Note: Descriptions are shown in the official language in which they were submitted.
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METHODS AND SYSTEMS FOR REPLICATING AN
INDEX WITH LIQUID INSTRUMENTS
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of U.S. Provisional Application No.
60/674,358, filed
April 22, 2005, and the benefit of U.S. Provisional Application No.
60/696,111, filed July 1,
2005. The entire contents of those two provisional applications are
incorporated herein by
reference.
BACKGROUND AND SUMMARY
Many bond indices contain a large number of securities, many of which are
illiquid or
simply not available in the secondary bond market. Consequently, simply
acquiring the indexed
securities in order to replicate the index is not feasible. Even if one could
buy the securities in
the secondary market, transaction costs would be prohibitive in obtaining
index returns. Thus,
bond index managers must find other ways to generate index returns while
minimizing risk.
Index replication is not just for passive managers of fixed-income portfolios.
Active
managers, managers of balanced fixed-income and equity portfolios, and plan
sponsors all may
wish to replicate the returns on, say, the Lehman Brothers U.S. Aggregate
Index or its sub-
components. (An overview of the Lehman Brothers U.S. Aggregate Index is
provided in
Appendix I.) While index replication has been of interest to a small group of
managers for a
number of years, there recently has been a substantial increase in interest in
replication strategies.
Though a desire to achieve index returns is a perfectly reasonable goal of
replication, demand for
replication strategies has been driven primarily by two very different needs.
First, low yields in fixed income markets and concerns over the likely future
performance
of equity_markets have spawned a "rush for alpha." (Alpha, as strictly defined
by the Capital
Asset Pricing Model, is the part of the return that is not explained by
exposure to the relevant
asset class.) This trend has manifested itself in a surge of inflows to hedge
funds, but a side
effect of this trend has been a broadening interest in "portable alpha"
strategies. Typically, a
portable alpha strategy involves the transfer of alpha from one asset class to
another. For
example, an equity manager uses equity futures to eliminate the "beta" from
stock market
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exposure, but preserves the alpha. The manager then uses non-cash instruments
to achieve the
desired bond market exposure (e.g., matching the Lehman Brothers Aggregate
Index).
Second, the increasing use of the Global Aggregate Index, a broad index of
investment
grade multi-currency fixed-income securities, has caused many managers to look
for strategies to
replicate its sub-components. A European-based manager may be adept at
managing European
credit and government securities, but may have less resources or expertise in
managing U.S.
fixed income. In particular, some non-U.S. managers may choose to refrain from
offering a
Global Aggregate product because they doubt their ability to manage U.S.
mortgage-backed
securities effectively. Since the Global Aggregate Index is fast becoming the
benchmark of
choice for many sponsors, such managers may be forced to forgo the possibility
of participating
in much of the growth in global fixed-income assignments. Instead, a strategy
of replicating
segments of the U.S. (and/or Global) Aggregate Index can allow such a manager
to offer a
Global Aggregate product. Indeed, derivatives can be used to create a
"portable alpha" strategy
for the Global Aggregate, in which the alpha from a 100% Euro fixed income
portfolio is
"transported" to a Global Aggregate Index.
There are additional reasons to replicate index returns. A U.S. fixed-income
active
manager who possesses skill in one aspect of fixed-income management (e.g.,
credit allocation)
may wish to offer the return of the Lehman Brothers Aggregate Index by
replicating the return
on the mortgage sector. Alternatively, this manager may, at any particular
time, wish to
eliminate the active risk in a given sector, either because the outlook for a
given sector is neutral
or because of a low level of confidence in a given view.
Plan sponsors engaged in asset allocation shifts are increasingly using
transition
managers to minimize implementation shortfall. Such transitions can involve
transactions in
multiple asset classes spread across more than a week. If the target portfolio
is fixed income, it
may be optimal to gain the desired exposure to fixed income at the beginning
of the transition,
before the liquidation of assets has even begun. If the legacy portfolio is
fixed income, there
may be a desire to retain fixed-income exposure throughout the transition. In
both cases, a
replicating portfolio of derivative instruments can achieve these objectives.
Similarly, asset managers may use replication strategies to manage portfolio
inflows and
outflows. For example, following an inflow, it may take days for new bonds to
be purchased. A
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replicating portfolio of derivatives can maintain market exposure on
uninvested cash. Similarly,
a replicating portfolio can maintain market exposure in the period between the
sale and
settlement of securities liquidated to meet a portfolio outflow.
Replication Methods
Methods of replicating bond indices generally fall into three categories:
replication with
cash instruments (i.e., bonds, not derivative instruments), replication with
derivatives, and total-
return index swaps.
Replication with cash instruments is an appropriate strategy in two kinds of
situations.
First, passive managers will generally use cash instruments to achieve very
low return deviations
from benchmark. This strategy makes sense for large portfolios with hundreds
of holdings, for
which the goal is pure indexation and the portfolio is fully funded. Second,
active managers may
wish to replicate that part of the benchmark for which they do not possess
skill. In this case,
however, using derivative instruments may be preferable, to permit the
managers to exercise skill
in other sectors (thereby generating alpha from 100% of portfolio assets).
Cash replications are
typically done using a stratified sampling approach, in which the index is
dissected into cells and
bonds are selected to represent the characteristics of each cell.
Managers who do not wish to use cash instruments, but also are not willing to
manage a
portfolio of derivative instruments, may choose to use total-return index
swaps. FIG. 1 shows
an example of a total return swap.
Under a total return swap, the investor is guaranteed to receive the total
return on the
index selected, in return for paying the counterparty floating-rate LIBOR,
plus a spread, to
compensate the dealer for the risk in hedging the index exposure. (Under the
swap, the basis risk
between a given replicating strategy and the index is effectively borne by the
dealer, who is
compensated for it by the investor.) This approach is appropriate for
investors with a high degree
of risk aversion or those with relatively long (one year and longer) time
horizons, owing to the
limited liquidity and higher transaction costs associated with a swap.
In most other situations, replication with derivative instruments is likely to
be preferable,
and this method is used in at least one embodiment of the present invention.
Derivative
instruments are highly liquid, have low transaction costs, and are unfunded
instruments. While
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there may be some basis risk between the derivative and underlying
instruments, this risk is
likely to be lower than the level of security-specific risk that a portfolio
of actively managed cash
instruments would typically possess.
Various methods of index replication for both U.S. and global indices are
known (see
references below). These include replication of the U.S. Aggregate and sub-
indices with futures
alone, as well as futures and swaps, replication of the U.S. MBS Index with
TBAs or large pools,
and replication of the Global Aggregate with both derivatives and cash
instruments. MBS stands
for Mortgage Backed Securities; TBA stands for To Be Announced, and refers to
the generic
forward market for mortgage backed securities. In this market, a coupon, par
quantity, agency,
maturity, and coupon characteristics are indicated, but the exact details,
such as specific pools,
are to be formalized at a later time. TBAs are discussed in more detail in
Appendix II.
Sources of Risk in the Lehman Brothers Aggregate Index
In considering the merits of various replication strategies, we should examine
the sources
of volatility in the U.S. Aggregate Index. Table 1 shows output from the
Lehman Brothers Risk
Model, which breaks down the sources of risk for the Lehman Brothers Aggregate
Index and
various sub-components. Details of the risk model are provided in "The Lehman
Brothers
Global Risk Model: A portfolio manager's guide", March 2005, accessible on
LehmanLive
Specifications of the MBS Risk Model and the Credit Risk Model are also
accessible from
Lehman Live.
Table 1: Sources of Risk in Lehman Brothers Indices, bp per month
Global Risk U.S. Aggregate U.S. Treasury U.S. MBS U.S. Credit
Factor
Yield Curve 150.03 141.78 77.65 150.91
Swap S reads 19.73 18.01 33.88
Volatility 7.34 0.06 10.33 0.30
Investment- 19.02 7.40 22.01 57.01
Grade Spreads
Treasury Spreads 0.79 7.40
Credit and 15.76 57.01
A enc S reads
MBS/Securitized 7.81 22.01
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CMBS/ABS 0.89
Systematic Risk 146.79 139.36 80.43 145.75
Idiosyncratic 2.74 0.61 2.83 7.89
Risk
Total Risk (bp 146.81 139.36 80.48 145.96
per month)
The Lehman Brothers Multi-Factor Risk Model quantifies the ex-ante tracking
error
volatility (the expected volatility of the return deviation) of a portfolio
versus its benchmark or
the absolute volatility of a portfolio or index. The model is based on the
historical returns of
individual securities in the Lehman Brothers Bond Indices, in many instances
dating back over
more than a decade. The model derives historical magnitudes of different
market risk factors and
the relationships among them. It then measures current mismatches between the
portfolio and
benchmark sensitivities to these risks and multiplies these mismatches by
historical volatilities
and correlations ("covariance matrix") to produce its output.
While tracking error volatility (TEV) is a measure of volatility, it can be
used (with
caution) to make forecasts of the likely'distribution of future relative
returns. For example,
assuming returns are normally distributed, a portfolio with a TEV of 25 bp per
month would be
expected to have a return within +/- 25 bp per month around the expected
return difference
between the portfolio and benchmark approximately two thirds of the time (and
underperformance of worse than -25 bp relative to the expected return
difference one-sixth of the
time).
