Note: Descriptions are shown in the official language in which they were submitted.
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METHOD AND DATA SYSTEM FOR DETERMIrIING FINANCIAL INSTRUMENTS FOR, AND TERM TO
MATURITY
OF, A LOAN
- 5
- INTRODUCTION
This invention concerns a method and a data processing
system/computer system for calculation of the type, the
number and the volume of financial instruments for funding of
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. In the method
according to the invention the term to maturity of the loan
is also determined at the beginning of each period of time so
that debtor's payments on the loan during the entire term to
maturity of the loan is within a band defined by a set of
maximum and minimum limits which can be fixed for each period
of time, and so that the term to maturity of the loan also is
within a band defined by a maximum and a minimum limit. To
the extent necessary, a rule for prioritization of the limits
regarding the payments on the loan and the term to maturity
is established. The results of the method according to the
invention may be used by the creditor, e.g. a financial
institution such as a mortgage credit institution, in order
to secure that such a loan is funded in such a way that
interest rate risks as well as imbalances in the payment flow
are avoided or minimized. Thus, by the use of the results of
the method according to the invention the creditor has the
- possibility of hedging lending and funding.
In the refinancing of a loan other financial instruments than
the instruments which have formed the basis of the principal
of the original loan may be used, which is the reason why, in
connection with refinancing, an adjustment of the interest
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rate on the loan may be necessary in relation to the interest
rate 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). An example of the financial
instruments is non-callable bullet bonds. In the following,
financial instruments are also called funding instruments,
just as funding volume is also used as a term for the
financial instruments constituting the principal.
BACKGROUND FOR THE INVENTION AND INTRODUCTION TO THE
INVENTION.
In the Danish mortgage credit market callable loans used to
be far the dominating type of loans, and, therefore, callable
bonds in a pure "pass through" form were also as dominating
as bonds. For a number of years, up to the withdrawal of the
permission to grant cash loans in 1985 by the Danish Ministry
of Housing, mortgage credit institutions also offered the so-
called loans with adjustable interest rates. The previous
loans with adjustable interest rates were characterized as
follows:
1) Long-term credit commitment.
2) Funding by the issue of bonds with a term to maturity of
1 to 5 years every fifth year.
3) Fixed interest rate in successive periods of 5 years.
4) The underlying bonds with a term to maturity of 1 to 5
years are non-callable. This gives the debtor the
possibility to terminate the loan at par prior to the
first occurring interest rate adjustment.
The Danish loans with adjustable interest rates did not turn
out very successfully, thus only per milles of the total
lending made by mortgage credit institutions was granted as
loans with adjustable interest rates. The reasons were,
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probably, that the call premium was insignificant at that
time due to a very large difference between market interest
rates and coupon rates and in addition, the investor market
did not pay as much attention to the problem as today.
. 5 Therefore, the difference in yields between callable and non-
callable bonds was not sufficiently large in itself to make
loans with adjustable interest rates attractive. Furthermore,
the product was not transparent seen from the debtor's side.
An aspect which might also have had some influence at times
was that a continued rise in the Danish interest rate level
was expected so that the debtor would not expect an interest
rate adjusted loan 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 success in these years.
In June 1993 certain Danish tax laws were changed so that the
mortgage credit institutions were, in reality, once again
given the opportunity to offer loans with adjustable interest
rates.
This offers the possibility of changing the long-term
mortgage market so that, in the future, the funding products
will be attractive to foreign investors. A precondition is,
probably, that bonds are offered in conformity with
international practice, e.g. as non-callable bullet bonds.
Thus, it has been of interest to examine whether variants of
loans with adjustable interest rates can be made attractive
to the debtors.
The traditional loans with adjustable interest rates involve
a risk for, in principle, unlimited upward jumps in the
interest rate. To many debtors, especially in the segment
comprising private customers, this risk is assumed to be
unacceptable, in particular with regard to the conseguences
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as to liquidity for debtor of an increase in the interest
rate to a very high level. Therefore, it is of interest to
examine whether the design behind loans with adjustable
interest rates may be combined with an adjustable term to
maturity, where a raising or falling, respectively, interest
rate only affects the payments on the loan to a predetermined
extent defined by a set of maximum and minimum limits,
whereas the term to maturity on the loan is varied in
accordance with the interest rate on the loan.
Typically there will be a maximum as well as a minimum limit
for the range wherein the term to maturity may vary that may
be determined by debtor, lender, public authorities or
legislation, respectively. Hence, a particular problem will
be to prioritize the maximum as well as the minimum limit for
the payments on the loan and the term to maturity of the
loan, respectively, and to establish a rule for determining
the payments on the loan and the term to maturity in cases
where the limits are incompatible with the given interest
rate on the loan.
Characteristic of the traditional loans with adjustable
interest rates was a match between the term to maturity of
the funding instrument having the longest term to maturity
and the period between interest rate adjustments, viz. 5
years. If this precondition is abolished a much wider range
of possibilities with respect to funding and interest rate
adjustment will, in principle, become available.
Thus, it becomes possible to secure a gradual adaptation of
the borrowing costs to the market interest rate with an
adjustment time depending on the maximum term to maturity of
the bonds applied and on the weight with which the individual
bond applied is included. This principle will, just as the
above-mentioned opportunity for adjustable term to maturity,
reduce the risk of substantial upward jumps in the payments
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on the loan which characterizes the traditional loans with
adjustable interest rates.
If the short-term interest rate is systematically lower than
the long-term interest rate it will be possible to reduce
5 debtor's long-term borrowing costs. Furthermore, borrowing
costs may, as mentioned above, be reduced compared to
callable bonds due to the absence of the call right and
through increased liquidity and internationalization of
sales.
Whether it is possible to counter a change in the interest
rates by varying the term to maturity of the loan depends,
apart from the determined maximum and minimum limits for
payments on the loan and the term to maturity, on the extent
to which the outstanding debts are adjusted to the market
interest rates at the time of interest rate adjustment.
Characteristic of the traditional loans with adjustable
interest rates was the 100 per cent adjustment of the
outstanding debts of the loan to the market interest rates
every 5 years. Partly by allowing other frequencies of
adjustment of the interest rate, and partly by allowing only
a partial adjustment of the interest rate of the outstanding
debts of the loan, larger shifts in the interest rate than in
the conventional loan design would be compatible with the
maximum and the minimum limits for the payments on the loan.
Thus, also a partial adjustment of the interest rate of the
term to maturity of the loan and other interest rate
adjustment frequencies should be compatible with 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 legislative regulation of the
activities of Danish mortgage credit institutions that the
institutions may undertake a limited interest rate and
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funding risk. On the face of it, designs of loans with
adjustable interest rates are contrary to these basic
principles as the funding side has a substantially shorter
term to maturity than has the lending side. The traditional
loans with adjustable interest rates are nevertheless
regarded as lying within the balance principle seen in the
perspective that debtor accepts to pay any interest rate that
may occur in connection with a future refinancing. In
principle, therefore, this is a "pass through" which does not
incur any risk onto the creditor.
In connection with funding of a new type of loans with
adjustable interest rates with a variety of e.g. non-callable
bullet bonds, four conditions must be fulfilled according to
Danish practice and legislation:
1. The volumes of each funding instrument must be determined
in such a way that the market price of the funding
instruments equals the principal of the loan.
2. The interest rate on the loan must be determined in such
a way that the interest rate equals the yield to maturity
of the funding portfolio, the yield being the interest
rate at which the present value of a future payment flow
on the funding instruments equals the remaining debt on
the loan.
3. The requirement as to maximum permissible imbalances
between payments from debtor and payments to creditor
must be fulfilled.
4. In addition, the legislative requirements to terms to
maturity and method of repayment must be met, also
regarding loans with adjustable interest rate and
adjustable term to maturity.
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Previously, when calculating the funding of conventional
loans with adjustable interest rates, the above-mentioned
requirement as to the interest rate on the loan was not taken
into consideration.
In the funding of conventional loans with adjustable interest
rates there was an unambiguous connection between the maximum
term 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 a 5
year period during which the interest rate on the loan was
fixed. The conventional loans with adjustable interest rates
were funded by debtor by issuing underlying bonds with a term
to maturity of 1 to 5 years.
However, this funding principle is not compatible with the
preferences for issuing a range of e.g. 10 non-callable
bullet bonds with terms to maturity of 1 to 10 years and at
the same time keeping the duration of the interest rate
adjustment period down at e.g. 1-2 years.
Thus, in Denmark there is a demand for a general funding
principle comprising funding with the above-mentioned range
of non-callable bullet bonds or other financial instruments
suitable for that purpose. In international financial markets
there is, at the moment, no tradition of a close connection
between lending and funding of loans. In spite of this,
however, the broad applicability of a principle that links
the loan to a range of financial instruments must be presumed
to give rise to also an international interest in a general
funding principle of the type described herein.
Thus, the funding principle may, inter alia, be used for a .
mark-to-market pricing of otherwise non-traded loans and
debts. By applying the principle it will be possible to
determine a portfolio of traded financial instruments with an
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equivalent cash flow based on which the non-traded loan or
debt may be priced in accordance with observed market prices.
Accordingly, the funding principle may find possible
applications in the risk management of loans and debts, since
the principle may be used for determination of a hedge
consisting of a portfolio of financial instruments and
pricing of such a hedge. During recent years the development
has focused on financial risks, including the possibility of
hedging these risks, so particularly in this field the
international attention towards the funding principle is
expected.
However, a technical problem in connection with such a
general funding principle has been that there was no
knowledge of an efficient general calculation method for a
computerized calculation of the volume of financial
instruments or funding volume for the funding of a loan where
at least a partial refinancing of the loan during the
remaining term to maturity of the loan is performed under the
condition that the calculation result must partly fulfil the
requirement that loan issuing institutions must not undertake
interest rate or funding risk or at least they must or will
not undertake such risks above a certain maximum, and must
partly be able to contribute to minimize costs of the debtor
so that the loan with adjustable interest rates gets as
inexpensive as possible within the given preconditions.
In Danish Patent Application No. 0165/96 and International
Patent Application No. PCT/DK97/00044 such a suitable
computerized method for calculation of the volume of
financial instruments or funding capital sums for financing
of the above-described type of loans is described.
BRIEF DESCRIPTION OF THE INVENTION
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The present invention is a further development of the
invention described in the above-mentioned patent
applications.
In connection with loans with adjustable interest rate it is
sometimes considered a problem that even short term increases
in the interest rate lead to substantial upsurges in the
payments on the loan, which puts a strain on lender's
economy. It would be desirable to provide the possibility for
performing the calculations of the loans in such a way that
instead of a raise in the payments on the loan, or in
combination with a small raise in the payments on the loan,
the term to maturity is prolonged so that the lender would be
able to manage the payments on the loan from his current
economy.
The invention relates to a method by which not only the
above-mentioned parameters may be determined, but in which
also conditions concerning maximum (or minimum) payments on
the loan for the lender through one or more periods of the
term to maturity of the loan may be laid down, the term to
maturity of the loan then, if required, being calculated
according to these conditions. On the other hand, it will be
possible in the method according to the invention to lay down
conditions for the maximum (or minimum) term to maturity of
the loan and then calculate adjusted payments on the loan.
As it appears from the following, the invention is a
technical enrichment of the field for data processing:
because of the number of variables to be calculated, the
financial conditions to be fulfilled and their mutual
relations, the problem to be solved is a complex simultaneous
problem. By the technical solution provided by the invention,
the complex simultaneous calculation problem is divided into
a sequence of processing steps, which makes it processable by
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sequential calculation in a data processing system/computer
system within realistic and effective computation times.
By the method according to the invention calculation results
of a high value may be obtained, which, among other things,
5 comprises a high level of stability of the size of the
payments on the loan calculated despite input of relatively
large fluctuations of the interest rate through the
individual periods of financing.
Thus, the invention relates to a method for determining the
10 type, the number, and the volume of financial instruments for
the funding of a loan with equivalent proceeds to a debtor as
well as the term to maturity and the payment profile on the
loan by means of a first computer system, the loan being
designed to be at least partially refinanced during the
remaining term to maturity,
requirements having been made to the effect that
the term to maturity of the loan is not longer than a
predetermined maximum limit or shorter than a
predetermined minimum limit,
- debtor's payments on the loan are within
predetermined limits,
- rules having been made as to how the two above-
mentioned requirements are mutually prioritized,
- requirements having been made as to 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,
- requirements having been made as to 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, on
the other hand, the principal of the loan,
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- and requirements having been made as to a maximum
permissible difference between the interest rate on the
loan and the yield to maturity of the financial
instruments applied for the funding,
which method comprises
(a) inputting and storing, in a memory or a storage
medium of the computer system, a first set of data indicating
the parameters: principal of the loan and repayment profile
of the loan,
(b) inputting and storing, in a memory or a storage
medium of the computer, a second set of data indicating
(i) a maximum and a minimum limit for debtor's payments
on the loan for each of a number of periods which
together cover the term to maturity of the loan,
(ii) a maximum and a minimum limit for the term to
maturity of the loan, and
(iii) rules for the mutual prioritization of, on the one
hand, the limits for debtor's payments on the loan input
under (i) and, on the other hand, the limits for the term
to maturity of the loan input under (ii)
(iv) and optionally a desired/intended payment on the loan
or a desired/intended term to maturity when there is
not equivalence between the maximum and the minimum
limit for the payment on the loan during the first
period (i) or when there is not equivalence between
the maximum and the minimum limit for the term to
maturity (ii),
(c) inputting and storing, in a memory or a storage
medium of the computer system, a third set of data indicating
a desired/intended refinancing profile, such as one or more
points) in time at which refinancing is to take place, and
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indicating the amount of the remaining debt to be refinanced
at said points) in time,
and/or said set of data indicating a desired/intended
funding profile such as the desired/intended number of
financial instruments applied for the funding, with their
type and volume,
(d) inputting and storing, in a memory or a storage
medium of the computer system, a fourth set of data
indicating 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,
(e) determining and storing, in a memory or a storage
medium of the computer system, a fifth set of data indicating
a selected number of financial instruments with inherent
characteristics such as type, price/market price, and 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
representing a first interest rate profile and either a first
term to maturity profile or a first payment profile on the
loan,
(g) calculating and storing, in a memory or a storage
medium of the computer system, a seventh set of data
representing
- a first term to maturity profile or a first payment
profile (depending on what was determined under (f))
corresponding to interest and repayment for debtor
- as well as a first remaining debt profile,
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the term to maturity profile or payment profile and the
remaining debt profile being calculated on the basis of
- the principal of the loan and the repayment profile
input under (a),
- the set of data input under (b),
- the refinancing profile and/or the funding profile input
under (c),
- and the interest rate profile and either the payment
profile or the term to maturity profile determined under
(f),
(h) selecting a number of financial instruments among the
financial instruments stored under (e), and calculating and
storing an eight set of data indicating said selected
financial instruments with their volumes to be applied in the
funding of the loan, which eight set of data is calculated on
the basis of
- the payment profile determined under (f) or calculated
under (g) and
the remaining debt profile calculated under (g),
- the refinancing profile input under (c) and/or the
funding profile input under (c),
- the set of data input under (b),
- the requirements input under (d), and
- when the calculation is for a refinancing where financial
instruments from an earlier funding have not matured yet,
the type, the number and the volume of these instruments,
if necessary performing one or more recalculations,
including, if necessary, selection of a new number of the
financial instruments stored under (e),
after each recalculation storing, in a memory or a storage
medium of the computer system
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- the recalculated interest rate profile,
- 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 conditions stated under (b) and (d) have been
fulfilled,
after which, if desired, the thus determined combination of
the type, the number and the volume of financial instruments
for funding the loan
- together with the calculated term to maturity,
- together with the calculated payment profile,
- preferably together with the calculated interest rate,
and
- preferably together with the calculated remaining debt
profile,
is read out, transferred to a storage medium or sent to
another computer system.
In addition to the input, determined and/or calculated data
being stored in a memory or on a storage medium, they may be,
e.g., output to a display or a printer. The memories applied
may, e.g., be electronic memories such as ROM, PROM, EEPROM
or RAM and the storage media may e.g. be tapes, discs or CD-
ROM.
It will also be possible to input data for use in or
resulting from the data processing according to the invention
in one set of memories or storage media which may be part of
the first or a second computer system and to transfer these
data to another set of memories or storage media which may be
part of the second or first computer system, these data being
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transferred, e.g., via a data transmission line or net
connecting the first and the second computer systems, or in a
wireless manner, e.g., electro-magnetically or optically.
Among other things, the method according to the invention
5 calculates the volumes of the individual financial
- instruments which are to be sold to finance the loan.
Normally, these volumes will not be whole or round figures,
and in certain cases they may be fairly small. The loan-
issuing institution solves the divisibility problem by adding
10 together many small loans when financial instruments are sold
in the market. It is, of course, crucial that the loan-
issuing institution makes an exact registration of the volume
of each individual financial instrument applied for the fund-
ing of each individual loan.
15 When it is stated in the present description and claims that
the volume, the type and the number of financial instruments
for the financing of a loan are determined by the method
according to the invention, this indicates that the
information which is the result of the method according to
the invention may, for instance, be applied as basis for the
actual physical action that the lender (for instance a
mortgage credit bank) issues/sells the instruments in
question. The information, which is the result of the method
according to the invention may, of course, also be applied
for pricing of a loan in connection with an offer for a loan
and/or for calculating lender's risk profile, for instance
with the aim of countering the risk through a hedge, without
actually issuing the said financial instruments.
It will be appreciated that the order of the above-mentioned
inputs/determinations/storage operations (a)-(e) is
arbitrary, and, therefore, that the sequence in letters does
not indicate an equivalent compulsory sequence in the steps.
Step (f) may also be carried out at an arbitrary stage in the
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sequence, unless, which is often preferred, it is chosen to
have the computer system calculate a first guess at the
interest rate profile and either a first term to maturity
profile or a first payment profile, in which case step (f)
will follow at least step (e). Instead of expressing that
data are input/determined and stored in the individual steps,
it may simply be expressed (and should be regarded as an
equivalent to the first expression form) that by means of the
computer system, the method according to the invention
calculates on the basis of stored input of data sets (a) -
(f). It will also be appreciated that these and other inputs
necessary for starting the individual calculations, for
instance the first interest rate profile and either the term
to maturity profile or the payment profile (f) as mentioned
above may be a guess or an initial value which may, of
course, also be made/determined by means of the computer
system according to predefined rules, and which is then
stored/used as initial value. Another example of data which
may either be input or guessed/calculated is the
desired/intended payment profile or the desired/intended term
to maturity under (b)(iv); if an initial value therefor is
not input/stored, the computer system will be adapted to
"guess" or calculate a value according to a predetermined
rule, for instance as an average of the values stored under
(b)(i) and (b)(ii).
It will also be appreciated that the condition in (b)(i) also
covers the case where no limits for the debtor payments have
been set for a period: in this case, the limits in (b)(i) are
zero and infinite.
A number of the inputs mentioned above apply for a
corresponding period. This is, for instance, the case for the
maximum permissible difference in balance and the limits on
the payments on the loan. In such a case the corresponding
period where the input in question applies is in itself
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input, or the period is already generally entered in the
computer system. For annuity loans, the period mentioned
under (b)(i) is preferably a refinancing period which
therefore normally will be default in the computer system,
but in principle it may be any period the lender may wish,
which is then normally input together with the limits
_ mentioned.
The requirement as to the maximum permissible difference in
balance is related to a period which, depending on the
legislation or practice on which the calculations are to be
based, may be a calendar year or any arbitrarily defined
year. In Denmark a balance requirement must be fulfilled per
calendar year.
In the calculation of data corresponding to the volumes of
the financial instruments applied for the funding, the re-
quirement as to a maximum difference in balance is, according
to present Danish rules for mortgage loans, given by a strict
balance, i.e. that no substantial difference in balance
occurs, or, expressed in another way, the difference is
practically zero. The method according to the invention may,
however, also be used in cases where a certain difference in
balance is tolerated or perhaps even intended, in which case
this tolerance or positive difference in balance will be
stored as a part of the data set in (d?.
In connection with the calculation according to the invention
both the requirement as to the difference in proceeds and the
requirement as to the difference in interest rates as well as
the requirement as to the difference in balance may be
indicated in different ways. For example, data may be input
which indicate a direct maximum permissible difference
between the total amount of the market price of the volumes
of the financial instruments applied for the funding and the
principal of the loan, and data may be input which indicate a
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direct maximum permissible difference between interest rate
and the yield to maturity of the financial instruments
applied for the funding, as well as data directly indicating
a maximum difference in balance allowed. The requirement as
to a maximum permissible difference in proceeds may also be
entered as data indicating a convergence condition for the
difference in proceeds and/or the requirement as to the
maximum permissible difference in balance between interest
rate and the yield may be given by inputting data indicating
a convergence condition for the difference in interest rates
and/or the requirement as to the maximum permissible
difference in balance may be given by a convergence condition
for the difference in balance.
When the loan is disbursed, the disbursement date and/or the
maturity date on the loan on the one hand will normally not
coincide with the settlement date of the financial
instruments applied on the other hand. Thus, the calculations
according to the method of the invention are preferably
adjusted for a possible difference between on the one hand
the disbursement date of the loan and/or the repayment date
and on the other hand the payment date of the financial
instruments in that a proportional adjustment is made for the
already past part or the remaining part of the payment period
and the redemption period, respectively. As an example, data
may be input or calculated which indicate an adjustment
factor for use in the calculation.
In practice it can be chosen to let the maturity of the loan
coincide with a debtor date of payment, which will normally
require that the last element in the term to maturity profile
is calculated accordingly to obtain this coincidence. Here it
is most suitable that the term to maturity is calculated
prolonged, which then implies that the payments on the loan
are reduced, for which reason the minimum limit for the
payments on the loan in the last period of financing is
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suspended in the calculation.
The calculation method according to the present invention may
also be used in cases where the data entered indicate that
more than one payment will be made by the debtor within one
bond settlement period.
In order to enable calculation of the volume of financial
instruments, information is input or present under (c) as to
at which points) in time refinancing is to be performed and
how much is to be refinanced at the said points in time. In
one case which is important in practice, the data input
indicate that full refinancing of the remaining debt is made
at the end of a predetermined period which period is shorter
than the term to maturity of the loan and in another
important case, the data input indicate that a refinancing of
the remaining debt is made by a fixed annual proportion.
The method according to the invention may be used for the
determination of the number and the volume of financial
instruments, the term to maturity profile and the payment
profile both in the situation where the loan is to be
calculated for the first time, i.e. the first funding
situation, and in the situation where a refinancing is to be
calculated. The expression funding thus comprises both "new
funding" and "refinancing". In the refinancing situation the
calculations include, in addition to the parameters mentioned
under (a)-(f), information on type, number and volume of the
financial instruments which have still not matured at the
time of refinancing. This information will often be stored in
the computer system from the previous calculation, but it is,
of course, within the scope of the invention to input this
information. Evidently, the parameters under (a)-(f) are
parameters relating to the funding situation in question, so
that in the cases where a refinancing is calculated, the
parameters will, of course, relate to the remaining debt of
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the loan as the principal of the loan and to the remaining
term to maturity as to the term to maturity.
When the patent claims refer to "remaining term to maturity"
or "term to maturity", this means - depending on the context
5 - either the remaining term to maturity or the term to
maturity which is the basis for the first calculation in the
financing period for which the calculation is made, or the
remaining term to maturity or the term to maturity that
results from a later calculation or recalculation for the
10 financing period to be calculated.
The result of the method according to the invention as
defined above is normally at least one set of data which may
be used in the first forthcoming financing situation, whether
this situation is the first financing period for the loan or
15 a later refinancing period.
To secure that the method according to the invention can
check that the data entered under (b) regarding payments on
the loan and term to maturity may be complied with, it is,
however, normally appropriate to make calculations for all
20 future financing periods until the maturity of the loan. This
implies that both some of the data constituting input for
calculations for later periods, and all the data being
calculated for these later periods are simulated data. The
simulations may of course be made on the basis of any desired
set of rules, but are suitably made based on an assumption of
an unchanged interest rate structure or based upon
observed/implicit forward interest rates. The financial
instruments in the simulations may for instance be defined
either as being unchanged or as being already existing
instruments, the term to maturity of the instruments being
adjusted. The simulations will typically be of importance to,
inter alia, the volume of the financial instruments and the
payments on the loan in the forthcoming financing period,
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21
which is why the simulations are an essential element of the
invention.
It is obvious that at any time of calculation it may,
instead, be chosen to have the same interest rate on the loan
through the simulations of all future financing periods.
The expression "term to maturity profile" is related to the
fact that in the method according to the invention an
estimate for the term to maturity is normally made for each
individual financing period or refinancing period, such as
mentioned above. However, at the end of the calculation, a
single well-defined term to maturity must of course result.
Thus, term to maturity profile means the sequence of terms to
maturity which in each calculation is assigned to the
respective refinancing periods.
Correspondingly, the expression "interest rate profile" is
related to the fact that in the method according to the
invention, an estimate of interest rates on the loan is
normally made for each financing period of refinancing
period; however, most importantly, at the end of the
calculation, a single well-defined interest rate on the loan
must result which applies for the nearest financing period
for which the calculations are made. Thus, interest rate
profile means the sequence of interest rates on the loan
which in each calculation is assigned to the respective
refinancing periods. As mentioned above it may, in a
simplified embodiment, be assumed that the interest rate on
the loan is constant throughout all simulated future
financing periods, whereby the interest rate profile will be
constituted by a sequence of elements of identical value. In
this case, the term to maturity profile will often be very
simple and simulations will then not be necessary.
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22
In the present description and claims, the expression "finan-
cial instruments" has the meaning normally used and thus
covers, inter alia, all types of interest rate related debts,
i.e. all types of bonds, including zero-coupon bonds and
derivatives like options, interest rate swaps, CAPS and
FLOORS.
When, in the method according to the invention, calculation
is made with derivatives which are not directly interest
bearing, it is suitable to begin by calculating the expected
payment flows in order to be able to calculate an internal
interest rate, whereby the payment flow or flows or the
likely payment flow/flows is/are expressed in parameters
corresponding to the above-mentioned parameters for interest
bearing debts, first and foremost a yield to maturity. Thus,
for instance, for an option which has a price of DKK 100 and
which has a probability of 50 per cent of resulting in
proceeds amounting to DKK 210 and a probability of 50 per
cent of resulting in proceeds amounting to DKK 0 after one
year, this can be done by a statistical calculation of the
average proceeds amounting to DKK 105, and expression of the
relevant parameters as a price of DKK 100, a quotation of DKK
100 and an interest rate of 5 per cent per year 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 as to maximum difference
in interest rates has been fulfilled. Then these parameters
may be entered in the computer system. Alternatively, and
often preferred, the data which are entered as
characteristics for the instruments in (e) above may be data
defining the said financial instruments directly, and the
computer system may be adapted to perform a recalculation
into parameters characterizing an interest bearing debt
according to predetermined principles. In case of CAPS or
FLOORS the same procedure may be used as the same payment
flows may be expressed by means of equivalent interest
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23
bearing instruments the characteristics of which may then be
stored as indicated in (e), or the computer system may
preferably be prepared to make a recalculation to the parame-
ters characterising an interest bearing claim according to
predetermined principles. It will be appreciated that in the
individual case a calculation may also be made on the basis
that in the individual funding or refinancing situation, a
combination of different types of financial instruments is
used, the characteristics to be applied for the calculation
being indicated for each type of instrument. In this
situation the fulfilment of the requirement as to a maximum
permitted difference in interest rates is preferably checked
on the basis of a total calculation which is based on the
total payment flows from all financial instruments applied.
