The_Barra_U.S._Fixed_Income_Risk_Model

The Barra U.S. Fixed Income Risk Model Asked to estimate the risk of a portfolio, a manager of fixed income securities might respond by giving the duration of the portfolio, perhaps relative to the duration of a standard benchmark. To estimate both spread risk and exposure to interest rate changes, the manager might also report the durations of various portfolio components (such as the Treasury, agency, corporate, and mortgage components) compared to the corresponding subindices of the benchmark. Duration is a well-understood, single number that identifies the portfolio’s exposure to the level of interest rates, which is the largest source of market risk. Using duration to measure risk, however, leaves several unanswered questions, including the following: • How do the various sectors interact (for example, how are mortgage spreads affected by a rise in interest rates)? • How does the risk impact of a one-year-duration overweighting of agency bonds compare to the risk impact of a one-year-duration overweighting of BBB corporates? • If finer measures of portfolio structure are used (such as sector-by-rating bins), how will the resulting long list of relative durations be used? • What is the benefit of diversification? For instance, how much is credit risk reduced by using 50 bonds, compared to using just 5, to replicate the corporate index? This paper describes Barra’s approach to answering these kinds of questions in the context of the U.S. fixed income market. The section entitled “Model Structure and Estimation” provides a detailed description of the components, estimation methodology, and quantitative results of the risk model. The section entitled “Model Tests” on page 21 describes test results for the model components. The section entitled “Summary” on page 28 summarizes the results and discusses the future of the model. The Barra U.S. Fixed Income Risk Model 1 Model Structure and Estimation Barra’s U.S. Fixed Income risk model1 (USFI) includes: • Marketwide interest rate and spread risk factors for taxable and tax-exempt fixed income securities (common-factor risk) • Heuristic models of specific risk for U.S. government bonds (Treasuries and agency bonds) and mortgages (asset-specific risk) • A credit migration model of specific risk for corporate, supranational, non-U.S. sovereign, and municipal bonds (issuer credit risk) In the parametric framework, the common-factor, asset-specific, and issuer credit risk models are combined to obtain a forecast of portfolio return variance: where x is the vector of portfolio factor exposures, F is the covariance matrix, wi is the portfolio weight for each asset or issuer, and is the corresponding return variance due to asset-specific factors or issuer credit migration.2 Common factors and asset exposures The common-factor model accounts for non-diversifiable, marketwide risk factors. The primary marketwide determinants of fixed-income portfolio risk are interest rates and yield spreads,3 and these are the factors considered in the U.S. model. Every fixed income security in the model is exposed to one or (usually) more of these factors. At the heart of the common-factor risk model is a covariance matrix that permits the calculation of forecasted return volatility given the portfolio’s exposures to the common factors. The exposures are the analogs of ordinary effective duration for the different factors. They include key rate durations for the spot rate factors and spread duration for the sector spread factors. The Barra U.S. Fixed Income Risk Model 2 Interest rates All fixed income securities except cash are exposed to taxable and municipal market interest rates at specified maturities. Whether taxable or tax-exempt, the exposures are the so-called “key rate” durations for each vertex. Key rate durations are calculated by applying symmetric, up and down “shocks” to the initial spot rate curve, revaluing the asset with the shifted curves, and approximating the partial derivative of the asset price with respect to the spot rate from the results. Figure 1 illustrates shocked term structure shapes for the first three key rates. The sum of the key rate durations is equal to the effective duration, since a simultaneous shock to all of the spot rates is equivalent to a parallel shift of the spot rate curve. (For this to work, the first and last key rate shifts are not tents, but are flat before the first vertex and flat after the last vertex.) Thus, key rate durations are also known as “partial” durations. Taxable debt Taxable securities have exposure to interest rate risk factors estimated with Treasury price data at eight spot rate maturities of 1, 2, 3, 5, 7, 10, 20, and 30 years. Tax-exempt municipal debt Tax-exempt (municipal) bonds have interest rate exposure to the curve estimated from AAA Government Obligation (GO) par yields using eight spot rates at maturities of 1, 2, 3, 5, 7, 10, 20, and 30 years (the same as the taxable rates). Figure 1. “Shocked” spot rate curves (dashed lines) corresponding to the first three key rate durations calculated for interest rate risk exposures Term (years) R at e 0 8 9 107654321 4 4.5 5 5.5 6 6.5 The Barra U.S. Fixed Income Risk Model 3 Spreads All fixed income securities except Treasuries and AAA municipals are exposed to a spread factor. There are nominally 61 taxable spread factors (1 agency spread, 55 corporate spreads, and 5 MBS spreads)— though not all of them can be estimated from market data—and 3 tax-exempt (municipal) spread factors. The magnitude of