The total volatility of a given index reflects the risk due to exposure to
various risk
factors and correlations between risk factors. Accordingly, the volatilities
are not additive. The
expected volatility of a given index can be expressed as a function of its
exposures to risk factors
and the volatility of those factors. The credit index (or an individual credit
security) will be
exposed to term structure risk, swap spread risk, credit spread risk
(together, "systematic risk"),
and idiosyncratic risk.
The risk characteristics of a given index determine which instruments can best
replicate
that index. For U.S. investment-grade fixed-income indices, term structure is
by far the
dominant source of risk. Therefore, a portfolio of treasury futures, matched
as closely as
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possible to the duration characteristics of the relevant index, should be able
to attain a reasonable
replication result. For mortgage-backed securities, swap spread risk is almost
as important as
MBS spread risk. Therefore, receiving fixed-rate interest rate swaps might be
expected to
achieve a better replication result than using treasury futures. For credit,
while swaps would also
be expected to achieve improved replication, additional instruments would be
needed to reduce
credit spread risk to achieve replication results closer to those of other
sectors.
In one aspect, the invention comprises a method for replicating a first index,
comprising:
constructing a basket of derivative financial instruments selected to
replicate said index; wherein
said basket of derivative financial instruments is constructed using key rate
duration matching
based on a plurality of instruments, and wherein said basket is reconstructed
on a periodic basis
approximately equal to that on which said index is reconstructed.
In various embodiments: (1) said plurality equals the number of types of
duration of
instruments in said index; (2) said first index is a fixed income index; (3)
said derivative
financial instruments comprise treasury futures; (4) said derivative financial
instruments
comprise interest rate swaps; (5) said derivative financial instruments
comprise CDX products;
(6) said derivative financial instruments comprise credit default swaps; (7)
said basket comprises
a second index; and (8) the method further comprises providing a total return
swap, wherein a
purchaser of said swap is guaranteed a return equivalent to that of said
index.
In another aspect, the invention comprises offering a total return swap for
sale, wherein
said total return swap is as described above.
In another aspect, the invention comprises a method comprising offering a
basket of
derivative financial instruments for sale, wherein said basket of derivative
financial instruments
is as described above.
In another aspect, the invention comprises a method for replicating a
portfolio of
securities, comprising: constructing a basket of derivative financial
instruments selected to
replicate said portfolio; wherein said basket of derivative financial
instruments is constructed
using key rate duration matching based on a plurality of instruments, and
wherein said basket is
reconstructed on a periodic basis approximately equal to that on which said
portfolio is
reconstructed.
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In various embodiments: (1) said plurality equals the number of types of
duration of
instruments in said index; (2) said first index is a fixed income index; (3)
said derivative
financial instruments comprise treasury futures; (4) said derivative financial
instruments
comprise interest rate swaps; (5) said derivative financial instruments
comprise CDX products;
(6) said derivative financial instruments comprise credit default swaps; (7)
wherein said basket
comprises a second index; and (8) the method further comprises providing a
total return swap,
wherein a purchaser of said swap is guaranteed a return equivalent to that of
said index.
In another aspect, the invention comprises offering a total return swap for
sale, wherein
said total return swap is as described above.
In another aspect, the invention comprises offering a basket of derivative
financial
instruments for sale, wherein said basket of derivative financial instruments
is as described
above.
Embodiments of the present invention comprise mathematical models, computer
components and computer-implemented steps that will be apparent to those
skilled in the art.
For ease of exposition, not every step or element of the present invention is
described herein as
part of a computer system, but those skilled in the art will recognize that
each step or element
may have a corresponding mathematical model, computer system or software
component. Such
computer system and/or software components are therefore enabled by describing
their
corresponding steps or elements (that is, their functionality), and are within
the scope of the
present invention.
The present invention comprises a methodology, described below, for
replicating a fixed
income index or portfolio. The indices include, but are not limited to:
The Lehman Global Aggregate Bond index and all of its subindices
The Lehman U.S. Aggregate Bond Index and all of its subindices
The Lehman Pan-European Aggregate Bond Index and all of its subindices
The Lehman Asia Pacific Aggregate Bond Index and all of its subindices
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The Lehman Global Treasury Index and all of its subindices
The Lehman Multiverse Index and all of its subindices.
A description of each index is provided in Reference 13.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 depicts a total return swap on the Lehman Brothers Aggregate Index.
FIG. 2 depicts option-adjusted spreads for the U.S. Credit and Mirror Swap
Credit Index.
FIG. 3 depicts option-adjusted spread of current coupon FNCL 30-year MBS
versus 5-
year swap spread.
FIG. 4 depicts a relationship between credit spreads and CDS.
FIG. 5a depicts realized return differences of MBS replication and credit
replication.
FIG. 5b depicts realized return differences of "full" aggregate replication
strategy.
FIG. 6 depicts changes in the sectoral distribution of the Lehman U.S.
Aggregate Index
over time.
FIG. 7 depicts the sectoral distribution of the Lehman U.S. Aggregate Index.
FIG. 8 depicts the distribution of the Lehman U.S. Aggregate Index by quality
(rating).
FIG. 9 depicts mechanics of a typical default swap.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
Preferred Derivatives Replication Strategy
An examination of the sources of risk in various indices indicates that a
replicating
portfolio that matches the systematic exposure of these indices might achieve
reasonable results
in delivering acceptably low levels of tracking error. However, there are at
least two categories
of choices in building such a portfolio: a choice of instruments and a choice
of replication
technique. See Table 2.
Table 2: Decisions in Forming a Replication Strategy with Derivative
Instruments
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Instruments
= Bond Futures
= Interest-Rate Futures
= Interest-Rate Swaps
= Mortgage TBAs
= Credit-Default Swaps
Replication Techniques
= Stratified Sampling (Cell Matching)
= Key-Rate Duration Matching
= Minimum Variance Hedge
Approaches to Replicating Exposures
There are three main approaches to replication:
= A stratified sampling approach divides the index into duration cells. A
derivative
instrument is selected for each cell, in an amount to match the duration
exposure of that cell.
= A key-rate duration (KRD) approach attempts to match the overall key-rate
duration
exposures of the index. Key-rate duration measures sensitivity to shifts at
specific "key-rate"
points along the yield curve (and can therefore measure the effect of non-
parallel yield curve
shifts), in comparison with "conventional duration," which measures
sensitivity to parallel yield
curve shifts.
= A minimum-variance hedge approach, with the help of a risk model, seeks to
minimize
the predicted tracking error of a replicating portfolio against its index.
Therefore, the replicating
portfolio reflects correlations between sectors and instruments in the
portfolio and index-for
example, between corporate and government bonds.
Previous replication studies have used a stratified sampling approach. Since
2001, KRDs
have been used, and in a recent study ("Replicating Index Returns with
Treasury Futures:
Duration Cells versus Key-Rate Durations," Global Relative Value, July 2004),
it was
demonstrated that such an approach has delivered modestly lower tracking
errors than the
stratified sampling approach. The regression hedge approach is more model-
driven and less
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transparent than the other two approaches. Furthermore, it is reliant on the
relationships between
different risk factors-for example, between term structure movements and
credit spread
changes, which change over time. At least some embodiments of the present
invention comprise
various replication strategies using the KRD-matching approach.
In the Lehman Brothers Yield Curve Model, there are six key rates (see Table
3). In
some cases, however, there are fewer than six instruments available for
replication (e.g.,
replication with Treasury futures, for which there are only four separate
instruments).
Accordingly, it is not feasible to match all six key-rate durations.
Table 3: Key-Rate Durations of Treasury Futures Contracts as of August 31,
2004
Key-Rate Duration
Contract 6-Mo 2-Yr 5-Yr 10-Yr 20-Yr 30-Yr
2-Year - 0.07 1.97 0.06 0.00 0.00 0.00
5-Year 0.00 0.70 3.55 0.00 0.00 0.00
10-Year 0.01 0.05 3.41 2.85 0.00 0.00
Long Bond 0.01 0.05 0.23 2.65 8.16 0.61
Replication Strategies
Replication with Treasury Futures
The number of bond futures contracts available-the 2-year, 5-year, 10-year,
and long
contracts-is not sufficient to achieve a perfect match of the six KRDs in the
Lehman Brothers
Yield Curve Model. There are two possible choices for dealing with this issue.
First, an
optimization can be established to minimize the sum of the squared differences
between the
respective index and the replicating portfolio KRDs. However, a preferred
embodiment uses a
second method, reducing the number of key-rates to equal the number of
available instruments in
order to achieve a perfect match, by combining the 6-month and 2-year key-rate
durations and
the 20- and 30-year KRDs. As Table 3 demonstrates, the keyrate duration
exposure of the bond
futures contracts is minimal for the 6-month rate, while only the long bond
contract has any
exposure to the 20- or 30-year rate. Nevertheless, there will still be an
unavoidable mismatch
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between the duration exposure of the futures replicating portfolio and the
Aggregate Index. The
sum of the KRDs of the 20- and 30-year vertices can be matched with a single
instrument, but
the KRD exposure of both vertices cannot be matched separately.
