the minimum limits, cf. (2.8)
3. the balance conditions are fulfilled, cf. (2.10),
(2.12 ) and (2 .13 )
4. the proceeds condition is fulfilled, cf. (2.I5) and
5. the payments on the financial instrument fulfil the
condition given by (2.14)
The number of variables in the model may be immediately
determined at m+3, m volumes, an interest rate on the loan, a
term to maturity, and the volume of the payments on the
financial instrument having to be determined. However,
payments from the financial instrument are conditional on the
term to maturity having reached its maximum or minimum allowed
value, thus no longer being adjustable. Actually, the number
of variables thus amounts to m+2.
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The balance condition must be fulfilled every year in the.
interest rate adjustment period and thereby defines m
equations. The proceeds condition defines another equation. If
there are no payments from the financial instrument, (2.15)
will determine the market price of the sold bonds. However, if
there are payments from the financial instrument, (2.14) which
was deduced on the basis of the proceeds condition will
determine the volume of the payments. Finally, (2.9) will
define a last equation determining the term to maturity such
that (2.3) and (2.8) are fulfilled. Thus, the model solves m+2
equations with m+2 variables.
In the degenerated model in which the term to maturity is kept
fixed, the number of variables and the number of equations are
reduced by 1, the term to maturity not constituting a variable
and (2.9) losing its significance. However, basically, the
problem will not be modified.
The problem has a simultaneous as well as a recursive
structure. The simultaneous structure appears when there are
no payments on the financial instrument.
For a given term to maturity, the determination of the bond
volumes will be determined by the proceeds condition as well
as by the payments on the loan via the balance conditions. If
one takes as a starting point any interest rate on the loan,
the payments may be calculated. With reference to the balance
conditions, the payments determine the volumes of the
underlying bonds. The proceeds of the sale of the bonds may
then be compared to the funding demand. If there is a deficit,
more bonds must be sold and the interest rate on the loan must
be increased, so that the large payments on the bond side are
covered by the payments of the debtor side. If, on the
contrary, there is a profit from the bond issue, the interest
on the loan may be lowered. It will always be possible to find
an unambiguous and positive interest rate on the loan, which
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solves the problem for the given term to maturity. This is, due
to the proceeds of the sale of bonds being a strictly rising
function of the interest on the loan, assuming that the prices
of the underlying bonds are positive.
When a solution has been found that complies with both the
balance conditions and the proceeds condition, the calculated
debtor payments on the loan may be compared to the limits for
payments on the loan, and a correction of the term to maturity
is performed. The determination of the interest rate on the
loan and the volumes of the underlying bonds is then to be
repeated. Similarly, it is always possible to find a term to
maturity that solves the problem, the payments on the loan
being a strictly declining function of the term to maturity.
In so far as the necessary correction of the term to maturity
causes the term to maturity to exceed the allowed interval,
the financial instrument is activated. Thereby the model
changes to a recursive structure. First, the interest on the
loan is determined by the requirement as to the payments on
the loan, one of the limits being binding. In the next step,
the payments on the loan determine an unambiguous value for
each volume. The proceeds of the sold bonds may then be
calculated, causing the payments on the financial instrument
to be determined at the same time.
The flow chart of the model appears from figure 7. As
indicated above, and as appears from the figure, the model is
divided into an inner and an outer loop. The inner loop
consists of steps 2 to 11. In these steps volumes of the
underlying bonds and the interest rate on the loan are
determined with reference to the balance conditions and the
proceeds criterion. The outer loop is constituted by step 1
and steps 12 to 16. The loop determines the term to maturity
with reference to the band for the payments on the loan.
Finally, steps 17 to 22 constitute the recursive model
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determining the volume of the financial instrument. The
individual steps are described in the following.
2.5.3 The steps of the model
Strip 1 - Determiae the term to maturity within the interval
from L"s° to L""'
In step 1 the outer loop of the model begins. In step 1 the
model determines an initial value for the term to maturity of
the loan.
At the dis~»rQ~m~~r r,f the loa_n_ the term to maturity is
determined on the basis of the input from the.debtor. It will
be natural to require that the debtor specifies an intended
term to maturity (the term to maturity is intended, a
realization of the term to maturity presupposing that future
rises and falls in interest rates cancel out) which may be
applied in the model immediately.
However, it may be argued that it is convenient that instead
the debtor has the possibility of determining an intended
payment on the loan, Which is easier to relate to the limits
for the payments on the loan. In this situation, the initial
term to maturity is determined as
1 _ Raoco~
( 2 .16 ) Lo = ~ ~oti)
1 +~
where YDo(1) is the payment on the loan during the first
integer year of the loan. The interest on the loan is
approximated initially by the yield to maturity of the last
maturing bond underlying the loan.
At adiustments of the interest rate on the loan the term to
maturity is basically maintained in relation to the term to
maturity at the preceding adjustment of the interest rate.
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Therefore, the initial value for L~ is determined by the input
vector which is projected through the lattice. cf. section
2.3.
gr the last adiustment of the intereer rmrp on the loan (J=M),
the term to maturity should be corrected such that the loan
matures at the same time as the underlying bonds. In so far as
it is possible, the correction is performed such that the
limits for the payments on the loan are observed. However, one
cannot preclude the possibility of the maximum limit being
exceeded by shortening the term to maturity to the next
creditor payment date, whereas the minimum limit is exceeded
', by prolonging the term to maturity. In this situation, the
term to maturity is prolonged, as a payment on the loan which
is too low must be considered a minor problem for the debtor
than a payment on the loan which is too high. If the payment
falls below the minimum limit, payments on the financial
instrument are not triggered in this instance. Payments from
the financial instrument presuppose that at the same time the
term to maturity conflicts with the minimum limit. With the
definition of the minimum limit (cf. subsequently in the
description of step 1), a shortening of the term to maturity
within the year will not solve such a conflict.
L" is determined according to the following directions.
1. Prolong LM if
2 5 ( 2 .17 ) YDx(.7)) it > ~n
where Lar denotes the term to maturity prolonged to the
i.aamediately subsequent creditor payment date. Otherwise move
on to 2.
2. Shorten L" to the imacdediately preceding creditor payment
date, if
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(2.18) ~x(7)~Zw'~ .
where y" denotes the shortened term to maturity. Otherwise move
on to 3.
3. Prolong the term to maturity to the next creditor payment
date.
Step 3 in the procedure ensures that a solution is always
found, so that the calculations do not continue infinitely.
The steps imply that it is necessary to operate with two terms
to maturity in the model, L~ and ;,,,,.respectively.
If the loan is a LAIR I for which the term to maturity is
fixed given by L"""=LJ=L'°'°, the term to maturity is naturally
determined in the model as the fixed term to maturity.
Step 1 being the first step in the model, all inputs are
further to be input to the model. These are constituted by
the funding sido
the number of bonds (and thus also the interest rate adjustment
frequency), the nominal yield of each instrument, the number of
annual creditor payment periods and creditor payment dates, and the
market price of each instrument.
tho d~sbtor sido
the disbursamaent date, the volume or the remaining debt, the number
of annual debtor payment periods and debtor payment dates, the
maximum and minimum limits for the payments on the loan and.
optionally, an intended payment on the loan or term to maturity, cf.
above .
Furthermore, values for inc. are input. inc are used in the
Gauss-Newton algorithm applied at the iterations in the model.
Finally, values are to be assigned to e. E is a set of
accuracy parameters specifying a minimum permissible deviation
from the various requirements.
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When inputting data, it is checked that .
d J : YD°~'" Z YD jia 2 0 and L"'s" Z Lmin ~ 0
Since the loan is required to mature at a creditor payment
date, it is required that the input values for L'"a"' and L"'in
comply with a creditor payment date. If the loan is disbursed
a . g . on 3 0 June, the model requires that L'"°" and L'"'° are
specified by an integer number of years +~.
Stop 2 - Determine the initial interest rate on the loan
The inner loop of the model is initiated by an initial
determination of the interest rate on the loan. The yield to
maturity of the last maturing bond is applied as an
approximation for the interest rate on the loan, the main part
of the issue being effected in this bond. The last maturing
bond has a term to maturity of m years, m being determined as
(2.19) m=max [the number of bonds initially; the remaining
term to maturity rounded up]
such that bonds with maturity later than the maturity of the
loan are not issued.
Step 3 - Calculate debtor payments on the loaa
Once an interest rate on the loan has been determined, the
debtor payments on the loan until the next adjustment of the
interest rate may be calculated both for
and R~ +inc
The repayment profile and the remaining debt profile until the
next adjustment of the interest rate are thereby also
determined.
Step 4 - Calculate the volumes
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The calculated payment profile and remaining debt at end of
I period permit the determination, via the balance conditions,
of the volumes of the underlying bonds. In a matrix form, the
balance conditions may be formulated
( 2 . 2 0 ) YDa1-RGa=1~ where
YDJ = ( YD~ ( 1 ) . YD,1 ( 2 ) , . . . , YD,1 ( m ) ) ,
~J - (o,a, . . .O,RG,,(m) ) ana
(H,,(1) ,H,,(2) ...,H,;(m) ) are mx1 vectors,
and where A is defined as an mxm upper triangular matrix given
by
1+REG~Rj(1} REGJR~(2) REG,aRj(3) . . REGoR~(m)
A= 0 1+RJ(2) R;;(3} R,N1(4) . RJ(m}
0 0 0 0 0 1 + R~(m}
The m bond volumes are found by
(2.21) 8J=[ATA] tAT[YDJ+Raal
In principle, [ATA]lATmay be replaced by A-' in (2.21) as long
as A is quadratic. The rewriting [ATA]lATis necessary only if
more bonds are introduced. Thus, the rewriting is only a
method by which non-quadratic matrices may be inverted
approximatively.
Stop 5 - ~t~e proceeds function
The difference between the funding demand and the market price
of the bond volumes defines a proceeds function given by
( 2 . 22 ) f(RJ) = RGa(0) - F.r K~(7)Ha(J)
j-s
The value of the proceeds function equals the payments on the
financial instrument, cf . ( 2 .14 ) . F,, and f(~) should however be
kept apart. The proceeds condition is fulfilled for f(R") - 0.
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Step 6 - Calculate adjustm~t to the intere8t rate on the loan
On the basis of the value of the proceeds function, an
adjustment to the interest rate on the loan is calculated. The
calculation of the adjustment is performed prior to the test
of the proceeds condition. The adjustment constitutes a better
measure as to whether the iterative procedure is to continue,
or whether the interest rate on the loan has converged. Thus,
a situation may arise in which the proceeds condition deviates
marginally between the funding demand and the market price of
the sold bonds, meaning that the model is imomediately to
continue, but a zero adjustment to the interest rate is
calculated, said interest rate having converged. The
calculation of the adjustment to the interest rate on the loan
follows the Gauss-Netaton algorithm.
(2.23) tIRJ=~IDTD1 'DTgr'~~'J~ ~ for
( 2 . 24 ) j" _ ( DlagJaTJa ~ ~
( 2 . 2 5 ) D = (~Je] ~ ~ ~J v7 v] °-1
(2.26) 9=~~~LJ)]~-'Jv 1
( 2 . 27 ) J. = [F(LJ) - F"(L~ + iac)] ~_ , inc°'i where
D'b is the Hesse matrix
g is the gradient
Ja is the Jacoby matrix
j" is the diagonal elements
A~B°'iis the Schur product of two matrices meaning that the
elements of the matrix are divided one by one.
(2.23) to (2.27) define the multi-dimensional Gauss-Newton
algorithm. In the specific example, the problem is
one-dimensional meaning that the expressions may be reduced.
D, Ja, and j" all have the dimension 11. Thus, it applies that
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(2.28) 7v ~ Ja
as j" _ [DiagJaTJa]'~ - [Ja']'~ = Ja. If j" = Ja is inserted in the
expression for D, the result is
D= J~Ja __ Ja =1
(2.29)
jv7v Jv
The expression for g may also be reduced to
( 2 . 3 0 ) g= 'T~'-'~R'-) = f(R~)
If the reduced expressions are inserted in (2.23), the result
is as follows
(2.31) ~~=~DTD~ 1DT~~~j=f(RJ) K In K
j" f(RJ)- f(RJ+ inc)
The calculation of the adjustment to the interest rate on the
loan by means of the reduced Gauss-Newton given by (2.31) is
illustrated in figure 8.
The basic idea of the algorithm is to apply the secant through
two points on the graph of f ( . ) for (R~,f(RJ)) and
'I ,~ ,,~ l
~Rj+II7C,~~+~G)~ respectively. In the figure, these points are
given by (6) and t~). When the secant has been determined, the
intersection of the secant with the'x axis (8) is determined.
The intersection is the next guess as to the interest rate on
the loan in the iterative process. If f(.) is strictly
declining, the algorithm will always reach a solution. This
', will be the case as a rising interest rats increases the
II proc~eds of the loan, and vice versa. The,figure shows that
ine is not to be seen as an accuracy parameter. 0n the
contrary, inc determines the step size in the iteration
procedure.
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gtep ? _ I8 the proceeds condition fulfilled?
The convergence on the interest rate is tested on the basis of
the adjustment. One of two conditions is to be fulfilled in
order for the interest rate to have converged.
(2.32)
E~R~
~ f(RJ)a f(RJ + ~RJ)a1 ~ <
(2.33)
f(RJ)a
If one of the conditions is fulfilled, the interest rate is
accepted and the model moves on to step 8. Otherwise the
interest rate is corrected in step 9.
Step 8 - Correct the interest rata on the loaa
The interest rate on the loan is corrected by the adjustment,
and steps 3 to 7 are repeated for RJ + ~R~
Stop 9 - Ara all volumes positive?
One cannot preclude the possibility that one or more of the
volumes are negative. The immediate interpretation thereof is
that the debtor is to buy bonds underlying the loan, which is
theoretically possible, but in practice this is not a viable
solution. The problem is solved by the loan changing
characteristics to a type F' loan in step 11.
However, if all the bond volumes are positive, the model moves
on to step 12.
Step 10 - Shift to F'
This step sends the model on to the F' variant which is
described in the appendix. Information concerning the loan is
input to the F' model.
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The negative bond volumes are produced as a result of the .
interest rate on the loan being significantly lower than the
coupon rate of the underlying bonds. Another possibility is,
therefore, that new coupon rates are determined for the bonds.
However, it is usually preferable, in practice, that the loan
is calculated as a type F'.
gtep 11 - Calculate adjustment to the term to maturity
On the basis of the relationship between the calculated debtor
payments on the loan and the limits for the payments on the
loan, the model calculates an adjustment to the term to
maturity in step l2. As in steps 6 and 7, the convergence of
the iteration is determined on the basis of the adjustment,
for which reason the adjustment is calculated before the
calculated debtor payments on the loan are compared to the
limits for the payments on the loan.
The calculation of the adjustment follows a procedure
corresponding to that applied in step 6. First, a function
f(.) is defined, which measures the distance between the
calculated debtor payments on the loan and the band.