the exposure to the spread factor is the spread duration. Spread duration is calculated analogously to effective duration, except that the option-adjusted spread (OAS), instead of the underlying interest rate curve, is subject to the shifts. In some cases, such as fixed rate bonds, this makes no difference to the calculated value. In other cases, including mortgages and floating rate bonds, the spread duration and effective duration are different. For floaters, the spread duration is comparable to the effective duration of a fixed rate bond of similar current coupon and maturity. The reason for the difference is that changing the OAS affects only the discount rate applied to future cash flows, while shifting the underlying spot rates also affects the projected cash flows. Note: Spread duration is never negative, while effective duration can be negative in exceptional cases (such as IOs and levered floaters). Agency bonds There is one spread factor to which every agency bond is exposed. Corporates, non-U.S. sovereigns, and supranationals The spread duration of a taxable corporate, non-U.S. sovereign, or supranational bond is mapped to a spread factor by its assigned sector and rating.4 The model includes up to 54 sector-by-rating spread factors (that is, the 9 sectors listed below times 6 rating categories from AAA to B). In addition, there is a single CCC rating spread factor across all sectors (there are not enough CCC bonds to adequately estimate spread changes for different sectors). The sectors are: • Canadian—Government, Provincial, Corporate • Financial—Bank, Insurance, Independent, Subsidiaries • Energy • Manufacturing, retail, consumer products, diversified industrial • Transport The Barra U.S. Fixed Income Risk Model 4 • Telecommunications • Utility—Gas and Electric • Yankee—Supranational • Yankee—Corporate and Sovereign (except Canada) Mortgage-backed securities The spread duration for a mortgage-backed security (MBS) passthrough is mapped by agency and program type to one of the five distinct mortgage spread factors below: • GNMA 30 year • Conventional 30 year • GNMA 15 year • Conventional 15 year • Conventional balloons Tax-exempt municipal spreads The spread duration of a tax-exempt (municipal) bond rated AA or below is mapped to one of the three muni rating spread factors. The three muni spread factors are spreads of AA, A, and BBB GO yields over the AAA muni spot curve. Swap spread The spread duration of any asset exposed to credit risk but not mapped to one of the other spread factors is mapped to the swap spread factor. Assets exposed to this factor include asset-backed securities (ABSs) and collateralized mortgage obligations (CMOs). The Barra U.S. Fixed Income Risk Model 5 Estimation of common-factor returns Two basic steps are required to estimate factor returns. 1 Calculate the Treasury spot rate curve for each month of the sample period. 2 Calculate spreads for everything else (agency, corporate, non-U.S. sovereign, and supranational bonds, and generic MBS passthroughs). Treasury curve The Treasury curve is estimated at the eight model maturities by solving for the spot rates that minimize the mean-squared pricing error of all regular Treasury coupon notes and bonds (that is, excluding STRIPs, TIPs, and the like). The Treasury spot rate factor returns are then: where is the time t spot rate at maturity vertex i.5 Note: The valuation algorithm accounts for option value if the bond is callable. Option-adjusted spread The purpose of the estimation of all the bond spread returns is to attribute the residual returns of non-Treasury bonds and MBSs to changes in marketwide OASs. Bond and mortgage OASs are calculated using algorithms based upon the one-factor, mean-reverting gaussian model of interest rates (also known as the Hull-White model). The parameters of the model are calibrated to historical interest rate movements, which typically results in an annual standard deviation of about 85 basis points for short-term interest rates, and a slightly lower standard deviation for longer-term rates. The algorithms are described in more detail in “The New Cosmos–U.S. Valuation Algorithms,” Barra Newsletter, Summer 1997.6 Corporate spread The Merrill Lynch Domestic Master and Merrill Lynch High Yield Master indices constitute the bond estimation universe for corporates. At each month end, the bonds’ OASs are derived from bond prices obtained from either Bridge Information Systems or Merrill Lynch Pricing Service. The bonds are grouped by sector and rating, and the spread change for a factor is taken to be the duration- weighted average OAS change of all remaining bonds in that sector-by-rating category.7 The Barra U.S. Fixed Income Risk Model 6 Agency passthrough spread The agency passthrough spread factors are estimated using securities in generic pools corresponding to the TBA (“to be announced”) market, rather than actual pools. The TBAs are identified as the lowest-priced generics with half-point multiple net coupon and reasonably large amount outstanding. If there is more than one lowest-priced generic of a given coupon, the one with lowest OAS is taken to be the TBA for that coupon.8 This usually results in identifying the TBAs with the most recently issued pools for each coupon. The mortgage spread changes are calculated in the same manner as the corporate spread changes, except that the weighting factor is the spread duration rather than effective duration. (Spread duration and effective duration are the same for fixed rate bonds under the valuation methodology