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WO2006116562A2 - Procedes et systemes permettant de reproduire un indice avec des instruments liquides - Google Patents

Procedes et systemes permettant de reproduire un indice avec des instruments liquides Download PDF

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Publication number
WO2006116562A2
WO2006116562A2 PCT/US2006/015946 US2006015946W WO2006116562A2 WO 2006116562 A2 WO2006116562 A2 WO 2006116562A2 US 2006015946 W US2006015946 W US 2006015946W WO 2006116562 A2 WO2006116562 A2 WO 2006116562A2
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Prior art keywords
index
replication
basket
credit
instruments
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PCT/US2006/015946
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English (en)
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WO2006116562A3 (fr
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Anthony Simon Gould
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Lehman Brothers Inc.
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Priority to CA002605466A priority Critical patent/CA2605466A1/fr
Priority to AU2006241069A priority patent/AU2006241069A1/en
Priority to JP2008508004A priority patent/JP2008538844A/ja
Publication of WO2006116562A2 publication Critical patent/WO2006116562A2/fr
Publication of WO2006116562A3 publication Critical patent/WO2006116562A3/fr

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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q40/00—Finance; Insurance; Tax strategies; Processing of corporate or income taxes
    • G06Q40/04—Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange

Definitions

  • 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.
  • 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.
  • 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.
  • 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.
  • 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, hi both cases, a replicating portfolio of derivative instruments can achieve these objectives.
  • 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 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 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.
  • 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.
  • 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.
  • FIG. 1 shows an example of 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.
  • 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.
  • Derivative instruments are highly liquid, have low transaction costs, and are unfunded instruments. While 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.
  • 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
  • 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.
  • TEV tracking error volatility
  • 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 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.
  • 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.
  • the invention comprises offering a total return swap for sale, wherein said total return swap is as described above.
  • 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.
  • 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.
  • 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.
  • the invention comprises offering a total return swap for sale, wherein said total return swap is as described above.
  • 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.
  • steps or elements of the present invention are 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 Asia Pacific Aggregate Bond Index and all of its subindices The Lehman Global Treasury Index and all of its subindices
  • 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.
  • 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.
  • 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.
  • 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.
  • 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 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.
  • 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.
  • 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.
  • the mortgage-backed securities (MBS) sector represents a large component of the Aggregate Index.
  • MBS mortgage-backed securities
  • 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 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
  • 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.
  • CDS Portfolio credit default swap
  • 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.
  • 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.
  • 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
  • TEV tracking error volatility
  • 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.
  • 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 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.
  • FIG 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.
  • 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.
  • 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.
  • CDX in the replication improves upon the replication with swaps alone.
  • 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.
  • 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.
  • futures would be expected to underperform cash treasuries, and swaps would outperform MBS (strong negative correlation).
  • MBS Index weak positive correlation
  • TBAs 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.
  • 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
  • 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.
  • the futures replication of the Aggregate Index attempts to replicate its term structure exposure, but cannot replicate its credit exposure.
  • Table 7 shows, the realized return differential of the futures portfolio to the Aggregate index is highly correlated with changes in credit spreads.
  • the return differential of the "full replication” strategy is not correlated with credit spreads.
  • 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.
  • 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 greatly increases the confidence in the forecast TEV suggested by these empirical results, accumulated over 25 monthly observations.
  • 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.
  • MBS spread risk and volatility risk which largely reflects the optionality of MBS
  • Spread risk factors are expressed relative to swaps, with the exception of Treasuries. Therefore, replicating credit or MBS using 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 investor may be subject to tracking error, as the performance of the " fallen angel” may not match that of the investment-grade credits.
  • 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.)
  • this risk can largely be eliminated if the investor buys single-name default protection for the downgraded issuer.
  • An all-derivatives portfolio does not require cash, outside of that needed to meet variation margin for futures or mark-to-market collateral calls for swaps.
  • 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).
  • the investor's risk "utility function" i.e., cost per unit of risk reduction
  • the degree of risk aversion is high, a total return swap may prove to be a desirable choice.
  • 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.
  • 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.
  • 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: 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).
  • RAI Replicating Bond Index
  • 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.
  • 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
  • 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.
  • 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 DVOl, will be the Spread DVOl 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.
  • OFS. Credit Index Option- Adjusted Spread
  • 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 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.
  • RBI baskets include the following: As the beta component in a portable alpha strategy
  • variable annuity providers To preserve market exposure during the course of an asset re-allocation. To hedge the market exposure of variable annuity providers
  • 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.
  • 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 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.
  • KRD key-rate durations
  • OAD option-adjusted durations
  • 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
  • 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 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 bucketing provides a reasonable view of yield curve exposure.
  • the buckets' duration contributions provide a view of yield curve exposure not too different from the ICRD profile.
  • 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.
  • 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.
  • 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.
  • 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.
  • Empirical test is perhaps the most effective way of gauging whether KRDs are indeed superior as a measure of yield curve exposure. Li 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.
  • 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 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.
  • 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,
  • 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.
  • 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. hi 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 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, hi at least one embodiment of the present 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).
  • 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.
  • Frequency Daily on a T+l 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.
  • Bid or Offer Side Outstanding issues are priced on the bid side. New issues enter the index on the offer side.
  • 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.
  • Unrated securities are included if an issuer rating is applicable
  • Unrated subordinated securities are included if a subordinated issuer rating is applicable
  • 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
  • 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).
  • a TBA to-be-announced
  • 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.
  • 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 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.
  • credit derivatives 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.
  • 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.
  • a standard credit default swap can be replicated using a cash bond and the repo market.
  • 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.
  • the buyer of protection receives a payment intended to compensate against the loss on the investment. This is depicted in FIG. 9.
  • the buyer of protection pays a fee.
  • 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.
  • 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 swaps (receiving fixed), it is possible to replicate, in unfunded form, the exposures of a portfolio of cash credit instruments.
  • 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.
  • CDX.IG $167,429,000 CDX Investment Grade 5yr 9/20/2009
  • 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.
  • all calculations preferably are performed by one or more computers.
  • 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.
  • all data preferably is stored in one or more electronic databases.

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Abstract

Dans un aspect au moins, l'invention concerne un procédé permettant de reproduire un premier indice, lequel procédé consiste à: construire un panier d'instruments financiers dérivés choisis pour reproduire ledit indice; ledit panier d'instruments financiers dérivés étant construit par appariement des durations des taux directeurs sur la base d'une pluralité d'instruments, et ledit panier d'instruments étant reconstruit sur une base périodique plus ou moins égale à celle sur laquelle ledit indice est reconstruit. Selon un autre aspect, l'invention porte sur un procédé permettant de reproduire un portefeuille de valeurs mobilières, qui consiste à: construire un panier d'instruments financiers dérivés choisis pour reproduire ledit portefeuille; ledit panier s'instruments financiers dérivés étant construit par appariement des durations des taux directeurs sur la base d'une pluralité d'instruments, et ledit panier étant reconstruit sur une base périodique plus ou moins égale à celle sur laquelle le portefeuille a été reconstruit.
PCT/US2006/015946 2005-04-22 2006-04-24 Procedes et systemes permettant de reproduire un indice avec des instruments liquides WO2006116562A2 (fr)

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