However, it is necessary to distinguish between the maximum
limit and the minimum limit for the payments on the loan when
f(.) is defined. If e.g. the debtor payments on the loan are
above the maximum limit, it follows from (2.9) that a
correction of the term to maturity is to be performed such
that the payments on the loan.correspond exactly to the
anaximum limit. If the payments on the loan become strictly
lower than the maximum limit, the correction is thus too
large. f(.) is to be defined such that both positive and
negative distances are measured at each limit. f(.) is given
by
~~.7)-YD"'J"' : YDJ(J) >
2 . 3 4 ) f(L,~) = 0 : YDJ~'" 5 YDa(J) s YD,~"
~.~J)-~n :YDJ(7)~~n
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For J=0 the payments on the loan are regulated in (2.34) such
that the term to maturity is evaluated on the basis of
payments on the loan during an entire year. The adjustment to
the term to maturity AL,, is calculated by means of the
Gauss-Newton algorithm meaning that
(2.35) ~~=~[DTD) lDTg~~'j~
where the notation is taken over from step 6. The problem is
one-dimensional and (2.32) is thus reduced to
2 . 3 6 ) ~,J = f(L,r) inc
f(L,,) - f(LJ+ inc)
In the calculation of the adjustment, account should be taken
of the special procedure for the determination of the term to
maturity at the maturity of the lean, i.e. for J=M. The
procedure determines that the term to maturity is prolonged to
Lx if the payment is not thereby lower than allowed. However,
the possibility that the term to maturity is to be further
prolonged in order to observe the maximum limit cannot be
precluded. f(.) is given by
(2.37) f(Lrr)=~~.7)~iN-for YDM(7)~i">YD~'
and otherwise f(Lx)= 0. Similarly, a shortening of the term to
maturity to L~ may be insufficient to ensure payments on the_
loan above the minimum limit. In this situation, f(.) is given
by
(2 .381 f(~x) _ ~r~(J)~j",-~n for YD~(,j)Iy" < YD~n
and otherwise f(j,~)=0. The last possibility is that
( 2 . 39 ) '~x(J)I~.~r < ~in ~d ~H(7)~L~ ~
In this situation, the term to maturity is prolonged to L~ and
it is accepted that the payments on the loan are lower than
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the minimum limit, causing payments to the financial ,
instrument to occur. Consequently, no adjustment to LM are to
be calculated, and f(L")=0 will apply.
The calculation of the adjustments at the maturity of the loan
ensures that subsequently it is not necessary to correct the
tezin to maturity plus the adjustment such that the loan
matures in a creditor payment period. This correction is built
into the adjustment.
It should be noted that in the calculation of the adjustment
to the term to maturity no account is normally taken of the
limits for the term to maturity. If an adjustment is
calculated which causes the term to maturity to exceed the
allowed limits, payments on the financial instrument are
triggered. Therefore, the range of the adjustments must not be
limited.
Even though L'"'°=L°'°" such that the loan has a
fixed term to
maturity, an adjustment is calculated. In this situation, any
AL,,~O implies that the fixed term to maturity is incompatible
with the limits for payments on the loan, for which reason
payments on the financial instrument axe required. If DL,, is
set equal to 0 without regard to the payments on the loan, the
model loses information as to when payments on the instrument
are necessary.
Step 12 - Are the payments on theVloan ~rithin the interval
frog YD~ to YD~""?
Step 12 determines whether the model is to continue the
iteration over terms to maturity in the outer loop, or whether
a solution has been found, which fulfils the requirements
imposed on the repayments on the loan.
The term to maturity has converged if f(LJ) fulfils one of the
convergence conditions
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~j < a
(2.40)
f f~l'rJ)a - f~LJ + ~LJ~2~
(2.41)
f(LJ)Z
If one of the above convergence conditions is fulfilled, the
term to maturity of the loan has converged and the
calculations of the debtor and bond aides of the loan are
completed. The model is finalized in step 13. If neither of
the conditions are fulfilled, however, the model moves on to
step 14.
Step 13 - The loan has been calculated!
The model reaches step 13 only if the payments on the loan and
the term to maturity are within the established limits. Thus,
in this case there are no payments from the financial
instrument, and the debtor and bond sides of the loan will
have immediate characteristics in common with a LAIR II with
respect to the volumes of the payments.
Step 14 - Is the corrected term to maturity v~ithin the
interval frcan L"''~' t0 L""?
If the term to maturity of the loan has not converged and the
debtor payments on the loan are thus outside the band, it is
assumed that the term to maturity is to be corrected.
However, a correction causing the term to maturity to exceed
the maximum limit will not bring the model closer to a
solution. In this case, the limits for payments on the loan
and term to maturity are mutually incompatible and the
financial instruaaent must be activated.
If the calculated correction causes the term to maturity to
exceed the maximum limit, the term to maturity is set to the
maximum limit, and the model moves on to step 16. In the
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reverse situation in which the correction causes the term to
maturity to exceed the minimum limit, the term to maturity is
set equal to the minimum limit and, similarly, the model moves
on to step I6.
If the corrected term to maturity does not exceed neither the
maximum limit nor the minimum limit, the model corrects the
term to maturity by the adjustment in step 15.
Step 15 - Correct the term to maturity by the adjustment
The term to maturity is corrected by the calculated
adjustment, after which steps 2 to 12 of the model are
repeated.
Step 16 - Determine the interest rate on the loan such that
the payments are within the interval
The model reaches step 16 only if the yield curve causes the
limits for payments on the loan and term to maturity to be in
mutual conflict. In the following steps, the payment on the
financial instrument is to be calculated. Thus, step 16 is the
first step in the recursive model structure.
In this part of the model, the proceeds condition no longer
constitutes a binding requirement in the determination of the
bond volumes. On the contrary, one of the limits for the
payments on the loan enters the model as a binding condition,
cf. the description of the conditions in the model.
In step 16 an interest rate on the loan is to be determined
such that the payments on the loan are exactly on the binding
limit. If the term to maturity has reached its maximum at the
same time as the maximum limit for the payments on the loan
being binding, an interest rate on the loan is to be
determined which produces the maximum permissible payment on
the loan. The interest rate must fulfil
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RJRG~(0) _ ~ for YD.~(J)~r.W.X > YDJ "
( 2 . 42 ) ~ L..x.,~ o
1-(1+R~)
where
is the elapsed part of the term to maturity.
L,,-~p is thus the remaining term to maturity of
the loan
may not be isolated in (2.42), but may simply be found by a
numerical method.
In the reverse situation in which a falling interest rate has
meant that the minimum limit for the payments on the loan
cannot be observed simultaneous with the minimum limit for the
term to maturity being observed, the interest rate is to
fulfil
~ ~ ~ min
( 2 . 43 ) RJRG ~ 0)t~..qp - ~in for YD,~(J)I r~
1-(1+R~)
The annuity payment is strictly rising in the interest rate.
This means that a solution will always be found to (2.42) and
(2.43).
Step 17 - Calculate the voluaaes
The calculation of the interest rate on the loan in step 16
permits the determination of the payment profile and the
remaining debt profile of the loan. As in step 4, the balance
conditions may subsequently be applied for the calculation of
the volumes in each bond. They are found as a solution to the
matrix equation
( 2 . 21 ) 8~ _ [ATA)-1 AT[YD J + RCS J)
where H,, is a vector of volumes, A is the payment matrix of the
I bonds, YDJ is a vector constituted by the annual debtor
payments on the loan, and RGJ is a vector in which the last
element is the remaining debt at the end of the interest rate
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adjustment period snd the other elements are 0 (zero), cf..
step 4.
step 18 - Are all volumes positive?
Once again it must be ascertained that no volumes are
negative, which will have unfortunate consequences for the
debtor. Negative bond volumes are treated as in steps 9 and 10
in the model for F'.
However, if all volumes are positive, the model moves on to
the step 20.
Step 19 - Shift to F'
If the presence of negative volumes is established in step 18,
move on to the model for F'. However, there will be a
difference between whether the model for F' is called from step
10 at which the financial instrument has not yet been
activated, or from step 19. The information as to the step
from which the model is coming is therefore input together
with the upper data in the F' model.
Step 20 - Calculate the proceeds
Step 17 determined the volumes of the underlying bonds. The
proceeds of the bond sale may thus be calculated as
m
( 2 . 44 ) F, K,r(j)HJ(j)
y
The proceeds of the bond sale must be compared to the funding
demand with a view to determining the payments on the
financial instrument. The payments are given by
m
(2.14) F~=RGJ(0)-F,IC,7(j)H~(j)
W
If the maximum limits for the payments on the loan and term to
maturity, respectively, are mutually incompatible F,, will
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assume a positive value, the bond volumes in step 17 having
been reduced, and vice versa.
Step 21 - The loan nad the instrument have been calculated!
With the determination of the payments on the financial
instrument, all variables on both the debtor and the funding
sides of the loan have been calculated for the interest rate
adjustment period in question.
The model has thereby found a solution which fulfils all the
formulated requirements.
The division of the model into a simultaneous and a recursive
structure, respectively, is necessary because of the design of
the financial instrument. Pointing to possible improvements of
the model, an integration of the inner and the outer loop in
the simultaneous structure of the model is possible. Thus, the
model will have to iterate simultaneously over the interest
rate on the loan and the term to maturity in a two-dimensional
Gauss-Newton algorithm.
An integration of the inner and outer loops in the model will
obstruct the interaction with the recursive part of the model.
The model reaches the recursive part when the interest rate on
the loan has converged, but a convergent solution for the term
to maturity is not found, and in a two-dimensional iteration
it cannot be taken for granted that the interest rate
converges prior to the term to maturity.
2.6 Tie model for a t~ III type P
For a LAIR type P, a partial adjustment of the interest rate
of the remaining debt of the loan is performed each year. It
is intended that the partial adjustment of the interest rate
constitutes a fixed fraction selected by the debtor at the
disbursement of the loan.
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On the funding side, the intended interest rate adjustment,
fraction is decisive for the range of the underlying bond
portfolio. If it is intended to have an adjustment of the
interest rate on the loan of 10 per cent annually, it is
convenient to sell bonds with terms to.maturity of up to 10
years. After 10 years, the loan will be fully interest rate
adjusted and at the same time the last of the bonds originally
issued will mature. If the number of bonds are called mfl, the
interest rate adjustment fraction may be expressed as mo. The
period until the loan has been fully interest rate adjusted is
termed the funding period.
Unlike a LAIR type F, in the case of a type P, the interaction
is close between the funding side of the loan from interest
rate adjustment to interest rate adjustment. The partial
adjustment of the interest rate implies that some of the
underlying bonds do not yet mature. In the determination of
the bond volumes, account should be taken of the bonds
previously issued as well as of the future issue. This
consideration complicates the model.
The interaction between the individual adjustments of the
interest rate on the loan means that the conditions of the
model have an interte~oral aspect. The conditions for a
solution of the model are described in section 2.6.1. In
2.6.2. the problem is formulated and the general structure is
described before the actual model is set up in section 2.6.3.
There are several methods for solving the problem for type P.
An alternative model is described in appendix B.
a.s.l coaaitio~as is the model
As in the model for type F, a solution of the P model is
subject to a number of requirements.
~ Firstly, the term to maturity of the loan must observe
the maximum and minimum limits for the term to maturity.
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Secondly, the debtor's payments on the loan must observe
the maximum and minimum limits for the payments on the
loan.
The conditions may be formulated as
(2.3) 'dJ: L"'~"SLaSL'°°"
(2.8) dJ: YD~nSYD,~(.)S
whereas corrections of the term to maturity as in the model
for type F must fulfil
( 2 . 9 ) La a argmin( ~LJ - Ls i ~ I YD~n ~ YDJ( . ) S YD~ ", L'°in <_
LJ 5 L""")
making (2.3) and (2.8) operational. However, the definition of
J in the conditions should be noticed. The annual adjustment
of the interest rate means that the requirements are to be
evaluated solely for the current interest rate adjustment
period, whereas the conditions for J+1, J+2,..., J+mo are not
evaluated.
If e.g. an upwards correction of the term to maturity is to be
calculated following a rise in the interest rate, it is
important that this correction is calculated solely on the
basis of the payments on the loan in year J (the annual
adjustments of the interest rate mean that there is
equivalence between year and J, Which will be applied in the
i
notation). If the years J+1, J+2,..., J+ma are also included in
the calculation of the correction, the latter will immediately
be larger, the rise in the interest rate gradually making
itself felt and increasing the payments on the loan. But the
larger correction implies that in year J, the payments on the
loan are lower than the maximum limit contrary to (2.9).
Therefore, it is necessary that the evaluation of the above
conditions is limited to the current interest rate adjustment
period, even though further corrections of the term to
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maturity etc. must be anticipated at the future adjustments of
the interest rate.
Thirdly, the payments on the funding side of the loan are
to balance the payments on the debtor side of the loan.
On the debtor side, the payments are constituted by the
payments on the loan and refinancing amounts, and on the
funding side the payments are constituted by coupon rates and
mature bonds.
The annual adjustment of the interest rate means that the
l0 balance condition is given another role in the model. At the
end of each year, the total payments on the funding side of
the loan are known. By determining the refinancing amount
residually, the balance condition is fulfilled by definition,
the extent of the bond sale being determinable on the basis of
a balance requirement.
The fact that the refinancing amounts are determined
residually means, however, that the actual adjustment of the
interest rate may deviate from what was intended. The
deviations from the intended adjustment of the interest rate
should naturally be limited, as the choice of the debtor
should be respected in so far as possible. The balance
condition may therefore be interpreted as a condition of
accordance between the actual and the intended adjustment of
the interest rate. Henceforth, this condition is termed the
interest rate adjustment condition.
A deviation between the actual and the intended adjustment of
the interest rate will result in one or more of the calculated
volumes being negative, which, as in the model for type F, is
not accepted. In this situation, said volumes are assigned the
value 0 (zero), causing the interest rate adjustment fraction
to increase in relation to the intended level.
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The payments in the interest rate adjustment condition are
known for certain only for one year at a time. Already at the
next interest rate adjustment, the payments on the loan must
be expected to change in line with the development in interest
rates. Consequently, the interest rate adjustment condition
will also change. The interest rate adjustment condition may
therefore be formulated as
(2.45) Year 1 YD,i(1)+REG~R~,o°~ =HJ(1)+REGJ~ R,H~(7)Ha(J)
jsi
Both mo and m denote the number of bonds in the underlying
portfolio. In general, mo and m will be in agreement. At the
maturity of the loan, m is gradually scaled down, whereas mo is
constant, for which reason it is necessary to distinguish
between the two parameters.
The issue of bonds must also be arranged with regard to the
remaining years in the funding period, so that the possibility
of the intended adjustment of the interest rate being
respected at the future adjustments of the interest rate is
not reduced. The expected future conditions may be formulated
as
m
(2 .46) Year 2 YDa(2)+REG~R~'o°~ =H~(2)+REG~F, R~(j)IiJ(,j)
ja
~m-1
( 2 . 47 ) Year m YD,~(m) + REGJ~ = H,r(m) + REG~ ~ R~(7 )HJ(m)
j.qn
where
is a regulation factor for the interest rate
adjustme~rt percentage. in the first year fol-
lowing the disbursement of the loan. In the
disbursement year, the interest rate ad-
justment percentage is written down depending
on the quarter after which the loan is disbur-
sed. Th~refore, the factor may assume the va-
lues ('~i,'1~,~.1} . For J>0 the factor will have
the value d.