Replication with Interest Rate Swaps
The fixed-rate leg of an interest-rate swap represents the average of forward
rates, which
reflect the credit quality of the panel of banks that set the LIBOR rates.
Therefore, the pricing of
interest-rate swaps reflects a credit risk premium, while their spread to
treasuries will also reflect
a liquidity premium. Accordingly, receiving the fixed component of an interest-
rate swap would
be expected to provide a better alternative to replicating the returns of non-
Treasury components
of the Aggregate Index. In addition, since the swap curve is effectively
continuous, one may
select six instruments to match exactly the key-rate duration profile of the
Aggregate Index.
The historical relationships between yields on various indices and on
portfolios of
duration-matched interest rate swaps can be examined using the Lehman Brothers
Mirror Swaps
Indices. The Mirror Swap Index is a portfolio of interest rate swaps
(receiving fixed) constructed
to match the key-rate duration profiles of various Lehman Brothers indices.
For more details,
see "The Lehman Brothers Swaps Indices," January 2002.
In addition, for investors who do not wish to enter several interest-rate
swaps, Lehman
Brothers offers a total-return swap on various Mirror Swap Indices. This also
eliminates the
need to rebalance the portfolio to bring duration exposures back in line as
the index changes
from month to month and swap instruments age.
Replication with Futures and Interest Rate Swaps
An extension of the futures and swaps replication is to use treasury futures
to replicate
the treasury sector, and swaps to replicate the non-treasury sectors. For this
strategy, the term-
structure replication error of the treasury component (see above) can be
eliminated using swaps.
Replication of the MBS Index with TBAs
The mortgage-backed securities (MBS) sector represents a large component of
the
Aggregate Index. The availability of liquid instruments to replicate the index
and a
straightforward method for doing so suggests that such an approach should not
greatly increase
the complexity relative to a futures-only or swaps-only replication. While
futures and swaps can
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replicate the yield curve exposures of the MBS index, they leave exposure to
MBS spread,
prepayment, and volatility effects. Using a mortgage product can improve the
replication
considerably by hedging these exposures as well. TBAs offer two key advantages
over MBS
pools in replication strategies: they are suitable for an unfunded strategy-
since no cash outlay is
required, prior to settlement a TBA is simply rolled from month to month; and
the back-office
aspects of investing in mortgages are much simpler for TBAs than for pools,
since monthly
interest payments and principal paydowns are avoided. The remaining risk in a
TBA replication
is essentially due to the difference in risk characteristics between new and
seasoned mortgages.
See Appendix II for more details.
Replication of the Credit Index with CDS and Interest Rate Swaps
While interest-rate swap spreads are at times highly correlated with credit
spreads, there
have been extended periods during which this relationship has broken down. In
such periods,
LIBOR spreads have typically remained quite stable while credit spreads have
been quite
volatile. For example, FIG. 2 shows that 2002 was a period of great volatility
for credit spreads,
while swap spreads, as measured by the Mirror Swap Credit Index, were
relatively stable. A
review of credit-default swaps is provided in Appendix III.
Portfolio credit default swap (CDS) baskets now provide a very liquid
instrument that
investors can use to take a long (or short) position in credit. Credit yields
can be broken down
into two constituents: the swap yield and a credit spread to swaps.
Accordingly, one can match
the exposure of credit to movements in swap yields using interest-rate swaps
and the exposure to
movements in LIBOR credit spreads by using CDS. The widely traded CDX.NA.IG
products
are baskets of 125 equally weighted CDS available in 5- and 10-year
maturities. In at least one
embodiment, 5- and 10-year CDX products are combined in proportions sufficient
to match the
spread duration and yield of the Credit Index. CDS and CDX are discussed in
more detail in
Appendix III.
Since these instruments have been available only since October. 2003, a period
of stable
credit spreads, it is difficult to gauge the benefits of including them in a
credit index replication
strategy. Therefore, one embodiment supplements the CDX data by valuing
portfolios of CDS
instruments constructed from the issuers that composed the CDX basket as of
October 2003, for
the period June 2002 to September 2003. A look-forward bias is introduced by
doing this.
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CDX-IG by construction comprises investment-grade-only issuers. In
constructing a basket in
October 2003 valued back to July 2002, one is certain to avoid some issuers
that were
downgraded over the period that may have been included in a basket actually
constructed in
2002. A large number of names in the basket (e.g., 125) mitigates this risk.
During the period,
EP and AHOLD were investment-grade issuers that were downgraded to high yield
that might
reasonably have been expected to have been included in a CDS basket. They
represented 0.4%
and 0.1% of the Credit Index, respectively, in the month prior to downgrade.
In addition to the
basis risk that exists between CDS and credit, there is an additional basis
that exists between
CDX and the underlying CDS.
Performance Summary of Replication Strategies
The key metric by which at least one embodiment measures the performance of
various
replication strategies is tracking error volatility (TEV). This is preferable
to using average out
(under) performance for several reasons. The volatility of returns tends to be
much more
persistent than the returns themselves; that is, history is a much better
guide for predicting
volatility than for predicting return. Also, it is unlikely that a period of
substantial
underperformance of a given replication strategy will persist, since this
would imply a secular
cheapening in a group of highly liquid derivative instruments, or a secular
trend in credit or MBS
spreads. Finally, the objective of any replication strategy is to replicate
the index, not
outperform. Outperformance is what active managers are paid for. Nevertheless,
mean
outperformance of each replicating strategy is reported herein, in order to
give a flavor for the
degrees of out (under) performance.
Table 4 shows the results of replicating the Lehman Brothers Aggregate Index
and
selected sub-indices using the approaches described above. The replication of
the Treasury
Index with treasury futures achieves an acceptable TEV of 10.6 bp per month.
Over this period,
the futures portfolio outperformed the Treasury Index. This is consistent with
prior studies that
showed mean outperformance of 3.1 bp per month over three separate time
periods. See
"Hedging and Replication of Fixed Income Portfolios," Dynkin et al., Journal
of Portfolio
Management, March 2002. This reflects two effects. This replication assumes
that cash is
invested at LIBOR, which over the past two years has had a 1.8bp per month
higher yield than
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treasury bills. The residual outperfomance suggests that the premium that long
futures positions
enjoy for being short the cash bond delivery option has been "too large" over
these periods.
Treasury futures fare less well, as expected, as instruments to replicate MBS
and Credit
Indices. While term structure risk is reduced, spread risk remains. Prior
studies have found that
interest-rate swaps delivered measurable reductions in tracking error compared
with Treasury
futures when replicating the MBS and Credit Indices. In the most recent
period, however, while
swaps deliver lower TEV against the Credit Index, they have a higher TEV for
replication of the
MBS Index compared with using Treasury futures.
Figure 3 shows that there has been a close relationship between mortgage
spreads and
swap spreads, so it might seem that swaps should have performed better than
futures. The
replication results suggest, however, that other factors are responsible for
this effect. In recent
years, swaps have been a favored tool for the convexity hedging of MBS
securities, and therefore
swap spreads have tended to behave directionally, tightening as Treasury
yields fall and
widening as they rise. Therefore, using swaps in a replication in place of
treasury futures may
increase the effective duration mismatch of the replication strategy. An
additional factor is the
optionality of MBS and futures. A buyer of futures is short a delivery option.
(There are
actually several delivery options, the value of all of which is positively
affected by interest-rate
volatility. The seller has the option to deliver one of a basket of cash
securities to the buyer.
Therefore, the futures buyer is short interest-rate volatility, as is the MBS
buyer. A combination
of swaps and swaptions would benefit from the correlation of swaps with MBS,
as well as the
exposure to interest-rate volatility.
Table 4: Index Replication Results (August 2002-September 2004, bp per Month)
Replication Method Mean Out- Tracking R2
performance Error
Volatility
a. U.S. Treasury Index Replication
Treasury Futures 4.5 10.4 0.997
b. U.S. MBS Index Replication
Treasury Futures 1.2 35.3 0.811
Interest-Rate Swaps -1.8 38.5 0.775
TBAs
0.3 4.3 0.997
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c. U.S. Credit Index Replication
Treasury Futures -25.1 62.7 0.878
Interest-rate Swaps -26.9 57.8 0.896
Interest-rate Swaps + CDX 2.5 29.1 0.974
d. U.S. A re ate Index Replication
Treasury Futures -5.2 22.7 0.972
Interest-Rate Swaps -7.4 17.5 0.983
Futures+Swaps -7.1 17.3 0.983
Futures+Swaps+TBAs -6.1 16.9 0.984
Futures+Swaps+CDX 0.7 10.9 0.994
Futures+Swaps+TBAs+CDX 1.6 9.4 0.995
Interest-rate swaps improve upon the replication of the Credit Index with
futures given
the credit exposure embedded in interest-rate swaps. FIG. 2 shows that swap
spreads have been
relatively stable during a period of volatility in credit spreads. The sharp
contraction in credit
spreads caused futures and swaps replications to under-perform the Credit
Index significantly, in
return terms. While swap spreads and credit spreads were relatively stable
following the fourth
quarter of 2003, the period prior to that was far from stable.