Alternatively, a weighted average of interest rates of the
individual instruments may be used.
Thus, in the method according to the invention, calculations
may be made on the basis of various types of financial
instruments or funding volumes, but in one case which is
important in practice, calculation is made on the basis of
bonds with a maximum term to maturity corresponding to the
refinancing period. The bonds are usually non-callable bullet
bonds, including also zero coupon bonds. However, as
explained above, the method according to the invention may
also advantageously be used for the calculation in connection
with other types of financial instruments like e.g. bonds
used for serial loans, bonds used for annuity loans, options,
CAPS or FLOORS.
As conventionally, meaning of the type of a financial instru-
ment means the combination of all basic information or basic
data together defining the said financial instrument unam-
biguously, thus, so for mortgage bonds, the nominal
principal, coupon interest rate, date of maturity, all dates
fixed for settlement of interest payments and the ex-coupon
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date, i.e. the deadline for the investor to receive the first
occurring yield on the bond, as well as possibly the day
count convention, i.e. the formula used for the calculation
of the payment flow of the bond to an annual yield. The
number of financial instruments indicates how many different
financial instruments are to be applied. The volume indicates
how many entities of the individual financial instrument or
how large a nominal sum of the individual financial
instruments is/are to be applied.
In accordance with normal practice, the expressions
"repayment profile", "remaining debt profile" and "payment
profile" indicate the development over time of repayment,
remaining debt and payments on the loan, respectively.
The repayment profile may reflect the annuity principle as
well as the serial principle and may also represent a bullet
loan. In addition to that, any arbitrary placing in time of
the repayments is of course possible. For types of loans
where the repayment profile depends on the interest rate on
the loan, the repayment profile may be determined either on
the basis of the interest rates 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.
The expressions "financing profile" and "funding profile",
respectively, indicate the type, the number and the volume of
the financial instruments applied for funding. In the present
description and claims, the expression may be used for both
the desired/intended funding profile which is entered and
stored under (c) and which can perhaps not be fulfilled, and
for the exact funding profile which is the result of the
calculations after the application of the method according to
the invention.
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The expression "refinancing profile" indicates at which
points in time and with which amounts the loan is to be
refinanced.
It should be noted that the desired/intended refinancing
5 profile which is stored as a second set of data under (c)
v above, may in some cases be rewritten as a funding profile,
i.e. as a number of financial instruments with their type and
volume. An indication of a desired annual interest rate
adjustment percentage of 100 may, for example, be rewritten
10 into an indication that the loan is desired to be funded -
exclusively through sales of bullet bonds with a term to
maturity of 1 year. It is evident that the invention also
comprises the case where such a rewriting has taken place in
the data stored under (c).
The method according to the invention may be carried out in
many different ways; thus, a simultaneous calculation of all
parameters to be calculated may be performed, or the
calculations may be grouped according to various criteria. In
a presently preferred embodiment, which is described in
detail in the following example section, the method is
divided into a "inner model" and a "outer model", where the
inner model has a certain term to maturity as input and then
calculates the other parameters to be calculated, including
the payment profile, and the outer model determines whether
the payments on the loan lie within the limits stipulated,
recalculation are being made as explained in the following
until all conditions are met.
It is obvious that if the limits stipulated for the payments
on the loan in the outer model are 0 and infinite, the
solution in the inner model will always meet the demands in
the outer model. The inner model will in this case constitute
an independent method for determination of, among other
things, the type, the number, and the volume of financial
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26
instruments in the situation where the loan has a fixed term
to maturity. This independent method is by large described in
Danish Patent Application No. 165/96, Danish Patent
Application No. 233/97 and International Patent Application
No. PCT/DK97/00044. The above comment is therefore of special
interest in relation to the below-described preferred
embodiment of the method, named Type P, where the
determination of the below-discussed factors for function
coefficients is made analytically, since this embodiment of
the inner model is considered an independent new method
according to the invention.
It is obvious that an embodiment in which the term to
maturity is calculated simultaneously with the other
parameters in "the inner model" will also be comprised by the
invention. For instance, this may be the case if the
dimension of the iteration procedure is increased by one. (In
a preferred embodiment, a Gauss-Newton iteration algorithm is
applied).
It is also obvious that there will be cases where a set of
data indicates limits for payments on the loan and term to
maturity that are not compatible. For instance, it is obvious
that a wish for calculation of data for a loan with proceeds
of DKK 1,000,000 and a maximum limit for the quarterly
payments on the loan of DKK 100 and a maximum limit for the
term to maturity of 30 years cannot be fulfilled. In this and
similar cases the method according to the invention will
either
- in accordance with the prioritization of the limits for
the payments on the loan and term to maturity calculate
data for a loan for which the highest priority limit is
met, and the lowest priority limit is exceeded to the
necessary extent
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or
- omit to calculate data for the loan, stating which inputs
are not compatible.
In the presently preferred embodiment of the method according
to the invention the calculations will meet the limits for
the term to maturity, and if necessary the limit for the
payments on the loan is then broken.
In accordance with the above-explained demand which the
present invention is to satisfy, the set of rules will most
often be one which to the greatest possible extent maintains
the payments on the loan within the given limits and then
adjusts the term to maturity to compensate for any
fluctuation in the interest rates. However, there may be
limits for the duration of the term to maturity; in Denmark,
there is at present, for certain market segments, a
legislative limit of 30 years for mortgages, and lender may
of course have set as a condition that the term to maturity
may not exceed a certain value. The set of rules will
suitably be structured in accordance herewith so that when a
maximum term to maturity (or minimum term to maturity) is
reached, a regulation of the payment on the loan to a level
above a maximum limit for the payment on the loan (or below a
minimum limit for the payment on the loan) will be
calculated. The set of rules may be structured in such a way
that the regulation of the payment on the loan is made only
for the last period(s), or it may be structured in such a way
that the regulation is started as soon as a change in the
interest rate is observed (or simulated). In this context
(and other contexts) it may be allowable for situations where
the actual calculation is, as regards time, still far from an
absolute term to maturity limit, to calculate, in the term to
maturity profile, with terms which are longer than the
absolute term to maturity limit.
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In the case where the limits for the term to maturity
stipulated under (b)(ii) are given a higher priority than the
limits for the payment on the loan stipulated under (b)(i)
and the term to maturity at which the payment profile is
within the limits stipulated under (b)(i) is not within the
limits for the term to maturity stipulated under (b)(ii), the
term to maturity ma be determined as for instance
(1) the term to maturity which is compatible with a
recalculated fixed payment on the loan until the
maturity, the recalculated fixed payment on the loan
being the smallest possible
(2) the limit for the term to maturity, among the limits
for the term to maturity stipulated in (b)(ii), which
would otherwise be exceeded,
(3) the term to maturity that is the shorter of either
the term to maturity at which the payment profile
is within the limits stipulated therefor under
(b) (i)
or
the limit for the term to maturit
y, among the
limits for the term to maturity stipulated under
(b)(ii), which is binding, that is, which would
otherwise be exceeded.
Among these possibilities it is normally preferred to chose
variant (3) as this variant implies that the calculated
payment on the loan will increase after an increase in the
interest rate and decrease after a decrease in the interest
rate, which would not generally be the case in (2). Another
advantage of variant (3) is that the calculated payment on
the loan normally will normally develop in a more stable
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29
manner at fluctuations in the yield to maturity for the
financial instruments applied than in variant (1).
In combination with the maturity of a loan, an immediate
result of the calculations may indicate that the maturity of
the loan is not coincident with the maturity of the last
maturing financial instrument considered applied. It is, of
course, possible to use such a result, but in a preferred
embodiment the maturity of the loan is adjusted so that it
coincides with the maturity of the last maturing financial
instrument applied. The adjustment comprises a determination
of whether the term to maturity is to be rounded up to a
settlement date on the funding side (the maturity date of a
financial instrument) or rounded down to the previous
settlement date on the funding side (a one period earlier
maturity date of a financial instrument). In this case the
adjustment of the maturity may suitably be performed as
follows:
When the set of data under (c) indicates that calculation is
to be performed for the case where a full 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, and the remaining term to maturity of
the loan is shorter than the period of time which according
to (c) passes between two consecutive interest rate
adjustments, and the remaining term to maturity does not
correspond to the maturity of the last maturing financial
instruments 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), then the
term to maturity can suitably be determined in the method
according to the invention as
(i) the term to maturity of the loan prolonged as little
as possible to a date of maturity of one or more of
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the selected financial instruments provided the
payment profile will not thereby be below the minimum
limit for the payment on the loan stated under
(b)(i), or
5 (ii) the term to maturity of the loan shortened as little
as possible to a date of maturity of one or more of
the selected financial instruments provided the
payment profile will not thereby be above the maximum
limit for the payment on the loan stated under (b)(i)
10 and provided the condition under (i) is not met, or
(iii) the term to maturity of the loan prolonged as little
as possible to a date of maturity of one or more of
the selected financial instruments provided none of
the conditions stated under (i) or (ii) are met.
15 When, on the other hand, the set of data under (c) indicates
that calculation is to be made for the case where a partial
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., so that the
20 refinancing is equivalent to a fixed fraction of the
remaining debt of the loan, and the remaining term to
maturity of the loan is shorter than or equal to a fixed
value, and it is desired that the loan matures no later than
the time of maturity indicated under (e) for one or more of
25 the financial instruments applied for refinancing of the
loan, then the term to maturity is suitably 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 financial instruments.
30 As mentioned above, it is at present preferred to carry out
the method according to the invention by the use of an "inner
model" and an "outer model", such as described in detail in
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the examples. Here, a more or less arbitrary term to maturity
will normally be determined first, whereupon the inner model
will be started (as described in the following) and as a
result will calculate the payment on the loan, which is then
is checked for whether it is within the desired/permissible
interval. If the payment on the loan is not within the
interval, the term to maturity is adjusted, and the inner
model is started with the adjusted term to maturity. This
process is repeated until the payment on the loan is within
the desired/permissible interval. Then it is checked whether
the term to maturity is within the desired/permissible
interval. If this is the case, the resulting data may be
used. If the term to maturity is not within the
desired/permissible interval, then the term to maturity is
adjusted so that it is within the interval, and the adjusted
term to maturity is used as input to the inner model, which
then calculates the payment on the loan for the now
determined term to maturity. In this way the payment on the
loan is found which is necessary for the term to maturity
criterion to be fulfilled.
It will be understood that this embodiment of the method
comprises a series of recalculations in the outer model,
where each of these recalculations will normally result in a
series of recalculations in the inner model. Every time, the
recalculations in the inner model are performed until the
conditions of the inner model are met.
It is a characteristic feature of the inner model of the
presently preferred embodiments that it has an interest rate
on a loan as a basis for its calculations. However, it will
be understood that here, also any mathematical expression may
be used which represents the interest rate on a loan, e.g.,
the remaining debt profile, the payment profile on the loan
or the repayment profile on an annuity loan or the payment
profile on the loan for a bullet loan or a serial loan, as
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long as the other calculation parameters are adjusted thereto
in accordance with current and obvious mathematical
principles. It is also evident that, in principle, a first
calculation of the number, the type and the volume of the
financial instruments may be performed before the first value
for the interest rate on a loan is determined, but even if
this should be the case, subsequent calculations and if
necessary recalculations of, among other things, the interest
rate on the loan will have to be performed according to the
principles stated above. Thus it would, for instance, be
possible under (f) to replace the interest rate profile on
the loan with the volume of the financial instruments and
under (g) to either calculate the interest rate profile on
the loan or recalculate the volume of the financial
instruments.
A possible equivalent way to express steps (f) and (g) could
thus in these two steps be to replace the interest rate
profile with the volume of the financial instruments, however
in such a way that the volume of the financial instruments
meets the proceeds criterion, and then calculate the interest
rate on the loan in step (h).
In the case where no instrument is selected in (e) for which
payment falls due within the period till the first point in
time where, according to the refinancing profile entered
under (c), a refinancing is to be performed, the calculations
in the inner model in the preferred embodiment concern a
situation where the resulting volume for at least one of the
financial instruments applied for the funding will be
negative, that is, the debtor should purchase one or more
financial instruments during the first to come period in
order for the balance requirement to be fulfilled. As it
appears from the following, it is presently preferred that
precautions are taken to adjust the calculations so that they
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33
do not result in negative volumes of the financial
instruments.
In cases where it is indicated in the refinancing profile
- input that remaining debt is to be refinanced in full, the
financial instruments applied for the funding in the inner
model may, for instance, be calculated in the same way as the
financial instruments used for the initial funding, in other
words, a new calculation may be made according to the method
of the volume of financial instruments for the funding of a
new loan, where the principal of the new loan is equivalent
to the amount to be refinanced.
In another embodiment of the invention for the inner model it
may be indicated in data input corresponding to the
refinancing profile that a partial refinancing of the
remaining debt is to be performed. Here, it will be possible
in the inner model to find a solution as to the volume of the
financial instruments constituting the principal, for
instance if the input indicates that a periodic refinancing
is desired with a predetermined period which is shorter than
the term to maturity of the loan. A solution may also be
calculated if it is indicated that a periodical refinancing
of a fraction of the remaining debt of the loan is to be
made, the denominator of the fraction corresponding to the
total number of years until the maturity of the last maturing
financial instrument applied at the disbursement of tre loan.
Here, the selected period may be, e.g., 1 year, but other
periods like 2, 4, 5, 6 or 10 years may be selected.
Furthermore, periods corresponding to a total number of
months, e.g. 2, 3, 4 and 6 months may be selected.
In connection with full or partial refinancing, it will
normally be necessary in the inner model to calculate with
one or more new refinancing instruments that are not
comprised by the range of initial financial instruments
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which, according to the data given, constituted the series of
funding volumes applied when the loan was disbursed or when a
previous refinancing of the loan was performed. Normally,
these new refinancing instruments will have such a term to
maturity that they mature on a later date than the dates on
which the range of initial financial instruments mature. In
case of partial refinancing, the refinancing in the inner
model may also include an additional volume of the financial
instruments applied 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
marginal funding.
The calculation method according to the present invention
will also be capable of providing a solution as to the volume
of the marginal funding. When calculating the volume of the
marginal funding, data comprising possible new refinancing
instruments within the range of selected financial
instruments must be entered. In case of calculation of
refinancing the requirement as to proceeds may, e.g., be
given as a requirement as to the difference between, on the
one hand, a funding demand based on the balance requirement
and, on the other hand, the sum of the market price of the
marginal funding.
As mentioned above, a refinancing may be performed by the
issue of new financial instruments as well as additional
issue of already applied financial instruments. However, it
will also theoretically be possible to repurchase already
applied financial instruments, but this would involve a
number of inconveniences, among other things, an extra
interest rate risk for lender and problems with the pledges,
which is why repurchasing is not performed in practice.
According to a preferred embodiment of the method, the volume
of the marginal funding will therefore be calculated taking
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into consideration the volume of the previously applied
financial instruments remaining at the time of refinancing.
In an embodiment of the inner model, in the following
detailed description also called Type F, the set of data
5 under (c) indicates that a calculation is to be made for the
case where a full refinancing of the remaining debt is to be
performed periodically with a predetermined period which
period is shorter than the term to maturity of the loan,
which method for determination of the indicated volumes of
10 financial instruments in step (h) comprises calculating the
difference in proceeds for the calculated volumes of the
financial instruments applied for the funding and/or calcula-
tion of an adjustment of the interest rate on the loan, said
adjustment of the interest rate preferably being calculated
15 taking into consideration the calculated difference in pro-
ceeds, it being calculated to whether the adjustment of the
interest rate is so small that it fulfils the requirement as
to a maximum permissible difference in interest rates or a
convergence condition of the difference in interest rates.
20 It should be noted that when a requirement as to a difference
in proceeds of 0 or very close to 0 is met, then the
difference in the interest rate will automatically be 0 or
very close to 0, which means in this case, the requirement as
to difference in the interest rate may suitably be left out
25 from the starting conditions input.
This is the reason why input about maximum permissible
difference in the interest rate is stated as optional, while
input about maximum permissible difference in the proceeds is
stated as compulsory in all cases. However, it should be
30 noted that a compulsory input about maximum permissible
difference in proceeds may be accomplished by inputting
information which is fully equivalent herewith, e.g., an
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36
interest rate input, and that such substitutions are, of
course, comprised by the present invention.
In case the requirements or conditions as to the difference
in proceeds or the difference in the interest rate is not
fulfilled, the recalculations in the inner model of Type F
include one or more interest rate iterations, each interest
rate iteration including
calculating and storing, in a memory or a storage
medium of the computer, data indicating a new interest rate
which is preferably based on the previous interest rate on
the loan and the calculated adjustment of the interest rate,
calculating and storing, in a memory or a storage
medium of the computer, data indicating a new payment profile
and a new remaining debt profile for debtor, which payment
profile and remaining debt profile are calculated taking into
consideration the new interest rate on the loan, the
principal of the loan, the repayment profile entered under
(a) and the refinancing profile and/or the funding profile
entered under (c), and
calculating and storing, in a memory or a storage
medium of,the computer system, of data indicating a new set
of volumes for the financial instruments applied for the
funding.
The interest rate iteration is preferably made by applying a
numeric optimization algorithm or by "grid search°.
As examples of numeric optimization algorithms may be
mentioned a Gauss-Newton algorithm, a Gauss algorithm, a
Newton-Ramphson algorithm, a quadratic hill climbing
algorithm, a quasi-Newton algorithm, a maximum likelihood
algorithm, a method of scoring algorithm and a BHI3H
algorithm. As it appears from the detailed description of
Type F, a Gauss-Newton algorithm has proved to be very
suitable.
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When the relevant requirements) as to a maximum permissible
difference in proceeds and/or the requirement as to a maximum
permissible difference in interest rates is/are fulfilled in
the Type F embodiment, it is appropriate to determine whether
. 5 all the calculated volumes of financial instruments are
positive. In case the calculated set of volumes comprises at
least one negative volume, an option is to apply the result
as such, meaning that the calculation indicates that the
debtor is to buy one or more financial instruments in order
to fulfil the balance requirement. As mentioned above, this
is normally not preferred, and therefore, in this case the
method according to the invention usually further comprises
either
i) selection of a new number of financial instruments
among the financial instruments stored under (e), in that one
or more of the instruments in the new number of instruments
is/are determined in such a way that the payments on
this/these instruments falls due relatively later in relation
to the original number of financial instruments, whereupon a
recalculation is made as indicated in connection with the
description of the Type F embodiment given here and in the
following, or
ii) the negative volume or the negative volumes is/are
set equal to 0, after which recalculation is made as
indicated in connection with the description of the Type F
embodiment given here and in the following.
In case the data entered indicate that a partial refinancing
is to be performed, the volumes of the financial instruments
applied for the funding or the refinancing will, in a
preferred embodiment for the inner model, be calculated so
that they reflect the remaining debt development given by the
remaining debt profile. This calculation may include the use
of a first function adapted to the remaining debt profile as
explained in the following. For example, in case the entered
data indicate a difference between on the one hand the
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payment date of the loan and/or the repayment date of the
loan, and on the other hand the settlement date of the
financial instruments, it is possible in the calculation
according to a preferred embodiment of the invention to
determine the volume or volumes of one or more financial
instruments in such a manner that this instrument or these
instruments does not/do not reflect the polynomial function,
but rather contributes) to a solution to the above marginal
conditions.
In a preferred embodiment of the invention, a determination
is made in the inner model as to whether the so calculated
volumes of financial instruments fulfil one or more
predetermined convergence condition(s). If such a condition
or such conditions is/are not fulfilled, one or more
iterations may be calculated until the new set of data of
financial instruments fulfils one or more convergence
condition(s).
In a preferred embodiment of the inner model, the function
coefficients will be calculated on the basis of a calculated
difference in proceeds and/or a calculated difference in
refinancing, preferably corresponding to the difference
between on the one hand a funding demand based on the balance
requirement and on the other hand a preferred refinancing.
The function coefficients may be found either analytically or
by iteration.
In case the calculation according to this embodiment of the
inner model determines that the calculated volumes of finan-
cial instruments applied for the funding or refinancing do
not fulfil requirements as to a difference in the interest
rates stipulated in the input data, then, in a preferred
embodiment, one or more recalculations in the form of
interest rate iterations will be made to determine or
calculate a new interest rate, after which a new set of
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financial instruments is calculated. An interest rate
iteration is made until the requirements as to a difference
in the interest rates is fulfilled. The discussion which
follows gives examples with a detailed explanation of this
embodiment. Both the situation where the function
coefficients of the function adapted to the remaining debt
. profile are found by iteration and the situation where the
function coefficients are found analytically are described.
The following is a detailed description of the case where the
set of data (c) indicates that calculation is to be made for
the case that a partial refinancing of the remaining debt is
made periodically with a predetermined period which is
shorter than the period being shorter than the term to
maturity of the loan, e.g., in such a manner that the
refinancing equals a fixed fraction of the remaining debt of
the loan. In this embodiment, generally termed P in the
following detailed description, some of or all of the
financial instruments applied for the funding are, in the
inner model, calculated in the first calculation in step (g)
so that they substantially reflect a shifted-level remaining
debt profile, whereupon, if necessary, recalculations are
made until all conditions indicated under (d) are fulfilled.
The adjustment to a shifted-level remaining debt profile is
suitably made by calculating the volume of some of or all of
the financial instruments in the calculation in step (h) and
possibly in one or more recalculations in step (h) using a
function which is adapted to a shifted-level remaining debt
profile. This function is suitably a polynomial function with
a maximum degree of 1 (one) less than the number of financial
instruments applied.
The polynomial function is suitably calculated using a
statistic curve fit method. It has been found that the least
squares' method is an appropriate statistic curve fit method,
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but other statistic curve fit methods like other maximum
likelihood methods or cubic splines methods are also
interesting methods for this use.
In a preferred embodiment of the invention a calculation is
5 made based on an analytic derivation (as opposed to
iteration) of one or more coefficients in the polynomial
function. The calculation is performed so the subsequent
determination of the marginal funding values, which are equal
to the difference between the value of the polynomial
20 function and the volume of the already issued financial
instruments, meet the requirements for maximum difference in
proceeds and, if possible, the demand as to maximum
difference in balance when at the same time the actual
refinancing percentage equals the intended refinancing
15 percentage.
In cases where coefficients in the polynomial function are
calculated so that one or more of the marginal volumes of the
financial instruments are negative, the said marginal volumes
are not applied, which is indicated by an adjustment of an
20 indicator function. The indicator function will, in this
embodiment, be an m-dimensional vector in which the elements
have either the value one or the value zero, and where the
value zero indicates that the said financial instrument is
not applied in the financing. Based on the adjusted indicator
25 function a new calculation of one or more coefficients in the
polynomial function is performed, the resulting marginal
volumes of the financial instruments are checked, and, if
necessary, the indicator function is adjusted again.
The calculation of the coefficients in the polynomial
30 function and the adjustment of the indicator function
continue until either
all increases in the financial instruments are non-
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negative (i.e. either positive or zero)
or
either the first element in 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 in the polynomial
function is calculated so that the resulting series of
marginal volumes of the financial instruments meets the
requirements as to maximum difference in proceeds; the
resulting refinancing will be defined by a residual
calculation in accordance with the demand as to maximum
difference in balance.
It is also possible to adjust only one element in the
indicator function at a time.
The above-discussed analytic method for determination of the
function coefficients in the polynomial function is efficient
with respect to calculation and hence a time-saving method.
This method is described in detail in section 3.2.3 with the
heading "the inner model for Type P - the analytic solution"
in the following detailed part of this description.
As mentioned above, an aspect of the invention concerns an
embodiment which applies the analytical determination of the
function coefficients in the polynomial function in
calculations which correspond to the above-explained
embodiment designated type P, but where the limits stipulated
for the payments on the loan in the inner model are zero and
infinite. Thus, this aspect of the invention concerns a
method for determining the type, the number, and the volume
of financial instruments for the funding of a loan with
equivalent proceeds to a debtor by means of a first computer
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system, the loan being designed to be at least partially
refinanced during the remaining term to maturity,
- requirements having been made as to 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,
- requirements having been made as to 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 on the other hand, the principal of the loan,
- and requirements having been made as to a maximum
permissible difference between the interest rate on the
loan and the yield to maturity of the financial
instruments applied for the funding,
which method comprises
(a) inputting and storing, in a memory or a storage medium
of the computer system, a first set of data indicating the
parameters: principal of the loan and repayment profile of
the loan,
(b) inputting and storing, in a memory or a storage medium
of the computer, a second set of data indicating the term to
maturity of the loan,
(c) inputting and storing, in a memory or a storage medium
of the computer system, a third set of data indicating a
desired/intended refinancing profile, such as one or more
points) in time at which refinancing is to take place, and
indicating the amount of the remaining debt to be refinanced
at said points) in time,
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(d) inputting and storing, in a memory or a storage medium
of the computer system, a fourth set of data indicating a
maximum permissible difference in balance within a
predetermined period, a maximum permissible difference in
proceeds and 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,
(e) determining and storing, in a memory or a storage
medium of the computer system, a fifth set of data indicating
a selected number of financial instruments with inherent
characteristics such as type, price/market price, and 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
representing a first interest rate profile,
(g) calculating and storing, in a memory or a storage
medium of the computer system, a seventh set of data
representing
- a first payment profile corresponding to interest and
repayment for debtor
- as well as a first remaining debt profile,
the payment profile and the remaining debt profile being
calculated on the basis of
- the principal of the loan and the repayment profile
input under (a),
- the set of data input under (b),
- the refinancing profile and/or the funding profile
input under (c),
- and the interest rate profile determined under (f),
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(h) selecting a number of financial instruments among the
financial instruments stored under (e), and calculating and
storing an eight set of data indicating said selected
financial instruments with their volumes to be applied in the
funding of the loan, which eight set of data is calculated on
the basis of
the payment profile calculated under (g) and
- the remaining debt profile calculated under (g),
- the refinancing profile input under (c) and/or the
funding profile input under (c),
- the set of data input under (b),
- the requirements input under (d), and
- when the calculation is for a refinancing where
financial instruments from an earlier funding have not
matured yet, the type, the number and the volume of
these instruments,
if necessary performing one or more recalculations,
including, if necessary, selection of a new number of the
financial instruments stored under (e),
after each recalculation storing, in a memory or a storage
medium of the computer system
- the recalculated interest rate profile,
- the recalculated payment profile,
- the recalculated remaining debt profile, and
- the selected financial instruments with their
calculated volumes,
until all conditions stated under (b) and (d) have been
fulfilled,
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after which, if desired, the thus determined combination of
the type, the number and the volume of financial instruments
for funding the loan
- together with the calculated payment profile,
5 - preferably together with the calculated interest rate,
and
- preferably together with the calculated remaining debt
profile,
is read out, transferred to a storage medium or sent to
10 another computer system,
the set of data (c) indicating that calculation is to be
performed for the case where partial refinancing of the
remaining debt is performed periodically with a predetermined
period, which period is shorter than the term to maturity of
15 the loan, e.g. in such a way that the refinancing is
equivalent to a fixed fraction of the remaining debt of the
loan, some of or all of the financial instruments applied for
the funding in the first calculation in step (h) being
calculated in such a way that they substantially reflect a
20 shifted level remaining debt profile, whereupon, if
necessary, recalculations are performed until all the
requirements mentioned under (d) are fulfilled,
the volume of some of or all of the financial instruments
being calculated, in the calculation in step (h), by applying
25 a function adjusted to a shifted level remaining debt
profile, the determination of one or more of the function
coefficients for the function adjusted to the shifted level
remaining debt profile being performed analytically.