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The payments in (2.46) and (2.47) are not known, and (2.46,)
and (2.47) may thus not be applied directly as conditions in
the model.
HJ(j) expresses the total bond volume in the jth bond. Thus, a
summing is performed over the issue in the bond in the current
and the preceding interest rate adjustment periods. One cannot
preclude that the issue in the jth bond will have a different
coupon rate depending on the date of the issue, for which
reason coupon rates are indexed by J. A summing is performed
over all issued bonds in the calculation of the total coupon
payments.
If (2.45) is considered by itself, the problem is manageable.
mo volumes are to be determined under one condition, assuming
for a moment that the proceeds condition is fulfilled. This
results in an infinite number of possible solutions.
If year 2 is considered, an arbitrary distribution of the bond
issue in the individual years will however cause problems. If
a large part of the bonds were issued as 2-year bonds, the
right-hand side of (2.46) will be large. A reduction of (2.46)
requires a negative sale of the now 1-year bond year, which is
not accepted. On the contrary, the left-hand side of (2.46)
must adjust to the right-hand side. The only possibility is
that the interest rate adjustment fraction is increased
causing the intended adjustment of the interest rate not to be
respected. If, by contrast, a limited volume of bonds in, the
originally 2-year bond was sold, the problem would simply
arise at a later stage.
In order to be able to respect the intended adjustment of the
interest rate also at the future adjustments of the interest
rate, the distribution of the bond issue in the individual
bonds must follow a dynamic strategy involving a long-term
perspective.
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The regard to the future interest rate adjustment fraction
stipulates that a decreasing fraction of the expected payments
on the debtor side is funded by bond issues in the current
period. Thus, the volume of the payments may be adjusted on a
current basis in line with the development in interest rates,
and the possibility of respecting the intended interest rate
adjustment is optimized.
In the model, the falling profile in the bond issue is ensured
by the marginal issue in each bond being determined by a trend
function which is estimated on the basis of the profile in the
intended adjustment of the interest rate and the bonds already
issued, so that negative volumes are avoided.
The trend function is adjusted to the interest rate adjustment
condition and, if possible, to the proceeds condition such
that the marginal issue fulfils these conditions. An
adjustment to the proceeds condition presupposes that the
proceeds condition is not already applied for determining the
payments on the financial instrument, cf. below. In such
cases, the trend function is adjusted solely to the interest
rate adjustment condition.
Fourthly, the proceeds of the sale of bonds and payments
on the financial instrument must balance the funding
demand.
If the financial instrument is active, the proceeds condition
defines the payments on the instrument as in the model for
type F. The payments on the financial instrument are thus
given by
m
(2.48) F~=FinJ(0)-~K~(j)M~(,j) where
W
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FinJio) denotes the funding demand at the beginning-of
the interest rate adjustment period
~(j) (not to be confused with M) denotes the margi-
nal issue of the period in the jth bond.
The notation has been changed as compared with (2.14). As in
the F model, the funding demand is constituted by the
disbursement of the volume of the loan. However, at the
interest rate adjustments, the funding demand is defined by
the balance condition of the preceding year. Therefore, the
funding demand may be formulated as
(2.49)
volume disbursement
Firt,r(0) _ ~ H~l(j)R~1(j).~",HJ i(1)-YD,~1(1) adjustment of the
interest rate
If the actual adjustment of the interest rate corresponds to
what was intended, it applies in (2.49) that
N
( 2 . 50 ) ~, H,m(j)R.m(7)+H.F-i(1)-YD,m(1) = RG,m(1)
mo
ji
Further, the notation has been changed, a distinction being
made between the marginal issue and bonds already issued. This
distinction is necessary, as the proceeds naturally
corresponds only to the market price of the marginal issue.
The marginal issue may be defined as
(2.51) Ma(j)=H~i(j)-H,~l(j+1)
i.e. as the adjustment to the jth bond.
If the financial instrument is inactive, corresponding to the
limits for payments on the loan and term to maturity being
compatible, the proceeds condition is given by
( 2 . 52 ) Fin~(0) _ ~ K,~( j)M,~(j)
W
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Fifth, the interest rate on the loan must correspond to
the yield to maturity of the underlying bond portfolio.
A LAIR type P is characterized in that there is not a
one-to-one connection between the debtor and the funding
sides. Consequently, it is not without ambiguity when the
payments of the debtor side in year j is funded by the issue
of bonds. In principle, this paves the way for an imbalance
which will make itself felt in the interest rate on the loan.
Therefore, a condition with respect to a connection between
the interest rate on the loan and the yield to maturity of the
underlying bonds must be imposed on the solution'to the model.
Full accordance is not required. Differences in the number of
payment dates on the debtor side and the funding side mean
that a certain difference between the interest rate on the
loan and the yield to maturity of the bonds will occur, said
difference not being due to imbalances.
In situations in which the financial instrument is active, the
interest rate on the loan is determined with regard to the
limits for the payments on the loan. It is not thereby
possible to maintain the condition with respect to a
connection between the interest rates on the debtor and
funding sides, for which reason the requirement is suspended.
2.6.2 The Qenarsl structure of the modal
The model for type P solves the problem
The pro~ble~ for type P
Determine the term to maturity of the loan. the interest
rate on the loan, the volumes of the underlyingr bonds, and the
volume of the payments on the financial instrument such that
1. the term to maturity is within the maximum limit and
minimum limits, cf. (2.3)
2. the payments on the loan are within the maximum and
minimum Limits, cf. (2.8)
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3. the interest rate adjustment condition is fulfilled,
cf. (2.45)
4. the proceeds condition is fulfilled, cf. (2.52)
5. the payments on the financial instrument fulfil the
condition given by (2.48), and
6. the interest rate on the loan fulfils the condition
with respect to a connection with the yield to
maturity of the bond portfolio.
The number of variables in the model constitutes m+2, m
volumes, an interest rate on the loan and a term to maturity
or a payment on the financial instrument having to be
determined.
On the basis of the two first conditions in the problem
formulation, (2.9) defines an equation to be fulfilled by the
solution with respect to the determination of the term to
maturity.
If, at first, a situation is considered in which the financial
instrument is inactive, the interest rate adjustment condition
and the proceeds condition each define an equation. However,
via the trend function which is adjusted to the two
conditions, m equations are formed which determines the
volumes of the underlying bonds. Finally, in this situation
the interest rate condition will constitute yet another
equation causing the conditions to lead to a total m+2
equations, including (2.9), as required.
In the situation in which the financial instrument is active,
the proceeds condition determines the payment on the financial
instrument, and, therefore, the condition is not included in
the adjustment of the trend function. The trend function is
adjusted solely to the interest rate adjustment condition
which, in this situation, leads to m equations. The interest
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condition is discontinued, causing the conditions to lead to
m+2 equations in this situation as well.
It appears from the above that the construction based on the
trend function i.s necessary in order to achieve equality
between the number of variables and equations in the model.
Without the trend function, the equation system would be
sub-determined with an infinite number of possible solutions.
The model for type P has the same simultaneous and recursive
structure as has the model for type F. If the financial
instrument is inactive, the model is simultaneous. In an outer
loop. iteration over the term to maturity is performed,
whereas an inner loop iterates over the interest rate on the
loan until the interest rate adjustment condition and the
proceeds condition have been fulfilled. The iteration in the
inner loop follows a two-step procedure. First, the bond
volumes are calculated via the trend function given the
interest rate, and then the convergence of the interest rate
is tested against the yield to maturity of the bond portfolio.
The inner loop is thus more extensive than in the model for
type F.
The recursive structure occurs if the limits for payments on
the loan and term to maturity collide under the given yield
curve and there are thus payments from the financial
instrument.
In the first step in the recursive structure, the interest
rate on the loan is determined as a function of the
established payments on.the loan. Then a new trend function is
defined Which estimates the remaining debt profile of the loan
under the recalculated interest rate. The trend function is
adjusted to the interest rate adjustment condition which is
thus also respected in a solution of the model with an active
financial instrument.
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Finally, the payments on the financial instrument are
determined on the basis of the proceeds condition. The flow
chart of the model is shown in figure 9.
2.6.3 The steps of the model
Step 1 - Determiae the term to maturity within the interval
from L~ to Lm°"
Step 1 determines an initial value of the iteration over the
term to maturity in the outer loop.
A_t the disbursement of the loan, the debtor's input determines
an initial value for the term to maturity. As in the model for
type F, the possibility of the debtor selecting an intended
payment on the loan must be kept open, in which case the term
to maturity is given by
1- xco(o>
( 2 .16 ) Lo = -In
1 +Ro
The interest rate on the loan is unknown and is approximated
by the initial interest rate on the loan determined in step 2.
A~~di ~ ne_nts of the interest rate on the loan, the term to
maturity is taken over from the preceding adjustment of the
interest rate. Therefore, the initial term to maturity may be
read in the input vector, cf. section 2.3.
At the maturity of the loan, the term to maturity is to be
determined according to special directions. The current issue
of bonds with maturity after the next adjustment of the
interest rate means that the term to maturity must not be
shortened uncritically when the loan is approaching the year
of maturity. A risk would thereby arise that the loan matures
before a volume of the underlying bonds, causing imbalances in
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the payments to be inevitable. Thus, when the model is
entering the last funding period of the loan for J>_ M- mo,.the
term to maturity must, in general, not be shortened.
Consequently, it is impossible in the case of a fall in
interest rates to maintain the debtor's payments on the loan
at the minimum limit by shortening the term to maturity. A
payment on the loan below the minimum limit will not
necessarily trigger payments to the financial instrument, as
this also requires that the term to maturity has reached the
minimum limit.
The adjustment of the term to maturity of the loan such that
the maturity coincides with a payment date on the funding side
may thus be effected solely as a prolongation of the term to
maturity of the loan to the next payment date of the
underlying funding.
However, it is inadequate first to adjust the term to maturity
for J=M. As soon as M is within the funding period, the model
funds payments in the last year. Therefore, it is necessary at
an earlier stage to correct the term to maturity.
Consequently, a parameter closing time is introduced, which
specifies when the model is to be adjusted to the time of
maturity of the loan for the first time, i.e.
(2 .53 ) dJZM-mo : LJ=LJ
Closing time must be determined on the basis of the type of
loan. Therefore, it will not make any sense to set closing
time > mo. Closing time = ~ will provide the most convenient
maturity, but implies, however, that the payment on the loan
may exceed the minimum limit for a long period.
If the loan has a fixed term to maturity, said fixed term to
maturity is immediately input.
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The total input to the model is input in step 1
the funding sidQ
the number of bonds (and thus also the interest rate adjustment
percentage), the nominal interest rate of each bond, the number of
annual creditor payment periods and creditor payment dates, and the
market price of each bond, volumes of bonds already issued.
the debtor aide
the disbursement date, the volume or the remaining debt, the funding
demand, the number of annual debtor payment periods and debtor
payment dates, the maximum and minimum limits for the payments on the
loan, the maxima~m and minimum limits for the term to maturity and,
optionally, an intended payment on the loan or term to maturity.
In addition, values for inc and closing time are input.
Finally, the validity of the limits is checked as in the model
for type F.
Step 2 - Determine the initial interest rate on the loan.
Determine m
As a start value for the iteration in the inner loop, the
following approximation is applied
~ m~ t~r~0, t~
(2.54 ) Ro = 'il ~ ~. t where
m
denotes the term to maturity of the underlying
bonds
r(O,t) is the yield to maturity at time 0 of the bond
with maturity at time t. At time 0, the yields
to maturity may be observed. In other situati-
ons, the yields to maturity are calculated on
the basis of the zero-coupon rate structure
estimated by the model.
(2.54) approximates the interest rate on the loan by a
term-weighted average of the yields to maturity of the
underlying bonds. (2.54) presupposes that a value for m is
determined. Typically the value is set to m=mo. In two
situations, however, another determination of m is necessary.
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Firstly, the interest rate adjustment fraction is written down
in the year of disbursement via the REGJ-faktoren factor in
relation to the quarter in Which the loan was disbursed.
Consequently, another year will elapse before the loan is
fully interest rate adjusted, and the funding period is thus
to be prolonged by one year. Therefore, a variable TILT is
introduced where
(2.55)
1 if the loan is disbursed in the period
TILT = from January to November
0 if the loan is disbursed in December
such that
( 2 . 56 ) m=nno+TILT
TILT also indicates that a special procedure is to be applied
in the determination of the volumes.
Secondly, m is to be adjusted such that bonds with maturity
later than the maturity of the loan are not issued.
Collectively, m is determined as
( 2 . 57 ) m = min [Ls - ~Q : mo + TILT]
where L~-cp is the remaining term to maturity of the loan.
Step 3 - Calculate the debtor payments on the loan
The model calculates a development in the payments on the loan
and the remaining debt on the basis of the interest rate on
the loan.
Stop 4 - Calculate the loan for m=1
If the number of bonds is limited to 1 - corresponding to the
loan maturing in one year - the problem is considerably
simplified. The funding demand is to be funded by sale of only
the 1-year bond. The marginal issue is given by
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M~(1)=F=! for J=M
(2.58)
K~(1)
As no further adjustments of the interest rate on the loan are
to be performed, the nature of the interest rate adjustment
condition changes. That the loan is not interest rate adjusted
at end of period may be interpreted as the interest rate
adjustment amount at end of period having to be 0 (zero) in
the condition. Thus, the condition has the same contents as
the balance condition, i.e. the payments on the loan are to be
determined as
(2.59) YDJ(1)=(1+RJ(1))H,T(1) for J=M
which determines an unambiguous interest rate on the loan.
The convergence of the interest rate is tested in step 12.
Step 5 - Define a tread function
For m~1 the bond volumes are determined by the trend function
to be defined in step 5. The trend function estimates the
future intended interest rate adjustment amounts corrected fox
the funding already issued. Therefore, the trend function must
have a functional form which is appropriate for estimating the
development in the remaining debt of the loan to be followed
by the interest rate adjustment amounts. A satisfactory
estimation is achieved by applying a polynomial of (q-1)th
degree, causing the trend function to have the form
i
(2 . 60 ) ao +ai(J -1)+aa(j -1)a + . . . +aQ-i(J -1)a-
for j=1,2,..., m. The trend function depends on (j-1) and not
on j. This is due to the fact that the intended adjustment of
the interest rate is given as a fraction of the remaining debt
at beginning of period. Therefore, the jth volume is decided
on the basis of the remaining debt at time j-1.