The use of CDX in the replication improves upon the replication with swaps
alone. As
FIG. 4 shows, CDS spreads tracked credit spreads closely over this period.
Also, the relative
advantage of CDS, compared to swaps alone, was much greater during the earlier
period of
volatility.
Table 5 demonstrates that the tracking error of the swaps-only strategy was
more than
twice as large that of the swaps+CDS strategy during the period of greater
spread volatility. An
additional benefit of CDS is the greater carry earned by the portfolio. In
return for accepting
default risk (which is reflected also in the credit index), the investor earns
that incremental carry.
As long as CDS spreads are sufficient to offset default losses, CDS will
increase expected return
and reduce risk.
Bringing together all of the various replication strategies listed in section
(d) of Table 4,
one can see how the tracking error of the Aggregate Index improves as more
replicating
instruments are added. The most notable improvement is adding CDS, which
reduced the
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volatility by 6.5-8.0 bp. While TBAs are greatly superior to other methods in
replicating the
MBS Index by itself (4.1 bp TEV versus 36 bp for replication with futures),
TBAs do not greatly
improve the replication of the Aggregate Index. Table 6 gives us some insight
into this.
Comparing the first two lines in the correlation matrix shows a substantial
negative
correlation between the MBS replication with swaps and the treasury
replication with futures.
There is a smaller, positive correlation between the MBS replication with TBAs
and the futures
replication. This reflects the volatility effect highlighted above. In an
environment of rising
interest-rate volatility, futures would be expected to underperform cash
treasuries, and swaps
would outperform MBS (strong negative correlation). In that same environment,
TBAs would
tend to underperform the MBS Index (weak positive correlation) as TBAs tend to
have higher
volatility exposures than the more seasoned issues in the index. The
correlation of the credit
replication strategy with the two MBS replication strategies is also notably
different. Rising
interest-rate volatility causes swaps to outperform MBS, while convexity-
hedging caused them
to underperform credit, demonstrating a negative correlation between the MBS-
with-swaps
replication and the Credit-with-swaps replication. An example of this can be
seen in FIG. 5,
which plots the return difference to benchmark of various replication
strategies. In July 2003,
the Aggregate Index fell by 3.36%, as yields rose 94 bp. Swap spreads widened,
causing swaps
to underperform duration-matched Treasuries, though they outperformed MBS.
Replicating
portfolios for both the credit index and the Aggregate index using swaps
substantially
underperformed, and so we see a negative correlation between these replication
strategies, and
the MBS replication-with-swaps strategy. During this same month, the TBA
replication strategy
also underperformed, a positive correlation with the non-MBS replication
strategies. Therefore,
a swaps replication strategy for MBS, while notably inferior for replicating
mortgages in
isolation, is little different from TBA replication as part of an Aggregate
Index replication
strategy.
FIG. 5b demonstrates that the return differential of the full Aggregate
replication strategy
is driven by the the performance of the Credit Index replication. Indeed, 91%
of the volatility of
the Aggregate replication strategy over this period can be explained by the
Credit Index
replication (as measured by r-squared).
Table 5: Credit Replication Tracking Error (bp per month) in Two Different Sub-
Periods
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8/02-9/03 10/03-9/04 Total Period
Swaps only 75.9 22.6 57.8
Swaps + CDS 34.7 19.0 29.1
Table 6: Correlations of Realized Return Differentials of Replicating
Strategies
Correlation Swaps for MBS TBAs for MBS Futures for UST Swaps for
Credit
Swaps for MBS 1.000 -0.533 -0.732 -0.268
TBAs for MBS -0.533 1.000 0.343 0.156
Futures for UST -0.732 0.343 1.000 0.364
Swaps for Credit -0.268 0.156 0.364 1.000
The replication "errors" of various strategies can be explained in some cases
by the
presence of a risk factor in the index, exposure to which cannot be reflected
in the replicating
portfolio. For example, the futures replication of the Aggregate Index
attempts to replicate its
term structure exposure, but cannot replicate its credit exposure. Not
surprisingly, as Table 7
shows, the realized return differential of the futures portfolio to the
Aggregate index is highly
correlated with changes in credit spreads. On the other hand, the return
differential of the "full
replication" strategy is not correlated with credit spreads.
These findings have important implications for the choice of replication
strategy.
Considered in isolation and given investor risk preferences, the choice of
strategy may be clear.
However, if this replication strategy is part of a larger portfolio, the
relationship between the
return difference of a given replication strategy and the returns of other
portfolio assets must be
considered. For example, an investor with sizeable equity exposure may prefer
a fixed-income
replication strategy using only futures, given the negative correlation with
equity returns shown
in Table 7. Falling equity prices have been correlated with rising credit
spreads and, therefore,
with excess returns to a credit replication strategy with bond futures (and
swaps).
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Table 7: Correlations of Selected Aggregate Replication Strategies with Credit
Spreads
and Equities
Futures Replication "Full" Replication*
Correl. w-ve change in -0.847 0.065
OAS Credit Index
Correl. w change in S&P -0.505 0.047
500 Index
* Replication with Futures, Swaps, TBAs, and CDX
Using a Risk Model to Forecast Replication Risk
While an empirical analysis is valuable in forecasting the likely tracking
errors of various
replication strategies, there are some drawbacks to this approach. Most
important, the
weightings and characteristics of the sectors within the Lehman Aggregate
index change over
time, and this will affect the relative success of each index replication
strategy. FIG.6 shows that
the sectoral distribution of the Aggregate Index has changed markedly over
time. Credit spreads
are the dominant source of risk in replication strategies. Accordingly, one
would expect that
replication performance would change depending on the weight of credit
instruments in the
Aggregate. There may, therefore, be some bias introduced into forecasts of
Aggregate
replication TEVs, by differences in the characteristics of the index over
time. The use of a risk
model can eliminate such biases.
The Lehman Global Risk Model forecasts the volatility of the return difference
(TEV)
between a portfolio and its benchmark. The TEV uses the current index weights
and the current
relative exposures between portfolio and benchmark (e.g., key-rate durations)
and the historic
volatilities and correlations of risk factors (e.g., yield changes).
Therefore, the Risk Model
approach generates a TEV forecast that is independent of changes in index
characteristics over
time.
Table 8 shows three replicating portfolios created to track the Lehman
Aggregate for
August 2004, using only Treasury futures, futures, and swaps, and a
combination of futures,
swaps, and TBAs. In each case, the forecast TEV is within 1-2 bp of the
empirically achieved
result. The risk model covariance matrix is constructed from many months of
data, which
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greatly increases the confidence in the forecast TEV suggested by these
empirical results,
accumulated over 25 monthly observations.
Table 8: Sources of Risk (Factor Volatilities) in the Lehman Aggregate and
Replicating
Strategies-Exponentially Wei hted Co-variance Matrix
Global Risk Factor Lehman Treasury Futures + Futures +
A re ate Futures Swaps Swa s+ TBAs
Yield Curve 150.0 3.2 6.0 2.7
Swap Spreads 19.7 19.7 1.8 0.8
Volatility 7.3 7.3 7.3 0.4
Investment-Grade Spreads 19.0 19.0 19.0 16.5
Treasury Spreads 0.8 0.8 0.8 0.8
Credit and Agency Spreads 15.8 15.8 15.8 15.8
MBS/Securitized 7.8 7.8 7.8 0.9
CMBS/ABS 0.9 0.9 0.9 0.9
Systematic risk 146.8 23.1 19.3 16.0
Idiosyncratic risk 2.7 6.4 3.1 3.3
Total risk (bp per month) 146.8 24.0 19.6 16.3
Empirically derived risk N/A 22.7 17.3 16.9
The risk model output also provides insight into the risks that are reduced
through various
replication strategies, as well as quantifying the exposures and risk factor
volatilities that remain.
Table 8 illustrates the importance of yield curve risk as part of the overall
volatility of the
Lehman Aggregate. Each replication strategy largely eliminates this source of
risk, leaving other
risk exposures. The risk of the futures replication strategy is not
surprisingly dominated by
credit and agency spread risk, while MBS spread risk and volatility risk
(which largely reflects
the optionality of MBS) also are significant. Using futures introduces
idiosyncratic risk,
reflecting the basis risk between cash and futures instruments. Spread risk
factors are expressed
relative to swaps, with the exception of Treasuries. Therefore, replicating
credit or MBS using
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swaps reduces the forecast TEV attributable to swaps spreads, but leaves the
TEV attributable to
credit and MBS spreads unchanged.
The risk model forecasts a reduction in TEV of 3.3 bp by using TBAs to
replicate the
MBS portion of the Aggregate, compared to using swaps. Empirical analysis
showed only a
reduction of 0.4 bp, however. This demonstrates the closer correlation between
swaps and MBS
during the past two years, than over longer periods during which the risk
model was calibrated.
This increased correlation caused swaps to perform almost as well as TBAs over
the period of
the empirical study. Using both empirical analysis and a risk model to
forecast replication
tracking errors allows investors to view the effect of changes in correlations
between
instruments. Using an exponentially weighted, or a simple-weighted covariance
matrix for ex-
ante tracking errors can also allow for the impact of changing correlations on
TEV.