Interesting and preferred embodiments of this aspect of the
30 invention appear from the patent claims depending on the
independent patent claim corresponding to the definition
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given immediately above; the more detailed explanations of
these embodiments are the same as the explanations, given
above and in the detailed part of this description, of the
corresponding embodiments where the limits stipulated for the
payments on the loan in the outer model differ from 0 and
infinity.
On the other hand, the function coefficients may also be
calculated by iteration as described in the in the
immediately following sections and as described in detail in
section 3.2.2 with the heading "The inner model for Type P -
the iterative solution" in the following detailed part of
this description.
In the embodiment designated Type P, recalculation of all of
or some of the data mentioned in (g) and (h), and/or one or
more function coefficients to the function representing the
shifted-level remaining debt profile, and/or the interest
rate in the inner model may be performed by iteration carried
out using numeric optimization algorithms or by grid search.
Also in this case, one of the optimization algorithms
mentioned above in connection with Type F may be used as an
optimization algorithm, and also in this case the
optimization algorithm is, suitably, a Gauss-Newton
algorithm.
In case the requirements as to the difference in proceeds
and/or the difference in interest rates and/or the difference
in balance calculated taking into consideration the
refinancing profile entered under (c) are not fulfilled, then
the recalculations in the inner model of the Type P
embodiment may include one or more iterations, each iteration
comprising
calculating and storing data indicating a new interest
rate and/or
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calculating and storing data indicating a new payment
profile and a new remaining debt profile for debtor, which
payment profile and remaining debt profile are calculated
taking into consideration the new interest rate on the loan,
the principal of the loan, and the repayment profile entered
under (a) the refinancing profile and/or the funding profile
. entered under (c), and the term to maturity and/or
calculating and storing data indicating a new set of
coefficients for the function which is adapted to the
shifted-level remaining debt profile, and/or
calculating and storing data indicating a new set of
volumes of the financial instruments applied for the funding,
which new set of volumes is calculated on the basis of the
financial instruments already determined for the funding, and
the new payment profile, and the new remaining debt profile
as well as the requirement as to the maximum difference in
balance.
In the example given below in the detailed part of this
description, section 3.2.2, it has been chosen to iterate as
to the proceeds requirement and the difference in balance
extended taking into consideration the refinancing profile
entered under (c), and only when the two requirements are
fulfilled, iteration is carried out as to the interest rate.
It will be understood that the iteration may be performed in
an arbitrary order, and that here also iteration of the
function applied, the so-called trend function, may be
included.
In this case, the method in step (h) in the inner model may
comprise determination of whether the calculated volumes of
financial instruments fulfil at least two of two or more
predetermined convergence conditions, which are preferably
calculated taking into consideration a calculated difference
in proceeds and a difference in balance calculated taking
into consideration the refinancing profile entered under (c),
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and in case the calculated volumes of financial instruments
do not fulfil these conditions, then the recalculations may
include 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 indicating two or more new
function coefficients for the function representing the
shifted-level remaining debt profile,
calculating and storing data indicating a new set of
volumes for the financial instruments applied for the fund-
ing, which new set of volumes is calculated taking into
consideration 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
predetermined convergence conditions, until the new set of
calculated volumes of financial instruments fulfils these
conditions. The new function coefficients) is/are suitably
calculated taking into consideration the calculated
difference in proceeds and a difference in balance calculated
taking into consideration the refinancing profile entered
under (c).
The difference between the interest rate on the loan and the
yield on the calculated volumes of the financial instruments
may be calculated in the inner model, it being calculated
whether the difference in interest rates is so small that it
fulfils the requirement as to maximum permissible difference
in interest rates or a convergence condition for the
difference in interest rates, and in case the determined
requirements as to the difference in interest rates are not
fulfilled, then the recalculations may comprise one or more
interest rate iterations, each interest rate iteration
including
calculating and storing an adjustment of the interest
rate, the adjustment of the interest rate preferably being
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calculated taking into consideration the difference between
the interest rate on the loan and the yield to maturity on
the calculated volumes of the financial instruments, e.g. by
use of a Gauss-Newton algorithm,
calculating and storing data indicating a new interest
rate which is preferably based on the previous interest rate
and the calculated adjustment of the interest rate on the
loan,
calculating and storing data indicating a new payment
profile and remaining debt profile for debtor, which payment
profile and remaining debt profile are calculated taking into
consideration the new interest rate, the principal of the
loan, the term to maturity and the repayment profile entered
under (a), and the refinancing profile and/or the funding
profile entered under (c), and
calculating and storing data indicating a new set of
coefficients for the function adapted to the shifted-level
remaining debt profile, and
calculating and storing data indicating a new set of
volumes for financial instruments applied for the funding.
It is also within the scope of the invention in connection
with the calculations in the inner model of Type P to
determine, at the same time, whether the calculated volumes
of financial instruments fulfil at least three of three or
more predetermined convergence conditions which are
preferably calculated taking into consideration a calculated
difference in proceeds, a difference in balance calculated
taking into consideration the refinancing profile entered
under (c) and a maximum permissible difference in interest
rates, and in case the calculated volumes of financial
instruments do not fulfil these conditions, then to have the
recalculations comprise one or more iterations, each
iteration comprising
calculating and storing an adjustment of the interest
rate, said adjustment of the interest rate preferably being
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calculated taking into consideration the difference between
the interest rate on the loan and the yield to maturity of
the calculated volumes of financial instruments,
calculating and storing data indicating a new interest
5 rate which is preferably based on the previous interest rate
and the calculated adjustment of the interest rate on the
loan,
calculating and storing data indicating a new payment
profile and remaining debt profile for debtor, said payment
10 profile and remaining debt profile being calculated taking
into consideration the new interest rate on the loan, the
principal of the loan, the term to maturity, the repayment
profile entered under (a) and the refinancing profile and/or
the funding profile entered under (c),
15 calculating and storing data indicating a new set of
coefficients for the function adapted to the shifted-level
remaining debt profile, and
calculating and storing data indicating a new set of
volumes for financial instruments applied for the funding,
20 which new set of volumes is calculated taking into considera-
tion 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 three or more
25 predetermined convergence conditions.
Also in this connection, the iterations may be performed by
the use of a numeric optimization algorithm, preferably a
three-dimensional Gauss-Newton algorithm.
In case the calculated set of volumes in the calculations of
30 the inner model of type P includes at least one negative
volume, the negative volumes) may suitably be set equal to 0
- to avoid negative volumes in the result, cf. the above
comments concerning the normally undesired situation where
debtor is to purchase financial instruments - after which the
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calculations continue on the basis of the thus determined
volumes of the financial instruments.
According to an embodiment of the invention it will also be
possible to determine the volumes of the financial
instruments applied for the loan in the cases where data are
entered corresponding to a funding profile desired by the
debtor and comprising desired financial instruments. In this
case the calculations may also comprise calculation of
whether the volumes of financial instruments in the funding
profile indicated fulfils the requirement as to maximum
permissible difference in proceeds, and in case the indicated
volumes do not fulfil this requirement, then, according to a
preferred embodiment of the invention, one or more
adjustments of the previously indicated volumes is/are
performed, adjustments being performed until the new set of
financial instruments fulfils the requirement as to a maximum
permissible difference in proceeds.
In addition to the calculation of the proceeds criterion, it
is preferred that a calculation is also made as to whether
the requirement for maximum permissible difference in balance
is fulfilled, and in case the calculated volumes do not
fulfil this requirement, one or more calculations of the new
financial instruments which do not fulfil the requirement as
to maximum permissible difference in balance is/are made.
Calculation of new financial instruments will here preferably
be made for one or more financial instruments to which
repayments are to be made in a period in which the
requirement as to maximum difference in balance is not
fulfilled. In a preferred embodiment, the calculation will be
performed for one or more financial instruments to which
repayments are to be made in the last period wherein the
requirement as to maximum permissible difference in balance
is not fulfilled. Preferably, the calculation of new
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financial instruments is based on the difference in balance
for the periods in which the corresponding previously found
financial instruments do not fulfil the requirement as to
maximum permissible difference in balance.
It applies generally to the method according to the invention
that in many cases it will be possible after a result has
been reached to make a new calculation on the basis of other
instruments in order to assess whether a cheaper loan is
thereby obtainable.
The range of financial instruments determined under (e) is
selected among a number of available financial instruments.
It will be appreciated that, if desired, this number of
instruments may be entered into a database in the computer
system or may be available via a net and that, if desired,
the determination may be performed automatically or
semi-automatically by means of the computer system according
to predetermined criteria or functions.
The invention also relates to a computer system for
determining the type, the number, and the volume of financial
instruments for the funding of a loan with equivalent
proceeds to a debtor as well as the term to maturity and the
payment profile on the loan, the loan being designed to be at
least partially refinanced during the remaining term to
maturity,
- requirements having been made to the effect that
- the term to maturity of the loan is not longer
than a predetermined maximum limit or shorter
than a predetermined minimum limit,
- debtor's payments on the loan are within
predetermined limits,
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- rules having been made as to how the two
above-mentioned requirements are mutually
prioritized,
- requirements having been made as to 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,
- requirements having been made as to 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, on the other hand, the principal of the loan,
- and requirements having been made as to a maximum
permissible difference between the interest rate on the
loan and the yield to maturity of the financial
instruments applied for the funding,
which computer system comprises
(a) means, typically input means and a memory or a storage
medium, for inputting and storing a first set of data
indicating the parameters: principal of the loan and
repayment profile of the loan,
(b) means, typically input means and a memory or a storage
medium, for inputting and storing a second set of data
indicating
(i) a maximum and a minimum limit for debtor's payments on
the loan for each of a number of periods which together
cover the term to maturity of the loan,
(ii) a maximum and a minimum limit for the term to maturity
of the loan, and
- (iii) rules for the mutual prioritization of, on the one
hand, the limits for debtor's payments on the loan
input under (i) and, on the other hand, the limits for
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the term to maturity of the loan input under (ii)
(iv) and optionally a desired/intended payment on the loan
or a desired/intended term to maturity when there is
not equivalence between the maximum and the minimum
limit for the payment on the loan during the first
period (i) or when there is not equivalence between the
maximum and the minimum limit for the term to maturity
(ii),
(c) means, typically input means and a memory or a storage
medium, for inputting and storing a third set of data
indicating a desired/intended refinancing profile, such as
one or more points) in time at which refinancing is to take
place, and indicating the amount of the remaining debt to be
refinanced at said points) in time,
and/or said set of data indicating a desired/intended
funding profile such as the desired/intended number of
financial instruments applied for the funding, with their
type and volume,
(d) means, typically input means and a memory or a storage
medium, for inputting and storing a fourth set of data
indicating 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,
(e) means, typically input means and a memory or a storage
medium, for determining and storing a fifth set of data indi-
cating a selected number of financial instruments with
inherent characteristics such as type, price/market price,
and date of the price/market price,
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(f) means, typically input means and/or calculation means
and a memory or a storage medium, for determining and storing
a sixth set of data representing a first interest rate
profile and either a first term to maturity profile or a
5 first payment profile on the loan,
(g) means, typically calculation 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
profile (depending on what was determined under (f))
corresponding to interest and repayment for debtor
- as well as a first remaining debt profile,
the term to maturity profile or payment profile and the
remaining debt profile being calculated on the basis of
15 - the principal of the loan and the repayment profile
input under (a),
- the set of data input under (b),
- the refinancing profile and/or the funding profile
input under (c),
20 - and the interest rate profile and either the payment
profile or the term to maturity profile determined
under (f),
(h) means, typically calculation means and a memory or a
storage medium, for selecting a number of financial
25 instruments among the financial instruments stored under (e),
and calculating and storing an eight set of data indicating
said selected financial instruments with their volumes to be
applied in the funding of the loan, which eight set of data
- is calculated on the basis of
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the payment profile determined under (f) or calculated
under (g) and
- the remaining debt profile calculated under (g),
- the refinancing profile input under (c) and/or the
funding profile input under (c),
- the set of data input under (b),
- the requirements as to maximum permissible difference
in balance, maximum permissible difference in proceeds
and maximum permissible difference in interest rates
input under (d), and
- when the calculation is for a refinancing where
financial instruments from an earlier funding have not
matured yet, the type, the number and the volume of
these instruments,
the means being adapted to perform, if necessary, one or more
recalculations, including, if necessary, selection of a new
number of the financial instruments stored under (e),
the means being further adapted to store, after each
recalculation,
- the recalculated interest rate profile,
- 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 conditions stated under (b) and (d) have been
fulfilled,
means for outputting the thus determined combination of the
type, the number and the volume of financial instruments for
funding the loan
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- together with the calculated term to maturity,
- together with the calculated payment profile,
- preferably together with the calculated interest rate,
and
- preferably together with the calculated remaining debt
profile,
or means for transferring the combination to a storage medium
or for sending it to another computer system.
A corresponding system which constitutes a separate aspect of
the present invention is adapted to the above-mentioned
aspect of the invention which concerns the type P-calculation
for the case where the limits for the payment on the loan in
the outer model is 0 and infinite and where the relevant
function coefficients are found analytically.
The system according to the present invention thus comprises
means for inputting and storing the necessary data for the
calculations. The input means may comprise a keyboard or a
mouse, a scanner, a microphone, a touch-sensitive screen or
plate or the like, but may also comprise means for electronic
input via a storage medium or via a data network. As
mentioned above, the means for storing may be electronic
memories such as ROM, PROM, EPROM, EEPROM or RAM, or erasable
or non-erasable plate or tape storage media such as tapes,
discs or CD-ROM.
The input means for may comprise a database or another
computer system and/or a data network from which the computer
system can enter data such as the necessary prices of the
financial instruments used for the funding, information about
lender and/or other information needed or desired for the
calculations or other purposes.
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The output means may comprise one or more data screens, one
or more printers, one or more telefax machines, one or more
voice generating devices and/or connection means which
electronically connect the computer system to a data network
designed to transfer data from the computer system to a data
transmission system that comprise or is connected to one or
more output means of the above-mentioned type. The output
means will typically be connected to a data transmission
system which, for instance, may comprise a computer close to
or remote from the computer system and which is connected
electronically to the computer system via the data network.
The output means may, for instance, be placed with an adviser
such as a real estate agent, a bank branch or a branch of
another financial institution, who or which via the data
25 network has access to the calculation means used for the
calculations and comprised by the computer system.
The data network may constitute or be a part of a local
network, which may be a part of or be connected to a wide
area network. The local network is typically adapted to
electronically distribute and/or collect data from a number
of units. A unit may, e.g., comprise a geographic district
which comprises a number of data systems/computers, and/or a
local area which also may comprise a number of data
systems/computers.
Furthermore the data network may comprise one or be one or
more ISDN connections or further developments hereof, one or
more telecommunication connections connected to the computer
system by means of at least one telecommunication connection
means such as a modem, the telecommunication connections
comprising one or more telephone connections, wireless
connections or other means of data connection. Typically, the
data network may combine the above telecommunication
connections and telecommunication connection means.
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For example, an ISDN connection (or a further development
hereof) or a telecommunication connection with modem may be
used as telecommunication connection between a real estate
agent and the computer system, or these connections may be
used to transfer, from central financial institutions, such
as stock exchanges, data for the financial instruments used
for the calculations, such as price and yield to maturity of
bonds.
Furthermore, the system comprises means for calculation
adapted to perform the calculations which are necessary for
the method. The means for calculation may typically comprise
one or more microprocessors.
A system according to the present invention may be a computer
system programmed in such a way that it is capable of
performing the necessary for working the method according to
the invention. In this connection it should be noted that
there may be different embodiments of the system,
corresponding to these different embodiments being designed
to perform the calculations indicated in the various
embodiments of the method according to the invention
mentioned above and in the claims.
The means for calculations may comprise one or more
electronic calculation circuits, such as a microprocessor
which is an integrated part of the computer system, or which
is connected to the computer system via the data network, so
that one or more external processors which may, e.g., be a
part of a calculation central, may perform the calculations
necessary for the method based on data transferred via the
data network from the data processing system.
The method according to the invention is typically performed
on the computer system according to the invention.
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Further embodiments and details of the method and the system
according to the present invention appear from the claims and
the detailed description given in connection with the
drawings and the examples.
5 BRIEF DESCRIPTION OF THE DRAWINGS
Fig. 1 shows an example of the variation in the bond yield as
a function of the term to maturity,
Fig. 2 shows an example of the determination of the interest
rate on a loan which is financed by issue/sale of financial
10 instruments in accordance with a preferred embodiment of the
invention at an increase in the yield to maturity of the
financial instruments,
Fig. 3 shows an example of the determination of a payment
profile on the loan and the term to maturity of a loan which
15 is financed by issue/sale of financial instruments in
accordance with a preferred embodiment of the invention at an
increase in the yield to maturity of the financial
instruments whereby a maximum limit input for the payment on
the loan becomes binding,
20 Fig. 4 shows a computer system which may be used in
performing the methods according to the invention,
Fig. 5 shows a characteristic example of limits for the
payments on an annuity loan,
Fig. 6 shows a characteristic example of limits for the
25 payments on a serial loan,
Fig. 7 shows characteristic shifts of the payments on the
loan as a function of the term to maturity,
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Fig. $ shows characteristic shifts of the payments on the
loan as a function of the term to maturity of a annuity loan,
Fig. 9 shows characteristic shifts of the payments on the
loan as a function of the term to maturity of a serial loan,
Fig. 10 shows characteristic jump-wise shifts of the payments
on the loan as a function of the term to maturity of an
annuity loan of Type F,
Fig. 11 shows characteristic jump-wise shifts of the payments
on the loan as a function of the term to maturity of a serial
loan of Type F,
Fig. 12 shows a flow chart describing a data processing
method for the outer model for the calculation of the volumes
of funding volumes of a loan in accordance with a first
embodiment of the invention, called Type F,
Fig. 13 shows a graphic representation of an example of a
function F, which indicates how much the payments on the loan
differ from the maximum and the minimum limit; it is taken
into consideration whether it is the maximum or the minimum
limit that is exceeded without adjustment of the term to
maturity,
Fig. 14 shows a graphic representation of a function F, which
indicates how much the payments on the loan differ from the
maximum and the minimum limit; it is not taken into
consideration whether it is the maximum or the minimum limit
. 25 that is exceeded without adjustment of the term to maturity,
Fig. 15 shows a step in the iteration routine in the
Gauss-Newton algorithm used for calculation of the term to
maturity of a loan,
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Fig. 16 shows a flow chart describing the inner model for the
data processing method for the calculation of the volumes of
volumes of funding volumes for a loan according to a first
embodiment of the invention, called Type F,
Fig. 17 shows a step in the iteration routine in the
Gauss-Newton algorithm used for calculation of the interest
rate,
Fig. 18 shows a flow chart describing the inner model for the
data processing method for the calculation of the funding
volumes for a loan according to an embodiment of the
invention called Type F+, which is a continuation of Type F
in cases where Type F leads to one or more negative funding
volumes,
Fig. 19 shows a characteristic example of a payment profile
on a annuity loan as a function of time,
Fig. 20 shows a characteristic example of a payment profile
on an annuity loan as a function of time, the payments on the
loan being,close to the maximum limit for the payments on the
loan at an increase in the interest rates,
Fig. 21 shows a characteristic example of a payment profile
on a serial loan as a function of time,
Fig. 22 shows a characteristic example of a payment profile
on a serial loan as a function of time, the payments on the
loan being close to the maximum limit for the payments on the
loan at an increase in the interest rates,
Fig. 23 shows a flow chart describing the outer model of a
data processing method for the calculation of the funding
volumes for the funding of a loan according to a first
variant of the embodiment called Type P,
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Fig. 24 shows a flow chart describing the iterative inner
model for the embodiment called Type P for the calculation of
the funding volumes of a loan,
Fig. 25 shows a characteristic example of a trend function
which can be used in the embodiment called Type P,
Fig. 26 shows a flow chart describing the analytical inner
model for the embodiment called Type P for the calculation of
the funding volumes of a loan, and
Fig. 27 shows a characteristic example of an initial trend
function and an adjusted trend function used in the
embodiment called Type P.
GENERAL DESCRIPTION OF LOANS WITH ADJUSTABLE INTEREST RATES
In the following is given a brief general description of the
new type of loans with adjustable interest rates where an
exact calculation of the funding is made possible by an
embodiment according to the present invention, the loans also
termed LAIR.
The debtor receives exact calculations of various
alternatives concerning the combination of a loan with
respect to term to maturity and repayment profile. For
example, debtor may decide for himself how often and when a
LAIR is to be refinanced and the part of the debt he prefers
to be refinanced.
An investor prefers, as an important element, to combine
portfolios where the return is known with a reasonable
certainty. Internationally the most well-known type of bond
is non-callable bullet bonds. They fulfil the demand for a
stable return which also is known beforehand. However, this
is, not the case with the Danish callable mortgage bonds.
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LAIR seeks to combine the preferences of the debtor and the
creditor. Debtor may want a loan with a term to maturity of
30 years, where the investor may have an maximum investment
period of 5 years. When a loan with a term to maturity of 30
years is funded only by means of a bond series of 30 years,
as is the case at the moment, it is difficult to combine the
preferences of the two parties.
The two parties meet, as LAIR permits funding with a range of
e.g. non-callable bullet bonds with maturities from 1 to 10
years, irrespective whether the debtor desires a loan with a
term to maturity of 10, 20 or 30 years.
In case of a yield structure, where the short-term interest
rate is lower than the long-term interest rate (see the
example in Fig. 1, in which the curve 1 shows the interest
rate as a function of the term to maturity of the loan), it
is cheaper for the debtor to provide funding by the issue of
short-term bonds instead of a long-term bond with a term to
maturity of e.g. 30 years. The example in Fig. 1 shows that
as of January 15 1996 the interest rate on a 1 year loan was
approx. 4 points lower than the interest rate of 30 years.
LAIR may also be used in connection with loans of e.g. 12
payment periods, i.e. a loan where debtor makes monthly
payments on the loan.
It is also possible to disburse bullet loans with maturities
of 1 or 10 years as well as various combinations of bullet
loans matching the funding need of the individual debtor.
When calculating the principals of a loan according to the
principle of LAIR it is intended that the payment profile and
the repayment profile reflect the profile on e.g. an annuity
loan with a term to maturity of 30 years irrespective of the
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refinancing percentage and the intervals between the refinancing.
In case the debtor wants a LAIR with a term to maturity of 30
years, this may thus be funded by issuing e.g. up to 10
bullet bonds with maturities of 1 to 10 years. Thus, as the
5 term to maturity of the debtor loan is longer than the
funding, the loan is to be fully or partially refinanced on
its way. However, a loan with more than 10 financial
instruments with maturities of more than 10 years may also be
selected.
10 Here, the debtor has different options. The debtor may choose
to refinance 10 per cent of the remaining debt each year, 50
per cent every second year, 100 per cent every fifth year
etc.
In cases where loans are refinanced by means of bullet bonds,
15 debtor's selection of refinancing percentage and refinancing
time determines the number of bullet bonds with which the
creditor is to refinance the loan. If, therefore, the debtor
selects a refinancing of 20 per cent each year, the creditor
will fund a LAIR by issuing 5 bullet bonds with maturities
20 from 1 to 5 years, if a refinancing of 10 per cent is
selected, the creditor issues 10 bullet bonds with maturities
from 1 to 10 years.
Figs. 2 and 3 show an example of a LAIR of DKK 1,000,000, the
debtor selecting a 10 per cent refinancing of the remaining
25 debt each year. Thus, at the time of a disbursement, the
creditor issues 10 bullet bonds with maturities of 1 to 10
years. In the example, the debtor repays a LAIR as an annuity
-~ loan with a term to maturity of 30 years. In the example the
interest rate structure is presumed unchanged during all 30
30 years and it is equal to the interest rate structure shown in
Fig. 1.
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Fig. 2 shows the development in the interest rate, and Fig. 3
shows the development in the quarterly payments on the loan
and the term to maturity of the same loan. The loan is a LAIR
with a refinancing of 20 per cent each year and with a
maximum and minimum limits for the payments on the loan of
DKK 23,000 and DKK 20,000, respectively. The loan is
disbursed with a term to maturity of 20 years.
An increase of 3 percentage points in the yield to maturity
of the financial instruments applied after a period of 5
years has been assumed. The increase in the yield causes the
payments on the loan to reach the maximum limit after 8
years, whereupon the term to maturity is adjusted. This
appears from Fig. 3. When the payments on the loan 4 reach
the maximum limit 3, then the term to maturity 5 increases
concurrently. At the maturity of the loan, the term is
prolonged to the settlement date corresponding to a term to
maturity of 22 years. At the same time as the prolongation,
the payment on the loan exceeds the minimum limit 3.
After the first year, the bullet bond with a term of 1 year
matures, and, at the same time, 20 per cent of the remaining
debt on the loan is refinanced. The refinancing of the loan
is carried out by issue of a new bullet bond with a term of 5
years and additional issue in the 4 bonds with terms to
maturity from 1 to 4 years. The same procedure applies to the
refinancing in year 2 and onward.
Hence, the LAIR is financed in 5 bullet bonds until year 18.
From year 18 till year 22 the number of bullet bonds is
reduced by 1 each year so that the loan may mature after the
above-mentioned 22 years. The number of bullet bonds is
reduced at the same time as the loan is refinanced.
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DETAILED DESCRIPTION
Below follows a detailed description of embodiments of the
method and the system according to the invention.
Where, in the following, the symbol x is applied in
connection with vector and matrix multiplication, the symbol
is not to be regarded as the outer product. Thus, the symbol
represents the inner product of vector and matrix
multiplication.
1. Adjustable term to maturity - the general problem
Characteristic for loans with adjustable interest rates and
adjustable term to maturity is that the payments on the loan
float within a band defined by a set of maximum and minimum
limits as the interest rate on the loan is adjusted to the
market rate . The limits are denoted YDJ°" and YDJi°,
respectively, YD denoting the payments on the loan in general
and J indexing the interest rate adjustment periods.
The fluctuations are kept within the band by adjusting the
term to maturity of the loan, when, otherwise, the payments
on the loan would exceed the limits of the band. Only to the
extent that the adjusted term to maturity exceeds a pre-
defined limit, the payments on the loan will exceed the
limits of the band. The limits for the term to maturity of
the loan are denoted, correspondingly, Ln°°"' and Lmi°, L
denoting the term to maturity in general.
It is not a requirement in the model that the term to
maturity is set as whole years or payment periods. Thus, the
possible adjustments of the term to maturity are defined on a
- continuous interval limited by L°"°"' and L°'i". This
is necessary
if it is to be possible to fix the payments on the loan
within a relatively narrow band at every interest rate adjustment.