The degree of the polynomial must not exceed the number of
bond volumes minus 1, whereby the polynomial would have too
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many degrees of freedom in the determination of the ~
coefficients. Therefore, the degree is limited by m. In the
special TILT procedure in which the number of volumes is
increased, it is not necessary also to increase the degree of
the polynomial, for which reason
(2.61) gym-TILT
Generally, the degree of the polynomial is maximized such that ~
q=m-TILT, causing a perfect estimation of the future intended
interest rate adjustment amounts to be achieved.
Step 6 - Determine the coefficients in the trend function
The actual estimation of the future intended adjustment of the
interest rate is performed in step 6 in which the coefficients
of the trend function are estimated. For each j=1, 2..., m the
value of the trend function is to balance the intended
adjustment of the interest rate less the band volume already
issued such that negative marginal issue does not occur. The
coefficients in the trend function must fulfil
( 2 . 62 ) ao + al( j -1)+ a2{ j -1)i + . . . + ag-i {.7 -1)~1 =
maxi REG,°iRG ~o-1) ~ H~(0 , J),
for j=1,...,m. (2.621 is solved by the matrix equation
( 2 . 63 ) {ao , al , . . . , aQ..l) _ [H B] B maxCREG,, mo a{ J)
DRG~~ H p, J
where {ao,ai,...,a~~) is a qxl vector, max( . ) forms an mx1 vector,
and 8 is an mxq vector given by
1 ji-1 Ui-1) ~ ~ (Jml)°-i
1 ja-1 ~~i"1)z ~ ~~2-1)~1
1 jm-1 (jm-1)2 . (j~-1)'ri
(2.64)
*rB
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1 0 0 . o
1 1 1 . 1 ' '
1 2 4 . 2 Q-i
1 m-1 (m-1)z . (m-1)~'
as (jo.j~~ ...,j,~)=(1, ...,m)
Step 7 - Determine Go and Gl
In step 7, the trend function is to be adjusted so that the
resulting bond volumes fulfil the interest rate adjustment
condition as well as the proceeds condition. The adjustment is
performed by two factors Go and Gl being added to the trend
function, said factors being determined on the basis of the
two conditions. The trend function appears as follows
(2.65) Goao+Giai(j-1)+aa(j-1)2+...+ag..l(j-1)x'1
Go shifts the trend function in the vertical plane, whereas G1
influences the slope of the function. Generally, it will thus
be possible to find a solution which fulfils both conditions.
The adjustment of the trend function is illustrated in figure
10.
The bond issue is to be arranged according to a long-term
strategy, cf. section 2.6.1. On the operational plane, the
long-term strategy may be interpreted as a declining part of
the expected future payments having to be funded by means of
bond issue here and now. The total volume outstanding which
determines the volume of the adjustment of the interest rate.
can thereby be adjusted such that the intended level is
realized. The long-term strategy is introduced in the model
via Go and G1. The primary task for Go is to adjust the issue
of the bond with the shortest term to maturity such that the
interest rate adjustment condition is fulfilled.
By adjusting the value of G1, the declining funding fraction of
the future payments is secured: G1 is determined primarily with
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a view to a fulfilment of the proceeds condition. Typically,
the slope of the curve is to be increased in order for the
proceeds not to exceed the funding demand. Thereby a declining
funding profile is also secured.
The trend function added Go and G1 may be divided into a (Go,GI)
variant part and a (Go,GI) invariant part. The variant part is
denoted X,,(j), whereas the invariant part is denoted YJ(j)
given by
Xa(7)=Goao+Glai(j-1) and
(2.66)
YJ(J)=aa(.7-1)2+aa(J-1)3+ . . . +ag-1(J-1)W
The marginal issue is determined as the value of the trend
function of the j in question corrected for bonds already
issued in the bond in question. However, M(j) is to be
positive. For j=1, 2,..., m is M,,(j) given by
M.~(7)=(0J Goao+Giai(J-1)+aa(J-1)2+
(2.67) , . . +a 1(j-1)ø'1-hiJ(0, 7)
g-
By means of (2.66) the expression may be formulated as
( 2 . 68 ) MJ(7) =~0 i X,r(J)+YJ(~)-H~(o, .i)1
Go and G1 cannot be isolated in (2.68) as long as the
expression comprises a max-function. Therefore, an indicator
function IJ(j) j=1, 2,..., m is implemented with the value 0
(zero) if the jth bond volume is to be assigned the value 0
(zero), and otherwise with the value 1. Thus, the indicator
function forms an m-dimensional vector.
The point of departure is that all bonds are to be applied,
for which reason IJ(j) is assigned the value
(2.69) IJ(j) - (1,1....,1)
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If negative volumes are produced, a new value is assigned to
IJ(j) in step 9. With the introduction of the indicator
function, (2.68) may be formulated as
r~(~)-IJ(~)IxJ(~)+YJo)-HJ{o, ~)l -
(2.70)
IJ(~)XJ(.~)+IJ(j)(YJ(J)-HJ(0. j))
If it is to be possible to fulfil both the interest rate
adjustment condition and the proceeds condition, at least two
bonds must be available. That is to say that at least two
bonds have not previously been assigned the value 0 (zero). At
the same time, one of the two bonds must be that with the
shortest term to maturity, as it is otherwise impossible to
fulfil the interest rate adjustment condition. This may be
formulated as
(2.71) ~,IJ(j)?2 and I,,(1) > 0
W
If both conditions in (2.71) are fulfilled, Go and G: are to be
found as the solution to both the interest rate adjustment
condition and the proceeds condition given by
(2 .45) YDJ(1)+REG,°iRm~~ =HJ(1)+REGJ~ R~(j)HJ(j)
j.1
where HJ(1)=xJ(o, i)+rlJ(1) and HJ(j)=MJ(j)+HJ(o, j) and
m
(2 . 52 ) FinJ{0) _ ~, KJ(j)MJ(j)
j.i
(2.70) may be inserted in (2.45) for MJ(1) and MJ(j).
YD,T(1)+REGJRC
(2.72) HJ(0, 1)+IJ(1)XJ(1)+IJ(1)(YJ(1)-XJ(0, 1))+
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REGJ~R,N~(,j)[IJ(J)XJ(J)+ra(J)(YJ(J)-HJ(0~ J))+xJ(0~ J)J
W
The expression is solved for X (j)
IJ(1)XJ(1)+REGJ ,F,, RJ(7)IJ(.7)XJ(J) _
~S1
(2 .73 ) ~J(1)+REG,°7RGm~(0) -HJ(0 ~ 1) _ IJ(1)(YJ(1)-HJ(0 ~ 1))-
REC~a~ RN(1)II,r(I~(YJO~ -H~(0,17) +H~(a,l~l
W
The right-hand side of (2.73) defines the variable Z:.
Correspondingly, (2.70) may be inserted in the proceeds
criterion. This leads to the expression
(2.74) Fins(0)=~KJ(,7)~IJ(.7)XJ(J)+IJ(.7)(yJ(J)-HJ(0~ J))J
j=i
(2.74) is solved for X,,(j)
(2.75) ~'rKJ(~)IJ(J)XJ(J)=FinJ(0)-~,KJ(J)IJ(J)(~'J(J)-HJ(0. J))
j=1 j=i
The right-hand side of this expression is denoted Z,. (2.73)
and (2.75) define two equations with two unknown quantities Go
and G1. The solution for the equation system depends on TILT.
If the interest rate adjustment fraction has been written
down, TILT has the value 1. In this situation the first volume
must be calculated explicitly. This is due to the fact that
the volume of the first adjustment of the interest rate may be
as small as '~4 of the other adjustments of the interest rate,
which may hardly be comprised by the trend function and the
adjustment thereof.
First, Go and G1 are found in the general situation in which
TILT has the value 0 (zero).
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XJ(j) is defined in (2.66) as that part of the trend function
which depends on Go and G,. In matrix form XJ (j) ~Y be
written as
(2.76) X=aa
where X=(XJ(1) , X,,(2) , . . .. XJ(m) ) is a lxm vector, G = (Ga, G1)
is a 2xl vector and a is an mx2 matrix given by
ao 0
ao ai
a = ao 2a1
ao (m-1)al
Then a matrix R is defined such that RxX, and thereby KxaxG
constitutes the left-hand sides in (2.73) and (2.75). K is
given by
(2 .77 ) R = IJ(1)(1+REG,rR~(1)) IJ(2)REGJR,~~(2) ::: I~(m)REGJRJ(m)
I,r(1)K.r(1) IJ(2)K~(2) IJ(m)KJ(m)
With the definition of K, the interest rate adjustment
condition and the proceeds condition given by (2.73) and
(2.75) may be written in a matrix form, in. which it applies
that the right-hand sides of (2.73) and (2.75) are given by Z1
and Zz .
(2.7$) Z=~
where Z = (Z1, Za) is a 2x1 vector. Ra forms an mxm matrix,
which is invertible. Therefore, Go and G, may be found as
(2.79) d = (Ra]-1Z
which determines the trend function. However, if TILT has the
value 1, MJ(1) must be determined explicitly as
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Ma(J)=(Ot (Wa(1). Goao+ai(7-1)+az(J-1)2+
t2.80)
. . . +aq..l(j-1)'rl)-Fla(0. J))
Again the trend function may be divided into a variant and an
invariant part given by
x.~(J) _ ~ G°a° ~3 ~ 1 and
W~(1) .1=1
(2.81)
=ai(.7 -1)+aa(.7-1)a+ . . . +aQ..i(.7 -1)r
1'J(J)
W,,(1) and Go must be determined such that the resulting volumes
comply with the interest rate adjustment condition and the
proceeds condition. It is not necessary to suspend the
conditions during the TILT procedure. The proceeds condition
is observed by definition, and the very objective of the TILT
procedure is to secure the interest rate adjustment condition.
The nature of the conditions does not change as a result of
TILT, and the conditions are thus, unchanged, given by (2.73)
and (2.75), which are not repeated here. The only change in
the conditions is the definition of X,,(j) and Y,,(j), which is
here given by (2.81). The right-hand sides of (2.73) and
(2.75) are denoted Ziand Z=, respectively. X,1(j) may be
expressed in the matrix form
( 2 . 82 ) R=a'G' where
0 1
(2.83) G' - (Go, WJ(1) ) and a'= a° 0
ao 0
Analogously to the above, (2.73) and (2.75) lead to the
expression
( 2 . 84 ) Z'=Ra'G'
where Z'=(Zi,Zi) and K are, unchanged, given by (2.75). As
before, the matrix equation is solved by
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(2.85) G~ - f~~1 'Z~
causing the trend function to be determined. However, if
either
(2.86) ~I(,j)<2 or I(1) - 0
W
only the proceeds condition can be fulfilled, whereas the
balance condition is suspended. It follows from (2.86) that
either the model has only one bond at its disposal or the
first bond is not available. In the first case, it is possible
i0 to fulfil only one condition in which case priority is given
to the proceeds condition. In the second case, observance of
the interest rate adjustment condition requires that the first
volume is negative, which is not accepted. Thus; a suspension
of the interest rate adjustment condition is necessary.
(2.86) having been fulfilled, Ga and G1 must, in principle, be
determined on the basis of only one condition resulting in an
infinite number of solutions. Therefore, a fixed value is to
be assigned to either Go or G1. It is immediately preferable
that Ga is maintained as the free variable. The adjustment to
the proceeds condition via G, could produce inappropriate
solutions in which e.g. the bond issue follows a rising
profile. A fixed value is thus assigned to Gl, said fixed value
being determined by numerical analysis.
The fixing of G1 implies that now G, belongs to the invariant
part of the trend function. The definition of XJ(j) and YJ(j)
is thus changed to
XJ ( j ) = Goao and
(2.87)
1'J(.7) = Giai(J -1) + aa(J -1)2 + . . . + sg-i(j -1)arl
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Go is determined in accordance with the proceeds condition.
which, solved for XJ(j), is given
(2 .75) ~ KJ(~)IJ(.~)XJ(~) ~ FinJ(0)- ~, KJ(~)IJ(~)(yJ(~)-HJ(0 ~ ~))
j=1
XJ(j) is one-dimensional which paves the way for a relatively
simple solution of (2.75). In the proceeds condition, the
expression for XJ(j) is inserted given by (2.87)
( 2 . 88 ) Goao ~ KJ(.7)IJ(.7) = FinJ(0) - ~ KJ( j)IJ(J)(YJ(J)-HJ(0 . .7))
ii j.l
Go may then be isolated
FinJ(0)- ~ KJ(7)IJ(J)(~'J(J)-HJ(0. 7))
(2.89) Go=
a0 ~ KJ(~)IJ(J)
ji
and thereby determined. By insertion of Go in the balance
requirement, the interest rate adjustment is calculated
residually in the first year. It is not necessary to determine
Go for TILT=1. TILT=1 will occur only at the disbursement of
the loan. At this point in time, the volumes of the already
issued bonds are all 0 (zero), causing the marginal issue not
to be negative. By definition, (2.86) will thereby not be
fulfilled.
step s - Determine the volumes
The marginal issue is determined by
MJ(7)=Goao+Giai(7-1)+aa(7-1)z+ . . .+
(2.90)
set-i(J -1)ø1-HJ(0. .7)
for TILT=0 and as
MJ(J)=(wJ(1)~ Goao+ai(J-1)+az(J-1)z+. . .+
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(2.91)
8Q..1(j -1)~l -Ho(0 . j))
for TILT=1. Both expressions are defined for j=1, 2,..., m. A
max-condition is not included in the expressions for M~(j):
Indeed, negative volumes are to be observable, so that
adjustments of the indicator function and Go and Gl are
possible.
Step 9 - Are all volumes po~itive?
Step 9 determines whether the calculated volumes are
applicable, or Whether an adjustment of the indicator function
and subsequent recalculations of the volumes are necessary. If
the marginal issue is negative for just one j. i.e.
(2.92) 3jE {1,2, ...,m} :Ma(j)<0
the indicator function is adjusted in step 10. Otherwise, the
model moves on to step 11.
Step 10 - Adjust the indicator function
For the bonds in which a negative marginal issue was
established in step 9, the indicator function is adjusted such
that
(2 . 93 ) IJ( j)= 0 dj :M~(j) < 0
1 dj :MJ(j) Z 0
Normally, nothing is gained by setting the volumes to 0 (zero)
one by one. When a volume is set to zero, fewer bonds in the
remaining years must be sold in order to fulfil the proceeds
condition. Thus; the volumes of all the remaining bonds will
typically decrease. Once a volume has turned negative, it is
likely to become even more negative for each operation of
assigning 0 values) via the indicator function.
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Step 11 - Calculate adjustment to the interest rate oa the
loan
According to the interest rate condition, there must be a
certain degree of accordance between the interest rate on the
loan and the yield to maturity of the underlying bond
portfolio. Therefore, an adjustment to the interest rate on
the loan is calculated in step 11 on the basis of the
deviation in relation to the yield to maturity of the bonds.
The condition does not require full accordance, differences in
the number of payment periods on the debtor and bond sides
having to be corrected for. Therefore, the interest rate on
the loan must be corrected for a "prepayment effect". At the
same time, there is a tradition that the interest rate on the
loan is calculated according to the money market convention.