The replication with futures, swaps, and TBAs is dominated by credit spread
risk.
Therefore, using CDS improves the replication, as the empirical results show.
Replication Details
A sample U.S. Aggregate replication portfolio is provided in Appendix IV, for
a portfolio
of notional size US $1 billion as at July 31, 2004.
Rebalancing, Re-investment, and Transaction Costs
In the empirical studies, all positions are assumed to be rebalanced monthly.
In practice,
most investors will make small adjustments monthly to positions to allow for
the changing
characteristics of the index and the aging of derivative positions. On a
quarterly basis, futures
will be rolled to prevent the exercise of the delivery option and swaps will
be rolled into the "on-
the-run" maturities. TBAs are rolled monthly to avoid pool delivery. New CDX
instruments are
created semi-annually, and it may be assumed that a roll into the new
instrument is executed with
that same frequency.
During the period between the creation of new CDX instruments, it is possible
that an
issuer will be downgraded, causing it to fall out of the Credit Index (but
remain in CDX).
During this period, the investor may be subject to tracking error, as the
performance of the
"fallen angel" may not match that of the investment-grade credits. Based on an
analysis of the
historic performance of fallen angels, in the months following a fall below
investment grade and
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the credit ratings of CDX, this risk is estimated to be 7 bp per month for the
credit index. (We
discuss the performance of fallen angels and distressed bonds in Portfolio and
Index Strategies
During Stressful Credit Markets, January 2004.) However, this risk can largely
be eliminated if
the investor buys single-name default protection for the downgraded issuer.
An all-derivatives portfolio, by definition, does not require cash, outside of
that needed to
meet variation margin for futures or mark-to-market collateral calls for
swaps. Cash is assumed
to be invested in 1-month LIBOR. In practice, investors will be required to
deposit initial margin
with the clearing firm (current CBOT initial margin requirements for 2-year, 5-
year, 10-year, and
long bond futures are $743, $810, $1,350, and $2,025 per contract,
respectively), which, for an
Aggregate Index replicating portfolio, currently averages 1.3% of the notional
portfolio amount.
However, both this and any variation margin can be posted in the form of T-
bills. As a result,
only a small portion of funds will be invested below LIBOR in practice.
Transaction costs will depend upon the choice of strategy and the frequency of
rebalancing. Table 9 displays estimated transaction costs, assuming monthly
rebalancing.
Table 9: Transaction Costs of Various Replication Strategies
Replication Strategy Cost (bp per month)
Futures 0.5
Swaps 0.3
Futures + Swaps 0.3
Futures + Swaps + TBAs 0.9
Futures + Swaps + TBAs + CDX 1.0
Replicating the Global Aggregate (or just the U.S. portion)
Using a combination of strategies can achieve the lowest tracking error for
replicating the
U.S. Aggregate Index. Whether this also holds true for the Global Aggregate
depends upon the
choices of strategies in the various currency "blocks," and whether these are
active or passive
replication strategies.
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Table 10 suggests that the choice of replicating strategy in the U.S. may be
correlated
with the strategy used for managing the Euro Aggregate component (of the
Global Aggregate).
In Table 10, the correlations between the return differences of two
replicating strategies (versus
benchmark) and returns on Euro credit (excess return) and Euro governments
(price return) are
shown. The return differences of the futures replication strategy turn out to
be strongly
negatively correlated with excess return to Euro credit. This is not
surprising, since the risk from
a futures-only replication strategy of the U.S. Aggregate is largely coming
from credit (see Table
8); the short U.S. credit exposure is negatively correlated with long Euro
credit. This may be
attractive for an active European-based investor if the value-added generated
is positively
correlated with European credit excess returns (and therefore negatively
correlated with the U.S.
replication strategy). However, for many investors, a low correlation will be
preferred, since the
overall risk of the portfolio will be reduced, whether the investor is short
or long Euro credit.
There is a modest improvement in tracking error contributed by replicating the
U.S. MBS
index with TBAs. This improvement would be reduced further in a Global
Aggregate-
benchmarked portfolio, since the weighting of the MBS index is much smaller,
and the higher
tracking error associated with replicating the index with swaps is diversified
away.
Different replicating strategies for the non-U.S. portions of the Global
Aggregate will
have different correlations with the U.S. replication. Table 10 suggests that
if Futures replication
is used for the U.S. portion, there will be a significant positive correlation
with a Euro replication
strategy that is effectively short Euro credit (e.g., a Euro-futures
replication strategy).
Fortunately, there are replication strategies that a Global Aggregate manager
can use to replicate
the Euro-Aggregate that mirror the techniques discussed herein for replicating
the U.S.
Aggregate. In particular, portfolio CDS products such as iTRAXX can be used,
together with
interest-rate swaps, to replicate the Euro-credit index. It is believed that
the use of iTRAXX,
together with a portfolio of interest rate swaps, can reduce the tracking
error associated with
replicating the Euro-credit index.
Table 10: Aggregate Replication Error (bp per month) in Two Different Sub-
periods
Futures Replication Full Replication
Correlation with Euro Credit Excess Return -0.69 -0.02
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Correlation with Euro Government Price Return 0.07 0.29
Choices
There are various considerations in choosing the appropriate replication
strategy.
Portfolio constraints may ultimately determine the choice of strategy, perhaps
restricting the
investor to a futures-only strategy or a combination not considered herein
(e.g., futures + TBAs).
In the absence of client constraints, the investor's risk "utility function"
(i.e., cost per unit of risk
reduction) will determine the choice of strategy. If the degree of risk
aversion is high, a total
return swap may prove to be a desirable choice. However, for large replicating
portfolios (e.g.,
above $300 million), sufficient liquidity may not exist to permit the use of
an index swap for the
entire portfolio.
The choice of replication method should not be considered in isolation but
rather in
combination with the overall strategy. It is not necessarily the case that the
lowest TEV strategy
is always preferable. For example, if the replication is part of a portable
alpha strategy, the
relationship of the expected return deviations from benchmark of various
replication strategies
should be considered relative to the expected alpha of the strategy. A
replication strategy for the
Aggregate Index using treasury futures will outperform during times of
widening spreads and
underperform in the opposite environment. The correlation of this performance
pattern to the
alpha strategy may actually make this a more attractive option than a
replication strategy that, by
itself, has a lower tracking error. The choice of replication strategy to be
used for the MBS
Index will depend upon whether the entire Aggregate index is being replicated
or just the
mortgage component.
Other Embodiments
At least one embodiment of the present invention comprises a computer-
implemented
method for creating a total return swap on a Replicating Bond Index (RBI)
basket (for example,
a total return swap on the Lehman Brothers U.S. Aggregate RBI basket). While
one embodiment
may be used to create RBI baskets for the U.S. Aggregate, and the U.S. Credit
index, other
embodiments, apparent to those skilled in the art, can be used to create RBI
baskets for the
Lehman Global Aggregate (of which the U.S. Aggregate and Credit Indices are
subsets). There
are several innovations related to this method. For example:
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1. The creation of a total return swap on a basket of instruments that
replicates a bond
index (there have been total return swaps on bond indices, but not on
replicating portfolios).
2. The creation of options on a basket of instruments that replicates a bond
index.
3. The creation of a structured note, the payment on which is linked to the
return of an
RBI basket.
4. The creation of a structured note or Special Purpose Vehicle that combines
an RBI
basket with an "alpha" source, such as a Hedge Fund of Funds
5. The process of constructing the RBI basket.
6. The use of Lehman Swap Indices, (or equivalents thereof) in a replicating
basket.
4. The use of Lehman fixed income indices in a replicating basket.
In one embodiment, a legal agreement for the transaction comprises a standard
total
return swap term sheet (Party A pays LIBOR + X b.p., Party B pays the return
on RBI basket),
and a "fact sheet" that describes the construction of the RBI basket. An
exemplary preferred fact
sheet is provided below.
Factsheet:
The Lehman U.S. Aggregate Index ("Aggregate Index") contains U.S. dollar
denominated securities that qualify under the index's rules for inclusion,
which is based on the
currency of the issue. The principal asset classes in the index are
Government, Credit and
Securitized bonds. The Aggregate Index was launched on January 1, 1976
The Replicating Bond Index (RBI) basket is an index designed to track the
return of the
Aggregate Index. Series 1 uses a combination of liquid instruments and Lehman
sub-indices to
track the Aggregate Index.
RBI Basket Construction: The components of the RBI basket will be adjusted
monthly in
order that the weightings to each index or instrument match the published
weightings of the
Aggregate Index. In Series 1, the sectors within the Aggregate Index will be
matched as shown
in Table 11.