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The fact that the possible adjustments of the term to
maturity are defined on a continuous interval opens up the
possibility of offering the debtor the option of fixed
payments on the loan as long as the term to maturity is
within Lm°" and L~i°. In the model, fixed payments on the loan
correspond to equal maximum and minimum limits for the
payments on the loan, hence
YI~X=Yl~ln
In the model, there is no need in general for a distinction
between loans with fixed payment and loans with payments
floating within maximum and minimum limits. The debtors,
however, are likely to view loans with fixed payment and
loans with floating payments as two different types of
products that require different computations, especially when
the loans are disbursed. Hence, a distinction can not be
completely avoided in the following.
1.1 Fixing the limits for the payments on the loan
The fixing of the limits for the payments on the loan is,
basically, the debtor's choice and is therefore considered an
input to the model.
Fixing the limits at an unreasonable level implies that the
model, at an early stage, will disregard the limits to
observe the limit for the term to maturity for the loan,
which will be discussed further in detail in the following.
Hence, unreasonable limits will automatically become non-
binding.
For annuity loans, it is reasonable to set the maximum and
minimum limits at a fixed level, hence
YDo ' : YD~ ..yD~ and yDo'~ :yD~.....yD"~
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For serial loans, corresponding limits with fixed values will
not be applicable unless the spread between the limits is
very wide. By definition, the payments on a serial loan
decrease over time - it will therefore be most reasonable to
fix similar decreasing maximum and minimum limits e.g, by
fixing the maximum and minimum limits as a spread to the
expected payment profile at the disbursement of the loan.
Adjusting the term to maturity to observe the limits for the
payments on the loan causes a special problem in regard to
serial loans. When adjusting the term to maturity, it is not
possible simultaneously to determine a level and a slope for
the payment profile. Therefore, it will not be possible, in
general, to observe the limits for the payments on the loan
for a multi-annual period (e.g. a full refinancing period) at
one time unless a very wide band is defined. In the model,
this problem is solved by observing the limits YD~°"and YDJ'"
only in the first year of each refinancing period.
Yet, the question of fixing the limits after the maturity of
the loan as originally expected is still to be answered. This
applies to a situation where increases in the interest rate
have caused the term to maturity to exceed the original term
to maturity which applied at the disbursement of the loan.
For annuity loans the model simply continues to observe the
defined limits which are at a fixed level and therefore well-
defined. For serial loans the gradually decreasing limits can
not just be applied onwards, since this would imply that the
maximum limit at one stage might be negative. For debtor, it
would be more appropriate if the maximum limit instead is
fixed at the level of the last well-defined maximum limit,
whereas the minimum limit is fixed at 0 (zero). Thus, the
loan may be continued as a serial loan without the limits
becoming binding per definition. An example of the limits for
the payment on an annuity loan is shown in Fig. 5, while Fig.
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6 correspondly shows an example of limits for a serial loan.
In both figures, the vertical lines denoted L~i° and Lm°"
illustrate the minimum and maximum limits, respectively, for
the term to maturity on the loan, whereas the lines denoted
5 YDJ°" and YDJ'n illustrate the maximum and minimum limits,
respectively, for the payments on the loan.
1.2 The concept of adjustable term to maturity
The financing of LAIR in more than one financial instruments
implies that the future interest rate on the loan will not
10 only shift, due to shifts in the yield to maturity of the
financial instruments applied, but will also shift due to
shifts in the distribution of the volume of the financial
instruments applied, provided the yield curve is not
horizontal.
15 Hence, to assume the yield curve to be constant does not
correspond to assuming constant interest rates on the loan.
On the contrary, at the time of each interest rate adjustment
a sequence of interest rates must be determined assuming
constant yields to maturity of the financial instruments
20 applied. In particular, this is relevant for the P-product,
since interest rate shifts penetrate gradually as the loan is
partially refinanced.
The non-constant interest rates complicate the concept of the
term to maturity for LAIR with adjustable term to maturity.
25 For each future refinancing period the payments on the loan
are calculated as a function of the term to maturity i.e.
YD(J,LJ) J=0,...,M
wherein
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YD(J,L,,) is the annual payments on the loan in the
period from the Jth refinancing till the
(J+1) refinancing.
LJ is the term to maturity at the Jth
refinancing.
denotes the future interest rate
adjustments. J is set to 0 (zero) at each
interest rate adjustment. Thus J=0 denotes
the time of the computations (either the
disbursement or a refinancing of the loan)
M denotes the last refinancing of the loan
before maturity. At the disbursement of a
F5 loan with a 20 year term to maturity, M
is set to 3 (J=0,1,2,3), since the loan is
not refinanced at the maturity. LM
determines when the loan has been repaid
and is, thus, the term to maturity of the
loan. M will shift continuously according
to the remaining term to maturity of the
loan.
If the payments on the loan are fixed or either YDJ°" or
yDJ'°
are binding, it is required, disregarding the situation where
also Lm°" or La'ln are binding, that
YD ( 0 , La ) =YD ( 1, Ll ) =...YD ( J, LJ ) =...=YD ( M, LM ) (
=YDT°"VYDJin )
For a non-constant interest rate on the loan, this will, in
general, imply that
Lo*L1*...*LJ*...*LM
Thus, the model must handle a sequence of terms to maturity.
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1.3 Fixing the limits for the term to maturity
The traditional concept of the term to maturity of a loan
corresponds to LM, and this will be the term to maturity
which debtor is informed of. In relation to the limits for
the maturity it will, therefore, be most natural to define
the limits relating to LM and not the other elements in the
sequence of the terms to maturity. Hence, it is accepted that
LJ>Ln'°" for J<M. This also implies that at each computation it
is necessary to simulate future refinancing periods to
determine whether the limits for the term to maturity are
observed.
In contrast to the limits for the payment on the loan, the
limits for the term to maturity will, typically, be
determined by external circumstances. Thus, the maximum limit
may be determined by legislative rules or credit policy
considerations, whereas the minimum limit may be determined
so that the debtor is not deprived of a capital loss tax
deduction. If the debtor prefers more narrow limits than
required by the external factors, nothing will prevent the
model from operating with these more narrow limits.
In the model the maximum and minimum limits for the term to
maturity are given a higher priority than the limits for the
payments on the loan. Only when the payments on the loan are
at the same level as the maximum or minimum limit, the term
to maturity is adjusted as mentioned above. If it is allowed
that Lu'°"' or Lmln is exceeded in order to observe the limits
for the payments on the loan, L~" or Lm'° will never be
binding and may, thus, be left out of the model. In addition
the nature of the limits for the term to maturity does not
allow that the limits are exceeded in practice.
The prioritization of the limits for the term to maturity and
the payments on the loan determines the fundamental pattern
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of shifts in the payments on the loan as shown in Fig. 7, in
which line 10 shows the fundamental pattern of shifts in the
payments on the loan.
The starting point is given by A. If the yield to maturity of
the financial instruments applied shift upwards, the payments
on the loan increase until YDJ°xis reached. When YD~" is
reached, the payments on the loan are held constant at this
level, whereas the term to maturity is prolonged. This
continues until the term to maturity reach the level L""°",
which must not be exceeded. Thus, the payments on the loan
must be increased to a level exceeding the maximum limit,
e.g. point B.
Accordingly, in case the yield to maturity of the financial
instruments applied falls, the payments on the loan will drop
to the level Y1?~in. Then the term to maturity is reduced so
that the payment on the loan is shifted horizontally in the
plane. The horizontal shift continues until L~'" is reached,
at which the shifts again are vertical, e.g., to the point C.
The fundamental pattern of shifts in the payments on the loan
imply that, within the band, the payments on the loan shift
correspondingly to a LAIR with fixed term to maturity i.e. in
the vertical plane. Thus, LAIR with fixed term to maturity
(the conventional LAIR) can be considered as the special case
where the maximum and minimum limits for the payments on the
loan are fixed at ~ (infinity) and 0 (zero) respectively, or
where the maximum and minimum limits for the term to maturity
are identical, in which case the limits for the payments on
the loan are disregarded.
A most obvious simplification of the model would be to make
the limits for the term to maturity apply constantly. In each
refinancing period it would only be necessary to check that
L,, is not higher than L°°" and not lower than
L'°'° respectively.
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Hence, it would not be necessary to simulate future
refinancing periods.
However, the simplification would incur unnecessary
deviations from debtor's intended payment profile, since an
expected future fall in the interest rate on the loan can not
be applied in the current period, cf. the above discussion.
When the debtor himself has fixed the limits for the term to
maturity more narrowly than required by the legislative
rules, this model is not applied.
Yet, problems regarding the legislative requirements can not
be ruled out, e.g., terms to maturity exceeding 30 years for
private debtors might not be accepted at any stage in the
payment profile, thus LJ<J for all values of J.
Correspondingly, it can not be ruled out that the debtor will
be deprived of a capital loss tax deduction if just one of
the elements in the sequence of terms to maturity is lower
than the term to maturity of the loan previously held by the
debtor. In the model it is thus necessary to introduce a
further set of limits for the term to maturity denoted Lr"~'
and LM=N, which limits must be observed by L~ at all times.
The simplified version of the model, in which each element in
the sequence of terms to maturity is tested against the
maximum and minimum limits for the term to maturity fixed by
the debtor, thus corresponds to putting Lr'~"' and LMIN equal to
L'°°" and L'°"', respectively. If it turns out that
L~"' and LMIN
are not necessary, the limits will be fixed at ~ (infinity)
and 0 (zero), respectively.
1.4 Deviations from the limits for the payments on the loan
According to the fundamental pattern of shifts in the
payments on the loan, the term to maturity is constant if
YD ( 0, Lo) <YDJ°". Yet, it can not be ruled out that
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YD ( 0 , Lo ) <YD~" and YD ( 0 , L,, ) >YDJ°" for LJ=La and lsJsM
corresponding to a situation where the current payments on
the loan are within the band, but it is expected that the
payments on the loan will exceed YD~"at a future refinancing
5 of the loan if the term to maturity is not prolonged compared
to Lo. In this situation YD(O,Lo) is maintained, whereas LJ is
prolonged so that
YD ( 0 , LJ ) =YDJ°"
Thus, the term to maturity is prolonged without the current
10 payments on the loan being increased to the level YDJ°".
Another solution will, however, result in unreasonable
payment profiles for loans with a long remaining term to
maturity.
The problem can be extended to a situation where LM will
15 exceed the maximum limit for the term to maturity even though
YD ( 0 , LJ ) =YDJ°" under the given yield curve .
The limit for the term to maturity may not be exceeded.
Instead the payment on the loan is increased, if necessary to
a level exceeding YDJ°". Yet, it is not completely trivial
20 when and by how much the payment on the loan should be
increased. At a first glance, it would seem obvious to
determine the payment on the loan as
(i) YD (J,L°'°") 'd J=O, 1,...,M
which ensures that the loan matures at the right point in
- 25 time. However, this solution ignores the maximum limit for
the payments on the loan fixed by the debtor, which is not
suitable. In addition, it can not be ruled out that Lo<Lm°"
even though LM>L'°°". In this situation, the payment on the loan
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~s
decreases for J=0 and will later increase beyond YD~". This
also implies that the solution in (i) is not suitable.
However, a more suitable solution is to determine the
payments on the loan based on a combination of YD(J,L,,) and
(i) so that the payments on the loan follow YD(J,L,,), as long
as this is larger than YD(J.Lm°") and, when this does not
apply, the payments on the loan are fixed at YD(J,L~"). In
mathematical terms, this can be expressed as;
Let J' denote the minimum value of J where
YD(J,L~"')>YD(J,LJ) b' J=0,1,...,M
For OsJsJ' the payment on the loan is determined as
YD(J,.)=YD(J,LJ)
and for J'sJ__<M
YD ( J , . ) =YD ( J , Lm°" )
The pattern of shifts in the payments on the loan will be as
illustrated in Fig. 8 and Fig. 9. Line 12 in Fig. 8 indicates
the pattern of shifts in the payments on the loan for an
annuity loan when it is necessary to exceed the limit YDJ°" to
observe the maximum limit for the term to maturity of the
loan. Analogeously, line 14 in Fig. 9 indicates the pattern
of shifts in the payments on the loan for a serial loan when
it is necessary to exceed the limit YDJ°" to observe the
maximum limit for the term to maturity of the loan.
In the analogous situation where the payments on the loan
will fall below YDJi" if the minimum limit for the term to
maturity is to be observed, the payment on the loan is
determined accordingly.
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Let J " denote the minimum value of J where
YD(J,Lmi°)~YD(J,LJ) b' J=0,1,...,M
For OsJsJ " the payment on the loan is determined as
YD(J,.)=YD(J,LJ)
and for J " sJsM
YD ( J , . ) =YD ( J , Lmin )
This solution is considered to be the most suitable. However,
other solutions are outlined below.
1) Firstly, the payments on the loan might be fixed at a
level corresponding to Lm°". In the model, this would
imply the definition of a new maximum limit at a level
either higher or lower than the previous maximum limit.
For serial loans it would imply the further problem
that the payments on the loan are decreasing in each
refinancing period. Thus, the payments on the loan
would have to shift upwards at each refinancing of the
loan and then, gradually, fall. Finally, the solution
would imply that increases in the interest rate in the
early stages of the loan would effect the payments more
than observed in the preferred solution.
2) A second possibility is to fix a new maximum limit for
the payments on the loan at a higher (lower) level,
and, gradually, let the payments on the loan rise
(fall) until the new limit is reached in the same
manner as the preferred solution. The limits fixed by
the debtor, are hereby ignored which is inappropriate.
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3) Finally, it is possible to determine the payments on
the loan based on the term to maturity L~" or Lmin
respectively as mentioned under (i). However, as
mentioned above, this implies the risk that the
payments on the loan will first decrease and then
increase and vice versa.
2. Type F
2.1 The general problem
LAIR type F is characterized by the fact that all the
remaining debt of the loan is refinanced at predetermined
points in time with a fixed interval.
The duration of the interval is determined by the debtor, yet
observing that the refinancing always takes place on 1
January. Hence, the duration of the interval will always be a
whole (interger) number of years except for the first period
where the duration depends on the date of the disbursement.
In the period between two consecutive interest rate
adjustments, the interest rate and thereby the payments on
the loan are fixed. Depending on the frequency of refinancing
desired by the debtor, the payments on the loan will shift
discontinously over time.
This is shown in Fig. 10 and Fig. 11. In Fig. 10, line 16
indicates the payments on an annuity loan shifting
intermittently, whereas line 18 in Fig. 11 denotes the
payments on a serial loan also shifting discontinously.
The model must find a solution which complies with a number
of requirements. The conditions may be assined to two
categories.
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Firstly, the solution must observe requirements defined for
the total period from the disbursement till the maturity of
the loan. These intertemporal conditions were the main focus
point in the above description of the general problem.
However, the conditions are briefly summarized in the
following.
1) The term to maturity must not be longer than the
maximum limit
LMs L'~"
2) The term to maturity must not be shorter than the
minimum limit
~zLm~n
3 ) If LM<Lm°", the payment on the loan is determined
observing the condition
YD ( J , LJ ) _< YDJ°"
20 and otherwise
YD(J,.)=YD(J,L,,) for OsJsJ'
(A)
YD(J, . )=YD (J,Lm°") for J'sJsM
25 where J' denotes the minimum value of J for which
YD (J,LJ)<YD(J,L'°°")
4) If LM>Lm'n, the payment on the loan is determined
observing the condition
YD(J,LJ) zYDJtn
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and otherwise
YD(J,.)=YD(J,LJ) fox OsJsJ "
5 (B)
YD(J, . )=YD(J,L~'n) for J"<JSM
where J " denotes the minimum value of J for which
YD ( J , L J ) >YD ( J , L°'i" )
The model observes these conditions by determining a sequence
10 of terms to maturity Lo, L1, ..., LM.
The additional conditions are defined for each refinancing
period, hence for each value of J and LJ. Thus, it is
necessary to alter the notation, since J and M are
insignificant in this context. Instead, it is necessary to
15 focus on each year in the refinancing period as denoted by
j=0,1,2,...,m, where m denotes the next refinancing of the
loan. Furthermore, the payment on the loan is denoted as a
function of j, i.e. YD(j).
First of all, the payments on the financial instruments
20 applied must match the debtor's payments on the loan as
required by the principle of strict balance in the Danish act
on mortgage credit loans. Thus,
Debtor's interest and repayments=
Bond redemptions+interest payments to the bond holders
25 In order to have the payments on the loan, which in addition
to payments of interest also comprise repayments, correspond
to payments on the bonds for each year, bullet bonds are
issued having maturities up to the duration of the
refinancing interval so that bonds mature each year until the
30 next refinancing.
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For each year (j') until the refinancing, the following shall
thus apply
(C) YD(j' )=H(j' )+ ~ RN(j)H(j) , j'<m
i.i,
where:
- 5 YD(j) is debtor's payments on the loan in year j.
H(j) is the jth bond principal at a given time
J.
R"(j) is the coupon interest rate of the jth
bond.
m (Not to be mistaken for M) is the number of
bonds at the beginning of the refinancing
period and also the time of the next
refinancing.
7 (Not to be mistaken for J) indicates years
within the refinancing period and, at the
same time, numbers the funding volumes. j
is thus set to zero after each refinancing.
In the notation there is a direct
compliance between each year and the bond
maturing that year.
That j at the same time may indicate years and funding
principals is solely due to the fact that the bonds only have
one annual settlement date on 1 January. If the number of
annual settlements is changed, the notation must be changed
as well.
As to the loan side, however, 1 or 4 annual payment dates may
be selected. Thus, j cannot also indicate debtor payment
dates. In order to facilitate the notation, debtor's
repayment and interest within one year is, nevertheless,
called YD(j), then summing up the payments made on payment
dates within the year. Let n indicate the number of debtor's
payment dates per year,
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R
YD{j)= ~ AFD(i)+ ~ R RG(i-1)
t.l i.1 n
where
R" is the interest rate on the loan
RG(j) is the remaining debt at the end of year j
AFD(i) is debtor's repayment on the payment date i
i indicates the payment dates within the
year, that is, i=1,2,...,n
The principle of strict balance is observed each year.
However, there are two years in which (C) is to be modified -
the first year and the year in which the loan is refinanced.
In the year of disbursement of the loan, the sum of debtor's
interest payments and repayments must be equivalent to the
sum of the principal payment on the bond having the shortest
term to maturity H(1) and interest payments on all bonds
applied in the financing of the loan. In addition, in year 1
an adjustment must be made so that the first settlement is
only paid in full by the debtor if the loan is disbursed
exactly on a settlement date.
Thus, for year 1 applies:
(D) YD(1) - H{1) + REG ~ RNH(j)
i
wherein
REG is a regulation factor determining how much of
the interest payments on the first settlement
date the bond holder should receive from the
debtor. REG is determined as the part of the year
in which the loan has existed since 30 November
on which date the bonds mature ex-coupon. Thus,
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REG can assume values between 1/12 (if the loan
is disbursed 30 November) and 13/I2 (if the loan
is disbursed 1 December, and thus, the first year
last 13 months).
In the year of the refinancing of the remaining debt (year m)
the total payments on the loan comprise, apart from debtor's
repayment and interest, the remaining debt at the end of the
year, corresponding to the refinancing amount. As to the bond
side, these payments match payments on the one bond which has
not yet matured. For the years in which the loan is
refinanced, (C) may therefore be formulated as follows:
(E) YD(m)+RG(m)=H(m)+RN(m)H(m)
Thus, the total balance conditions may be written as (seen
from the point in time 0):
Year 1: YD(1) - H(1) + REG ~ RN(j)H(j)
i
Year 2: YD(2) - H(2) + ~ RN(j)H(j)
2
Year m: YD (m) + RG (m) - [ 1 + RN (m) ] H (m)
Besides observing the balance condition, the market price of
the bonds issued must exactly match the principal of debtor's
loan. In the following, this condition is referred to as the
proceeds criterion, i.e..
RG(0) - ~ K(j)H(j)
i
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wherein
K(j) is the price of the jth funding instrument.
RG(0) is the remaining debt at the beginning of the
refinancing period, which at the disbursement of the loan is
equivalent to the principal of the loan, and, at the
refinancing of the loan, complies to the refinancing amount
since the remaining debt in full is refinanced.
The problem to be solved by the model for each refinancing
period comprises the m equations that resulted from the
balance condition as well as the one equation which resulted
from the proceeds criterion. However, the problem is
simultaneous. This results from the financing of the loan
being determined both by the debtor's payments on the loan
and the proceeds criterion. The problem is solved by fixing
the interest rate on the loan. When starting off with an
"arbitrary" interest rate, debtor's repayment and interest
payments may be determined. They determine the funding with
regard to the balance condition. The proceeds from the issue
of the funding may then be compared to the desired proceeds
of the loan. In case of a deficit more bonds must be sold,
and the interest rate must be raised so that the larger
payments on the bonds - more bonds are sold - are covered by
the payments on the loan. If, on the other hand, there is a
surplus from the issuing of the bonds, the interest rate may
be lowered.
Thus, it will always be possible to determine an unambiguous,
positive interest rate solving the problem for each
refinancing period. The explanation hereto is that the
proceeds from the sale of the funding instruments is a
strictly growing function of the interest rate:
higher interest rate ~ larger payments on the loan ~
larger funding principals ~ higher proceeds.
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The structure of the model reflects the structure of the
conditions that the solution must observe. In a outer model
the intertemporal conditions, i.e. the conditions defined for
the total period from disbursement to maturity of the loan,
5 are applied. The remaining conditions are applied in an inner
model.
The structure in the model is that the outer model applies a
iteration routine involving the inner model. When, for each
refinancing period, the volume of the financing instruments
10 applied, the interest rate and payments on the loan have been
determined in the inner model, this solution is tested
against the conditions in the outer model. If the conditions
are not observed, the term to maturity is adjusted, and
subsequently the inner model once again computes volumes of
15 the financing instruments applied, interest rate and payments
on the loan etc.
A attractive characteristic of the model is a close
compliance between the interest rate on the loan and the
yield to maturity of the portfolio of financing instruments
20 applied. The compliance is obtained by the principle of
balance, since the compliance between the payments on the
bonds and on the loan only allows the interest rate on the
loan to deviate from the yield to maturity as a result of
difference in the timing of the payments within the year.
25 Thus, it is not necessary to advance explicit conditions to
the compliance between the interest rate on the loan and the
yield to maturity of the portfolio of financing instruments
applied.
2.2 Method
30 For the sake of good order the general problem is repeated
below:
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A sequence of terms to maturity must be determined observing
that:
in each refinancing period, the payments on the loan
are within the band defined by the maximum and minimum
limits for the payments on the loan
stipulated that the total term to maturity is within the band
defined by the maximum and minimum limits for the term to
maturity and stipulated that the payments on the loan, in
each refinancing period, observe:
the balance condition (equations (C) - (E)) and the
proceeds criterion.
2.2.1 The outer model of type F
As mentioned above, the problem is solved in an inner and an
outer model. The outer model is shown in the flow chart in
Fig. 12, which will be described thoroughly in the following.
It appears that the solution is obtained in 12 steps, which
will be described below.
It should be noted that steps A to E are repeated for each
refinancing to follow i.e. for J=0,1,...,M. Thus, the model
determines a full payment profile which, period for period,
observes the YDJ°" and YDJi° conditions .
Thus, the model also determines a value of LM. When the full
payment profile has been calculated, it is therefore to be
tested that Lmi"sLMsL°'°". The test is applied in steps F and I.
If LM>_L'"°", a new payment profile must be determined according
to (A). YD(J,LJ) has already been calculated in the model,
but YD(J,L"°") must be calculated in the inner model, which
takes place in steps J and K. If, on the contrary, LMsLmi", the
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term to maturity is adjusted complying with (B) in steps G
and H.
If L'°'nsLMsL'°°" the model continues in step K, where
the
calculations are finalized.
Step A - Determine initial LJ
Tn step A the model determines an initial term to maturity
which may be considered as a first guess in the iterative
routine.
First, an initial value for J=0 is determined. The following
steps determine, inter alia, the payments and interest rate
on the loan and a final value of Lo_ Having determined these,
the model returns to step A to determine an initial value of
L1 etc. Thus, each time step A is applied an initial term to
maturity is determined.
If J>0 the initial value of L,, is relatively simply
determined as LJ_1, LJ-1 being, of course the final value -
after the necessary iterations - of the term to maturity in
the preceding refinancing period. This procedure is applied
since the term to maturity is only adjusted if, otherwise,
the payments on the loan would be outside the band. Thus, as
a starting point it is assumed that the term to maturity
needs no adjustment.
If J= , the determination depends on whether the (physical)
time of calculation coincides with the disbursement of the
loan or a refinancing of the loan.
At the disbursement of the loan, the term to maturity is
. determined based on input from the debtor which may comprise:
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1) The debtor selects an annuity loan with fixed payments
(fixed payments rule out serial loans). One could
imagine that the debtor prefers that the payments on
the loan in the first year of each refinancing period
is determined according to a particular pattern.
However, this possibility would be comprised by the
intended payments on the loan under 3 ) i . a . YDJ°"-YDJin
An initial term to maturity must be calculated. The
initial guess of Lo may be determined applying the
annuity formula
YD(O,Lo). RG(0)RR
1- ( 1.R R) - ~~-~ ~
which, solved for Layields
1 1- RG ( 0 ) R'~
YD(O,L )
0
( F ) Lo'- In ( 1+R K) '~P
where YD ( 0 , Lo ) =YDJ°"=YDJin,
and where
cp is the number of years since the disbursement of
the loan. Thus, Lo-cp is the remaining term to
maturity to be applied in the annuity formula.
Yet, R" has not been determined. To reduce the number
of iterations, R" is set equal to the yield to maturity
of the last maturing financing instrument applied.
2) The debtor selects an intended term to maturity and
maximum and minimum limits for the payments on the
loan.
In this situation, Lo is simply set equal to the
intended term to maturity. After the disbursement of
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the loan, the intended term to maturity is of no
consequence and is, thus, not stored in the model.
3) The debtor selects an intended payment on the loan and
maximum and minimum limits for the payments on the
loan.
Analogously to the situation in which the debtor
selects fixed payments on the loan, Lo is determined by
applying (F) so that YD(O,Lo) is the intended payment
on the loan. However, for serial loans a similar
formula must be developed. The first year's payments on
the loan are given by
YD(O,Lo)- nRG(0)-n(n-1) RG(0) RR+n RG(0)
2 n(Lo cp) n n(Lo c~)
which, solved for Lo yields
RG(0)-RG(0)R~ 2n
L~ YD(O,Lo)-RG(0)RR-W
After the disbursement of the loan, the model
disregards the intended payment on the loan, which thus
is not stored in the model.
Other situations can be imagined, among which that the debtor
may want a loan with fixed payments and an intended term to
maturity of, e.g., 25 years. This would imply that YDJ°"=YDJi"
has to be determined so that LM=25. However, to solve this
problem a further iterative routine is required, which would
complicate the model to an extent not commensurate with the
possible gain.
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All inputs to the model (both the inner and the outer model)
are entered in step A. These inputs are:
The bond side
Number of financial instrument applied (and thus the
5 frequency of interest rate adjustments), coupon
interest rate on each financial instrument applied,
number and date of settlements each year and price of
each of the financial instruments applied.
The loan side
10 Date of disbursement, principal, principle of
redemption, number and date of payments on the loan,
maximum and minimum limits for the payments on the
loan, maximum and minimum limits for the term to
maturity on the loan, and, possibly, an intended
15 payment on the loan or term to maturity.