The adjustment is calculated on the basis of the function
" RNRGJ(i-1)-~GJrp where
(2.94) f(Ro)= ~r n R" G
is the yield to maturity of the bond port-
folio
n is the number of payment periods per year
i indexes the payment periods within the
year
If (2.94) is inserted in the reduced Gauss-Newton algorithm,
the adjustment may be calculated as
a _ inc
( 2 . 95 ) ~J -' f(R J) _ f(R~+ inc)
Step 12 - Bas the iaterest rate on the loan corwerrged?
The mathematical convergence of the interest rate on the loan
is tested by
(2.32) E~~) <e
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f(RJ)2 -f\RJ+~J~2 ~ .
(2.33) <s
f(R~)s
where merely one of the conditions must be fulfilled in order
that the interest rate has converged and the model can proceed
to step 14. Otherwise, the model proceeds to step 13.
Step 13 - Adjust the iatereat rate on the loan
The interest rate on the loan is adjusted by the calculated
adjustment. Then the model repeats step 3.
Step 14 - Calculate adjustment to the term to maturity
The adjustment to the term to maturity is calculated as a
function F(LJ) of the deviation of the payments on the loan
from the established limits.
If the payments on the loan are within the band defined by the
maximum and the minimum limits, no adjustment to the term to
maturity is calculated, corresponding to DLJ=0. If the payments
on the loan exceed their maximum permissible value, a positive
adjustment to the term to maturity is calculated on the basis
of the relationship between the calculated payments on the
loan and the maximum limit. However, if the payments on the
loan are too law, a negative adjustment is calculated as a
function of the relationship between the payments on the loan
and the minimum limit. f (LJ) is given by
~n~(~)-3rd"' : Yn~(~) > 3rn~ "
( 2 . 3 4 ) f(La) = 0 : YD~" S YDJ( j ) S YD°~°"'
YDa(.7) - YD ~i" : YD,r(J) < Y
for J<M-mo where all payments on the loan are for integer
years. The adjustment is calculated by use of the Gauss-Newton
algorithm which may be written in its reduced form as
( 2 . 3 6 ) phi = f(LJ) inc
F(Lo) - f(L.r+ i.nc)
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The term to maturity must not be shortened for JZ M - mo. The
definition of ffLJ) is thus changed to
(2.96) f(Lo)= YDa(7)-~ : YD~(j)>Y~
YD,,(~) s 'YD~'n
whereas the adjustment is calculated, unchanged, by (2.36). If
the term to maturity of the loan is fixed, an adjustment to
the term to maturity is calculated after all according to the
above directions. This is the only way of determining whether
the payments on the loan are compatible with the fixed term to
maturity.
Step 15 - Are the payareats on the loan within the interval
fro~n YD~ to YD~'?
The calculated debtor payments on the loan are to be compared
with the limits for the payments on the loan. This is done by
testing the mathematical convergence of the adjustment to the
l5 term to maturity given by
E'~j~ < a
(2.40)
f(Lv)2 - f(L~+dLJ)a ~ < E
(2.41)
f(L~)2
where ~LJ must fulfil merely one of the conditions for the term
to maturity having converged.
Step 16 - The loan has been calculated!
If the term to maturity has converged in step 15, all
variables on the debtor and funding sides of the loan have
been calculated. Step 16 presupposes that a solution has been
found which fulfils the limits for payments on the loan as
well as term to maturity. Thus, the payments on the financial
instrument are 0 (zero).
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Step 17 - Is the corseet~l term to maturity within the
interval frown L~° to L'a'?
Step 17 determines whether payc~nts on the financial
instrument are necessary in order that the limits for payments
on the loan as well as term to maturity can be observed.
In step 15 it was concluded that the term to maturity has not
yet converged. If the term to maturity corrected by the
adjustment calculated in step 14 is within the allowed limits,
the adjustment is applied, and the debtor and funding sides of
the loan are recalculated in the simultaneous model in which
the financial instrument has the payment 0 (zero) by
definition.
If a correction of the term to maturity by the calculated
adjustment will cause the term to maturity to exceed the
allowed value, the model continues in the recursive structure
instead.
Step 18 - Correct the term to maturity by an adjustment
The calculated adjustment may be applied, and steps 2 to 17 of
the model are repeated with the corrected term to maturity.
Step 19 - Determine the interest rate on the loan such that
the payments on the loan are ~rithin the interval
Step 19 initiates the recursive part of the model in which the
financial instrument is active. The interest rate on the loan
must be determined such that the payments on the loan
correspond to the relevant limit. That is to say that the
interest rate on the loan is determined by a numerical method,
so that the relevant condition of the following two conditions
is fulfilled.
(2.42) R'RG'(0) _ for ~.~~)~~ >
1-(1+RJ)~
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RJRGa(0) = YD°in for YD~(.7)~rr« < YD~"
(2.43) J
1- ~y ~r~~-r
It should be noted that throughout the recursive model, TILT
has the value 0 (zero), as the loan is presumed to be within
the limits for payments on the loan and term to maturity at
the disbursement . Sima.larly, REGJ and REC3D may also be
omitted.
Step 20 - Define a trend function
As in the simultaneous model, the basis of the determination
of the bond volumes is a trend function estimating the
intended interest rate adjustment profile. First, the trend
function is defined, unchanged, as
( 2 . 60 ) ao + auJ -1)2 + . . . + aQ-i(j -1)øi
for j=1,2,...,m and q Sm
Step 21 - Determine the coefficients in the trend function
(a°, al, . . . , aq_1) must be estimated on the basis of the
intended interest rate adjustment profile which may be
calculated on the basis of the recalculated interest rate on
the loan. The coefficients are found by the matrix equation
$~,$ 1B1,~R~iJy HJ(y ~)~
( 2 . 97 ) (ao . ai . . . . , aa.i) C J mo
(2.97) has been changed as compared to (2.63). as REG° may be
omitted since RBGn= 1 by definition. B is, unchanged, given by
(2.64) .
Step 22 - Dotezmiae G° such that the interest rate adjus~aent
ca~d~.tion is fulfilled
In the following step.in the model, the proceeds condition is
to be applied to determine the payments on the financial
instrument. Therefore, the trend function is only adjusted to
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the interest rate adjustment condition, for which reason only
one factor Go may be determined.
Alternatively, Go could be maintained, whereas G1 is used as an
adjustment variable. G1 determines the slope of the trend
function, which has a greater impact on the long-term bonds in
the portfolio than on the very short-term bond which is
decisive for the interest rate adjustment condition. Thus, by
adjusting G1 to the condition, there is a risk of large
imbalances occurring in the issue of long-term bonds.
The trend function is divided into a G~ variant part and a Go
invariant part given by
X~( j) = Go ao and
(2.98)
Y.~(.7)=Giai(j-1)+aa(j-1)2+ . . . +a~l(7-1)ø1
for j=1,2,...,m. G1 is maintained in the invariant part. By
numerical analysis, a fixed value for G, may be determined
providing the trend function with an appropriate slope. It is
not given in advance that G, is to be assigned the same value
as in step 7. Firstly, the trend function is to be adjusted to
another payment profile and secondly, there will be a
difference in the value of Go, depending on whether said value
is determined on the basis of the interest.rate adjustment
condition as is the case here, or the proceeds condition as in
step 7.
The marginal issue is given by
(2.99)
M,i(j)=max(0: Goao+Gisi(j-1)+aZ(j-1)a+
. . . +aø1(j-1)~'1-x~(0, j))
which by use of (2.98) may be formulated as
(2.100) M~(,7)=I~(J)Xa(j)+IJ(J)(Y~(.7)-H,~O, J))
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where I,,(j) is the indicator function as defined in step 7. The
interest rate adjustment condition is given by
(2 .102 ) YDJ(1)+ RGmo(0) =H~(1)+ ~ RJ(j)H,r(j)
The interest rate adjustment condition is inserted in (2.100),
which produces the expression
IJ(1)XJ(1)+ Fr RJ(J)IJ(7)~.7) = YDJ(1) + RGmoO) -HJ(a ~ 1)
im
(2.103)
-IJ(Z)II'J(1)-HJ(~, 1)I-FrR~(J)II~(J)(1'J(.7)"HJ(~, j))+Ho(~, J))
W
X,~( j) = Goao is inserted in the expression
m
(2 .104) I~1)Goao+ ~ RJ(j)Ia(j)Goao = Z'"
i-~
where Z" is the right-hand side of (2.103). Solved for Go,
(2.104) appears as follows
(2.105) Go= m
Ia(1)ao + ,F,, R~(j)I,r(j)ao
j=1
The determination of Go permits the calculation of the
distribution on the individual bonds in step 23.
Step 23 - Determine the volumes
The marginal issue is given by
MJ( j) = Goao + Giai(j -1) + sa (j -1)2 + . . .
(2.90)
+sa..i(7 -1)ø1-H~(0 , j)
Step 24 - Are all volumes positive
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(2.90) paves the way for negative marginal volumes for one or
more bonds. If
(2.92) 3je {1.2. ...,m~ :M,r(j)<0
the trend function is adjusted in step 25. Otherwise, the
model proceeds to step,26.
Step 25 - Adjust the indicator function
The indicator function is adjusted according to the following
directions
(2 .93 ) Iatj)= 0 b~j :MJ(j) < 0
1 b'j :MJ(j) ~ 0
Step 26 - Calculate the proceeds condition
The proceeds of the determined marginal issue of bonds may be
computed to
m
(2 .106 ) ~, Ka(J)MJ(J)
f-i
However, in relation to the funding demand, one may record
either a loss in proceeds if the maximum limits for payments
on the loan and term to maturity were incompatible, or a gain
in proceeds in the opposite situation in which the minimum
limits are incompatible.
The loss or gain in proceeds defines the payments on the
financial instrument which is thus given by
m
( 2 . 48 ) Fs s Fina(0) - ~ K~7)Ma~7)
FinJ(0) is given by (2.49).
Step 27 - ~ load and the 3astrume~nt have been calculated!
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All variables on the debtor side as well as the funding side
of the loan have been determined for the interest rate
adjustment period in question.
A variant of the model steps 5 to 10 is described in appendix
B. In terms of formulae, the variant is easier, but it is more
difficult to calculate. This is because in the model, Ga and G,
are determined by an iterative procedure which, on the one
hand, stipulates smaller requirements as to formulae but
which, on the other hand, requires more calculation time.
2.7 Suaamary
A LAIR III introduces a band for the debtor's payments on the
loan. The band is defined by a maximum limit and a minimum
limit which are determined according to the choice of the
debtor.
To begin with, the limits for the payments on the loan are
defended by the adjustable term to maturity. If the payments
on the loan are floating outside the band, the term to
maturity is corrected. The corrections of the term to maturity
are determined such that the payments on the loan only just
moves within the band . The term to maturity is thus adjusted
as little as possible.
If the term to maturity reaches either its maximum or minimum
value, the possibilit~.es of adjusting the term to maturity are
exhausted. If the interest rate further rises or falls,
respectively, the limits for the payments on the loan are
secured via the financial instrument.
The payments from the financial instrument are thereby
conditional on the maximum limits for payments on the loan and
term to maturity having both been reached. Similarly, payments
to the instrument will occur only if the minimum limits for
payments on the loan and term to maturity are incompatible.
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Thus, the adjustable term to maturity functions as a buffer,
which reduces the price of the financial instrument.
The method described implies that the payments on the
financial instrument are determined either as the gain or loss
in proceeds at the beginning of every interest rate adjustment
period. The loss or gain in proceeds is produced by the model
funding, via a bond issue, a payment on the loan within the
band. If the limits are incompatible, the proceeds of the bond
issue does not correspond to the funding demand. The residue
defines the payments on the financial instrument.
The residual calculation of the payments on the financial
instrument implies that the model will fulfil by definition
the balance principle of the Danish Mortgage Credit Act
without imbalances. The volume of the product, is not thereby
limited by the provisions in the law.
In the section a model is set up for calculating the debtor
side as well as the funding side of the loan. If the limits
are compatible, the model has a simultaneous structure, all
variables being mutually dependent in the model. However, if
the limits are incompatible, the model changes to a recursive
structure, cf. the residual calculation of the payments on the
financial instrument.
A special situation arises if the model calculates negative
bond volumes. Negative bond volumes are not accepted, as the
debtor is to buy bonds in this situation. The situation is
dealt with in different ways, depending on the interest rate
adjustn~nt pattern of the loan. P'or type P negative bond
volumes may imply that the intended interest rate adjustment
is not respected.
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3. The pricing of the financial instrument
3.0 Introduction
In this section the financial instrument is to be priced. The
price may be both positive, 0 (zero) and negative. Thus, the
economic interpretation of prices is applied. In the preceding
sections it was described how the yield curve is modelled,
them how the payments on the financial instrument are found as
a function of the interest rate. With these conditions in
order, the final pricing of the instrument is possible.
The price of the financial instrument is calculated as the
present value in the lattice of the payments on the
instrument. By pricing the instrument as the present value of
the payments, two significant assumptions are implicitly made.
~ Firstly, it is implicitly assumed that the market for the
financial instrument is characterized by no arbitrage.
~ Secondly, it is assumed that the market discounts the
payments on the instrument in the same way as does the
vendor of the instrument.
If the above conditions are not fulfilled, the market will
either calculate another present value or price the instrument
with a deviation in relation to the present value. In both
cases the price of the market will deviate from the price
predicted by the model set up.
Therefore, the price that may be calculated in the model is to
be considered primarily as a theoretical price. In practice,
especially factors such as credit risk and liquidity premiums
are to influence the price mechanism of the instrument, for
which reason the theoretical price must be corrected before it
is applied for practical purposes.
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The correction is difficult to model as it rests on factors
that are not dependent on the interest rate. To a great extent
the correction should also be based on observations of credit
and liquidity premiums in the market. In the following, the
attention is thus focused on the determination of the
theoretical price.
Section 3.1 describes a method for calculating the theoretical
price of the instrument in the lattice. The method applies a
backward induction principle which is immediately easier to
handle than is the forward,induction principle previously
described.
There are obvious advantages connected to the instrument
having the price 0 izero) at the disbursement, said advantages
being described in greater detail in section 3.2. This means
that the limits are to be quoted as it is known e.g. from swap
interest rates, etc. In section 3.3 a model for quoting the
limits is set up.
The price of the instrument is not only relevant at the
disbursement of the loan. In the case of a prepayment of the
loan, a price of the instrument is also to be computed in the
calculation of the prepayment amount. The computation of the
prepayn~nt amount is discussed in section 3.4.
3.1 alet'hod for pricing the fiaaacial instr~nt
The method for pricing the financial instrument is that all
payments on the instrument in the interest rate lattice are
discounted to time 0 (zero). cf. the introduction.
In each node the payment on the financial instrument is
determined by the model which was set up in section 2. If the
node coincides with an adjustment of the interest rate on the
loan, the payments on the instrument depend on the mutual
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compatibility of the limits under the given yield curve in the
node.