Table 11
Sector Index/Instrument
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Treasury Lehman U.S. Treasury Index
Mortgage Lehman U.S. MBS Index
Credit Lehman Mirror Swap U.S. Credit Index + CDX..N.A.IG 5yr and l0yr
Agency Lehman Mirror Swap U.S. Agency Index
ABS Lehman Mirror Swap U.S. ABS Index
CMBS Lehman Mirror Swap U.S. CMBS Index
Lehman Mirror Swap indices provide published total returns of a portfolio of
interest-rate
swaps constructed to match the key-rate durations of major Lehman bond
indices. The Lehman
Brothers U.S. Credit Index ("Credit Index") is replicated using a combination
of the Mirror Swap
Credit Index and the most current investment grade CDX instruments with 5 and
10 year
maturities. The allocations to CDX are computed in order that the weighted
average Spread
DV01, will be the Spread DV01 of the Credit Index and the weighted average
spread to LIBOR
of CDX will equal the differential between the Option-Adjusted Spread (OAS) on
the Credit
Index and the OAS on the Mirror Swap Credit Index, values as reported on
LehmanLive.
Table 12: Pricing and Related Issues
Issue Index/Instrument
Pricing Frequency Daily on T+1 basis
Timing of pricing 3:00 pm New York time
Bid or Offer Outstanding issues are priced on bid side. New issues enter on
the
offer side
Sources Lehman trading desks
Verification All prices are checked against a blend of multiple contributors
by our
quality control group. Variations are analyzed and corrected if
necessary
Reinvestmen of Index cashflows are reinvested at the start of the month
following their
cashflows receipt
Interest on cash Mirror Swap indices assume that cash is invested at 3mth
LIBOR
End of Factsheet
Reference 3, cited below, discusses replication of the Global Aggregate index.
That
paper describes a number of different approaches to replicating bond indices.
Embodiments of
the subject invention comprise at least two new innovations to these
approaches to replication.
The first innovation, described below, provides an improved approach for
matching the interest-
rate sensitivity of a given index. Previous approaches split the index into
duration "buckets"
(e.g., 0-3 year duration, 3-5 year, etc.), and matched the interest-rate
sensitivity of one future to
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one bucket. At least one embodiment of the present invention comprises
matching the full
duration profile of the index, using a Key-rate duration approach. A further
embodiment of the
present invention comprises utilizing total return swaps on certain components
of a bond index in
addition to swaps on baskets or replicating instruments in order to replicate
a broad index.
Applications
Applications of RBI baskets include the following:
As the beta component in a portable alpha strategy
To express an investment view by creating or eliminating broad exposure to a
market
index
In management of portfolio cash inflows and outflows
To preserve market exposure during the course of an asset re-allocation.
To hedge the market exposure of variable annuity providers
To create enhanced index products, for example, by combining an RBI basket
with a
portfolio of floating-rate assets.
Replicating Index Returns with Treasury Futures: Duration Cells versus Ke, -y
Rate
Durations
Since the rediscovery of duration in the late 1970s, investors have been
looking for better
ways to measure interest-rate sensitivity. Duration proved to be a useful
measure of price
sensitivity to parallel shifts in the yield curve, though managers recognized
that for nonparallel
shifts, additional information was needed to gauge interest-rate risk
properly. Many managers
sliced their portfolios and indices into maturity buckets and used duration
distribution across
these buckets. As managers switched from government/credit benchmarks to
aggregate
benchmarks, with a high percentage of callable securities, duration buckets
replaced maturity
buckets.
In recent years, partial durations have become increasingly popular as a
measure of yield
curve sensitivity. Instead of a single duration number, a vector of partial
durations describes the
sensitivity to yield curve twists. The sensitivity of a given bond to a non-
parallel yield curve
movement is a function of the distribution of its cash flows. If a portfolio
is constructed from
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bullet bonds, the present values of whose cash flows are largely distributed
within a narrow
maturity "window" (e.g., bullet securities), duration bucketing should give a
reasonable view of
yield curve risk. However, where the present value contributions from bonds'
cash flows are
distributed more evenly across the curve (e.g., amortizing securities such as
MBS), duration
bucketing is likely to be less satisfactory.
KRD is related to partial duration. Certain points on the par curve are
selected as key
rates. For maturities between the key rates, it is assumed that rates move
according to linear
interpolation. For example, in Lehman's model, six key rates are used-6-month
, 2-year, 5-
year, 10-year, 20-year, and 30-year. A 5-year KR shift assumes no shift in the
2- or 10- year
rate, and interpolated shifts between the 5- and 2-year and the 5- and 10-
year, a so-called tent
shift. The 5-year KRD of any bond is then the sensitivity of the bond price to
a 100 bp shift in
the 5-year key rate with an appropriate tent shift in the term structure
between two and ten years.
The durations and key-rate durations (KRD) of a Treasury security and an MBS
security
are shown in Table 13. Both securities have near-identical option-adjusted
durations (OAD), but
very different interest rate profiles. Accordingly, a long position in one
security, offset by a
short position in the other, will be sensitive to non-parallel interest-rate
movements. The
Lehman Brothers global risk model can quantify the yield curve risk arising
from this KRD
mismatch. (Risk is a function of the exposure (the key rate duration mismatch)
and the historical
volatility of that exposure.) Examining just the term structure risk due to
the KRD mismatch
(excluding risk due to convexity or sector mismatches), this is found to be
7.8 bp of return
volatility per month.
Table 13: Option-Adjusted Duration and Key-Rate Durations of U.S. Treasury and
Mortgage Security
Key Rate Duration
OAD 6-Mo 2-Yr 5-Yr 10-Yr 20-Yr 30-Yr
UST 6.5% 2/10 4.74 0.02 0.10 4.05 0.57 0.00 0.00
FNMA 5.5% 2003 4.73 0.15 0.57 0.99 1.77 1.15 0.11
It is often argued that duration bucketing should provide a reasonable picture
of interest
rate exposure for diversified portfolios and indices. The reasoning is that
while some securities
may indeed be placed into duration buckets that do not reflect their true
interest-rate sensitivities,
perhaps these errors are reduced in large portfolios. To examine this
assertion, in Table 14 the
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duration profiles of the Lehman Brothers Intermediate Treasury Index and the
U.S. MBS Index
are compared.
Table 14: Comparative Duration Exposures for the Intermediate Treasury
and Mortgage Indices, as of May 31, 2004
Duration
0-2 Yr 2-4 Yr 4-6 Yr 6-8 Yr 8-10 Yr 10+ Yr
Market Value (%)
Intermediate Treasury 35.07 29.13 19.84 12.94 0.00 0.00
MBS 5.30 40.04 50.60 4.07 0.00 0.00
OAD Contributions
Intermediate Treasury 0.54 0.86 0.90 0.93 0.00 0.00
MBS 0.09 1.28 2.52 0.26 0.00 0.00
Key-Rate Durations 6-Mo 2-Yr 5-Yr 10-Yr 20-Yr 30-Yr
Intermediate Treasury 0.14 0.96 1.37 0.98 0.00 0.00
MBS 0.16 0.56 1.02 1.53 0.78 0.09
A comparison of the duration contributions with the KRDs shows that for bullet
bonds,
duration bucketing provides a reasonable view of yield curve exposure. For the
Intermediate
Treasury Index, the buckets' duration contributions provide a view of yield
curve exposure not
too different from the KRD profile. However, for the MBS Index, duration
buckets present a
somewhat misleading picture. If the Treasury Index is viewed as a portfolio
and the MBS Index
as its benchmark, a duration-bucketing view would suggest that the portfolio
has a large yield-
curve mismatch compared with the index. In particular, the portfolio would
seem to have a
substantial underweight in the 4- to 6-year duration bucket, almost fully
offset by an overweight
in the 6- to 8-year duration bucket. Accordingly, a hypothetical portfolio
manager might
conclude that the portfolio was exposed to yield curve flattening and choose
to reduce risk by
increasing exposure to the 6- to 8-year bucket. However, the KRD exposures
tell a very different
story. The portfolio is overweighted to 5- and 10-year yield curve points and
underweighted to
the 20-year point. Therefore, a hypothetical manager is actually exposed to a
yield-curve
steepening.
As an exercise, the return effect of a particular yield curve shift is
examined. In Table 15,
instantaneous shifts of plus and minus 25 bp are applied to the 5-year key
rate and every security
in each index is revalued. By the definition of key-rate shifts, the move in
the 5-year will not
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affect the par rates shorter than two and longer than ten years, but will
affect intervening
maturities at a declining linear rate.
Table 15 shows that the Treasury Index is more sensitive to a shift in the 5-
year rate than
the MBS Index. This is consistent with the sensitivities indicated by the KRD
profile in Table
14, but is not consistent at all with the duration bucketing pattern.
Table 15: Projected Total Returns under Instantaneous Yield Curve Shifts Total
Return (bp) under Scenario
Index 5-Year KR Down 25 bp 5-Yr KR Up 25 bp
Intermediate Treasury 34.6 -34.3
MBS 26.8 -30.1
Empirical test is perhaps the most effective way of gauging whether KRDs are
indeed
superior as a measure of yield curve exposure. In particular, one can test
whether a strategy that
seeks to replicate a given index by matching KRDs is superior (i.e., results
in a lower tracking
error) to one that matches the index by duration bucketing.