In addition, values of inc must also be entered. In each
iteration routine, two values of each function are calculated
- a function value of x and a function value of x+inc - x
denoting the variable subject to the iteration routine, e.g.,
20 the term to maturity. Hence, in the model the calculation of
the adjustment of x is based on two observations. inc is thus
a set of parameters indicating an increment in the iterative
routine. Unless otherwise stated all elements in inc have the
value 0,00001.
25 Finally, also a must be assigned a value. a is a set of
accuracy parameters indicating a maximum permissible
deviation from the conditions allowed.
When entering the inputs, it is tested that
YD'"°x>YD'°inZO and Lmax)Lmin)0
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Since the loan is to mature on a settlement date, it is
required that the entered values of L'°°" and L'"'n comply with
a
settlement date. For instance, if the loan is disbursed at 30
June,. the model requires that L'"°" and La'ln are indicated by an
integer number of years +1/s.
Step B - Adjust LM, Determine M
In general, it is necessary to adjust the term to maturity in
the refinancing period preceding the maturity of the loan
(J=M). This is due to fact that the bonds financing LAIR
mature at the end of the year. Hence, it is suitable that the
loan also matures at the end of the year, amongst other
considerations because of the interest rate on the loan.
Thus, the term to maturity in the refinancing period
preceding the maturity of the loan (LM) is either prolonged
or shortened to the nearest settlement date.
If possible, the term to maturity must be adjusted in
compliance with YDM " and YDM'". However, it cannot be
precluded that it may be necessary to violate one of the
limits, in particular if the limits are identical. If
violating one of the limits is necessary violating the
minimum limit, and thus prolonging the term to maturity, is
most suitable. Therefore, adjusting the term to maturity
proceeds in three steps:
1) LM is prolonged till the nearest settlement date if
YD (M, LM) >YDMin
LM indicating the prolonged term to maturity. Otherwise
move on to 2.
2) LM is shortened till the nearest settlement date if
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YD (M, ~) <YDM°"
_Lr, indicating the shortened term to maturity. Otherwise
move on to 3.
3) The term to maturity is prolonged till the nearest
settlement date and YDMi°is suspended.
Step 3 ensures that a solution will always be found, thus,
the calculations will not continue infinitely. The steps
imply that for J=M two different terms to maturity, L,M and I,M
respectively, must be input into the inner model.
In step B the value of M is also determined. Depending on the
term to maturity, the model simply determines M as the round
down value of the remaining term to maturity divided by the
duration of the refinancing periods. This is particularly
important when the model arrives at step B from step E with a
new term to maturity.
Finally, also L~'"'" and LMIN are input into the inner model to
test if these limits are binding for the payment on the loan.
Step C - Calculate adjustment of I~
Based on the terms to maturity input the model has calculated
the payments on the loan. Step C calculates an adjustment aL,,
of LJ based on the ratio between the payments on the loan
(for a full payment period) on the one hand, and the limits
for the payments on the loan on the other hand. Depending on
the time of the calculations, the first payment on the loan
may be fractioned, in which case the first payment is
calculated proportionally. Thus, the model allows the first
payment after the disbursement of the loan to violate the
band.
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In step B, three and four terms to maturity were determined
and input into the inner model. Thus, the inner model has
.. returned with 3 or 4 different payments on the loan i.e.
YD ( J , Lt's" ) and YD ( J , LMIN ) f o r a 11 J
YD(J,LJ) for all J<M and
YD (M, LM) and YD (M, LM) for J=M
If either
YD ( J . L~"' ) zYDM " Or
YD ( J , L,MIN ) cYDmin
L''""' respectively LM=N is binding, thus
OL,,= 0
i.e. the calculated adjustment to LJ is 0 (zero). On the
other hand, if
YD ( J , L~xN ) >YDMi° and YD ( J , L~"" ) <YDM "
the adjustment of the term to maturity is determined by the
limits for the payments on the loan.
IfIf J<M a function F is first defined as the deviation of the
payments on the loan from the maximum and minimum limits. The
function is given by
1) For YD(J.LJ)>YDJ°"
(i) F(LJ)=YD(J,L,,)-YDJ°"
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2 ) FOr yDJinsyD (J, LJ) __<YDJaX
(ii) F(LJ)=0
3 ) and for YD(J.LJ) <YDJi°
(iii) F(LJ)=YD(J,LJ)-YDJin
For a serial loan, YD(J.L,,) is defined as the payments on the
loan the first year of the refinancing period. The function
may appear as shown in Fig. 13, in which the lines (i), (ii)
and (iii) refer to the corresponding formulae derived above
(However the graph F is conventionalized, as a linear
relation between YD(J,LJ) and LJ will only appear as a special
case).
At a first glance, it may seem more relevant to define F as
indicated by line 20 in Fig. 14. However, this would imply
that the model loses information on how large adjustment of
LJ required to exactly match YDJ°" or YDJln as prescribed by the
fundamental pattern of shifts. If an F-function according to
Fig. 14 is applied, it will only be possible for the model to
calculate an adjustment of LJ, which results in the payment
on the loan to be within the band, but not necessarily an
adjustment, which results in the payment on the loan being
equal to the relevant limit. Hence, a method corresponding to
the method illustrated in Fig. 14 can not be applied in the
present invention.
Whether the adjustment of the term to maturity is calculated
according to (i), (ii) or (iii) is determined the first time
the calculations are performed in step C for each value of J.
Subsequently, this is not altered. If the model were to
decide, whether (i), (ii) or (iii) were relevant each time
the calculations are performed, there would be a risk that
the model shifts from, e.g., (i) to (iii), thus the model
would aim at the wrong limit when calculating the term to
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maturity. From starting off with too high payments on the
loan, the model would in this case end with payments on the
loan equal to the minimum limit, which is, of course, not
acceptable.
5 Based on the value of F(LJ) an adjustment of LJ is to be
calculated. The adjustment is calculated applying the Gauss-
Newton algorithm, which however is slightly modified.
If F (LJ) =0, YD (J, L,,) is within the band and thus LJ is not to
be adjusted i.e.
10 OLJ=0
If F(LJ)#0, the adjustment is calculated applying
DL,,=C [ DTD ] lDTgJo j ~ 1
fOr jv= [DiagJaTJa] ~~5
D= [ JaJn ] O [ j vJ V ] o_~
g= [JaF (LJ) ] Oj°'1
where Ja= [ F ( LJ ) -F ( LJ+inc ) ] oinc°-1
and where
D''D is the Hesse matrix
g is the gradient
J~ is the Jacoby matrix
is the diagonal elements
inc is a parameter with a standard value of 0,0001.
The influence of inc is elaborated in the following. At this
point it should only be mentioned that the idea of inc is to
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have two observations of F(L,,) and F(LJ+inc) from which to
calculate OL,, .
AoB°-lis the Schur product of two matrices meaning that the
elements of the matrix are divided one by one.
In appendix 1 the elements of the matrix are written in full.
However, in this case the problem is one-dimensional, which
is the reason why the Gauss-Newton algorithm may be reduced,
as D, J and j~ are all of the dimension 1x1. Thus, it applies
j "=JA
as j"= [DiagJTJ] °-5= [J~] °-5=J . If j"=J is placed in the
expression
for D it applies
D. JTJ _ JZ _ 1
wJ v w
At the same time, the expression for g is reduced to
JTF(L )
9' _ J J _ F(LJ)
v
If the reduced expressions are introduced in OL," the follow-
ing applies
DL''. [D TD]-1D T3
a
Jv
~LJ - F(L~) inc
F(L~)-F(L~inc)
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The method may be illustrated graphically. In Fig. 15 the
graph of F is indicated by line 24, shown for YD(J,LJ)>YDJ°"
i.e. (i) applies.
The algorithm approximates the value of LJ fulfilling the
condition YD~"'=YD(J,L,,) if the payments on the loan start off
exceeding YDJ°". Correspondingly, the algorithm will
approximate the term to maturity fulfilling YD(J,LJ)=YDJi° if
the payments on the loan start off below YDJ'n.
The basic idea of the algorithm is to use the secant 26
through two points 30 on the graph 24 of F for (LJ,F(LJ)) and
(LJ+inc,F(L,,+inc)), respectively. When the secant has been
determined, the intersection 32 of the secant with the x-axis
is calculated. The intersection 32 is the next guess as to
the interest rate in the iterative routine. If the graph
1S F(LJ) is strictly declining, the algorithm will always reach
a solution. This will be the case, since a prolongation of
the term to maturity always yields lower payments on the loan
and vice versa.
Fig. 15 shows that inc should not be regarded as an accuracy
parameter. inc determines the step size in the iteration
routine.
In this connection it is not necessary to apply the matrix
apparatus, but the same method is to be applied later on.
Therefore, the formula apparatus is now introduced.
For J=M the F-function is to be defined according to the
three steps in the adjustment procedure in step B.
If YD (M, LM) >YDMl", the term to maturity is prolonged to Lf, at
first hand. However, it can not be excluded that the term to
maturity has to be prolonged even more, corresponding to
YD(M,LM)>YDM°", due to an upward shift in interest rates.
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Hence, it is necessary to calculate an adjustment relative to
the maximum limit. F is therfore defined by
( i ) F (L,,,) =YD (M, LM) -YDM " for YD (M, LM) >YDM°" and
(ii} F(LM)=0 otherwise.
If YD (M, LM) <YDMl" and at the same time YD (M, L~.,) <YDM ", the term
to maturity is immediately shortened to the nearest
settlement date. Analogously, the possibility that the term
to maturity must be further shortened can not be excluded.
Thus, an adjustment relative to the minimum limit is
calculated. F is defined by
( i } F (L,) =YD (M, Ice} -YDMin for YD (M, ~) >YDMl" and
(ii) F(LT)=0 otherwise.
Finally, if YD (M, Lr,) <YDMl° and at the same time YD (M, ~)
>YDM°"
the term to maturity is prolonged to Lr,. Thereby, the
payments on the loan fall below the minimum limit, which just
has to be accepted since, at the term to maturity L," the
payments on the loan exceed the maximum limit, which must be
regarded even worse, and a term to maturity between ~ and Lr,
does not comply with the condition that the loan must mature
at a settlement date.
Step D - Has F(LJ) converged ?
In step D it is tested whether F(LJ) has converged so that
the payments on the loan are within the band defined by the
maximum and minimum limits. If not, the model applies the
calculated adjustment.
Whether F(LJ) has converged is determined by testing the
mathematical convergence of the adjustment. This is done by
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evaluating the size of the adjustment of the term to maturity
which was determined in step C against the value of F(L,,). If
the adjustment is very small, cf. the conditions mentioned
below, there is no reason to continue the iteration routine:
The model can not get any closer.
Mathematical convergence is defined by the conditions
(i) I~L~I ~ E
E I LJI
( ii ) ~ ~F(L') 2 F(L'~'~LJ) 2l I < E
F(LJ) a
where E is a very small figure. If just one of the conditions
are fulfilled, LJ is accepted as the term to maturity at the
Jth refinancing. The convergence may also be tested by
applying
(iii) IF(LJ)I<E
where E is fixed at 0,00001. Thus, a maximum deviation from
the maximum or minimum limit of DKK 0,00001 is accepted.
The definition of F(L,,) in step C implies that the
convergence conditions are identical irrespective of the term
to maturity for which the value of F is calculated.
Step E - Apply the adjustment of LJ
YD(J,LJ) was rejected in step D.
Therefore, L,, is to be adjusted and subsequently, in the
inner model the payments on the loan for LJ+pLJ are
calculated. However, at first step B is applied so that the
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term to maturity is rounded for J=M.
Step F - IS I~<L~n ?
In step F, it is checked that the full term to maturity is
not shorter than the minimum limit. If
LM<Lm"'
the model moves on to step G in which the payments on the
loan are adjusted according to (B). Otherwise, the model
proceeds in step F.
It should be noted that steps F and I are the only steps
which are not applied for every value of J.
Step G - Calculate the payments for LJ=Len
If LM<L'°"' the payments on the loan are adjusted according to
(B) which was formulated as
YD(J,.)=YD(J,LJ) for 05J<_J "
(B)
YD (J, . )=YD (J,L"'n) for J' ' SJSM
where J " denotes the minimum value of J for which
YD ( J , LJ ) >YD ( J , L'°in ) .
YD(J,LJ) has already been calculated for J=0,1,2,..., M, whereas
YD(J,Lmln) is to be calculated in the inner model. However,
the transition from step G to the inner model is not simple.
At first, (B) is calculated for J=0. Depending on whether
YD(O, . )=YD(O,Lo) Or YD(O, . )=YD(O,Lmin)
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the remaining debt at the end of the period may be
determined. Based on the remaining debt, YD(1,L'""') is
- calculated and YD(1,.) can be determined and thus the
remaining debt at the end of the period etc. Hence, the link
- 5 from step G to the inner model is successive.
However, it will never be necessary to recalculate YD(J,LJ)
in a corresponding manner. From the point where YD(Lmin)
determines the payments on the loan, this will be the case
until the maturity.
Step H - Adjust the payments according to minimum condition
Having determined the value of YD(J,.) for each J, it only
remains to compound a payment profile. Information on
funding, interest rate on the loan etc. are compounded in
compliance with the payments on the loan determined, of (B).
When the payment profile in full has been determined, the
model proceeds in step L.
Step I - Is L~>L"°" ?
Fully analogously to step F, it is to be checked that LM does
2D not exceed L'~°". If LM>Lm°", the payments on the loan are
adjusted in step J. Otherwise, the calculations are completed
and the model can proceed to step L.
Step J - Calculate the payments for LJ=L'°"
This step is fully analogous to step G except from the fact
that LM>Lm°". Thus, the payments on the loan are adjusted
according to (A).
YD(J,.)=YD(J,L,,) for OsJsJ'
(A)
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YD(J, . )=YD (J,L°'°") for J'sJ<_M
where J' denotes the minimum value of J for which
YD(J,LJ)<YD(J,Lm°"). Just as in step G the link from step J to
the inner model is successive.
Step K - Adjust the payments according to maximum condition
Analogously to step H, a payment profile based on the
payments on the loan for J=0,1,2,...,M determined in step J is
to be compounded. Subsequently, the model continues in step
L.
Step L - The calculations are complete. The result may be
applied
A compounded payment profile has been determined, the term to
maturity LM has been calculated, and in the inner model the
volumes of the financial instruments applied and the interest
rate on the loan have been determined. Thus, the computation
of the loan is complete and the result may be applied.
2.2.2 The inner model of type F
The model solves the problem in 10 steps as shown in Fig. 26.
In the following, the model is discussed in detail.
The model applies an iterative routine. On the basis of a
start value of the interest rate, the funding is determined
in steps 2 to 4. In step 5, an adjustment of the interest
rate is determined. If the proceeds criterion is not
fulfilled in step 6, the interest rate will be adjusted, and
step 2 is repeated. If the proceeds criterion is fulfilled,
the model will make a final test that all funding principals
are positive. If this condition is fulfilled also, the
results of the model may be applied.
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If on the other hand that one or more funding principals are
negative, the model proceeds in the inner model for type F'.
Step 1 - Determine initial interest rate
At first all inputs are entered from the outer model. In the
inner model, LJ is considered an input just like the other
inputs. Thus, every time the inner model is applied, the
value of LJ is fixed.
In step 1, an initial value of the interest rate is
determined. To minimize the number of iterations which are to
be carried out later, it is expedient to set the first
interest rate equal to the yield on the funding instrument
having the longest term to maturity.
On the basis of the number of funding instruments entered and
the remaining term to maturity, a value for m is determined
as
m=max(number of funding instruments initially;
remaining term to maturity rounded up to the next
integer?
so that the model does not issue bonds with maturity after
the maturity of the loan.
Step 2 - Determine payments on the loan
Based on the interest rate, the principal input, and the term
to maturity, the payments on the loan until the next
refinancing may be calculated for both R" and R"+inc.
The model will return with the following information
concerning debtor's payments: interest rate, repayments,
payments on the loan, and the remaining debt profile.
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Step 3 - Determine funding volumes
Provided that the payments on the loan YD(j) (j<m) and the
refinancing amount at the end of the period RG(m) are
determined, the individual volumes of the financial
instruments applied may be found applying the equation
system resulting from the the balance requirements. The
equation system may be written in a matrix form
(G) YD+RD=AxH,
where YD=(YD(1),YD(2),...,YD(m)),
RG=(0, 0,...,RG(m) ) and
H=(H(1),H(2),...,H(m)) are vectors with the
dimension mx1
and where A is defined as a mxm upper triangular matrix:
[1.REC3RN(1) ] RF~RN(2) RECRN(3) . . REC~RN(m)
0 1.RN(2) RN(3) RN(4) . RN(m)
A-
0 0 0 0 0 [1.RN(m) ]
The A-matrix is designed in such a way that the balance
equations appear immediately, if (G) is calculated.
In the A-matrix the number of columns corresponds to the
number of financial instruments applied, whereas the number
of rows corresponds to the number of years in the refinancing
period. If the number of settlements within each year are
increased, the dimension of the matrix will increase by the
same factor.
By this rewriting, the solution as to the m volumes of the
financial instruments applied in matrix form (the top sign T
means that the matrix is transposed) is as follows:
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(H) H= [ATA] -lAT [YD+RG]
In principle, [ATA]-lAT may be replaced by A-1 in (H) , as long
as A is quadratic. Only if more funding instruments are
applied, the rewriting [ATA]-lAT is necessary. Thus, the
rewriting is only a method by which non-quadratic matrices
may be inverted.
The funding for R"+inc is determined by the same method.
Step 4 - Determine the proceeds function
A function F is defined as the difference between the
proceeds required and the market prices of the bond prin-
cipals calculated under step 3.
F(RK) - RG(0) - ~ K(j )H(j ) ,
Hence, the proceeds criterion is fulfilled for F(R")=0. The
values of F(R") and F(R"+inc) are calculated given the volumes
of the financial instruments calculated in step 3.
Step 5 - Calculate adjustment of the interest rate
Then an adjustment DR" of the interest rate guess is
calculated. The adjustment will not be implemented until step
7 and only if the convergence condition in step 6 is not
fulfilled. The adjustment is calculated applying the Gauss-
Newton algorithm described in step C in the outer model. Thus
OR"=C [ DTD ] -lDTg~o j ~'
where j~, D and g are defined as in B. Ja is defined by
J4=[F(R")-F(R"+inc) ]oj~l
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Also in this case, the expression may be reduced which
implies
DR K . F ( R R) inc
F(R x) -F (R R inc)
The solution is illustrated in Fig. 17. Also here, the basic
idea of the algorithm is to use the secant 42 through two
points 44, 46, on the graph for F, (RK,F(RK)) and
(RK+inc, F (RK+inc ) ) respectively. lnThen the secant 42 has been
determined, the intersection 48 with the x-axis is
determined. The intersection 48 is the next guess as to R" in
the iteration routine. If the graph 40 for F(R") is strictly
declining, the algorithm will always reach a solution. This
will be the case, as an increase of the term to maturity
results in decreasing payments on the loan.
The algorithm approximates an adjustment of R" so that
F(R"+~R")=0. F(R") is strictly declining, since an increase in
the interest rate on the loan implies that (strictly) more
bonds are sold, and thus the proceeds from the funding of the
loan are increased (strictly) as mentioned in the
introduction.
Step 6 - Is the proceeds criterion fulfilled?
In this step, it is tested whether the interest rate on the
loan complies with the proceeds criterion.
Once again, this may be determined by the mathematical
convergence of the calculated adjustment. Thus, one of the
two conditions below must be fulfilled.
(i) I~RxI ~ E
E.IR xI
(H) H= [ATA] -lAT [YD+RG]
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(ii) ~ ~F~Rg)2-F~R~*~RR)2~ ~ < E
F(RR)2
where a is a very small figure. It is also possible to test
the proceeds criterion directly. This may be formulated as
- (iii) ~F(R")~<E
Where E is fixed at 0,00001. Thus a maximum deviation from
the proceeds criterion of DKK 0,00001 is accepted.
If (iii) is fulfilled, the requested interest rate on the
loan (R") has been found, and the model can move on to step
8.
If none of the conditions are fulfilled, the computations
proceed in step 7.
Step 7 - Adjust the interest rate
The interest rate rejected in step 6 is now adjusted with the
adjustment factor DR" calculated in step 5. The computations
in the model proceed in step 2, where the payments on the
loan are recalculated applying the adjusted interest rate
( R"+OR" ) .
Step 8 - Are all funding volumes positive?
In certain cases, one or more of the calculated volumes of
the financial instruments applied may be found to be
negative. Negative funding volumes correspond to a debtor
purchasing a number of the bonds applied in the financing of
the loan. In principle, there is nothing which excludes this.
However, a number of considerations, including tax
considerations, indicate that negative funding volumes should
be avoided.
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The negative funding volumes may occur as a result of bond
market prices above par. An obvious way to avoid negative
funding volumes is thus to change funding instruments to be
applied, which in this connection would mean the opening of
bonds with a lower coupon rate and thus lower market price.
Alternatively, the negative funding principals may be avoided
by treating the loan as a LAIR Type F' until the next
interest rate adjustment is performed or even after that.
Thus, if negative funding principals occur, the model
continues in step 9.
If, on the other hand, all funding principals are positive,
the model continues in step 10.
Step 9 - Shift to F'
In step 9 the model transfers computation of the loan to the
inner model for type F+. Information concerning the loan,
i.e. principal, term to maturity, amortization principle etc.
is entered in the F'model from the F model.
Step 10 - Interest rate, funding, and payments on the loan
may be applied!
An interest rate on the loan as well as a number of positive
volumes for the funding instruments fulfilling the given
conditions have been found and the calculations are therfore
transferred for the given term to maturity. Thus, the model
may return to the outer model with information of payments on
the loan, the interest rate on the loan, and the volumes of
the funding instruments applied.
2.3 The inner model for type F'
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First, it should be noted that the link to the outer model is
identical for type F and type F'. In general, the inner model
for type F' may thus be considered exclusively as a
replacement of type F when the volumes of the financial
instruments applied would otherwise be negative. Hence, is is
only to keep a clear overview that types F and F+ are
separated in the model.
If the bond with the longest term to maturity in a LAIR with
100 per cent interest rate adjustment has a price over 100,
the nominal issue is less than the principal of the loan.
This will affect the balance principle in the last year of
the refinancing period when the bonds applied mature at price
100 at the same time as the remaining debt of the loan is to
be refinanced.
If the difference in the nominal bond volume and the
principal of the loan exceeds debtor's repayment in the
period until the next refinancing, the balance principle can
not be fulfilled when the interest rate on the loan is
constant. The problem is solved in that the loan is financed
solely in one bond and that a so-called minimum refinancing
is introduced.
The basic idea in the minimum refinancing is to transfer
payments from the previous years to the last year in the
refinancing period by covering a deficit in the payments on
the loans by a new bond issue with the same term to maturity
at the end of each year so that the balance principle is
complied with. Thereby the volume of bonds is increased and
the surplus the last year is reduced. Normally, the
issue will not only have the same term to maturity, but also
the same bond ID code year after year. Yet, adjustments of
the statutory minimum coupon interest rate and the like may
imply that the coupon interest rate of the issue is changed.
In the following, this possibility is disregarded, however.
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A particular problem is connected to the determination of the
interest rate. As a consequence of the minimum refinancing,
the interest rate is not constant during the whole
refinancing period, but will, on the contrary, vary from one
year to another in the refinancing period.
This means that the limits for the payments on the loan might
be violated in the refinancing period. Typically, however,
the fluctuations in the interest rate will be moderate, since
the minimum refinancing is minimal, as the term suggests.
Thus, the consequences will be negligible in practice. The
present method only ties up the interest rate during the last
year of the refinancing period. For the other years, an
arbitrary determination of the interest rate will only change
the minimum refinancing. However, an unsuitable determination
of the interest rate will result in a substantial change of
the interest rate the last year in order to comply with the
balance principle. An aim is, therefore, to choose a method
for the determination of the interest rate which results in a
stable course. The present method is thus a suitable method
among others. For example the interest rate could be
determined as the bond yield or the like.
Funding in only one bond considerably facilitates the
calculation of the funding, the funding principal for the
bond with the longest term to maturity being unambigously
determined by the proceeds criterion. Thus, the funding may
be determined already in the first step of the model.
When the funding has been determined, the interest rate may
be determined by applying the principle of strict balance.
The interest rate must be determined subject to the condition
that the total payments on the bond side equal to the total
payments on the loan side to the next ordinary refinancing.
However, the remaining debt profile depends on the interest
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rate on the loan, thus, it is necessary to iterate over the
interest rate on the loan.
The inner model for type F' comprises 7 steps as shown in
Fig. 18. In the following, the model will be described in
detail.
In the model, it is necessary to distinguish between ordinary
refinancing and the minimum refinancing. The annual minimum
refinancing imply that the model is to be applied not only at
the refinancing times (e. g. every fifth year for a type F5),
but each year.
Step 9 of the F model
The F+ model can only be initiated from step 9 of the inner
model for type F. Therefore, input has already been entered
into the F model and is transferred to the F' model without
changes.
Step 1 - Calculate the funding volume
The funding is determined at the disbursement of the loan or
immediately after an ordinary refinancing as
H(m) - RG(0) H(J)-0 far j<m
K(m)
m is unchanged in relation to the inner model of the F model.
If the loan is refinanced every fifth year, m has the value 5
irrespective of the issue being in one bond only.
For the other years until the ordinary refinancing the
- minimum refinancing is given by
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M,(m) . Fin(~~) for J'=1,2,...,m
K~, (m)
wherein
Fin (j') generally designates the funding demand
year j'. In this connection Fin(j') is thus
the minimum refinancing.
M~,(m) (Not to be mistaken for M) designates the
marginal funding in the mth financial
instrument in year j', i.e. the part of .
H(m) to be issued at the end of year j'.
K~,(m) is the market price of the mth funding
instrument in year j'.
It should be noted that j gets a slightly different meaning,
as j is only set to zero in connection with ordinary interest
rate adjustments. Thus, funding is issued at times j=1,2..,m
etc, and not only at the time j=0 as in the F model.
The minimum refinancing is determined as the difference
between the annual payments on the loan side and on the bond
side.
Fin(j')=RN(m)J(O,m)-YD(j') for lsj'<_m-1
wherein
H(O,m) designates the already issued funding, i.e.
the volume of the mth funding instrument
before the minimum refinancing.
Step 2 - Determine initial interest rate
The iterative routine is initiated with an initial value for
the interest rate on the loan. The initial value is
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RK . H(m)-RG(j') a H(m)RN(m)
'~ (m-j') RG(j ~) RG( j')
Step 3 - Determine payments on the loan
Based on the resulting interest rate on the loan, the
- payments on the loan, and in this context, the remaining debt
profile are calculated.
Step 4 - Calculate adjustment of the interest rate
The adjustment of the interest rate is calculated according
to the same principles as in the steps 4 and 5 of the inner
model for type F. First, a function measuring the difference
between payments on the loan side and on the bond side is
defined.
F(RR)-~ YD(j).RG(m)-[H(m)..(m-j')H{m)RN(m) ]
After this, the adjustment may be calculated by use of the
reduced version of the Gauss-Newton algorithm.
ERR- F(R~) F{R k)-F(RRinc)
It should be noted that F(RK) is strictly increasing,
ensuring that the algorithm reaches a solution.