If the maximum limits far payments on the loan and term to
maturity are mutually incompatible, the payments on the
instrument are positive seen from the point of view of the
debtor. If both the maximum and minimum limits are fulfilled,
there are no pay~nts on the instrument, whereas minimum
limits in mutual conflict trigger negative payments on the
instrument. The possible pattern in the signs of the payments
is shown in figure 11. The figure is simplified, the
underlying yield curve being assumed to be flat, causing the
lattice to be symmetrical around the initial interest rate. It
is further assumed that the limits are symmetrical. Finally,
the step size in the lattice corresponds to an interest rate
adjustment period.
At all other points in time at which the nodes do not coincide
with an adjustment of the interest rate, the payments on the
instrument is by definition 0 (zero).
The adjustable term to maturity means that, in principle, the
size of the lattice is not known at the disbursement of the
loan. In the lower section of the lattice, the term to
maturity of the loan - and thus the term to maturity of the
financial instrument - will be relatively close to Lmln
whereas the term to maturity in the upper section of the
lattice will be relatively close to L'~". That the instrument
has matured means, in practice, that there are no payments on
the instrument. It is therefore not necessary to allow for the
date of maturity in the pricing, the payments of 0 (zero) not
influencing the pricing on the assumption of no arbitrage.
Thus, the lattice must be dimensioned such that all h contain
the maximum terra to maturity of the loan.
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In the construction of the interest rate lattice, a forward
induction principle is applied. This is due to the interest
rate being forward path dependent, corresponding to the
interest rate tomorrow depending on the interest rate today.
Similarly, the projection of input to the model in section 2
is performed by forward induction, the input being known at
time 0 (zero) and being calculable successively for g=1, g=2,
etc.
This method cannot be applied here. Since it is the price at
time 0 (zero) which is to be determined, a method based
thereon is inapplicable.
By contrast, a method will be used in the following which is
based on a backward induction principle. By following a
backward induction principle, the method is based on the end
nodes of the lattice. i.e. on the nodes in which g assumes its
maximum value. There are no future payments on the financial
instrument in the end nodes. Under no arbitrage,. the
theoretical price of the financial instrument must correspond
to the payments in the node, as there are no future payments
on the instrument.
In the immediately preceding nodes, the value may then be
found as the discounted expected value in the next period plus
payments in the node. The expected value is calculated by
applying the probabilities of the different branching
structures, and the interest rate in the node is used in the
discounting. Thus, the pricing will be in accordance with the
assumption of no arbitrage. An expression of the price may
then be deduced successively for the preceding period, etc.
until the price in (0,0), has been determined.
The backward induction principle may thus be interpreted as
the value of the instrument today depending on the value
tomorrow.
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The pricing by use of the backward induction principle may be
formalized. First, the payments on the financial instrument
are to be defined. In the model in section 2, said payments on
the financial instrument were determined as
m
(2.48) Fs=FinJ(0)-~K~(j)MJ(.7)
j=i
In the following, the notation is changed, the payments having
to be related to a node in the lattice. Furthermore, the
payments on the financial instrument may at an advantage be
seen as a function of the initial term to maturity of the
loan, the limits for payments on the loan and term to
maturity. Thus, the notation is changed to
3 .1 ) F(o. ~) (Lo ~ YD~n . Y~ , L°'ia ~ Lmax ) ~ F J
The basis of the pricing is the end nodes for g=g'"a". If P,9,h1
denotes the theoretical price of the instrument in the node
(g, h), it applies in the end nodes that
( 3 . 2 ) p(o""', h1 = F(Q"°x, h)(Lo ~ Yin ~ YD j'~' , L°~in ~
L,'0.°u)
For g=g~°"-l, i.e. in the immediately preceding nodes, the
expected discounted value thereof is given by
( 3 . 3 ) (Pcp(c+i, b+1+k) '~ pmp(a+i, htx) + pnp(W , h-1+x))e ret
r is the At period interest rate, causing e-'°' to form the
discounting factor from time t+Ot to time t. Po, Pm and P
denote the probability of an upwards, middle, or downwards
branching. The parameter k is seen to be included in the
expression. From section 4 it is remembered that k shifts the
branching upwards or downwards. The parameter thus determines
the nodes over which the expected price in the next period is
to be calculated.
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The pricing according to the backward induction principle is
illustrated in figure 12.
The backward induction implies that in each node only one
expected value of three nodes is to be calculated in the next
period. In the trinomial lattice, there are always three
possible branching structures from each node, whereas one node
may be reached from a varying number.of nodes. This further
implies that in the backward induction argument it is adequate
to operate with the simple P probabilities rather than the
more complex q probabilities which were deduced in section
2.3.
The price of the instrument in (g,h) further depends on the
payment in nodes and is thus given by
p(o. ~ = Fta. a>(Lo ~ YD~i" ~ , Lmin ~ I,"'~u').i'
(3.4)
(PoPta+i. n+i+x~ + pmPt~l, t~.x) + Pn ptQ+i, h-i+x~)e-me
The expression (3.3) applies in general. Provided that P,9,l,n,
has been determined for all h, P,9,h, may be found for all h by
means of (3.3). Thus, (3.3) is to be applied successively for
g=g'""'-1, g"~"-2 , . . . , 0
until a theoretical price of the financial instrument has been
calculated in all nodes.
(3.4) is seen not to depend on the step size in the lattice ~t
which is thus only bound by 0t not being allowed to exceed the
length of the interest rate adjustment period of the loan.
A low value of ~t will increase the accuracy of the pricing.
By assigning a low value to fit, the value of At falls
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simultaneously, cf: section 4.2.3. (At, Or) will thereby span
a tighter lattice in which the distance between each node is
very small. The discrete distribution of the interest rates
for each t will thus approach the continuous distribution.
On the other hand, a low value of L1t reduces the calculation
rate of the model. 0t does not directly affect the number of
adjustments of the interest rate to be calculated by the
model, but via Or a shorter step size will indirectly increase
the number of calculations significantly. The determination of
At is thereby a trade off between, on the one hand, the
accuracy in the pricing and, on the other hand, the
calculation rate of the model.
3.2 Quoting the limits in the model
In principle, the limits are to be selected by the debtor and
are thus seen as exogenous in the model. Depending on the
selection, a price of the financial instrument may be fixed by
the method described above. The debtor must accept this price
as a result of the selected limits.
However, there are several good arguments in favour of
applying a combination of maximum and minimum limits ensuring
a price of 0 (zero) on the financial instrument at the
disbursement of the loan.
Firstly. one cannot preclude the possibility that a funding of
an optionally positive price could not be performed by the
issue of mortgage credit bonds. The rules of determining the
volume of loans and the underlying bond issue are presently
being revised, and it cannot be precluded that the result will
be a close linking between the lending limits and the bond
issue. In that case, it will not be possible to equalize a
positive price of the instrument by selling a large volume of
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bonds. Thus, the debtor will not be able to have the same
proceeds as in another type of mortgage loans.
Secondly, there will be a risk that the debtor fixes the
limits at unrealistic levels and does not, therefore, benefit
fully from the product unless the limits are quoted.
Thirdly, an initial price of 0 (zero) binds the future
prepayaaent costs to a certain extent. This argument is
elaborated on in section 3.4.
A possible procedure for determining the limits was outlined
in section 2.1.2. The procedure will here briefly be repeated.
~ The initial term to maturity is determined by the debtor.
~ The maximum limit for the term to maturity is determined
as the legislative or credit policy maximum.
~ The minimum limit for the term to maturity is determined
as the initial term to maturity of the loan.
The maximum limit for the term to maturity is determined
such that the price of the instrument is 0 (zero).
The procedure means that the maximum limit is determined at
the lowest level possible given the initial term to maturity
of the loan. By following the procedure, the maximum limit is
a function of the initial'term to maturity. Thus, there is a
trade off between, on the one hand, the debtor s payment here
and now and, on the other hand, the risk of future increases
in the payments on the loan.
3.3 lloc~el for qu~otiaQ the mauimo~m limit for the payments on
the loan
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The quoting of the maximum limit for the payments on the loan
is impeded by the complex connection between the limits for
the payments on the loan and the payments on the financial
instrument.
Thus, the path from the limits to the payments on the
financial instrument goes via the modelling of the debtor and
funding sides in section 2 and the pricing as described above.
Hence, a simple functional connection between the limits and
the payments on the instrument for use in the quoting cannot
be deduced.
The complex connection means that an iterative procedure is to
be included in the quoting. The iteration may be outlined as
follows.
For given values of the other outputs, a maximum limit for the
payments on the loan can be guessed. On the basis of the
guess, a price of the instrument is calculated. Then an
adjustment to the maximum limit is determined on the basis of
the deviation of the price from 0 (zero). If the adjustment
has converged, the iteration may be ended - otherwise the
iteration continues with a corrected maximum limit for the
payments on the loan. The structure of the iteration may be
described by the flow chart in figure 13.
The steps of the model are described in the following.
Stop 1 - Determine the initial maxiao~a limit for the payments
2 5 oa th,e loan
The iteration over the maximum limit is initiated by an
initial value for YDs"'. Each iteration involves a pricing of
the financial instrument. Thus, great importance is to be
attached to the initial value being determined such that the
number of iterations are limited, as the pricing is difficult
to calculate, cf. the preceding section.
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The quoting of the maximum limit for the payments on the loan
depends on the initial term to maturity and the maximum limit
for the term to maturity, if the other limits are determined
in accordance with the directions in section 3.2. Further, the
yield curve will also affect the quoting.
The plurality of factors combined with the complex connections
in the model makes it difficult to deduce a simple
approximative connection to be applied in all situations. In
substitution a connection may be estimated by an empirical
method on the basis of pricing operations.
Step 2 - Calculate the tl~vretical price
The theoretical price is calculated for the determined value
of YD~"' In each node in the lattice, the payments on the
instrument are calculated by the model in section 2. The
payments may be inserted in (3.4), and the price of the
instrument may be calculated.
Stop 3 - Calculate adjustment to the maximum limit for the
paym~eents oa the load
In step 3 an adjustment to is calculated such that the price
is approached to 0 (zero). The calculation follows the
Gauss-Newton algorithm. The first step in the calculation of
an adjustment is to define the function
(3.5) f('~')=p(o.o)-0
~asuring the deviation of the price from 0 (zero).
AYD~" is found by the reduced Gauss-Newton expression
( 3 . 6 ) DYD~"" = f(YD'~ ") f(~ _ f ~ + inc)
inc denotes the step size in the iteration. Thus,, if the
iteration is to converge by a reasonable velocity, inc must
not be valued very low in relation to ~. Conversely, a
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value of inc which is too large could mean that the iteration
becomes.divergent. In the model inc is determined as
(3.~) inc=0,0001RGot0)
i.e. as 0,1 per miller of the volume of the loan.
Step 4 - 8as the maxima limit for the payments on the loan
convetQ~
Step 4 determines whether the iteration has converged and the
price is sufficiently close to 0 tzero), or whether, the
iterative process is to be continued. The question is
determined by the mathematical convergence of the adjustment.
One of the following two conditions must be fulfilled in order
for the iteration to have converged.
t3.9) ~ ' <E
(3.10) f~~)~-g~+DYD;~")~ ~ <s
Step 5 - Correct the maximsum liaait by an adjustment
If the adjustment to YD~'~" has not converged, the adjustment is
applied. In step 3, a new price of the financial instrument is
calculated for the correcting maximum limit
~"' +Om;"'
Step 6 - The maximaurm limit has been quoted
If the adjustment to the maximum limit has converged and the
price is thus sufficiently close to 0 (zero). the calculations
of the model have been completed. The maximum limit has thus
been quoted.
A maximum limit for the payments on the loan is quoted in the
model, whereas the notation invites the limit to vary from
interest rate adjustment to interest rate adjustment via the
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indexation J. Thereby, the model aims primarily at the
situation in which
yD~'~=~_ . .. =YI
which must also be considered the most interesting situation.
It will not be possible to quote the limits independently of
each other. Thus, if the limits are determined independently,
there will be an infinite number of solutions to the quoting
problem. The number of solutions may be reduced.to a singleton
by determining the limits independently of each other such
that if YD~"' is well defined, b'J> 0 : YD~"x is also well
defined. The quoting problem is thereby determined, and it
will be possible to quote the limits by the method described
above.
It must be expected that the model will always be able to find
a maximum limit for the payments on the loan, which ensures a
theoretical price of 0 (zero) of the financial instrument.
Firstly, the bond issue is a strictly rising function of the
maximum limit for the payments on the loan provided that the
limits for payments on the loan and term to maturity are
incompatible. This connection follows from the balance
condition.
Secondly, the payments on the financial instrument is a
strictly declining function of the bond issue and thus of the
maximum limit for the payments on the loan. This follows from
the proceeds condition.
Finally and thirdly, it follows from (3.3) that the price of
the financial instrument is a strictly rising function of the
payments on the financial instrument. The theoretical price is
thus a strictly declining function of the maximum limit for
the payments on the loan.
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3.4 The debtor's prnpaYment costs .
Normally, the bonds underlying a LAIR are, non-callable.
Therefore, the debtor cannot prepay his LAIR at par: However,
a LAIR may be prepaid by handing in the underlying volume of
bonds, or in cash, the lending institution calculating the
prepayment costs as the market price of the underlying bonds.
However, it is possible to prepay interest rate adjustment
amounts at par in cash.
That the bonds are non-callable means that the market price of
the underlying bonds is not bound by a maximum limit. However,
the short term to maturity.and resulting short duration of the
bonds reduce the market price sensitivity of the remaining
debt, for which reason the prepayment amount typically
fluctuates less than in the case of traditional types of
loans. Hence, the debtor's interest rate risk is limited.
When prepaying a LAIR III, the underlying bonds as well as the
financial instrument must be prepaid if imbalances in the
payments are to be avoided. Thus, the price of the financial
instrument at the date of prepayment will be included in the
computation of the debtor's prepayment costs together with the
current market price of the underlying bonds. Normally, this
precludes other forms of prepayment than prepayment in cash.
If the interest rate has risen to a level in the upper section
of the lattice, the payments on the instrument will be
positive seen from the point of view of the debtor. The
debtor's position in put options is thereby in the money, and
the debtor will thus be able to obtain a positive price of the
instrument, the opposite party avoiding future payments.
If, by contrast, the interest rate is in the lower section of
the lattice, prepmyment of the financial instrument will be
connected with costs on the part of the debtor. In this
situation, the debtor's short position in call options will
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thereby be in the ~neY and the payments on the instrument,
will be negative, the opposite party selling future payments.
Thus, 'the price of the financial instrument will follow the
same pattern as the market price of the underlying bonds.
Thereby. the fluctuations in the debtor's prepayment costs
increase with the introduction of the financial instrument.
Ensuring that the debtor's payments on the loan are within a
band may be interpreted as a fixing of the interest rate
during the term to maturity of the loan, which increases the
duration of the loan. Thus, the fluctuations in the prepayment
amount will also increase.
The quoting of the limits at the disburseament of the loan such
that the initial price of the financial instrument is 0 (zero)
will, to a certain extent, limit the fluctuations. Thus, the
interest rate is to move away from the initial level in order
for the instrument to have a positive or a negative price.