In a series of studies dating back to 1997, techniques for replicating returns
of popular
Lehman Brothers indices with baskets of Treasury futures were developed. These
techniques are
popular with asset managers engaged in portable alpha strategies or in active
tactical asset
allocation. See Replicating Index Returns with Treasury Futures, Lehman
Brothers, November
1997; Replication with Derivatives-The Global Aggregate Index and the Japanese
Aggregate
Index, Lehman Brothers, March 2001; "Hedging and Replication of Fixed-Income
Portfolios,"
Dynkin, Hyman, and Lindner, The Journal of Fixed Income, March 2002. As part
of these
studies, the tracking errors associated with replicating various indices were
examined using a
duration-bucketing approach. Typically, the relevant index is divided into
four duration cells: 0-
3 year, 3-5 year, 5-7.5 year, and 7.5 years and higher, with the exception of
mortgages. (For the
MBS Index, given the lack of long-duration securities, we eliminate the 7.5 +
duration bucket so
that the third bucket becomes 5+ year duration, which is replicated with 10-
year note futures
contracts.) For a given target portfolio size, the number of 2-year, 5-year,
10-year, and long
bond futures contracts required to match the dollar duration of each cell is
calculated. At the end
of each month, this calculation is performed on the forward-looking
("statistics") universe of the
index, and the numbers of futures contracts are adjusted as appropriate. Once
a quarter, the
contracts are rolled to avoid the possibility of the exercise of the delivery
option.
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As discussed above, an alternative to the duration bucketing approach is KRD-
matching,
which minimizes the differences between the KRD profiles of a given index and
the replicating
futures position. Because there are six KRDs in Lehman's term structure model
and only four
futures contracts, it is not possible to achieve a perfect match. Therefore,
an optimization that
minimizes the sum of the squared differences between the respective index and
replicating
portfolio KRDs is set up, subject to the constraint that the sum of the KRDs
must be identical.
The cash is assumed to be invested in 1-month LIBOR.
Table 16 shows the results of replications of the U.S. Treasury Index, the MBS
Index,
and the Credit Index, using the duration bucketing approach and the KRD-
matching approach.
Table 16: Comparison of the KRD Replication Approach with Duration Bucketing,
Monthly Rebalancing, June 2000-Apri12004
Monthly Tracking Error Volatility (bp)
Index Duration Buckets KRD Matching Difference Percent
Treasury 10.7 8.6 -2.1 -19.10
MBS 38.3 36.9 -1.4 -3.60
Credit 87.9 86.7 -1.2 -1.40
The KRD-matching approach does improve tracking in the replication of all
three
indices. The biggest improvement, in both absolute and percentage terms, is
achieved in
replicating the Treasury Index. This is not entirely unexpected, since yield
curve exposure is the
only important source of risk, where the advantage of KRD matching matters
most. On the other
hand, the Credit Index shows the smallest improvement because of the magnitude
of other risk
exposures.
A replication strategy using duration buckets was developed in 1997. In mid-
2000, key-
rate durations for U.S. fixed income securities and for bond futures were
generated. Recent
analysis suggests that using key-rate durations to replicate indices leads to
a small improvement
in the performance of replication strategies using futures.
The second innovation, described above, combines separate replication
instruments
previously used separately, and also uses a relatively new instrument (CDX).
Mirror swap indices were first created by Lehman Brothers in 2002 and use a
key-rate
duration approach to match the term structure exposure of a given index with a
portfolio of
interest-rate swaps. See reference 8, cited below. In at least one embodiment
of the present
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invention, in constructing an RBI basket, use is made of a number of different
techniques
outlined herein, to replicate sub-sectors of various indices. Additionally, in
the case of certain
subsectors, the RBI basket may include the index itself (e.g., the U.S.
treasury index).
Bibliogrraphy of Relevant Publications
1. Replicating Index Returns with Treasury Futures, November 1997
2. The Global Aggregate: Return Replication with Derivatives, September 2000
3. Replication with Derivatives: The Global Aggregate Index and the Japanese
Aggregate Index, March 2001
4. Tradable Proxy Portfolios for the Lehman Brothers MBS Index, July 2001
5. The Replication of the Lehman Global Aggregate Index with Cash Instruments,
August 2001
6. L. Dynkin, J. Hyman, and P. Lindner (2002), Hedging and Replication of
Fixed-
Income Portfolios, The Journal of Fixed Income, March, pp. 43-63.
7. Replicating Index Returns with Treasury Futures: Duration Cells versus Key-
rate
Durations, July 2004
8. The Lehman Brothers Swap Indices, January 2002
9. Swaps as a Total Return Investment, April 2003
10. Simulating Portable Credit Strategies with CDS and Mirror Swap Indices,
October
2003
11. Credit Derivatives Explained, March 2001
12. Replicating the Lehman Global Aggregate Index with Liquid Instruments,
August
2005
13. A Guide to the Lehman Brothers Global Family of Indices, March 2006
All publications referenced above may be accessed from the Quantitative
Portfolio
Strategy site under Global Strategy on LehmanLive (live.lehman.com), except 10-
11 (Lehman
Quantitative Credit Research). Also, 3 is included in provisional application
no. 60/674,358 as
Appendix C; 6 is included as Appendix B; and 8 is included as Appendix F.
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APPENDIX I THE LEHMAN U.S. AGGREGATE INDEX
The U.S. Aggregate Index contains U.S. dollar-denominated securities that
qualify under
the index's rules for inclusion. See FIGS. 7 and 8, and Tables 17-20 below.
Inclusion is based
on the currency of the issue, and not the domicile of the issuer. The
principal asset classes in the
index are Government, Credit (including corporate issues), and Securitised
bonds. Securities in
the index roll up to the US Universal and Global Aggregate Indices. The U.S.
Aggregate Index
was launched on January 1, 1976.
Table 17
Access to the Index
Index Client Website = Index and constituent-level data
www.lehmanlive.com = Performance time series
= Index turnover reports
KEY FEATURES = Fully customisable views
= Standardised market structure reports
= Guides to indices and portfolio strategies
Bloomberg Page LEHM = Total Return Index Value: LBUSTRUU
= Market Value: LBUSMVU
Tickers for Key Data Series = Yield to Worst: LBUSYW
= Mod. Adj. Dur. (Returns Universe):
LBUSRMD
= Average OAS: LBUSOAS
= Maturity: LBUSMAT
POINT (Portfolio and Index Tool) = Performance attribution
Accessible for selected clients via = Market structure reports
www.lehmanlive.com = Index constituents
= Portfolio upload/analysis
KEY FEATURES = Multi-factor global risk model
= Tracking error optimiser
= Automated batch processing
= Fully customisable
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Table 18
Pricing and Related Issues
Frequency Daily, on a T+1 basis. If the last business day of the month is a
holiday in the U.S. market,
then prices from the previous business day are used.
Tiniing 3:00 pm New York time.
Bid or Offer Side Outstanding issues are priced on the bid side. New issues
enter the index on the offer side.
Sources Lehman trading desks.
Methodology Multi-contributor verifications: The Lehman price for each
security is checked against a
blend of alternative valuations by our quality control group. Variations are
analyzed and
corrected, as necessary.
Reinvestment of Index cashflows are reinvested at the start of the month
following their receipt. There is no
Cashflows return on cash held intra-month.
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Table 19
Rules for Inclusion
Amount Outstanding $250 million as of July 1, 2004
Quality A minimum bond level rating of Baa3 from Moody's Investors Service or
BBB-
from Standard & Poor's Ratings Group.
= The lower of the'two agencies' ratings applied for qualification purposes
= Where a rating from only one agency is available, that rating is used to
determine the bond's index rating
= Unrated securities are included if an issuer rating is applicable
= Unrated subordinated securities are included if a subordinated issuer rating
is
applicable
Maturity = One year minimum to final maturity on dated bonds, regardless of
put or call
features
= Undated securities are included in the index provided their coupons switch
from
fixed to variable rate. These are included until one year before their first
call
dates, providing they meet all other index criteria
Seniority of Debt Senior and subordinated issues are included. Undated
securities are included
provided their coupons switch from fixed to variable rate.
The following types of fixed to variable-rate security structures will also
qualify for
the index
= If the holder has the option to force the issuer to issue preference shares
post the
call date
= If there are other economic incentives for the issuer to call the issue,
such as the
removal of tax benefits after the first call date
Fixed rate perpetual capital securities which remain fixed rate following
their first
call dates, and which provide no economic incentives to call the bonds, are
excluded.
Currency of Issue US dollars
Market of Issue US public debt market
Security Types Included: Excluded:
= Fixed rate bullet, puttable = Bonds with equity-type features (e.g.,
and callable warrants, convertibility to equity)
= Soft bullets = Private placements are excluded
= Floating rate issues
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Table 20
Rebalancing Rules
Frequency Statistic (projected) Universe: Daily.
Returns Universe: Monthly, on the last business day of the month.
Methodology During the month, all indicative changes to securities are
reflected in both the statistics
(projected) universe and returns universe on a daily basis. This would include
changes to
ratings, amounts outstanding, or sector. These changes affect the
qualification of securities
in the statistics (projected) universe on a daily basis, but only affect the
qualification of
bonds for the returns universe at the end of the month.
Timing Qualifying securities issued, but not necessarily settled, on or before
the month-end
rebalancing date qualify for inclusion in the following month's returns
universe.