Step 5 - Is the balance requirement fulfilled?
Step 5 determines whether the routine is to continue or
whether an interest rate has been found which fulfil the
balance requirement. As in step 6 of the F model, the
question may be determined by the mathematical convergence of
the iteration, but it is more obvious to evaluate the actual
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balance condition. This means that the routine stops and the
model continues in step 7 if the condition
YD(J)~RG(m)-[H(m).(m-j')H(m)RN(m) ) «
._p,
has been fulfilled for E=0.00001. If the condition has not
been fulfilled, the model continues in step 6 instead.
Step 6 - Adjust the interest rate
The interest rate did not fulfil the balance requirement and
must therefore be adjusted with the adjustment DR". Then step
3 is repeated.
Step 7 - The model is complete. The result may be applied
If the balance condition has been fulfilled, the funding and
the interest rate fulfil all conditions and may, together
with the payments on the loan, be used in the outer model.
Thus, the inner model returns to the outer model with
information on volumes of the financial instruments applied,
interest rate and payments on the loan.
3. Type P
3.1 T'he general problem
In LAIR type P, a part of debtor's opening remaining debt is
refinanced each year. The debtor chooses an intended annual
refinancing percentage which at the same time determines the
number of financial instruments to be issued and thus the
duration of the funding period.
If debtor chooses, e.g., an annual refinancing percentage of
10 per cent, it will take 1/10 per cent = 20 years before the
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loan is fully refinanced. The funding period is therefore 10
years and the number of funding instruments 10. At every
refinancing, a funding instrument matures. In order to keep
the refinancing proportion at the intended level, a new
funding instrument with a term to maturity of mo years is,
therefore, issued. This continues until the number of funding
- instruments is gradually reduced when the loan approaches its
maturity.
The relation between the number of funding instruments and
the proportion of the remaining debt that is refinanced each
year implies that the latter can be expressed as 1/mfl. When
debtor chooses the value mo, the part of remaining debt that
is refinanced is chosen at the same time.
The gradual adjustment of the interest rate on the loan
implies that the effects of an increase or decrease in
interest rates can be noted in the payment profile for a
period of mo years. Thus, the need for continous adjustments
of the payments on the loan is intensified compared to type
F.
A payment profile for a LAIR type P at increasing interest
rates may be as shown in Figs. 19 to 22. Fig. 19 shows a
payment profile 52 for an annuity loan at increasing interest
rates where the payments on the loan 52 remain within the
given limits, whereas Fig. 20 shows a payment profile 54 at
increasing interest rates where the payments on the loan 54
exceed the maximum limit.
Analogeously, Fig. 21 shows a payment profile 56 for a serial
loan at increasing interest rates where the payments on the
loan remain within the given limits, whereas Fig. 20 shows a
- 30 payment profile 58 at increasing interest rates where the
payments on the loan 58 exceed the maximum limit.
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If the payments on the loan are close to the maximum limit,
already when the rise in interest rates occurs, the model
will soon be in a situation where the payments on the loan
exceed YD~" in the remaining part of the funding period.
However, L,, is not adjusted for this reason, as this would
imply that YD(J,LJ) would be too low. On the contrary it must
be accepted here and now that the future payments on the loan
- also within the funding period - exceed YD'J" and are not
adjusted via the term to maturity until the model proceeds to
later points in time. In other words, the conditions
stipulated for the payments on the loan are only to be tested
for 1 year at a time. Thus, the model will accept a payment
profile 54, 58 as shown in Figs. 20 and 22, respectively,
well aware that the term to maturity must be adjusted for the
next J.
Thus, J=0,1,2,...,M should not be interpreted as funding
periods, but as 1 year refinancing periods. Thus a condition
stipulated on, for instance, YD(J,LJ) will only be binding in
one year.
The intertemporal conditions may be formulated as in the
model for adjustable term to maturity for type F. As a matter
of form the conditions are repeated below.
1) The term to maturity must not be longer than the
maximum limit, that is,
~SL~x
2) The term to maturity must not be shorter than the
minimum limit
3 0 LMZLmin
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3 ) If LM<L'°°", the payment on the loan is determined
observing the condition
YD (J, LJ) __<YDJ°"
and otherwise
YD(J,.)=YD(J,LJ) for 0<_JsJ'
(A)
YD(J, . )=YD (J,Lm°") for J'sJsM
where J' denotes the minimum value of J for which
YD(J,LJ)<YD(J,L'°°")
4) If LM>Lm'n, it is required that the payment on the loan
is above the minimum limit, that is
YD ( J , LJ ) 2YDJin
and if the limit for the term to maturity is violated
YD(J,.)=YD(J,LJ) for OsJsJ "
(B)
YD(J, . )=YD (J,L"'ln) for J' ' SJSM
where J " denotes the minimum value of J for which
YD ( J , L,t ) >YD ( J , L'°in )
The model observes these conditions by determining a sequence
of terms to maturity Lo, L1, ..., LM. This is done in an outer
model.
Like in the model for type F, it is required that the
- principle of strict balance and the proceeds criterion are
observed in each refinancing period. In addition, an explicit
condition is stipulated on the relation between the interest
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rate on the loan and the yield to maturity of the portfolio
of the bonds applied. These conditions are handled in the
inner model.
However, the inner model may be solved both applying an
iterative routine and analytically. The iterative solution
has the advantage of very stable funding profiles. The
analytical solution has the advantages of a more simple
structure and shorter computation times. Both an iterative
inner model as well as an analytical inner model are
described in the following.
The annual refinancing implies that the balance principle
plays a slightly different role. At the end of each year, the
total payments on the debtor side and on the bond side are
known. By determining the interest rate adjustment amount
residually, the principle of strict balance has been
fulfilled by definition. It is, however, not necessarily so
that the intended refinancing percentage corresponds to the
actual one.
Therefore, the problem to be solved in the model is to adjust
the funding so that the intended and the actual proportions
of the remaining debt that are refinanced correspond, at the
same time as the proceeds criterion is fulfilled and at the
same time as all the volumes of the financial instruments
applied are positive (z0). Upon the disbursement of the loan
the problem is described by the equations
Year 1 YD(1).REGDR~ )-H(1).REG~RN(j)H(j)
Year 2 YD(2)+RG(1) _g(2).REG~RN(j)H(j)
mp ~-2
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Year m YD(m)~RG(m1)_H(m).REGRN(J)Ii(j)
wherein
REG° is a regulation factor for the refinancing
percentage in the first year. REG°can assume the
values {'~,'h,3~a, 1} which will be elaborated on in
the following.
In the following, these equations are designated as the
balance requirements.
20 If year 1 is considered alone, the problem seems easy to
solve. The task is merely to determine the volumes of the
financial instruments applied which lead to the intended
payments and the intended proceeds. In principle, the
distribution of the volumes of the financial instruments
applied is of minor importance, so there is an infinite
number of solutions to the problem.
The problems start to arise when the interest rate is
adjusted by the end of the year. If an arbitrary distribution
of the funding was chosen at the disbursement of the loan, it
is not certain that it is possible to hit the desired
refinancing profile again, if at the same time all the
volumes of the financial instruments applied must be positive
(>_0). If most of the funding of the loan was placed in H(2)
at the disbursement of the loan, the right side of the
equation will be large. Thereby the refinancing amount will
be large too, if the balance requirement is to be fulfilled.
Thus, it is not possible to keep the part of the remaining
debt that is refinanced down at the intended level unless
bonds are redeemed.
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If only a very small part of the funding was placed in H(2),
it may, on the other hand, be difficult to reach the intended
refinancing percentage. This may, however, be partly saved by
placing a large part of the proceeds from the refinancing in
the bond now having a term to maturity of 1 year. The result,
however, will be an unstable funding profile.
If the balance requirements with the intended refinancing
percentage every year are to be fulfilled, it is necessary to
formulate a dynamic strategy for the placing of the funding
also taking into consideration the long term perspective.
An arbitrary strategy for the funding will also raise
another, just as essential problem. At each refinancing, the
model cannot merely place the entire funding in bonds
maturing in a single year. As it will be remembered, the
actual volume of bond redemptions corresponds to the actual
refinancing amount (apart from premium or discount), which on
its side is determined by the refinancing percentage and the
remaining debt on the loan. When the newly issued bonds are
redeemed, the remaining debt is smaller, but the refinancing
amount is of fairly the same size as currently. Thus, the
future refinancing percentage will be higher than intended.
Therefore, the model must continuously have the possibility
of placing the funding in bonds with different terms to
maturity.
In the iterative inner model as wells as in the analytical
model, the dynamic strategy implies that the funding is
adjusted each year in such a way that the payments of the
bonds correspond to a decreasing proportion of the payments
from the debtor side as time is progressing. At each
refinancing, issue is further performed in each financial
instrument applied - thus, gradually there will be increasing
accordance between the bond side and the debtor side given
the part of the remaining debt that is refinanced. At the
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disbursement of the loan, the problem being solved via the
model can, therefore, be formulated as
Year 1 YD(1).REG°R~ ).H(1).REG~RN(j)ht(j)
Year 2 F(YD(2).RG(1) ).H(2)..~RN(j)H(j)
-"0 ~-2
...
Year m F(YD(m).RG(m-1) }_~1~RN(m)]H(m)
ma
wherein F is a declining function of j.
In principle, the expressions may be generalized so that the
bonds maturing in year j may have different coupon interest
rates. However, this is excluded in order not to
unnecessarily complicate the notation.
In both models the funding profile follows a so-called trend
function which is estimated based on the intended refinancing
percentage and the funding already issued H(O,j).
The trend function is adjusted in such a way that to the
extent possible, the actual refinancing percentage correspond
to the intended refinancing percentage while at the same time
the proceeds criterion is fulfilled and strictly negative
volumes of the financial instruments applied are avoided. The
adjustment is applied either by iteration or analytically,
thus, at this stage the two inner models differ.
When these conditions are met, the interest rate on the loan
is subject to iteration in an outer routine until the
interest rate equals the yield to maturity of the portfolio
of bonds applied for the financing of the loan. An adjustment
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is made for differences in the number of settlements on the
debtor side and the bond side. Typically, the number of
settlements on the debtor side exceeds the number of
settlements on the bond side, which means that the debtor
pays in advance compared to the one annual settlement on the
debtor side. Thus, in order for the total payments to be
equal the interest rate on the loan must be a little higher
than the yield to maturity of the bonds applied.
Hence, the inner model is solved in a 2-step-procedure and
not simultaneously as for the F model.
There is a special problem attached to the date of
disbursement of the loan. If the loan is disbursed in
December, the first refinancing will be performed well over
one year later. The rule set out above can thus be applied.
On all other dates during the year, the refinancing
percentage at the first refinancing is, however, written down
in relation to the quarter in which the loan has been
disbursed. This is indicated by the regulation factor REG°
defined above. This means that the funding period is
prolonged by 1 year, and the number of funding instruments is
increased by 1. At the disbursement of the loan the following
applies
m=mo+ 1
Already at the first refinancing, the general rule with mo
funding instruments is followed. Thus, no new instrument is
issued at the first refinancing. In the model, there is a
special procedure in connection with the extra funding
instrument. The procedure is indicated by a variable, TILT,
being given the value 1.
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In general, the notation is the same as in the F model.
However, the continuos issue of bonds makes it necessary to
differentiate between three definitions of funding volumes.
M(j) designates the marginal funding, i.e. the volume
of bonds issued in the jth year when the current
issuing of bonds is performed.
H(O,j) are the bonds already issued before the present
issuing of bonds.
H(j) is the total amount of bonds in the jth year.
From the definitions the following relation appears
M(j)+H(O,j)=H(j)
At the disbursement of the loan, it is obvious that H(O,j)=0,
and thus it applies that M(j)=H(j). In the notation, it is
therefore not necessary to differentiate between the disbur-
sement of the loan and the interest rate adjustment in this
respect.
Correspondingly, the funding demand at the disbursement of
the loan as well as at the refinancing will be indicated by
Fin(j). Thus, Fin(j) may either by the principal of the loan
or the current amount of refinancing. Also in this respect,
it is unnecessary to differentiate between disbursement and
refinancing.
As a consequence of the fact that no differentiation is made
between disbursement of the loan and refinancing, REG and
REG° will be a part of expressions which also apply to
refinancing. Here, REG=1 and REG°=1 in accordance with the
definition.
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3.2 Type P - method
A sequence of terms to maturity is to be determined observing
that in each refinancing period, the payments on the loan are
within the band defined by the maximum and minimum limits for
the payments on the loan, stipulated that the total term to
maturity is within the band defined by the maximum and
minimum limits for the term to maturity, and stipulated that
the payments on the loan, in each refinancing period, fulfil:
actual refinancing percentage=intended refinancing
percentage
and
the proceeds criterion
and
adjusted interest rate on the loan=yield to maturity of
the portfolio of financial instruments applied.
3.2.1 The outer model of type P
Like the type F model, this model is divided into an outer
and an inner model. tnThile the differences in the inner model
for the two types are substantial, the outer models differ
less. The outer model is illustrated in Fig. 23. The general
structure in the outer model is not altered. Furthermore,
several steps are identical. To facilitate the overview of
the model, the steps are repeated here, but with an
indication that the content is the same as in the type F-
model.
The procedure applied when the loan matures is altered.
A substantial part of the funding is issued with a term to
maturity exceeding the refinancing period. When the loan
approaches maturity, the term to maturity must, therefore,
not be shortened without due consideration - this would imply
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the risk of the loan maturing before one or more of the
financial instruments already issued. Thus, in the model when
computations are carried out for the last funding period,
i.e., for JAM-mo, the minimum limit for the payments on the
loan is to be suspended so that the term to maturity will not
be shortened.
At the same time, the maturity of LAIR type P must coincide
with a bond settlement. However, it is not sufficient to take
this into account in the last refinancing period for J=M. As
soon as M is within the funding period, debtor payments in
the last year before maturity are funded in the model. Hence,
it is necessary to adjust the term to maturity at an earlier
point in time. Therefore a parameter termed "closing time" is
implemented designating when the model is to start
adjustuding the term to maturity, i.e., for which J L,, is to
be adjusted. The adjustment of the term to maturity is
carried out like in step B in the type F model, but, as will
be understood, not only for J=M as in the type F model.
Closing time must be assigned a value based on the type of
loan. Thus, it would not make sense to assign closing
time>ma. Closing time=mo will lead to the most suitable
maturity of the loan, but on the other hand, implies that the
payment on the loan deviates from the intended pattern of
shifts in the payments for an unnecessary time period. Thus,
assigning the value of closing time is a trade off between
considerations relating to the maturity of the loan and
considerations relating to the payment profile until then.
As an alternative to implementing closing time, one could
choose to fund the refinancing in the last mo years in the
bond with the second longest term to maturity. Thereby, the
full remaining debt on the loan would be refinanced one year
before the maturity of the loan. Then, in the last year
before maturity, the term to maturity may be prolonged to the
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nearest settlement date without any problems in the last year
before maturity.
Step A - Determine initial LJ
As in the type F model, the term to maturity is assigned in
step A, an initial value which is to be considered a first
guess in the iterative routine.
If J>0, the initial value of L,, is relatively simply
determined as LJ_1.
If J=0 and the calculations coincide with a refinancing of
the loan, Lo is assigned an initial value based on the
sequence of terms to maturity as calculated at the preceding
refinancing of the loan. The point in time defined by J=0
corresponds to J=1 at the preceding refinancing of the loan
since J is set to zero. Thus, Lo is to be assigned an initial
value equal to L1 at the preceding refinancing of the loan in
the current refinancing.
If J=0 and the calculations coincide whit the disbursement of
the loan, the term to maturity is determined based on input
from the debtor which may comprise:
1) The debtor selects an annuity loan with fixed payments,
thus YDJ°"=YDJi°, and an initial term to maturity is to
be determined. The initial guess as to Lo may be
determined applying the annuity formula
YD(O,Lo). RG(0)Rg
1- ( 1~R R) -~~-~~
which, solved for Lo. yields
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1 1_ RG(0)R~
~(o,Lo)
( F ) Lo._ ~~P
In ( 1aR x)
where YD ( 0 , Lo ) =YDJax=YDJin
and where
cp is the number of years since the disbursement of
the loan at the point in time of the
calculations. Thus, Lo-c~ is the remaining term to
maturity to be applied in the annuity formula.
However, R" has not been determined. To reduce the
number of iterations, R" is set equal to a weighted
average of the yield to maturity of the financial
instruments applied.
1 t r(0, t)
Rx m m
t
~.1 m
where
t is the term to maturity for each financial
ins trument . t= { 1, ... , m?
r(O,t) is the yield to maturity of the bond which
matures at t.
2) The debtor selects an intended term to maturity and
maximum and minimum limits for the payments on the
loan.
In this situation, Lo is simply set equal to the
intended term to maturity. After the disbursement of
the loan, the intended term to maturity is of no
consequence and is, thus, not stored in the model.
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3) The debtor selects an intended payment on the loan and
maximum and minimum limits for the payments on the
loan.
Lo is determined analogously to the situation in which
the debtor selects fixed payments on the loan, that is
by applying (F) where YD(O,Lo) is the intended payment
on the loan. However, a similar formula must be derived
for serial loans. The payments on the loan for the
first year are given by
YD(O,L°). nRG(0)_n(n-1) RG(0) Rein RG(0)
2 n ( Lo cp ) n n ( Lo cp )
which, solved for Lo yields
RG(0)-RG(0)RR 2n _
L° YD(O,L°)-RG(0)RK ~
After the disbursement of the loan, the model suspends
the intended payment on the loan, which, thus, is not
stored in the model.
All inputs to the model (both the inner and the outer model)
are entered in step A. These inputs are:
Inputs related to the funding
Number of financial instrument applied (and thus the
refinancing percentage, coupon interest rate on each
financial instrument applied, number and date of
settlements each year and price of each of the
financial instruments applied.
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Inputs related to the debtor
Date of disbursement, principal, principle of
redemption, number and date of payments on the loan
each year, maximum and minimum limits for the payments
on the loan, maximum and minimum limits for the term to
maturity on the loan, and, optionally, a preferred
payment on the loan or term to maturity.
In addition, also the values of inc and a are input. Finally,
the value of closing time is to be input which may depend on
the refinancing percentage of the loan. Thus, closing time is
a set of parameters comprising one element for each possible
refinancing percentage (at present 9 elements)
When inputting the inputs, it is tested that
YD'J"°xZYDJinZO arid Lm°">_jmin>O
Since the loan is required to mature on a settlement date, it
is required that the values of L°'~" and Lmin entered comply
with a settlement date. For instance, if the loan is
disbursed on 30 June, it is required that Ln""' and Lmin are
indicated by an whole (integer) number of years +~.
Step B - Adjust L~, Determine M
As in the type F model, the term to maturity is to be
adjusted when the loan approaches maturity. However, it is
not sufficient to adjust the term to maturity just in the
last refinancing period before maturity. On the contrary, the
. 25 term to maturity must be adjusted already when the funding
period - closing time stretches to the expected maturity of
the loan, cf, section 3.1. On the other hand, the adjustment
- is facilitated by the fact that only a prolongation of the
term to maturity to LM is to be considered, whereas the term
to maturity will not be shortened cf. section 3.1. Thus, the
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adjustment needs not follow a step-by-step procedure as in
the type F model.
In mathematical terms, the adjustment of the term to maturity
is given by
For J>_M-closing time, the term to maturity is prolonged
to LJ, i.e., the nearest bond settlement date,
and otherwise not.
Inputs to the inner model is thus
LJ for J<M-closing time or
LJ for JzM-closing time and
L'""' and LMIN
Additionally, in step B the value of M is determined based on
the term to maturity.
Step C - Calculate the adjustment of LJ
Based on the terms to maturity input, the model has
calculated the payments on the loan. Step C calculates an
adjustment OLJ of LJ based on the ratio between the payments
on the loan (for a full payment period) on the one hand and
the limits for the payments on the loan on the other hand.
Depending on the time of the calculations, the first payment
on the loan may be fractioned, in which case the first
payment is calculated proportionally. Thus, the model allows
the first payment after the disbursement of the loan to
violate the band.
As in the type F model, the adjustment of the term to
maturity is set to zero if either L''"~' or LMIN is binding.
Thus, if either
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YD(J.L''""') >_YD~" Or
YD { J , LM=N ) 5 YDMl° then
~L,,=0
If neither L"~" nor LMIN is binding, the model calculates an
adjustment based on the limits for the payments on the loan
by applying a function F(LJ), which. like in the type F
model, measures the deviaion of the payments from the band.
However, there is a number of points in time for the
calculation where the minimum limit (YDJin) is to be
suspended.
If the funding period comprises the maturity of the loan,
hence JAM-mo, the term to maturity must not be shortened. To
prevent the model from continuing the routine endlessly, YDJin
is suspended.
Correspondingly, the minimum limit must be suspended for
JzM=closing time so that the term to maturity can be
prolonged to the next bond settlement date. However, since
closing timesmo this is comprised by the above-mentioned
suspension of the minimum limit. Thus, when defining F(LJ) it
is not necessary to take closing time into consideration.
F(LJ) is thus defined as, for J<M-mo,
1 ) For YD ( J . LJ ) >YDJ°"
(i) F{LJ)=YD (J,LJ)-YDJ°"
2 ) FOr YDJl"<_YD ( J, LJ) <_YDJ°"
(ii) F(LJ)=0
3) and for YD(J.LJ)<YDJ'n
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(111) F(LJ)=YD (J,LJ)-yDJin
and for JAM-mo (where LJ is replaced by LJ when the model
reaches the closing time period)
1) For YD (J,LJ)>YI7~"
(i) F(L,,)=YD (J,LJ)-YDJ°"
2 ) FOr YDJin<YD (J, LJ} <_YD'J"°"
(ii) F(LJ)=0
For a serial loan, YD(J.LJ) is defined as the payments on the
loan the first year of the refinancing period.
Based on the value of F(LJ), an adjustment of LJ is to be
calculated like in the type F model. The adjustment is
calculated applying the reduced version of the Gauss-Newton
algorithm.
If F(LJ)=0, the payment on the loan is within the band and
thus L,, is not adjusted, corresponding to
DL,,=0
If F(LJ)~0, the adjustment is calculated applying
~L~ - F(L~) inc
F(LJ) -F(L~inc)
Step D - Has F(LJ) converged ?
(identical with step D in the type F model)
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In step D it is tested whether F(LJ) has converged so that
the payments on the loan are within the band defined by the
maximum and minimum limits. If not, the model applies the
calculated adjustment.
Whether F(LJ) has converged is determined by testing the
mathematical convergence of the function. This is done by
evaluating the volume of the adjustment of the term to
maturity which was determined in step C against the value of
F(LJ). If the adjustment is very small, cf. the conditions
mentioned below, there is no reason to continue the iterative
routine: The model can not get any closer.
Mathematical convergence is defined by the conditions
{i) ~~LJ~ < E
E I L,,I
( ii ) I IF(LJ) 2-F(L~~LJ) z1 I < E
F{LJ) a
where E is a very small figure.
If just one of the conditions is fulfilled LJ is accepted as
the term to maturity at the Jth refinancing. In principle,
the convergence may also be tested by applying
(iii) ~F(L,,)~<E
where E is fixed at 0,00001. Thus, a maximum deviation from
the maximum or minimum limit of DKK 0,00001 is accepted.
The definition of F(L,,) in step C implies that the
convergence conditions are identical irrespective of the term
to maturity for which the value of F is calculated.
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Step E - Apply the adjustment of L,,
(identical with step E in the type F model)
YD(J,LJ) was rejected in step D. Thus, LJ is to be adjusted,
after which the inner model calculate the payments on the
loan for LJ+pLJ. First, however, the model must pass step B so
that the term to maturity is adjusted for J=M.
Step F - Is LM<Lmin
(identical with step F in the type F model)
In step F, it is checked that the full term to maturity LM is
not shorter than the minimum limit. If
~~Lmin
the model proceeds in step G in which the payments on the
loan are adjusted according to (B). Otherwise, the model will
proceed in step F.
It should be noted that steps F and I are the only steps
which are not applied for every value of J.
Step G - Calculate the payments on the loan for LJ=Len
(identical with step G in the type F model)
If LM<Lmi°, the payments on the loan are to be adjusted
according to (B), which stipulated
YD(J,.)=YD(J,LJ) for 0_<J<_J "
(B)
YD (J, . )=YD (J,Lmin) for J' 'sJsM
where J " denotes the minimum value of J for which
YD (J, LJ) >YD (J, Lain) .
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YD(J,L,,) has already been calculated for J=0,1,2,...,M, whereas
YD(J,L°'1°) must be calculated in the inner model. However,
the
link from step G to the inner model is not simple. At first,
(B) is calculated for J=0. Depending on whether
YD(O, . )=YD(O,Lo) Or YD{O, . )=YD(O,L'nin)
the remaining debt at the end of the period may be
determined. Based on the remaining debt, YD(l,Lmin) is
calculated and YD(1,.) can be determined and thus the
remaining debt at the end of the period etc. Hence, the link
from step G to the inner model is successive.
However, it will never be necessary to recalculate YD(J,L,,)
in a corresponding manner. From the point on where YD(L°'i")
determines the payments on the loan, this will be the case
until the maturity.
Step H - Adjust the payments according to minimum condition
(identical with step H in the type F model)
Having determined the value of YD(J,.) for each J, it only
remains to compound a payment profile. Information on
funding, interest rate on the loan etc. are compounded in
compliance with the payments on the loan to be applied, cf.
(B) .
When the payment profile in full has been determined, the
model proceeds in step L.
2 5 Step I - Is L~Lm°"?
(identical with step I in the type F model)
Analogously to step F, it is tested that LM does not exceed
Lm°" . I f
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~> LAX
the payments on the loan are to be adjusted in step J.
Otherwise, the calculations are complete and the model can
proceed in step L.
Step J - Calculate the payments on the loan for LJ=Lm°"
(identical with step J in the type F model)
This step is completely analogous to step G except from the
fact that LM>L~'°", so that the payments on the loan are to be
adjusted according to (A).
YD(J,.)=YD(J,LJ) for OsJsJ'
(A)
YD(J, . )=YD (J,Lm°") for J'sJ<_M
where J' denotes the minimum value of J for which
YD (J, L,,) <YD (J, L°"°") . As in step G, the transition
from step J
to the inner model is successive.
Step K - Adjust the payments according to maximum condition
(identical with step K in the type F model)
Analogously to step H, a payment profile based on the
payments on the loan for J=0,1,2,...,M determined in step J is
to be compounded. Then, the model proceeds in step L.
Step L - The calculations are complete. The result may be
applied
(identical with step L in the type F model)
A compounded payment profile and the term to maturity LM have
been calculated. In the inner model, the volumes of the
financial instruments applied and the interest rate on the
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loan have been determined. Thus, the computation of the loan
is complete and the result may be applied.
3.2.2 The inner model for type P - the iterative solution
The inner model for type P is not simultaneous, but rather
consits of an outer and an inner loop in a two-step
procedure. Thus, together with the routine in the outer
model, the total model comprises three levels.
In Fig. 24 a flow-chart for the model is shown. The two-step
procedure does not appear clearly form the flow-chart and is,
therefore, briefly outlined in the following.