An obvious possibility is that a maximum limit for the price
of the instrument in connection With a prepayment is built
into the instrument. The maximuan limit could optionally be
combined with a minimum limit in order to reduce the costs
related to the maximn~m limit .
However, it will be difficult to price this facility. In this
situation, the price should reflect the conversion behaviour
of the debtor which must be modelled, if need be, on the basis
of the observed prepayment rates under different yield curves.
However, thm experience in this area is very limited, LAIR
having existed as a product only since October 1996. At the
same time; the yield curve in this period has been very
stable.
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A pricing of a financial instrument with a maximum limit for
the prepayment costs as a facility could thus be arbitrary.
3.5 SuemarY
The theoretical price of the-financial instrument is
determined as the discounted value of the future payments at
time 0 tzero). The discounting is performed in the interest
rate lattice, causing the price to be in accordance with the
observed yield curve. The theoretical price is thereby no
arbitrage.
The theoretical price may be seen as a function of the mutual
relationship between the limits of the loan, the term to
maturity of the loan at the disbursement, and the yield curve.
Therefore, it is possible to influence the pricing via the
determination of the limits.
There are good arguments in favour of the instrument obtaining
a price 0 (zero) at the disbursement via the determination of
the limits. Firstly, the debtor is thereby secured the full
proceeds of the loan and secondly, the limits will be
determined at a sensible level. Hence, the limits are guoted
as it is known from swap interest rates, etc.
The price of 0 (zero) at the disbursement does not imply that
the financial insstrument will have the price 0 (zero) at all
future points in time. In the upper section of the lattice,
the financial instrument will have a positive price seen from
the point of view of the debtor, whereas the price in the
lower section of the lattice will be negative. If the debtor
wishes to prepay his LAIR, the prepayment amount will further
fall if the interest rate has risen, and vice versa. This is a
consequence of the fact that securing the debtor's future
payments on the loan increases the duration of the loan.
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Appendix A Type F+
A. 0 Iatsoductiasa
It will not be always possible to calculate a number of
positive bond volumes fulfilling the proceeds and interest
rate adjustment conditions of a LAIR type F.
If the last maturing bond underlying a LAIR type f has a price
over 100, the nominal issue is less than the volume of the
loan on the debtor side. Thus, the debtor obtains a capital
gain which affects the baia~nce between the payments in the
last year in the interest rate adjustment period when the
remaining debt of the loan is to be interest rate adjusted
simultaneous with the bonds maturing at price 100.
If the difference in the volume of the nominal issue and the
volume of the loan is larger than the debtor's total
repayments up to the adjustment of the interest rate, the
balance condition cannot immediately be fulfilled, the
interest rate on the loan being unchanged. In this situation,
the payments of the bond side are larger than the payments of
the debtor side.
One possibility is that the interest rate on the loan is
determined in accordance with the balance condition in the
last year of the interest rate adjustment period, but the
interest rate on the loan will then have to float within each
interest rate adjustment period, making the loan an interest
rate shift loan. 'Pursuant to the present provisions of the
Danish Gains on Securities and Foreign Currency Act, interest
rate shift loans are treated asymmetrically for fiscal
purposes. This road has thus been barred.
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Another possibility is~that the loan is funded solely in the
bond with maturity in year m simultaneous with the
introduction of a so-called minimum refinancing.
The idea of the minimum refinancing is to transfer payments
from the previous years to the last year of the interest rate
adjustment period in which a surplus in the payments on the
debtor side waS imaaediately established. This is carried out
by a deficit in the debtor payments on the loan being funded
by a new issue in the bond with maturity in year m such that
the surplus is gradually reduced. If the bond is closed, e.g.
as a result of an adjusted minimum interest rate, the issue is
carried out in a new bond with maturity 'sn year m.
A special problem is connected to the determination of the
interest rate on the loan. In connection with the minimum
interest rate adjustment, the interest rate on the loan is
recalculated year by year, i.e. also within each interest rate
adjustment period.
The minimum refinancing is calculated each year as the residue
between the payments of the debtor and funding sides.
Therefore, the interest rate on the loan is only bound in the
last year of the interest rate adjustment period. In the
remaining years, an arbitrary determination of the interest
rate will merely change the minimum refinancing. However, the
consequence of an unfortunate determination of the interest
rate is a drastic adjustment of the interest rate in the last
year in order to fulfil the balance condition. One objective
is, therefore, to select a method for determining the interest
rate on the loan, which results in a stable development. The
specified method is thus one appropriate method among others.
The interest rate could e.g. be determined as the bond yield,
etc.
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The recalculation opens up the possibility of the payment .
breaking during the interest rate adjustment period, which
does not trigger payments from the financial instrument,
however, unless the term to maturity of the loan is
incompatible with the limits for the payments on the loan.
Howetrer, the fluctuations in the interest rate are typically
limited, as the minimum refinancing is indeed minimal.
It is necessary to distinguish between two situations in the
model for type F' .
If the model is called from step 10 of the F model, the
financial instrument is inactive. In that situation, the
proceeds condition may be applied, in a relatively simple
manner, to the determination of the volume of this one bond in
which the loan is funded. Thus, the model has a recursive
structure based on the proceeds condition.
If, by contrast, the model is called from step 19 in the model
for type F, the proceeds condition is to be applied to the
determination of the payments on the financial instrument
which is now active. Therefore, the structure of the F" model
must be altered such that the balance condition is applied in
the determination of the bond volume. Again, the structure of
the model will, however, be recursive.
In the following description, a distinction is made between
two situations.
A.l Tie type F' ss~l for co~stible limits
First, a model is set up for calculating the debtor and
funding sides of the loan in a situation in which the limits
of the loan are compatible, and the financial instrument thus
inactive. Hence, the starting point is step 10 of the F model.
The flow chart of the model is shown in figure 14.
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It appears that the F' model returns to step 11 of the F model
when the bond volume, the interest rate and payments on the
loan have been calculated. Thus, the F' model is covered by the
iteration over the term to maturity in the outer loop in the F
model. Therefore, the term to maturity may be seen as given in
the F' model.
The steps of the model must be applied both at the ordinary
adjustment of the interest rate and at the minimum
refinancing. A distinction between the two situations is made
in the individual steps.
Step 10 of the F' m~lel
All information concerning the loan, including the term to
maturity. is input from step 10 of the F model.
Step 1 - Calculate the volume by means of the proceeds
co~ditioa
The volume of the mth bond is determined at the date of
disbursement of the loan, or immediately following an ordinary
adjustment of the interest rate as
(A.1 ) H,l(m) = RGa{0) and H,i(j) = 0 for j < m
K,r(m)
The volume of m is taken over from the F model. This does not
affect m, as the issue is made in only one bond. In the
remaining years until the ordinary adjustment of the interest
rate, the minimnxm refinancing is given by
.------ for =1, 2 , . . , m-1
(A. 2 ) 1~~,(m) - ~~j) j
It~,,(m)
such that the proceeds condition is fulfilled. The top sign j
indexes the minimum refinancing. j runs until m-1, a minimum
adjustment of the interest rate not being performed in
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connection with the ordinary adjustment of the interest rate
at the end of the period.
It should be noted that j gets a slightly different meaning, j
being set to zero only in connection with the ordinary
adjustments of the interest rate. Thus, bonds are issued at
time j=0.1,2,.., m-1 in connection with the minimum
refinancing, and not just at time j=0 at the ordinary
adjustments of the interest rates.
The funding demand in (A.2) is given by
(A.3) FinJ(J)=RJ(m)Ii~(J.m)-Y~~~J) for 15j5m-1
where HJ(j,m) is the bond volume already issued at time j.
Step 2 - Calculate the 3.nterest rate on the loan by means of
the balance condition
The interest rate on the loan must be determined on the basis
of a condition for a global balance between the total payments
of the debtor side over all m years and the payments of the
funding side in the same period. Hence, the interest rate on
the loan must fulfil
(A. 4 ) ~, YDa(i)+l~G~(m) _ (m-.7)R~(m)C~JW)+H~(.7 ~ m))
ti+~
ll~~(m)+H,~(j,m) constitutes the total bond volume issued in the
mth year following the minimum adjustment of the interest
rate. The interest rate on the loan is included in the
payments on the loan as well as in the remaining debt and
cannot immediately be isolated. Therefore, the interest rate
is found by a numaerical.method. It should be noted that (A.4)
does not allow for the future minimum refinancing. It is the
very idea that the interest rate is determined at a level
which is too low such that a deficit occurs, and hence a
minimum re f ina~nc ing .
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9th 3 - Calculate tha debtor pats on the loan
The determined interest rate determines an unambiguous profile
of the debtor's payments on the loan and remaining debt which
are calculated in this step.
Step 4 - Sh3.ft to step 11 of the F model
All variables on the debtor and funding sides are determined,
for Which reason the model may return to step 11 of the F
model in which an adjustment to the term to maturity of the
loan is calculated.
A.2 The type F' model for incompatible limits
If the limits for payments on the loan and term to maturity,
respectively, are incompaatible, the financial instrument is
activated.
The minimum adjustment of the interest rate means that several
bonds are issued during the interest rate adjustment period,
causing the payments on the loan to increase until the last
year of the interest rate adjustment period in which the
capital gain is realized and the payments on the loan
decrease.
The rising payments on the loan constitute a problem in the
situation in which the maximum limit for the payments on the
loan is binding. Thus, it will not be possible to determine a
payment on the loan which constantly corresponds to the
maximum limit. Exceeding the limits cannot be built into the
model at an advantage either. Indeed, the basic idea of the
minimum refinancing is to transfer payments to the interest
rate adjustment year. At the sate time, an expansion of the
financial instru~nt to include these payments as well makes
the characteristics of the instrument difficult to co~rehend.
The frequency of payments on the instrument will thus vary -
not just in lane with whether the limits may be fulfilled -
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but also in line with the model shifting between F and F".,
Therefore, exceeding the limits during the interest rate
adjustment period must be accepted. At the same time, the
extent to which the limits are exceeded will be limited.
In connection with the minimum refinancing, payments on the
financial instrument are thus not to be calculated, causing
the structure for j>0 to be the same as with compatible
limits. The model is shown in figurel6.
Step 1 - Calculate the interest rate on the loan such that the
paymeaats on the loan are ~rithin the interval
The interest rate on the loan is to be determined such that
the payments on the loan up to the first minimum adjustment of
the interest rate correspond to the binding limit. The
interest rate on the loan must fulfil the relevant one of the
conditions
( A . 5 ) R~RG ~ 0) ~ = Y~ f or YDa( 0 ) ~ L~.x > YD~ ~'
1-(1+RJ)
(A.6) ~~R~R~)~~~ =~n for YD,7(~)I ~1~ > YDjin
1-(1+R~)
Step 2 - Calculate the volute by means of the balaace
camditiaai
The volume of the mth bond is determined on the basis of a
requirement With respect to balance in the payments of the
debtor and funding sides, the payment profile and the
remaining debt profile on the debtor side being calculated
with an intere~at rate determined in step 1. The volume must
fulfil
( A . 7 ) ~ YD,r( j ) + RGJ(m) = mR,"~(m)H°J(m)
W
Step 3 - Calculate the prc~c
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On the basis of the volume, the proceeds may be computed,,
Which also defines the payment on the financial instrument.
The payment on the financial instrument is given by
( A . 8 ) Fs ~ Fsn,t( 0 ) - x°,~m)H~(m)
The variables of the loan have thereby been calculated for
j=0. In step 4, the model continues the calculation of the
minimum refinancing for j=l, 2,.... m-1
Step 4 - Calculate the margiaal issue by means of a proceeds
condition
As there are no payments on the financial instrument in
connection with the minimum refinancing, the proceeds
condition may be applied to the determination of the marginal
issue . . M~J(m) must fulfil
Fin,~(J )
(A.2) MjJ(m)= H~~) for j~1,2, . . .,m-1
where Fin~~~ is given by (A.3)
Step 5 - Calculate the interest on the loan by means of the
balance condition
The interest rate on the loan is calculated by the expression
(A.4) F,, YD~(3)+RaJ(m)=(m-j)R~~(m)~M~m)+H,r(j.m))
fj+1
for j=1,2,...,m-1 which is solved by a numerical method.
Step 6 - Calculate the debtor payments on the loan
Eventually, the payments on the loan on the debtor side may be
calculated.
step 7 - The ion and the fin~anaial 3.astru~t have been
calculated!
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When steps 1 to 3 have been applied for j=0 and steps 4 to-6
have lien applied for j=1, 2,.... m-1, all variables have been
calculated. In this situation, it is.not necessary to return
to the ~uodel for type F.
The ircg~lementation of the model in the trinomial lattice with
a view to the pricing of the financial instrument is not
entirely without probiesns.
In the lattice, the debtor and funding sides of the loan are
calculated in the nodes which coincide with an ordinary
adjustment of the interest,rate on the loan. The calculation
is performed partly to determine payments on the financial
instrument and partly to determine the development in the
remaining debt and term to maturity of the loan with a view to
the calculation of the loan at the next adjustment of the
interest rate. This information is projected through the
lattice.
If e.g. a LAIR III type F5 is considered, the loan is to be
calculated in the lattice at intervals of 5 years. This means
that the projection of the relevant information is performed
regularly over five years.
However, if the loan changes its characteristics to a LAIR III
type F5' because of the yield curve, the debtor and funding
sides of the loan mn~st be calculated each year as a result of
the minimum refinancing. The minimum refinancing is dependent
on the interest rate and should thus be performed in the
lattice in accordance with the modelled development in
interest rates.
However, a calculation at intervals of 1 year means that in
the lattice, projection is performed over periods of both 5
years and i year. As far as we know, no methods exist for
dealing with this situation.
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The problem is gist obviously solved by a calculation of the
debtor and funding sides being performed at each ordinary
adjustment of the interest rate for the whole period until the
next ordinary adjustment of the interest rate, as is the case
for type F. This requires an assumption concerning the
development in the interest rate of the m'th bond.
A yield curve is determined in each node, which yield curve
also defines an illicit forward rate. As an approximation of
the development in the lattice, it is convenient to apply the
implicit forward rate in the determination of KJ(m). The
approximation should not affect the pricing of the financial
instrument, as the minimum refinancing is indeed minimal.
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Appendix B -Tl~~ variant of the model f or type P
8.0 Introduction.
Appendix B contains a description of an alternative modelling
of the debtor and funding sides of a LAIR III type P.
The alternative modelling may replace s ups 2 to 13 in the
model for type P. Thus, it is only the inner loop in the
simultaneous model structure (the part of the model in which
there are no payments on the financial instrument) which is
different in the two variants of the model. The link to other
steps in the model is thus the same.
The alternative modelling has several characteristics in
common with the model in section 2.
Firstly, the interest rate on the loan is applied as an
iteration variable. For a given interest rate on the loan, the
debtor payiaents on the loan and volumes are calculated such
that the proceeds condition and the balance condition are
fulfilled. Then the convergence of the interest rate
determines whether the debtor payments on the loan and volumes
are to be recalculated in a new iteration.