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APPENDIX II REPLICATING THE LEHMAN MBS INDEX
Mortgage securities constitute a significant portion of the Lehman Brothers
Aggregate
Index and the Lehman Global Aggregate Index (35.5% and 14.2% of market value,
respectively,
as of September 30, 2004). To track these indices, it is desirable to take
exposure to the U.S.
mortgage market. To some global investors, the U.S. mortgage market is
enigmatic and
intimidating because of its arcane terminology and highly variable cash flows.
However, while
achieving outperformance in this market indeed requires considerable knowledge
and
experience, the MBS Index is easier to track.
The Lehman MBS Index consists of tradable fixed-rate mortgage pass-through
securities,
and is limited to conforming pools guaranteed by the U.S. government (Ginnie
Mae) or by
government-sponsored enterprises (Fannie Mae and Freddie Mac). In lieu of
buying a pool, an
investor can buy a TBA (to-be-announced) contract that is a forward contract
to buy MBS pools
of a given agency/program and coupon. The specific pools that the investor is
buying are
unknown until two days before settlement. Because it is a forward contract, no
cash outlay is
required until settlement. For example, in October 2004, an investor could
agree to buy a 30-
year FNMA 5.5% TBA for delivery and settlement on November 15, 2004. The
investor could
choose to take delivery of the security, or roll the TBA, by selling the same
TBA prior to
settlement date, and purchasing a TBA for December delivery. By purchasing a
portfolio of
TBAs, an investor can maintain exposure to the MBS market without ever taking
delivery of any
pools.
Generally, buyers and sellers of TBA contracts on current production mortgage
coupons
implicitly assume average attributes of the pools likely to be delivered. In
other words, a TBA
contract corresponds to a large pool of recently issued loans or a current
production index
composite. Because there is ample supply of new production to deliver against
the TBA contract
and little prepayment history to help identify pools with potentially highly
idiosyncratic
prepayment behavior, it is likely that a current coupon TBA contract will
closely track the
current production index composite.
TBAs offer two key advantages to investors. First, they are suitable for an
all-derivative
mortgage-replication strategy, since no cash outlay is required. Second, the
TBA strategy greatly
simplifies back-office processing because there is no physical delivery of
pools, and therefore
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there are no monthly interest and principal payments. There also are some
disadvantages. A
change in the prepayment quality of TBA deliverables versus the rest of the
MBS market can
lead to underperformance of TBAs, even if the investor rolls their TBAs from
month-to-month.
Since the seller of a TBA has the option to deliver any mortgage pool, he will
generally deliver
the least attractive pool, which is reflected in the pricing of TBAs. The
investor can also at times
earn significant return from rolling TBAs due to imbalances in the current
month's supply and
demand for a particular mortgage coupon.
A detailed description of the construction of TBA portfolios to replicate the
MBS Index
is provided in the paper on "Tradable Proxy Portfolios for the Lehman MBS
Index," listed in the
bibliography herein.
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APPENDIX III CREDIT DEFAULT SWAPS
The primary purpose of credit derivatives is to enable the efficient transfer
and
repackaging of credit risk. "Credit risk" encompasses all credit related
events ranging from a
spread widening, through a ratings downgrade, all the way to default. In their
simplest form,
credit derivatives provide a more efficient way to replicate in a derivative
form the credit risks
that would otherwise exist in a standard cash instrument. For example, a
standard credit default
swap can be replicated using a cash bond and the repo market. Alternatively, a
cash credit
instrument can be replicated by combining a credit default swap with the fixed
receipt of an
interest-rate swap.
A default swap is a bilateral contract that enables an investor to buy
protection against the
risk of default of an asset issued by a specified reference entity. Following
a defined credit
event, the buyer of protection receives a payment intended to compensate
against the loss on the
investment. This is depicted in FIG. 9. In return, the buyer of protection
pays a fee. Usually,
the fee is paid over the life of the transaction in the form of a regular
accruing cash flow. The
contract is typically specified using the confirmation document and legal
definitions produced by
the International Swap and Derivatives Association (ISDA).
Some default swaps define the triggering of a credit event using a reference
asset. The
main purpose of the reference asset is to specify exactly the capital
structure seniority of the debt
that is covered. The reference asset also is important in the determination of
the recovery value
should the default swap be cash settled. In many cases, following a default,
the protection buyer
will deliver a defaulted security for which it will receive par from the
protection seller. The
maturity of the default swap need not be the same as the maturity of the
reference asset; it is
common to specify a reference asset with a longer maturity than the default
swap.
CDX.NA.IG is a static portfolio of 125 equally weighted credit default swaps
on 125
North American reference entities that are rated investment grade; it is
available in a range of
maturities. Every six months a new set of CDX instruments is created, though
existing
instruments will continue to trade. Like individual CDS, they are unfunded
instruments. A
credit event triggered by a reference asset will be settled by the physical
delivery of a deliverable
defaulted security in exchange for par. By combining CDX with a portfolio of
interest rate
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swaps (receiving fixed), it is possible to replicate, in unfunded form, the
exposures of a portfolio
of cash credit instruments.
This appendix draws on material from the Lehman publication "Credit
Derivatives
Explained" (cited herein).
FIG. 9 depicts mechanics of a typical default swap. Between trade initiation
and default
or maturity, protection buyer makes regular payments of default swap spread to
protection seller.
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APPENDIX IV REPLICATING PORTFOLIO AS AT JUL-31ST 2004
Table 21
Sec- Identifier Position Description Cou- Maturity
tor Amount pon Date
Cash
USD $1,000,000,00 CASH - U.S. Dollar
0
Sector: FUTURES
(4 positions)
TUU4:CBT $98,800,000 2 year Treasury Notes
FVU4:CBT -$1,800,000 5 year Treasury Notes
TYU4:CBT $75,800,000 10 year Treasury Notes
USU4:CBT $42,700,000 30 year US Treasury Bonds
Sector: INTEREST_RATE_SWAP
(6 positions)
IRD_9327 $72,227,000 IRSwap USD 1.965 LIBOR 1/31/2005
6M
IRD_9332 $144,781,000 IRSwap USD 3.087 LIBOR 7/31/2006
2Y
IRD_9335 $135,037,000 IRSwap USD 4.199 LIBOR 7/30/2009
5Y
IRD_9338 $67,731,000 IRSwap USD 4.99 LIBOR 7/30/2014
10Y
IRD_9341 $21,536,000 IRSwap USD 5.535 LIBOR 7/30/2024
20Y
IRD_9344 $16,971,000 IRSwap USD 3.0 LIBOR 7/30/2034
30Y
Sector: MORTGAGES (2
positions)
FNC044QG $43,608,833 FNMA Conventional Interm. 4.5
15yr
FNC050QG $39,688,972 FNMA Conventional Interm. 5.0
15yr
FNC054QG $13,943,931 FNMA Conventional Interm. 5.5
15yr
FNC060QG $18,488,638 FNMA Conventional Interm. 6.0
15yr
FNA054QG $42,630,967 FNMA Conventional Long 5.5
T. 30yr
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FNA060QG $38,007,295 FNMA Conventional Long 6.0
T. 30yr
FNA064QG $54,389,948 FNMA Conventional Long 6.5
T. 30yr
FGB050QG $31,962,198 FHLM Gold Guar Single 5.0
F.30yr
FGB054QG $27,945,910 FHLM Gold Guar Single 5.5
F.30yr
GNA064Q $22,404,484 GNMA ISingle Family 30yr 6.5
G
GNA060Q $13,129,784 GNMA ISingle Family 30yr 6.0
G
GNA054Q $1,961,961 GNMA ISingle Family 30yr 5.5
G
GNA050Q $10,934,080 GNMA ISingle Family 30yr 5.0
G
Sector: CREDIT DEFAULT
SWAPS (2 positions)
CDX.IG $167,429,000 CDX Investment Grade 5yr 9/20/2009
2/09 #2
CDX.IG $76,671,000 CDX Investment Grade 10yr 9/20/2014
2/14 #2
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Embodiments of the present invention comprise mathematical models, computer
components and computer-implemented steps that will be apparent to those
skilled in the art.
For ease of exposition, not every step or element of the present invention is
described herein as
part of a computer system, but those skilled in the art will recognize that
each step or element
may have a corresponding computer system or software component. Such computer
system
and/or software components are therefore enabled by describing their
corresponding steps or
elements (that is, their functionality), and are within the scope of the
present invention.
For example, all calculations preferably are performed by one or more
computers.
Moreover, all notifications and other communications, as well as all data
transfers, to the extent
allowed by law, preferably are transmitted electronically over a computer
network. Further, all
data preferably is stored in one or more electronic databases.
Various embodiments described herein are not intended to be mutually
exclusive; those
skilled in the art will recognize that various combinations of these and other
embodiments are
within the scope of the invention.
While particular elements, embodiments, and applications of the present
invention have
been shown and described, it should be understood that the invention is not
limited thereto, since
modifications may be made by those skilled in the art, particularly in light
of the foregoing
teaching. The appended claims are intended to cover all such modifications
that come within the
spirit and scope of the invention.
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