The model begins with a guess at an interest rate on the
loan. On the basis hereof, the payments on the loan and a
first guess at the funding are determined. Then the model
estimates the funding in an inner loop in step 7 to step 10.
The estimation is performed in an iterative routine. The loop
is left when the funding fulfils the requirement as to the
part of the remaining debt that is refinanced, and the
proceeds criterion.
However, a situation may arise for the model where it is not
possible to observe the intended refinancing percentage and
the proceeds criterion at the same time. If both criteria are
not met within 30 iterations in the inner loop, it is
presumed that a solution does not exist and the model leaves
the inner loop. Then, the funding is adjusted to observe the
proceeds criterion.
In the outer loop, it is tested that the interest rate on the
loan corresponds to the yield to maturity of the funding
portfolio, which, however is adjusted for possible
differences in the number of payment dates on the debtor side
and on the funding side, respectively. If this is not the
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case, the guess at the interest rate is adjusted, and the
model again recalculates a debtor payment profile etc.
Thereby, the inner loop is also called again with the new
interest rate.
Thus, the iterative procedure concerning the interest rate
constitutes the outer loop.
Only when all the requirements have been fulfilled does the
model leave the outer loop too, and the final result of the
calculations in the inner model is available. Then the model
returns to the outer model with information on funding,
interest rate on the loan, and payment profile.
The same notation as in the inner type F model is applied.
Thus, in the inner model for type P, J and LJ are fixed,
whereas the individual years are indexed by j.
Step 0 - Determine m and TILT
Before actual calculations are performed, it must be input to
the model whether the loan is disbursed in December with m=mo
funding instruments or at any other time of the year with
m=ma+1 funding instruments.
This is done by assigning TILT eE(0,1} a value.
Set TILT=1 and m=mo+1 if the loan is disbursed in January-
November
Set TILT=0 and m=m0 in all other cases.
The primary function of TILT is to indicate that an adjust-
ment in the number of funding instruments at the disbursement
of the loan is performed, where it will generally apply that
m=ma+TILT
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In connection with the maturity of the loan, m is also to be
adjusted to ensure that no financial instruments with
maturity later than the maturity of the loan are applied.
Thus, m is assigned a value according to
m=min [Lo-Q; mo+TILT]
where Lo-~ designates the remaining term to maturity of the
loan.
Step 1 - Determine initial interest rate
In this step an initial interest rate on the loan is
determined. In order to minimize the number of iterations to
be carried out later on it is expedient that the
determination of the initial interest rate on the loan is not
arbitrary.
In the model, a guess is made that the interest rate on the
loan is a weighted average of the yield on the individual
funding instruments, thus
1 t r(0, t)
RR ~ m
1t
~1 m
as in step A.
Step 2 - Determine the payments on the loan
In this step the model begins the outer loop. When a guess
at the interest rate has been made, the payment profile may
be calculated.
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Step 3 - Is m=1, m=2 or au2?
The remaining term to maturity on the loan is decisive for
the further operations in the model. The calculations in the
model may proceed in three different ways.
If m=1, the funding and the interest rate are calculated
rather simply in step 3a.
If m=2, the calculations proceed in step 3b, wherein the
funding and the interest rate on the loan are calculated by a
method which is related to the F model. LAIR type P50 will be
calculated in this step apart from at the issue, wherein m
may be 3 depending on the date of the disbursement of the
loan. In addition, in the last year, the loan will be
calculated in step 3a.
If m>2, calculations are continued in the model proper in
step 4.
This means that in step 3 the type P loan is, in general,
first calculated in the model proper, then in step 3b, and
finally in step 3a.
Step 3a - Determine funding and interest rate for m=1
If m=1, the funding can be determined on the basis of the
proceeds condition as
M(1)-Fia(0)
K(1)
Since no more refinancing occur, it follows from the strict
balance principle that
YD (m)-[1.RN(m) ]FI(m)
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which determines an unambiguous interest rate on the loan.
4~hen the calculations in step 3a are complete, the
calculations proceed in step 14.
Step 3b - Determine funding and interest rate for m=2
If m=2, the balance requirement in the first following year
can be written as
(I) YD(1).REGDR~ ) -(1.REG RN(1) )H(1).REG RN(2)H(2)
and the proceeds criterion as
(J) M(1)K(1).M(2)K(2)-Fin(0)
These are sufficient conditions for determining the funding
analytically as the solution to two equations (I and J) with
two unknown variables (M(1) and M(2)).
In order to do this, (I) and (J) are written into a matrix
form in the following way
CxM=D
Wherein
M=(M(1),M(2)) is a 2x1 vector and
D~~RDGD ~ ).YD(1)-~1~RIJ1GRN(1)~H(0,1)-RDGRN(2)H(0,1),Fin(1)
is a 2x1 vector too, and
2~ C, 1~REG RN(1) REG RN(2)
K(1) K(2)
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C is a square matrix and therefore can be inverted which
yields the solution
M=max [ 0 ; C-1D]
as strictly negative funding volumes are not accepted.
The REG and REGD factors only influence loans which are
either disbursed with a term to maturity of 2 years or type
P50,0 the first year.
The model continues in step 11, where it is checked that the
funding criterion has been observed. This check is necessary
if C-1D yields negative funding volumes. In this situation,
the max-condition implies that the model is making an excess
of funding when determining the funding.
If a funding principal is adjusted because it would otherwise
be negative, a deviation in the current refinancing
percentage arises in relation to the intended refinancing
percentage. Therefore, the analytic solution does not provide
certainty that the intended refinancing profile may be
respected.
Step 4 - Define a trend function
In step 4, a trend function is defined, which trend function
estimates the size of the refinancing amounts as a function
of the time t. The trend function is shown in Fig. 25.
In principle, the trend function may have any functional
form. A excellent estimate for the development of the remain-
ing debt can be obtained by applying a polynomial of the (q-
1)th degree.
Thus, the trend function has the form
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ao+al t+a2 t2+...aQ_l tq-1
wherein 0<_t<_m. Furthermore, a limitation must be set on q so
that the degree of the polynomial does not exceed the number
_ of funding instruments minus 1, whereby the polynomial would
have too many degrees of freedom. This means that
qsm
Thus, for a LAIR type P20,0 a polynomial of at the most the
4th degree is estimated. If q=m, the trend function will
estimate the refinancing amounts perfectly.
If the loan is disbursed with m=mo+1 in the special TILT
procedure, the degree of the polynomial is not increased.
Therefore, the exact restriction on q can be expressed as
qsm-TILT
Step 5 - Determine trend function coefficients
Then the coefficients of the trend function must be estima-
ted. From step 2 the development of the remaining debt is
known given the guess at an interest rate on the loan and
given the term to maturity of the loan.
At times of interest rate adjustment t=1,2,..,m the trend
function must correspond to the intended refinancing of the
_ loan. At the same time negative marginal funding must be
avoided. Therefore the trend function must be estimated so
that the function value of each t corresponds to the maximum
of either the intended refinancing or the funding already
issued in the bond with maturity on the date t.
aa.alt.a2t2~....a ølt ø1 ~nR t ,~p, t 1,
mo
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for t={O,l,...,m-1}.
Thereby, the coefficients can be determined by the matrix
equation
~ao , al , ... , aø1~B TB~ -1B T REG D Rd t~ , H(0 , t~1~
wherein (a°, al, . . . , aq_1) is a qx1 vector
B is a mxq matrix and
max[.,.] is a mxl vector
The matrix B is given by
1 0 0 0 . 0
1 to to . . to 1 1 1 1 1 . 1
1 tl ti . . tø1 1 2 4 8 . 2ø1
_ . 1 3 9 27 . 3ø1
z øiJ ,
1 t~l t~l . . t~l
~1 m-1 (m-1)z . . (m-1~~
The matrix is produced by assigning values from 0 to m-1 to t
in the trend function. The first row comprises the values of
t°, tl, t2, ..., tq-1 for t=0 . The second row comprises the similar
values for t=1, etc . The expression [BTB] -iBT is an
approximation of B-1 based on least squares. The approximation
is necessary when B can not be inverted. B will always be of
full rank. However, if q<m - as in the special TILT procedure
- B will not be a square matrix and thus can not be inverted.
Step 6 - Guess at an increment to the two coefficients
The trend function must be adjusted in the following steps,
so that the balance requirements correspond to the intended
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refinancing and so that the proceeds criterion is fulfilled.
Initially, however, the trend function must be adjusted.
The adjustment is generally performed by means of two factors
Go and G1 in the following way
G~a~+Glal t+a2 t2+...aq_1 tq-1
Go effects a parallel shift of the trend function up and down
in the (H(j),j) plane, whereas G1 influences the slope of the
trend function.
Already before the iteration in the inner loop is started,
the factors are assigned the values
Go=1.25 and Go=1
corresponding to a shift upwards. The idea of the shift is to
obtain better information concerning the relationship between
the individual marginal funding principals.
In the example, H(0,4) is shown disproportionately large so
that according to the trend function, M(4) will be 0. If the
marginal funding volumes are generally increased in a later
step, the model, however, does not have much information as
to how much the volume of the other marginal funding
instruments must be increased before M(4) is increased as
well. By a parallel shift of the trend function upwards this
information is obtained.
If TILT=1 and the model thus operates with an extra funding
instrument, only one G-variable is introduced in step 6. That
is, because of the special refinancing profile, the volume of
the first funding instrument is explicitly estimated by a
variable Z which is explained in detail in the next step.
Provisionally, set Z=1.
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Step 7 - The funding is determined
The inner loop of the inner model starts in this step.
On the basis of the trend function, the marginal funding is
determined by
M ( j ) - max [ 0 ; [Goao+Glalt+azt2+ . . , . +aq_lt9-1 ] -H ( 0 , j )
wherein t=j-1={O,l,...,m-1} as before. At the disbursement of
the loan, H(j)=M(j)+H(O,j) and H(O,j)=0 are again applied.
That the loop starts in step 7 and not in step 5 or 6 implies
that ao, al, . . . , aq_1 are only estimated once for every guess at
an interest rate. Therefore, in the following step
ao, al, . . . , aq_1 are constant. Correspondingly, (Go,G1) _ (1.25, 1)
is solely an initial guess.
For TILT=1 the volume of the first instrument M(1) is
determined as
M(1)=Z-H(O,j)
This is necessary, since a polynomial would have
difficulties in estimating M(1) which constitutes down to one
fourth of the volume of the other funding instruments as a
consequence of the reduced refinancing percentage.
When the model in the following steps estimates Z on the
basis of the balance requirements and the proceeds criterion,
M(1) is explicitly determined so that the criteria are
fulfilled with certainty at the first occurring refinancing.
The other funding volumes are determined as above. Generally,
therefore, the following applies
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M ( J ) =max [ 0 ; [ Z. Goao+alt+a2t2+ . . . . +aq_ltq-1 ] -H ( 0 . J ) ]
Analogously, the funding for
( ( Go+inc , G1 ) , ( Go , G1+inc ) )
- is determined. From now on the notation (Ga,Gl)+inc is simply
applied. The standard is that inc has the value 0.00001.
Step 8 - Calculate the proceeds and balance criteria
In step 8 it is assessed whether the factors Go, and G1 are
determined so that the balance requirement, given the
intended refinancing profile, is observed and the proceeds
criterion is fulfilled.
The balance requirement is given by
~(1)~REGDR ~ ) W(1)~REG~ RN(j)H(j)
and the proceeds criterion is given by
Fin(0)-~K(j)M(j)
,.
Only the balance requirement for the first year must be
checked. Of course, the funding must also be arranged taking
into consideration the remaining years, but this is ensured
by the initial determination of the trend function.
Step 9 - Calculate adjustment of increment
On the basis of the calculated values for the balance requir-
ement and the proceeds criterion, respectively, an adjustment
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of (Go,GI) is determined on the basis of the Gauss-Newton
algorithm.
Already in step C in the type F model a general formula
apparatus was set up. In the F model an adjustment to the
term to maturity of the loan was determined, whereas in step
9 an adjustment to the factors in the trend function is
determined - however, to a large extent, the method is the
same.
The most important difference si the definition of the F
function. F ( . ) is here defined as a function of (Go, G1 )
F{Go,Gl)-[Fin(0)-~K(j)M(,j) .
~.i
H(1).REG ~RN(j)H(J)-YD(1)-RECD 1 RG(0)
J.1 m0
Also here, the value of F(.) is determined for both (Go,Gl)
and (Go,Gl)+inc. The adjustment of (Go,Gi) is referred to as h
and is defined as
D TD 1D T9
h O {Go, Gl).
J~
wherein D and g are defined as in the F model. It must,
howver be noted that
1 Fl~Go, Gl~-Fl~Go.inc, Gl~ Fl~Ga, Gl~-Fl~Go, Gl.inc)
inc F2~Go, Gl~-Fz~Go.inc, Gl, FZ~Go. Gl~-FZ~Go. G1+inc~
The subsign for F states whether it is the first or the
second part of the expression for F which is estimated. I.e.,
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F1(Go,Gl) determines the value of the proceeds criterion for
(Go,Gl) .
If TILT=1, an adjustment to (Z, Go) is determined instead.
It must be noted, however, that step 9 only allows 30
attempts. This means that if the model during 30 attempts has
not been able to estimate an adjustment of the trend function
and thus the funding satisfactorily, the loop is abandoned,
and the model continues in step 11. However, the model may
hereafter return with a new guess at the interest rate.
Step 10 - Are the balance and the proceeds criteria
fulfilled?
On the basis of the calculated adjustments, it is to be
determined whether the trend function determines the volumes
of the financial instruments applied in such a way that the
balance requirement and the proceeds criterion have been
fulfilled.
In step D of the type F model were introduced the criteria
for mathematical convergence which are to be used here.
If h fulfils just one of the conditions
(i) ~h~ <e
E ~ Go, G1I
(ii) F'(Ga.Gl)2-F( (Go.Gl)4h) <E
F'(Go~ Gl) 2
h has converged, and (Go,Gl) and the matching funding can be
applied.
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If none of the conditions has been fulfilled, the adjustment
h is added to (Go,Gl) and step 7 is repeated.
Also here, the requirement as to mathematical convergence can
be replaced by the specific conditions - the proceeds
criterion and the balance requirement. This means that the
funding may be applied if
(111) Fin(0)-~K(j)M(j) <e
~-i
(iv) in{1)-REGDRG(0) <E
~'b
Step 11 - Is the proceeds criterion fulfilled?
Step 11 aims at the situation in which the model has not been
able to determine a trend function satisfactorily. Thus, the
funding does not fulfil both the balance requirement, or more
specifically, the intended refinancing profile, and the
proceeds criterion for the given interest rate on the loan.
It cannot be excluded that no solution which fulfils both
requirements exists, so the model necessarily has to conti-
nue.
The proceeds condition is, however, an indispensable require-
ment. Therefore, it is checked whether the proceeds condition
is fulfilled. This is done by the condition
in{0)-~K(j)M(j) <e
~-i
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If this is the case, then the model continues in step 13 -
otherwise the volumes of the financial instruments applied
are adjusted in step 12.
Step 12 - Adjust the funding
The model only reaches step 12 if it has not been possible in
the inner loop at the same time to meet the intended
refinancing profile and the proceeds criterion. Thus, the
volumes of the financial instruments applied must be adjusted
with the sole object of fulfilling the proceeds condition.
Therefore, appropriate adjustment of each volume is
performed.
In principle, three situations may occur
1) The model calculates an excess of funding, i.e. the
funding volumes must be reduced
Fin(1) M for M=(M(1),M(2),...,M(m)
K(J)M(J)
J-1
2) The model makes a deficiency of funding, i.e. the
funding volumes must be increased. Here M(1) is
maintained so that the refinancing percentage at the
next refinancing does not increase more than what is
absolutely necessary. M(1) is determined as
1~ Fi.n(1)-K(1)M(1) M for M=(M(1),M(2), ...,M(m)
K(J ) M(.7 )
2
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3) Finally, the funding principals may sum up to zero.
Here the entire funding is placed in the instrument
with the shortest maturity, i.e.
M(1)-F111~ and M(j)=0 for
j={2,3,...,m}
After the adjustment, the proceeds condition is fulfilled,
and the model continues in step 13.
Step 13 - Calculate an adjustment of the interest rate
In this step, an adjustment to the guess at an interest rate
on the loan is to be calculated. The adjustment is ca~_culated
in relation to the yield of the portfolio of funding
instruments, with correction, however, for the typically
different numbers of settlements within each year on the
debtor side and the funding side, respectively. Thus, the
adjustment of the interest rate on the loan is calculated as
the difference between the corrected interest rate on the
loan and the yield to maturity of the funding portfolio.
Firstly, the function F(.) is defined as
~~ R ~ RG ( i-1 ) -~G r p
n RG(0)
wherein
rp is defined as the yield to maturity of the
funding portfolio.
n is the number of debtor payment dates per year in
the next refinancing period.
With the definition of F(R") the Gauss-Newton algorithm may
be applied.
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OR R D TD 1D Tg
Jv
The definitions are not to be repeated here. It should only
be noted that
J._ F(RR) - F(RRinc)
inc
The algorithm can be simplified analogously to step C in the
type F model.
Step 14 - Has the interest rate converged?
Step 14 determines whether the outer loop is to continue or
whether the model has reached a satisfactory result. The
criterion is the extent of the accordance between the
interest rate on the loan and the yield on the portfolio of
funding instruments.
The interest rate on the loan has converged and can thus be
accepted if
_R'~ RG ( i-1 ) -~ r ~e
i.i n RG(0)
wherein a in the model is set to be 0.00001,
is fulfilled. This means that the calculations in the inner
model are finished, and can be finalized in step 16.
Otherwise, the calculations proceed in step 15, wherein the
guess at the interest rate on the loan is adjusted.
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Step 15 - Adjust the interest rate
The guess at the interest rate on the loan which was rejected
in step 14 is adjusted with DR''. Then the model continues in
step 2, wherein the payments on the loan are determined for
the new guess at the interest rate.
Step 16 - Calculations are complete. The result may be
applied
The model has determined an interest rate and a number of
positive funding volumes. Thus, the calculations are complete
for the given value of the term to maturity.
The model may return to the outer model with information on
the payments on the loan, the funding, and the interest rate
on the loan.
3.2.3 The inner model for type P - the analytical solution
To a large extent, the analytical solution is based on the
same structure as the iterative solution. Thus, the
analytical solution divides the inner model into an inner and
an outer loop, where the outer loop comprise an iterate
routine targeting the interest rate on the loan, whereas the
funding profile is determined in the inner loop.
Yet, the analytical solution of type P differs from the
iterative solution by the fact that in the inner loop, the
model does not determine the funding volumes by applying an
iterative routine, but rather analytically. Primarily, this
facilitates the computations in the inner model.
However, the analytical solution also implies that the model
will always reach a solution. In the iterative solution there
was a risk that the model would have to discontinue the
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iteration after 30 guesses in which the model did not succeed
in finding a solution fulfilling both the proceeds criterion
and the intended refinancing at the same time. The model then
adjusted the funding volumes in the outer loop (steps 11 and
12 ) .
Thus, the fact that a solution will also be found has the
consequence that a similar adjustment of the funding volumes
will not be necessary in the outer loop. Apart from this, the
outer loop is identical with the outer loop in the iterative
solution.
The complete structure of the model is shown in Fig. 26. In
the following, each step is described. For the sake of
clearness, the steps that are identical with the iterative
solution are included, with an indication of the identity.
The analytical solution implies that the inner loop comprises
four steps.
First, (Go,Gl) is determined in step 6 as the factors in the
trend function which comply with both the proceeds criterion
and the intended refinancing in the first year. Based on the
adjusted trend function the funding volumes are calculated in
step 7 as the difference between the value of the trend
function and the funding volumes previously issued.
In advance it cannot be excluded that one or more of the
calculated funding volumes are negative, which is not
accepted, since it would imply that the debtor has to buy
bonds. If it turns out in step 8 that one or more funding
volumes are negative, these are set to 0 (zero) by means of
an indicator function. The indicator function forms a m-
dimensional vector, in which the jth element is zero if the
jth funding volume is set to zero, and otherwise one. The
adjustment of the indicator function is performed in step 9.
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lnThen the indicator function has been adjusted, the
calculation of (Go,Gl) is repeated based on the proceeds
criterion and the intended refinancing in the first year, but
with the reduced number of funding instruments. This process
is performed until no further negative funding volumes are
revealed in step 8.
It cannot be excluded that only one funding instrument is
left at the end. In this situation, it will generally not be
possible to observe both the proceeds criterion and the
intended refinancing requirement for the first year. Thus,
the latter must be suspended as the proceeds criterion is
given the higher priority. In this situation, a value of Go
is determined in compliance whit the proceeds criterion in
step 6.
Correspondly, in the procedure it may often occur that the
volume of the first funding instrument is set to 0 (zero), in
which case it will generally not be possible to observe the
intended refinancing requirement in the first year. Hence,
the same procedure as above where only Go is determined in
step 6 is applied.
Step 0 - Determine m and TILT
(identical with step 0 i the iterative model)
TILT is assigned a value according to the following rules.
Set TILT=1 if the loan is disbursed in ,Tanuary-
November
Set TILT=0 in all other cases.
At the same time, the value of m is temporarily determined as
m=mo+TILT
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In connection with the maturity of the loan m is also to be
adjusted to ensure that no financial instruments with
maturity later than the maturity of the loan are applied.
Thus, m is assigned a value according to
m=min [Lo-6; mo+TILT]
where Lo-~ designates the remaining term to maturity of the
loan.
Step 1 - Determine initial interest rate
(identical with step 1 i the iterative model)
The initial interest rate on the loan is assigned according
to
1 t r(0, t)
RK ~ m
1t
~1 m
Step 2 - Determine the payments on the loan
(identical with step 2 i the iterative model)
In this step, the model begins the outer loop calculating the
payments on the loan given the interest rate on the loan.
Step 3 - Is m=1 or aul?
The calculations in the model may proceed in one of two
different ways depending on the value of m.
If m=1 the calculations are continued in step 3a, where the
funding is determined very simply since only 1 funding
instrument is applied.
If m>1, the calculations proceed in step 4 in the general
model. There is no need to distinguish between the situations
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where m=2 or m>2 as in the iterative model, since the
determination of the funding volumes are based on the same
principles in the analytical model. Thus, step 3b is left out
of the model.
Step 3a - Determine funding and interest rate for m=1
If m=1, the funding can be determined on the basis of the
proceeds condition as
M(1)_ Fin(0)
K{1)
Since no more refinancing occur, it follows from the strict
balance principle that
YD(m)=[1+RN(m)]H(m)
which determines an unambiguous interest rate on the loan.
When the calculations in step 3a are finished, they proceed
in step 14.
Step 4 - Define a trend function
(identical whit step 4 in the iterative model)
As in the iterative solution the trend function is defined as
a polynomial of the degree q-1
a~+al t+a2 t2+...aq_ 1 tq-1
wherein Ostsm and qsm-TILT so that the degree of the
polynomial is not increased in the speciel TILT-procedure.
Step 5 - Determine trend function coefficients
(identical whit step 5 in the iterative model)
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The coefficients in the trend function ao, al, ..., am are
determined by applying
~ao , al , ... , aø1~B TB~ ~1B T REG D Rdt~ , H(0 , t~1~
mo
wherein (ao, al, . . . , aq_1) is a qx1 vector
B is a mxq matrix and
max(.,.] is a mx1 vector
The matrix B is given by
1 0 0 0 .
0
1 to to . 1 1 1 1 .
. to 1
1
1 tl ti . 1 2 4 8 .
. t 21
l
1 3 9 27 .
31
1
1 t~l t~l . -
t~l
1 ml (rrE1)2 .
. (m-1~
Step 6 - Determine Go and Gl
In step 6.the trend function is adjusted in accordance with
two conditions. For one thing, the funding must fulfil the
proceeds criterion and for another, the funding must fulfil
the intended refinancing in the first year given by the
balance requirement.
In general, the adjustment is performed by introducing the
parameters Go and G1 so that the trend function appears as
follows
Goao+G1 al t+az t2+...+aQ_1 tq-1
The trend function may immediatly be divided into a variant
part and an invariant part in regard to Go and G1. The variant
part is denoted X(j) and the invariant part Y(j), thus
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X ( j ) =Goao+Glalt and Y ( j ) =azt~+...+aq_ltq-1
for t=j-1.
The marginal volume of the funding instrument with a term to
maturity of j years, M(j), is determined as the value of the
trend function - the volume of the funding instrument already
issued, H(O,j). However, the volume must not be negative,
thus
M ( j ) =max [ 0 ; G9ao+G~alt+a2tz+...+a9_itq-1-HO , j )
which by applying X(j) and Y(j) may be simplified into
M(j)=max[0;X(j)+Y(j)-H(O,j)]
The expression must be further simplified. This is necessary,
since Go and G1 cannot be isolated as long as the expression
comprises a max-function. Thus, an indicator function, I(j)
j=1, 2 , ..., m, is implemented with the value zero if the j th
funding instrument is to be set to zero, and otherwise with
the value one. Thereby, the indicator function forms an m-
dimensional vector.
When step 6 is first applied the values of the indicator
function is set according to
I(j) - (1,1, ...,1),
After this, the value of I(j) will be determined in step 9.
When implementing the indicator function, the marginal
funding volumes are determined according to
(I) M(j)=I(j)[X(j)+Y(j)-H(O,j)]=I(j)X(j)+I(j)(Y(j)-H(O,j))
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If both conditions
~I(j)z2 and I(1)>0
j-1
are fulfilled, corresponding to the first funding volume
exceeding 0 (zero) and, at the same time, the number of
remaining funding volumes being greater than or equal to 2,
the marginal funding must observe both the proceeds criterion
and the intended refinancing in the first year.
Thus, Go and G1 are calculated solving
(i) YD(1).REG°RG(0) , H(0,1).M(1)+REG~Rn(J)~M(J)~H(O.J)~
~0 j.l
and
(ii) Fin(0)-~K(j)M(j)
j-1
wherein (i) is the balance requirement and (ii) is the
proceeds criterion.
The expression (I) may then be inserted into (i) for M(1) and
M(j)
YD(1).REG°R ~ ) -H(0,1).I(1)X(1).I(1) (Y(1)-H(0,1) )
F~~Rn(7)~1(J)X(j)sI(j) (Y(j)-H(O.J))~H(0.J)~
1
The expression is solved for X(j)
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(J) I(1)X(1).REG~RN(j)I(j)X(.7)-~(1)~REGDR~ )-H(0,1)
~a
-I (1)~Y(1)-H(0, 1)~-~.~ Rn(J)~I(j)~Y(.7)-H(O.j)~+H(O.J)~
,7-1
In the following, the right-hand side of the expression is
designated by the variable Z1.
Analogously, the expression (I) may be inserted in the
proceeds criterion. This leads to the expression
Fin(0)-~ K(J)(I (j)X{j)~I (j)(Y(j)-H(0.J)
j-i
The expression is solved for X(j)
(K) ~K{j)I(j)X(j)-Fin(0)-~K(j)Z(j)~Y(j)-H(O,j)
~-i .i.i
The right-hand side of this expression is denoted Z2.
Together, (J) and (K) define two equations with to unknown
variables Go and G1. However, the solution of the equation
system depends on the value of TILT.
If TILT eauals 0, the general solution is applied.