Secondly, the voiumts are determined by use of a trend
function. However, the adjustment of the trend function to the
proceeds and balance conditions is performed in an iterative
procedure in which the factors Go and G, are iteration
variables.
The iterative procedure for determining Ga and G, facilitates
the alternative modelling in terms of formulae. Conversely,
the iterative procedure reduces the calculation rate of the
model.
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s.l The Qeaeral structure of the mor3e3.
The determination of Gn and G, by iteration means that another
loop is added to the model.
In the outer loop. iteration is performed over the term to
maturity. Tn the central loop, the model iterates over the
interest rate on the loan, and in an inner loop, the model
iterates over Go and G, until the proceeds and interest rate
adjustment conditions are observed
The structure of the model appears from the flow chart in
figure 16. The flow chart shows steps only for the part of the
model in which iterations over the interest rate on the loan
and over Go and G, are performed, the other steps being, as
already mentioned, identical to the model for type P which was
described in section 2. iFurthermore, certain of the steps in
figure 1fi are identical to the steps in the model in section
2. However, these steps are included, as the description would
otherwise be difficult to comprehend).
The handling of negative bond volumes is different in the two
variants of the model. In this variant of the model, the
iteration continues in the inner loop until both the proceeds
condition and the balance condition are fulfilled for positive
volumes. If a solution is not found, the model will end the
iteration after 30 attempts, after which a correction of the
volumes is performed such that the proceeds condition is
fulfilled. This appears from the flow chart in that the model
may leave step k and move directly to step m without checking
the conditions in step 1 after 30 iterations.
B.a The steps of the model
step a - Deter the initial interest rate on the loss.
Determine m
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On the basis of the term to maturity which has been determined
in the outer loop, the model determines a value for m as
( B .1 ) m = min [L,r- ~; mo + TILf'J
Lo - ~ denoting the remaining term to maturity of the loan.
TILT specifies that a further bond is to be applied at the
disburs~aent of the loan due to the reduced interest rate
adjustment percentage at th: first adjustment of the interest
rate. At the disbursement of the loan, TILT has the value
(B.2)
if the loan is disbursed in the period
TILT = 1 from January t o November -
0 if the loam is disbursed in December
for J=0. For J>0 TILT=0.
Tha initial value of the interest rate an the loan is
determiaed as a weighted average of the yield to maturity of
the underlying bond portfolio.
E~.t.r(o, t)
( B . 3 ) R~ °~'~ ~ y t
Step b -Calculate the debtor payments oa the loan
The payment profile and the remaining debt profile on the
debtor side of the loan may be calculated as a function of the
term to maturity and the interest rate.
step a - Is mil, mesa, or m~2?
The model may continue in three different ways.
If m=1, the volume of the 1-year bond and an interest rate on
the loan are calculated in a relatively simple manner in step
d.
If m=2, the model moves on to step a in which band volumes and
an interest rate on the loan are calculated without the use of
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iteration. A LAIR Type P50 will be calculated in this step
until the maturity of the loan is only 1 year away. Other
types of loans are calculated in step a when the remaining
term to maturity is 2 years.
If m>2, the model moves on to step f in which a trend function
is defined.
Step a - Determine the volumrs of mil
If m=1, the funding may be determined on the basis of the
proceeds condition as
(8.3) ~i)=Fig
K,r(1)
FinJ(0) denotes the current funding demand at the time of
calculation. which funding demand is defined by the payments
in the preceding interest rate adjustment period such that the
balance condition is fulfilled.
(B.4)
Volume disbursement
Fin,T(0) _ ~ H~1~~)R~1~~)+H~1~1)- YD,~-1(Z) adjustment on the
interest rate
m
Step a presupposes thaat J=M, for which reason the first part
of the definition is irrelevant here but is nevertheless
included far later use.
As the loan is not to be interest rate adjusted at a later
stage, it follows froaaa the balance condition that
( B . 5 ) YDx(1) _ (1 +Ra~(1)]Fls(1)
which determines an unambiguous interest rate on the loan.
Step f - Dete~sime tth~e volun~sa of n~a
If m=2, the interest rate adjustment condition may be
formulated as
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(B.6) Yt7s(1)+REG~ ~~) =H,~(1)+REGJ~(1)HJ(1)+REG,,RJ(2)x~(2)
and the proceeds condition as
(B . 7 ) M~(1)K,r(1)+M,rt2)K~(2) = FinJ(0)
which collectively define two equations with two unknown
quantities. (8.6) and t8.7) may be described by the matrix
equation
B . 8 ) Ca~~.r = Da
where l~= (M,, l 11 , M,, ( 2 ) ) is a 2x1 vector, and
(B.9) DJ~~YDJ(1)+REG,°~~ ° -(1+REG~R~(1))Ha(0, 1)-REGaRJ(2)H~(0,
1), F.in,l(0)~
is also a 2x1 vector. In the setup of (B.9), the relation is
applied.
xJ(j) =Ms(j)+H~(0, ~)
CJ is a 2x2 matrix given by
i
(B.10) Ca= 1+~G~R'(1} REGoRJ(2)
Ks(1} Ko(2)
C is quadratic and may thus be inverted, which produces the
solution
(B.11) ~=~(0: Cs Da}
strictly negative fundi.n~ volumes (<0) not being accepted. The
max-condition may mean that the volumes do not fulfil both
conditions. If a negative volume is assigned the value 0
(zero) in the max-condition, the model will overfund the
funding demand, causing the proceeds condition not to be
fulfilled. If it is the volume of the 1-year bond which is
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assigned the value 0 (zero), nor the interest rate adjustment
condition v~ill be fulfilled. Therefore, the model moves on to
step m in which the proceeds condition is checked.
Step f - Define a tread ftu~etioa
The volumes are determined by means of a trend function v~hich
is deffined as
( B .12 ) ao + ai(j -1)+ aa(j -1)a + . . . + a~.i( j -1)°rl
for j=1,2,...,m, where q Sm - TAT as in the model in section 2.
Step g - netecoefficients in the trend function
The coefficients of the trend function are estimated such that
the value of the trend function corresponds to the volumes
given the intended adjustment of the interest rate and the
bond volume already issued in each bond.
ao+al(j-1)+aa(j-1)z+ . . . +aQ.l(j-1)W =
(8.13)
~~~jRG,yj-1) ~ HJ(0, j)J
A max-condition is included in the expression, negative
volua~s not being accepted. The coefficients are determined by
the matrix equation
Ha(0. .7),
2 0 ( B .14 ) (ao . ai . . . . , ay) _ [Bz"B] B max REGJ mo
The matrix B is given by (2.64)
Std h - ti~sss at an iacreas~t to the coefficieats
In this step, tv~o factors are added to the trend function. The
factors function as an increment to the trend function, so
that the trend function is adjusted to the. interest rate
adjustment condition and the proceeds condition.
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Go and G, are added to the trend function as factors to ao :and
al.
(B.15 ) Dodo +GianJ -1)+as(j -1)a + . . . +a~-1(J -1)ø1
Go effects a vertical parallel shift of the trend function in
the (HJ(j). j) place, whereas G, influences the slope of the
trend function.
As a start value for the iteration,
(B.16) Go=1,25 and G~=1
are set.
The trend function is thereby shifted upwards. This supplies
the model with information concerning the relationship between
the marginal isssue in the individual bonds. If the marginal
volume is assigned the value 0 (zero) in a bond, the model
does not i~nediately obtain information as to the extent to
which the trend function is to be changed for the volume to be
positive once again. By shifting the trend function initially
in the iteration, this information is provided. The principle
is shown in figure 8.2
HJ(0,4) - in the figure (9) - is disproportionately large, so
according to the trend function, M3(4) will be 0 (aero). If the
marginal issue in all bonds is increased (10), the model
determines the relationship between MJ(4) and the other
marginal volusses .
If TIZ~T=1, cad the model thus operates with an extra bond,
only one G variable is introduced in step h. Dug to the
special interest rate adjustment profile, the first funding
volume is est~ted explicitly by a variable Z which is
described in more detail in the next step.
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Step i - The volts are ~terstined
In this step, the inner loop of the model is initiated, in
which inner loop the final values of G~ and G, are determined
by iteration such that the trend function is adjusted to the
interest rate adjustment condition and the proceeds condition.
The fact that step i initiates the inner loop implies that ao,
al,..., aq_, are only estimated once for each iteration over the
interest rate in the central loop. Similarly, tGo, G,)=(1,25,
1) is marely an initial guess.
On the basis of the trend Function, the marginal issue is
determined by
) _ [0; [G~aa + Gla, (j-1) + a~ (7-1)' + . . . . +
(B.17)
aQ_1(j-1)q~' 1- HJ(O,j)1.
for j=l, 2,..., m. At the disbursement of the loan, it is once
again applied that Ho ( j ) =M" ( j ) +Ho ( 0 , j ) and Iio ( 0 , j ) =0 .
For TILT=1, the first volume Mo(1) is determined as
(B.18) H"(1)=z-H"co,j)
The rest of the marginal issue is determined by the trend
function in which only G" is included. The total marginal issue
is thus given by
Mo(7) _ ~[0: [Z,Goao + al(j-1) + a2(j-1)2 + .... +
(8.19)
aa_,(7-1)q~'. 1- Ho(0~7)1
for j=l, 2,..., m. In parallel thereto, the volumes are
determined for
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(B.20) ( (Ga+inc,G,) ~ fGo,Gl+inc) . .
for use in the Gauss-Newton algorithm.
Step j - Calculate proceeds and balaace co~.tions
On the basis of the calculated volumes, the interest rate
adjustment condition and the proceeds condition are evaluated,
so that a possible adjustment to Go and G, may be calculated in
the next step. The interest rate adjustment condition and the
proceeds condition are given by
f B . 21 ) YD~t1)+REGJR- G~ = H,rtl)+REGJ~, RJ(j)H,rtJ) ~d
m
(8.22 ) F'ins(0) _ ~, IC,~(j)1Ks(3)
respectively.
Step k - calculate ad~ustme~t of iacresyent
In step k an adjustment of the increment variables fGo,G,) is
calculated on the basis of the interest rate adjustment
condition and the proceeds condition which were calculated in
step j.
A function f(Go, G,) is defined, which measures the deviation
from the conditions given (Ga,G,).
ftGo, Gi)=~Fir~tO)-~,Kati)M.Ttj).
(8.23)
xy)+xa~s~ RJt~)x~(~)-~~(1>-~~R-=~~
The adjustment of (Go, G; ) is given by
... IDT DI CDT Q
(B.2~) nc~o, Gl)-- .-
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where D, g, j" fulfil (2.24) to (2.27), and the Jacoby matrix
is given by
(B.25) J,
1 fuGo~ Gi)-W{Go+inc~ W ) fi{Go, Gi)=~a~Go~ W +inc)
inc[ f~{Go ~ Gi)- f={Go + inc, Gl) fZ{Go . Gi)
where the subsign on ff.) specifies the part of the argument
in ( B . 23 ) which is evaluated. That is to say that f 1 ( Go. G1 )
determines the value of the proceeds condition for (Ga,GI).
For TILT=l, an adjustment to (Z, Go) is determined instead.
Step 1 - Are the conditions fu3.filled?
Step 1 determines whether (Go, G,) has converged such that the
interest rate adjustment condition and the proceeds condition
are fulfilled, or whether the iteration in the inner loop is
to continue.
The mathtical convergence for (Go, G,) is determined by
(8.26) ~G°' Gl) <e
E Qo , Gu)
8 . 27 ) ~Go ~ Gx)~ - fE{Go ~ Gi)+A{Go ~ Gi))~ ~ ~ E
f{Oo , Gi)a
where merely one of the conditions must be fulfilled in order
that (Go, G, ) haos converged and the volumes may be applied.
Stop m - Zs the proceeds coa~d3.tion fulfilled?
Step m reaches the a~del in three ways.
Firstly, the model may reach step 1 when a convergent solution
has been found. In this situation, the solution fulfils the
proceeds condition, and the model moves on to step o.
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Secondly, the model may reach step m after 30 iterations in
the inner loop without convergence. In this situation, the
model will not fulfil both conditions, but it cannot be
precluded that the proceeds condition is fulfilled. This is
clarified before the model moves on to either step o or step
n.
Finally and thirdly, the model may reach step m from step a
for m=2.
The precondition for moving on to step o is given by
(B.28) I FinJ(0)-~Ka(j)MJ(j)~ <e
If the condition is not fulfilled, the volumes are corrected
in step n.
Step a - Adjust the volumes
Step n adjusts the volumes such that at the proceeds condition
is definitely fulfilled. In principle, three situations may
occur
1. The model overfunds; i.e. the volumes must be reduced
~ FinJ(0)
E Ka~7)M~J)
for
(B.29)
1',1= ~~1). MJ(2). . . . . Mom))
~S = (~J(1) ~ ~',r~2), . . . , ~'.r~m))
where M* marks the adjusted funding volumes.
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2. The model underfunds, so that the volumes must be ,
increased. The first volume is maintained such that the
interest rate adjustment percentage is not increased more than
absolutely necessary. The other funding volumes are adjusted
by
(B.30) ~= 1~ Fin,r(1)-KJ(1)Mo(1) ~
E Ka(.7)Ma(j)
where M and M' are deffined as above.
3. Finally, the volumes may sum up to 0 (zero). Then the
interest rate adjustment is funded solely in the bond with the
shortest term to maturity, i.e.
FinJ(0) =1
(B.31) M',~(j)= K.~(1) ~ 3
0 : j=2,3, ...,m
Step o - Calculate adjustment to the interest rate on the loan
An adjustment to the interest rate on the loan is calculated
according to the same directions as in the other variant of
the model.
The adjustment is calculated in relation to the yield to
maturity of the portfolio of underlying bonds corrected for
the effect of the typically different number of payment
periods on the debtor and creditor sides. First, the function
f(.) is deffined as
R J RG~( i -1 ) - REGJrp
( B . 3 2 ) f(R,,) _ ~ n
ii RG~(0)
The adjustment is calculated by the reduced Gauss-Newton which
was deduced in section 2.
(B.33) ~~= inc
f(R~) - f(R J + fine)
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Step p - xas t~ ~tereet rate oa the loan converged ?_
It is to be determined in step p whether the interest rate on
the loan has also converged, or whether the iteration in the
central loop is to continue.
The mathematical convergence of the interest rate is examined
by
(8.34) ~~<E
x
J
( f(RJ)2 - f(R~+~a)Z ~ c E
(B.35)
f(R,i)a
where merely one of the conditions need to be fulfilled in
order for the interest rate to have converged. In that case,
the model continues in step r.
Step q - Correct the interest rate on the loan
If neither (8.34) nor (8.35) were fulfilled in step p, the
interest rate on the loan is corrected by the calculated
adjustment.
Step r - The loan has been calculated
If the interest rate has converged, all variables have been
calculated for the given term to maturity. Having been
implemented in the model for a LAIR III type P, the model
moves on to 14 from here.