X(j) was defined as the part of the trend function variant as
to Go and G1. In a matrix form X(j) may, therefore, be
described as
X=axG
wherein
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X=(X(1),X(2),...,X(m)) is a 1xm vector and
G= (Go, G1) is a 2x1 vector and
a is a mx2 matrix given by
ao 0
ao
a = ao 2 al
ao (m-1 ) al
A matrix K is then defined so that KxX and thus KxaxG forms
the left-hand sides in (L) and (K).
K I(1)~1~REGRN(1)~ I(2)REG.RN(2) ... I(m)~,RN(m)
I(1)K(1) I(2)K(2) ... I(m)K(m)
Defining K, the two equations (L) and (K) may be written on a
matrix form, utilizing that the right-hand side of (L) and
(K) are given by Z1 and Z2.
Z=KxaxG
wherein Z=(Z1,Z2) is a 2x1 vector. Kxa forms a mxm matrix that
may be inverted. Go and G1 may thus be determined as
G= [Kxa]'1Z
If, on the other hand if TILT=1, M(1) is determined
explicitly as in the iterative solution. The marginal funding
is determined in accordance with
M ( J ) =max [ 0 ~ ( X ( 1 ) : Goao+al t+a2 t2+...+aq_1 tq-1 ) -HO , J ) ]
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Again the trend function may be divided into a variant and an
invariant part defined as
GQao dj>1
X ( J ) - X ( 1 ) j-1 and Y~J~-~ t+~ t 2+ . . . +a~l t ø1
for t=j-1
X(1) and Go must be determined in compliance with the
proceeds criterion and the intended refinancing in the first
year. This defines the equations (J) and (K) identical to
above - X(j) and Y(j), however, being defined by the above
definitions.
(J) I(1)X(1)+RDG~RN(j)I(j)X(j)-YD(1).REGDR~ )-H(0,1)
j.l
-I(1)(Y(1)-H(0.J)~-REG~, Rn(.7)~I (.7)~1''(J)-H(0..7)~+H(0.7)~
J.1
(K) ~K(J)r(J)X(.7)-Fin(0)-~K(j)I(.7)~3.'(.7)-H(O.J)
J.1 J.1
The right-hand sides of (J) and (K) are designated Zi and Zz,
respectively.
X(j) may be written on the matrix form
X=a' xG'
where in G'= ( Go , X ( 1 ) ) and
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0 1
as 0
a'
ao 0
Analogously to the above, (J) and (K) lead to the expression
Z*=Kxa.xG.
wherein Z"=(Zi,Zz) and K is, unchanged, defined as
I(1)~1.REGRN(1)~ I(2)REG.RN(2) ... I(m)REG.RN(m)
K =
I(1)K(1) I(2)K(2) ... I(m)K(m)
The matrix equation is solved by
G~= [Kxa'] -1Z~
If, on the other hand,
~I(7)<2 or I(1) - 0
j.l
only the proceeds criterion may be fulfilled, and the balance
requirement is suspended. In principle, then Go and G1 must be
determined on the basis of only one equation which yields an
infinite number of solutions. Therefore in this situation, G1
is assigned a fixed value which, of course, must be suitable
so that the refinancing the first year does not differ to an
unnecessarily large extent. Numerical analysis has shown that
G1 = 1
leads to suitable results.
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The fixing of G1 implies that G1 now is comprised by the
invariant part of the trend function, thus
X ( j ) - Goao and Y ( j ) =Glal t+a2 tz+...+aq_1 tq-1
It remains to determine Go in compliance with the proceeds
S criterion. The proceeds criterion is given as
Fin(0)-~K(J) (I(J)X(j),I(j) (Y(J)-H(O..J) ) )
j.l
Solved for X(j) this leads to the expression
(K) ~K(J)I(J)X(J)-Fin(0)-~K(J)-t(J) (~.'(J)-H(O.J))
~.i ~.a
When the right-hand side of (K) is denoted Zz*, Go is
determined by solving
G..= ~K..Xa..~ _lZ.=
wherein
G'i=Go and
K*~=[I(1)K(1),I(2)K(2),...,I(m)K(m)] and
0
ao
a' and
ao
Zf~-Z**
2
On the other hand, it may also be utilized that X(j)=Goao,
which inserted in (K) yields
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Goao~K(J)I(.7)-Fin(0)-~K(J)I(J) (1'(J)-H(0.J))
701
Solved for Go
Fin(0)-~K(J)I(.7) (1'(.7)-H(0.J)1
G - j.i
0
ao~K(.7)I(J)
~.i
and thus Go is determined. By insertion of Go in the balance
requirement, the refinancing in the first year may be
calculated residually.
When both Go as well as G1 (or X(1) for TILT=1) have been
determined, the calculations in the model continue in step 7.
Step 7 - The funding is determined
In step 7 the marginal volumes of the funding instruments are
determined as the difference between the value of the trend
function for each respective j and the volumes previously
issued. The marginal funding volumes are determined either as
M ( J ) =Goao+Glalt+a2tz+...+aq_ltq-1-H0. J )
for TILT=0 or as
M ( j ) _ ( X ( 1 ) ; Goaa+Glalt+a2t2+...+aq_ltq-1 ) -H0 , j )
The implementation of the indicator function is the reason
why the marginal funding volumes are determined without
applying a max-condition which would involve that negative
funding volumes are set to 0 (zero) in this step. If M(j~) is
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negative, I(j~) is set to zero, and thus the next time the
step is applied, M(j~) equals zero.
Step 8 - Are all funding volumes positive?
In step 8 it is determined whether the calculated funding
profile may be applied, or whether it is necessary to perform
the calculations in the inner loop once again.
If
M(j)~0 b'jE(1,2,...,m}
the model moves on to step 10 i the outer loop.
On the other hand, if
~jE(1,2,_..,m) M(j)<0
is fulfilled, the model proceeds in step 9.
Step 9 - Adjust indicator function
If the model reaches step 9, one or more of the marginal
funding volumes are negative, which is indicated by assigning
the indicator function the value 0 (zero) for the particular
j. Thus, I(j) is adjusted as follows
I(j)=0 for all j where M(j}<0
I(7}=1 otherwise.
Alternatively, I(j) might be set to 0 (zero) for one funding
instrument at a time. If so, the particular funding
instrument should be the one with the numerically greatest
volume among the negative funding volumes. When a negative
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funding volume is deleted, the proceeds criterion implies
that fewer bonds are sold in total. Thus, it is only in
speciel cases that the marginal funding will not decrease for
every j when a negative funding volume is set to 0 (zero).
Thus, if the marginal volume of a particular instrument is
negative, it is not very likely that subsequent calculations
will yield a positive funding volume for the particular
instrument.
TnThen the adjustment has been performed, the calculation of Go
and G1 is repeated in step 6.
Step 10 - Calculate an adjustment of the interest rate
(identical with step 13 in the iterative model)
Based on the extent of convergence relative to the yield to
maturity of the funding portfolio, an adjustment of the
interest rate on the loan is calculated in step 10.
A function F(R") is defined as
Rx RG(i-1) -~G rp
~'~-1.1 n RG~o)
The function is applied in the Gauss-Newton algorithm, thus
~R'= DTD lDTg
J"
After this, the model continues in step 11
Step 11 - Iias the interest rate converged?
(identical with step 14 in the iterative model)
If the adjusted interest rate on the loan equals the yield to
maturity of the funding portfolio, the interest rate on the
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loan has converged and may be applied. The criterion for this
is
Rg RG(i-1) -~ rp <E
;.-~ n RG ( 0 )
wherein E is assigned the value 0.00001.
If the criterion is met, the iterative routine in the inner
loop may end, and the model returns, via step 13, to the
outer loop. Otherwise, the model proceeds to step 12.
Step 12 - Adjust the interest rate
(identical with step 15 in the iterative model)
The interest rate on the loan is adjusted applying the
adjustment calculated in step 10, and the model continues in
step 2.
Step 13 - Calculations are complete. The result may be
applied
(identical with step 16 in the iterative model)
The model returns to the outer model with information on the
funding profile, the interest rate on the loan and the
payments on the loan.
3.3 General comments on LAIR type P
From the outset, it is not certain that the model is able to
respect the intended refinancing percentage each year, even
though the model determines the funding according to a long-
term strategy as mentioned in the introduction.
The long perspective in the determination of the funding is
introduced via the adjustment of the trend function. Firstly,
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the trend function is estimated on the basis of the intended
refinancing profile for the whole funding period. This is,
however, not a very good strategy for the funding, since the
model can easily end up in a situation wherein too many bonds
with maturity in a single year have been issued. In the inner
loop of the model, however, an adjustment is made to al,
which controls the slope of the function. In Fig. 27 is shown
an example of the adjustment of the funding. Line 78 is the
initial trend function, whereas line 80 is the adjusted trend
function.
Thus, only a part of the future refinancing amount is funded
now, the rest of the funding will take place later pari passu
with the development in the interest rates.
It is exactly the determination of the trend function that
justifies the special procedure applied when the loan
matures. The adjustment of the trend function is based on the
refinancing percentage in the first year and the proceeds
criterion. Without applying a special procedure there would
be a risk that the term to maturity on the loan is determined
as, e.g.,.13 months on the loan side, whereas payments
according to a term to maturity of two years are funded on
the bond side. The consequence would be a substantial rise in
the interest rate on the loan in the last year before
maturity. By introducing the closing time parameter, a term
to maturity of two years is calculated with also on the loan
side.
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Appendix 1 Detailed description of the matrix in the
Gauss-Newton algorithm
F(x)=(fl, fz)ixz
The suffixed subsigns (1x2) designate the size of the matrix
(number of rows x number of columns)
fi fi ( 1 ) fi fi (2 )
inc inc
J
f2 - f2 ( 1 ) f2 - f2 ( 2 )
inc inc
F(x+inc)=(f;(1),f2(2))lxz, for x+inc=(x(1)+inc,x(2))
=(fi (2) , f2 (2) )lxz~ for x+inc=(x(1) ,x(2)+inc)
N J2(1.1)~(.TT(1,2)~J(2,1) ) w
J
J2 (2, 2 )+ (JT(2,1) ~J(1, 2) )
wherein
w=JT(2,1) ~ J(1,1)+JT(2,2) ~ J(2,1) .
g= ( gi . gz ) iXz
gl=JT(1,1)fl+JT(1,2)fz
gz=JT ( 2 , 1 ) f 1+JT ( 2 , 2 ) f z
J"=(J~(1).J~(2))iXa
JV(1). JTJ(1,1)
J°(2)- JTJ(2,2)
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J~(1)2 x
J"J"T = x
for x = J~(1) J"(2)
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EXAMPLE OF LAIR WITH LIMITS FOR THE PAYMENTS ON THE LOAN
In the following, an example of calculation of a LAIR type P
with limits for the payments on the loan and limits for the
term to maturity is given. The calculations in the model are
based on the inputs entered, these are listed below. In
addition, the model comprises a number of parameters that
have fixed values in all the calculations performed. .In
principle, these values may be altered from one calculation
to the other, but for practical reasons they are kept
constant. The parameters are also listed below.
Fixed value parameters:
Maximum difference in balance (e) . DKK 0.0000001
Maximum difference in proceeds (e) . DKK 0.0000001
Maximum difference in interest rates (e) . 0.00001
inc . 0.0001
Input values:
Principal of the loan . DKK 1,000,000
Intended term to maturity . 20 years
Maximum limit for the payments . DKK 23,000
Minimum limit for the payments . DKK 20,000
Maximum limit for the term (L") . 30 years
Minimum limit for the term (L'l") . 0 years
Technical maximum limit for the
term (L~"") . 100 years
Technical minimum limit for the . 0 years
term (LMIN)
Number of debtor payments pr. year . 4
Number of bond settlements pr. year . 1
Repayment principle . annuity
Type of LAIR . P20 (20 per cent
of the remaining
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debt is refinanced
each year)
Date of the disbursement of the loan . 01.01.1997
Closing time . 2 years
Shift in yield structure in . year 5 (rise in
interest rates of 3
percentage points)
In addition, the model requires information on the bonds
applied for the funding of the loan. Since the loan is a LAIR
type P20 where 20 per cent of the remaining debt is
refinanced each year, the loan is funded in six bonds. The
relevant information about the six bonds is listed in table
1.
Table 1
Coupon
Date of
interest Price
maturity
rate
01.01.1998 6 101.92
01.01.1999 6 102.62
01.01.2000 6 103.42
01.01.2001 7 106.28
01.01.2002 7 106.23
To better illustrate the understanding of the functionality
of the model, a permanent rise in interest rates in year 5
(subsequent to the 4th refinancing - before the 5th
refinancing) is assumed in the example. The rise in interest
rates causes the yield to maturity of all bonds to rise 3
percentage points and remain at this level until the loan
matures. This implies that the valid bond prices subsequent
to the 4th refinancing are as stated in table 2.
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Table 2
Coupon
interest Price
rate
6 99.07
6 97.13
6 98.06
7 95.98
7 93 . 94
The yield structure entered into the model when the loan is
disbursed is applied both for the disbursement and for the
refinancing until the year in which the yield structure is
altered as stated above.
Subsequent to the second refinancing of the loan, a new bond
with the term to maturity of 5 years is required in the
model. As the price of this bond is not known when the loan
is disbursed, the model "issues" a bond with a term to
maturity of 5 years with data identical to the bond with a
term to maturity of 5 years at the disbursement of the loan.
In other words, the model reuses the bond data entered at the
time of disbursement of the loan until the year in which the
yield structure is shifted. From then on, the model reuses
the new bond data in the remaining term to maturity.
The results of the model may be related to the debtor side
and the bond side, respectively, the debtor side and the bond
side being linked together by the balance requirement and the
proceeds criterion.
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Repayment of the loan:
The debtor side:
The model outputs a complete amortization profile
in accordance with the repayment principle entered. In
addition, the amortization profile observes the limits for
the payments on the loan which are given a lower priority
than the limits for the term to maturity.
The amortization profile includes the payments on the loan
comprising interest payments and repayments, and the
remaining debt at the end of each year and the interest rate
on the loan.
The bond side:
Correspondingly, the model outputs coupon payments on the
bonds, bond redemptions, and remaining debt at the beginning
of each settlement period.
The amortization profile is shown in table 3.
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o, o~ o~ o~ 0 0 0 0 0~ o~ o~ a~ ,o ~o
h h h h
N N N N rl e-1,- 1 M M M M lf7 In
-i
M M M M M M M M M M M M M M
r1 1J
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N 4J
11 S-I
1.1
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(~ 1~
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f~
rl ~1
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'
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l~
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p ~r ~ ~ o rn ov ov oo ,-~.-i,~ r h h
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e-iri e~ r-I ri
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dy N N
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CA 02282642 1999-08-25
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N N N N d~ d~ d~ d~ N N
lf1U7 1D l0 l~ l0 01 O~ O~ O~ sy~ ~ ~ y W
ri tJ lf1L(1tO ~O ~O ~D O O O O r r r r pp t0
M M M M M M tD ~O l0 ~O r r r r CO
r4 U1 -i '-I~ ri r-1 v-1wi wl '-ie1 ~-1 r1 ,~
N N
l~ H
?a N
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I~ 1W1
-ri S..I
M M Lf1Lf1lf1 Lf1M M M M O O O O O O
M M O~ O~ O1 O~ N N N N O O O O O O
L11U'1tf1Ln Lf1 Lf101 01 01 Ov 00 OD d0 aO O O
O O O O O O d v-1r-In-iN N N N M M
N N N N N N N N N N N N N N N N
N
b a ~
-ri
N
JJ M QO O OD CO Ov N CO tfiV1 lf1 t0 O t0 O~ 00
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N ~r m ~ r co o, -i m o ao u wo co rn er
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a
N
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CA 02282642 1999-08-25
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N N tf1Lf1In U1 N N N N M M M M Ov T
QO OD CO OO d0 OD M M M M N N N N O O
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CA 02282642 1999-08-25
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01 O~ CO W CO c0 N N N N lf 7 U1 tf1
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CA 02282642 1999-08-25
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V~ O O O O N N N N If1tf1 U'1L( W ~p
p
lf1t11 GO c0 CO OO 01 01 01 01 a0 c0 OD W O O
r r r r O O O O T O~
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CA 02282642 1999-08-25
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0 o co co ao co r r r r
o, ov ~ .1 .-~.1
N N M M M M 01 O~ O1 p~
~-1 N N N N N N N N N N
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ro ~
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1J d~ N 1I1r1 d~ V' r N CO r
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ra
CA 02282642 1999-08-25
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The debtor side as well as the bond side are specified in
detail in two additional tables.
The debtor side:
In table 4 the refinancing percentage, the interest rate on
the loan, the remaining term to maturity, the total term to
maturity, the actual payments on the loan, and the maximum
and minimum limits for the payments on the loan are shown on
a yearly basis. Based on the table it may be verified whether
the model observes the requirements and specifications
entered and, in addition, how the model adjusts the term to
maturity to comply with the limits for the payments on the
loan.
CA 02282642 1999-08-25
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O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O
N N N N N N N N N N N N N N N
.~ o ro
-, w C1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
O O O O O O O O O O O O O O O
w y~
M M M M M M M M M M M M M M M
N N N N N N N N N N N N N N N
ro w x
ro
r-i ~J
01 01 d~ M l(1M O O O O O O O O O
~ I~ M M O~ N O O O O O O O O O
M N 'd~t(1 tf1O~ CO O O O O O O O O
H ~ ~ O O O O O O r-IN M M M M M M ~
H N N N N N N N N N N N N N N N
ro
~ v
U ~ ro
ro u' a
O O O O O O O t0 QO Lf7(~ M
O 1~J N N N N N N N OD I~ M d~ eh N p~ dp
tD N d~ l~ lf1O N V~
J~-~ ~ M l~ O~ O .~ N N N
ro ~-. O O O v-1,1 ,.-~,--~
~ N N N N N N N N
l~ 1~ F.,
O Ov a0 f' WD lf1~ OD c0 U1 l~ M 01 01 l~
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.r~i O ~ N V~ t~ 1ttO N V~
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1J
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ro ~
v v ro
s
co o, r u-, o o, u~ ao o m N co ,~ cv co
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S.1 ro M N M C~ ~ d~ O tf100 O N M er t11I17
O lf1lf1lf7lf1 lJ1tD l~ I~ L~ OD CO 00 OO CD 00
1.1 r1
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ro
v v v
v a
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
N N N N N N N N N N N N N N N
W d1
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v
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w ~r
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w
O
O O O O O O O O O O O O O O O
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N N N N N N N N N N N N N N N
a.~ M W W t1~ L4 W (1,W LL 4L L~ W p~ ?~ LL
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ri
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0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
C'a ~ ~.I
N N N N N N N
r-I .-1
O ft1
W a
O O O O O O O
O O O O O O O
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M M M M M M M
N N N N N N N
ri ,T;
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, O O O O O ~ 10
~ O O O O O O to
M
.-I N Ci M M M M M Lf1U1
O
Q H N N N N N .~ H
t~ rt N
U W d ,~
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N ~O d' OD DO N N
H 01 eh ~--1.-1N N
O 11 lU l0 f a0 Ov
yJ rl N N N N N
r-i ~ 1
1It -~. H e-iri r-ir-1
~
!J y.~ N N N N N
yl
O N 10
W t~
N l0 ~ OD 00 N .~
b1 '-iO~ ~' m-Iri
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-r-1 O N N N N N
1.1
Ci J-1
rl
rl ~.i ~O Lf1V~ M N
~-1 1~
a~ v ro
~ s
CO N O W tf1M ~f1
-i
N H N ~O V~ N N Ov
yJ N f e-i Yn t0 f l0
Iy 0 l0 f CO e-1N ~O
O OD c0 CO CO 01 O W
1
1J r1 1.1 H
.
N
N N N
N W
1~
~ H
o a
0 0 0 0 0 0 0
N N N N N N
ri d1
U c0
C
N
-rl U
N N
s~ a
w
0
0 0 0 0 0 0 0
a~ x
a H O O O O O O O
iJ W LL Pa W W LL W
f CD O~ O .1 N
~-1H r1 r1 N N N
IC
N
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Table 4 shows that the intended refinancing profile is
complied with each year.
Based on the intended term to maturity the model calculates
the payments on the loan. Given that the payments on the loan
comply with the limits entered, the model continues to
calculate with this term to maturity until the model finds
that the limits for the payment can no longer be observed.
Table 4 shows that at the disbursment of the loan, the
intended term to maturity of 20 years corresponds to payments
on the loan within the maximum and minimum limits of DKK
23,000 and DKK 20,000, respectively. Thus, the model
continues to use the term to maturity of 20 years as long as
the limits for the payments on the loan are observed.
The interest rate column shows that the model applies the new
bond data from the 5th refinancing and onwards. A LAIR type
P20 is characterised by the fact that the full impact of a
rise in interest rates takes 5 years. This is recognized in
table 4 where the interest rate on the loan has in fact risen
about 3 percentage points during a 5-year period.
Furthermore, table 4 shows that before the full impact of the
rise in interest rates has manifested itself, it is necessary
for the model to prolong the term to maturity beyond the
intended 20 years. In year 8, the term to maturity is
prolonged to 20.3688 years in order to reduce the payment on
the loan to the level of the maximum limit of DKK 23,000. As
the interest rate on the loan rises further after year 8, the
term to maturity must be continuously prolonged so as to
maintain the payments on the loan within the band. The terms
calculated by the model during the term to maturity of the
loan are designated as technical terms to maturity, since it
is only the term to maturity in the last year that indicates
how long the loan actually exists.
CA 02282642 1999-08-25
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This process continues until the closing time period is
reached. By definition, the loan is required to mature on a
bond settlement date, thus, when the loan is refinanced the
last time, the remaining term to maturity must be exactly one
year. This leaves a choice of rounding an odd term to
maturity either up or down to a whole year. To avoid a
situation where bonds issued mature later than the loan, the
term to maturity of a LAIR type P is always rounded up.
Closing time designates the number of years prior to the
maturity of the loan where the remaining term to maturity is
to be rounded up. In this particular case, closing time is 2
years. Table 4 shows that the total term to maturity in year
is 21,2918 years. In year 21, the model reaches the
closing time period, and the term to maturity is rounded up
15 to 22 years. The procedure described may also be perceived
from the point of the remaining term to maturity.
Because of this procedure of rounding up the term to
maturity, it may become necessary to suspend the minimum
limit for the payment on the loan. This is shown in table 4
20 where the payment on the loan is lower than the minimum limit
in the last two years (the closing time period)
The bond side
The model outputs the initial funding and the prices of the
funding instruments applied. Table 5 shows that the model
initially funds the loan in 6 bonds with prices according to
the input. The columns in table 5 show data for the bonds
with a maturity of 1, 2, 3, 4, 5, or 6 years, respectively,
for each refinancing. The rows in table 5 show, for each
refinancing, the volume of bonds already issued of the bond
with the term to maturity in question, the price of the
bonds, and the marginal volume issued at the time of the
refinancing in question.
CA 02282642 1999-08-25
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O N M
[.-
N
O O
r1 O
?i
O
10
,~
O M 00 M M O O M tD
N N L~ N O~ O N N
W O w-i Lf7 tD N O l0 N
O ~ O O M O M
v-101 CO ri O rl N
N ro ~' M N m
O
ui
.~2
O CD 10 00 CQ M v-1 QO N
N u1 N N L~ t~ N tf1
W O O~ .-1 l0 M c0 10 t0
O N Lf1 O r-I M O CD
u-il~ 01 rl M CO '-Il0
O W -1 c'1 00 l7D
N ro ~ V~ M
O
d'
.~
O N N ~O N Ov ,-~ N tf1
V' d~ ti1 ~ t0 O V' e-1
M O~ 01 M \O lf1 M cr
O tf1 N O O~ 1D O ~D
e-1rl l~ rl LC1 N v-1e-~
ro
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Table 5 shows that when disbursed, the loan is funded as:
H(1): 219,535.99
H{2): 161,258.79
H(3): 155,159.42
H(4): 148,729.56
H{5): 141,951.28
H(6): 134,805.73
Thus, given the bond prices the proceeds criterion is
fulfilled:
{(219,535.99*101.92)+(161,258.79*102.62)+(155,
I59.42*103.42)+(148,729.56*106.28)+(141,951.28*
106.23)+(134,$05.73*104.92)}/100=1,000,000.00
Then, information on the marginal funding issued at each
refinancing is shown. In the particular example, the funding
at the first refinancing comprises:
H(1)=at the disbursement H(2): 53,168.11
H(2)=at the disbursement H(3): 45.881,59
H(3)=at the disbursement H(4): 38.596,69
H(4)=at the disbursement H(5): 31.313,73
H(5)=at the disbursement H(6): 24.032,98
Thus, when the first refinancing of the loan has been
performed, the total funding volume is:
H(1): (161,258.79+53,168.11)=214,426.90
H(2): (155,159.42+45,881.59)=201,041.01
H{3): (148,729.56+38,596.69)=187,326.25
H(4): (141,951.28+31,313.73)=173,265.01
H(5): (134,805.73+24,032.98)=158,838.71
At the second refinancing, the first funding instrument is
redeemed and, thus, is not be found in table 5. At this
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refinancing a new bond with a term to maturity of 5 years is
required. The model uses the same price for this bond as for
the bond with a term to maturity of 5 years when the loan was
disbursed. Table 5 also shows that the volume previously
issued is zero for this bond.
The funding proceeds (the refinancing amount) may be
calculated at each refinancing. When the loan is refinanced
for the first time, the refinancing amount is
{(53,168.11*101.92)+(45,881.59*102.62)+(38,596.69*103.4
2)+(31,313.73*106.28)+(24,032.98*106.23)}/100=200,000
This amount corresponds to 1/5 of the remaining debt being
refinanced. At the next refinancing the remaining debt has
been reduced and the refinancing amount is DKK 194,213.75.
Still, 1/5 of the remaining debt, at this point DKK
971,068.75 cf. the amortization profile, is refinanced.
Debtor side and bond side:
It is also possible to list the total payments on the debtor
side as well asd on the bond side to check that the balance
condition is fulfilled.
The model is solved by calculating the interest rate that
equals the payments on the debtor side with the payments on
the bond side, respectively, at the same time as all other
requirements are met. Given an interest rate on the loan of
5.31 per cent and a term to maturity of 20 years, the first
year's payments on the loan are DKK 81,477.32 cf. table 1.
On the bond side, the payments on the loan must equal the
coupon payments and the bond repayments. These are shown in
table 1, but may also be calculated explicitly on the basis
of the other result tables. The bond repayment in the first
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198
year is equal to the volumes of bonds redeemed at the first
refinancing of the loan minus the bonds issued at this
refinancing. Based on table 5 the bond repayments may be
calculated as
219,535.99-(53,168.11*101.92+45,881.59*102.62+
38,596.69*103.42+31,313.73*106.28+24,032,98*106.23)/100
=19,536.00
The coupon payments in the first year is also calculated on
the basis of table 5. The total coupon payments are
calculated by multiplying the volume of bonds issued and the
matching coupon interest rates.
219,535.99*6~+161,258.79*6~+155,159.42*6~+
148,729.56*7~+141,951.28*7~+134,805.73*7~=61,941.32
When adding the total bond repayments and coupon payments in
the first year, it appears that the total payments match the
payments on the loan in the first year exactly.
19,536.00+61,941.32=81,477.32
Thus the balance condition is fulfilled.