used, so the bond spreads are actually also weighted by spread duration in the average.) Bond and MBS spread factor returns are then: where is the spread at time t of the kth bond in sector-by-rating category i, and is the weight applied to the bond, based on duration and callability. Muni spread Municipal securities are exposed to the muni spot curve (estimated from the AAA GO yield curve) and possibly also to credit spreads if the securities are rated below AAA. The muni spot curve is estimated by constructing a series of synthetic par bonds (one for each annual maturity out to 30 years) and then using the same error-minimization methodology as is used for the Treasury curve. Rather than “bootstrapping” the reported yields, this more cumbersome procedure is used, because muni yields are conventionally reported for callable bonds at maturities beyond 10 years. Were the par yields bootstrapped and longer callables priced using that spot curve, they would have significantly higher yields than reported.9 This methodology is applied to yield curves for AAA, AA, A, and BBB GO bonds. The AA, A, and BBB spread levels are then calculated at five years, and the changes in these spreads constitute the muni spread factor series. The AAA muni spot rate factor returns then are calculated analogously to the Treasury spot rate factor returns, while the muni spread factor returns are produced in the same manner as the swap spread return series. The Barra U.S. Fixed Income Risk Model 7 Swap spread The swap spread change is calculated as the change in the spread of the five-year swap rate over the five-year Treasury yield. This factor is used as a catchall proxy for the spread risk of any “spread product” not exposed to one of the corporate, MBS, or muni factors. This includes ABSs, CMOs, and LIBOR- and swap-based derivatives. The swap spread factor return is then: where now denotes the five-year swap spread at time t. Covariance matrix construction Once factor return series for all of the risk factors are available, statistical estimation and forecasting methods can be applied to them. In particular, a forecast can be produced of the covariance matrix of future factor returns. The covariance matrix can then be used to forecast the return volatility of a portfolio given the exposures of its assets to the risk factors. Barra historical data are used to estimate the current best-guess covariance matrix of the factors. As discussed in the following sections, this entails using the following two procedures: • Weighting recent data more heavily than older data • Employing an effective method for handling gaps in the factor return series Half-life estimation Greater importance is attached to more recent market events than to older ones by exponentially weighting the factor series in the covariance calculation. Each factor return is weighted by a multiplier l (between 0 and 1) raised to the power of the number of elapsed months since the return. The value of l can be related to the half-life of the weighting scheme, that is, the number of months n such that ln=1/2. This approach is tested by evaluating the out-of-sample performance of covariance matrices constructed using different half-lives n. The probability of observing each month’s factor returns given the prior month’s covariance forecast (for a given half-life) is calculated, and the probabilities for successive months are multiplied together. This provides the correct probability for the entire series of returns, assuming that factor returns (that is, spot rate and spread changes) in different months are independent. The Barra U.S. Fixed Income Risk Model 8 The probability numbers are kept intelligible by reporting the negative logarithm of the likelihood. These quantities are also normalized by subtracting the minimum value over the range of tested half- lives. An increase in this statistic by 1 over the minimum corresponds to a decrease in probability by a factor of 1/eª0.37.10 Figure 2 shows this analysis applied to Treasury spot rates, with an in-sample period of January 1996 through December 1999. Data from 1988 through 1995 were used to construct the covariance matrices, but they were not included in the log-likelihood calculation. The curve shows that the negative log-likelihood of the observed spot rate changes as a function of half-life. There is a minimum of approximately two years, but longer half-lives are not strongly excluded. Half-lives under approximately one year, however, are clearly rejected. (If data back to 1983 were used in the covariance matrix estimation, longer half-lives would be more strongly rejected. The best value remains around two years.) Figure 2. Negative log-likelihood of observed Treasury spot rate monthly changes as a function of half-life in the covariance calculation. The best fit is about two years. 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 8 Half-life (yrs) -lo g( lik el ih oo d) The Barra U.S. Fixed Income Risk Model 9 Figure 3 shows the same analysis applied to agency and corporate spreads, MBS spreads, muni spot rates, credit spreads, and swap spreads. In each case, the curve shows an average of the negative log- likelihoods for the factors in each category. The optimal forecasting half-life is long for all but the high-grade corporates, for which the best fit is a fairly short four months. Unfortunately, it is difficult to accommodate different half-lives for different factor series in a single covariance matrix. A single half-life must be selected based upon both the results of the tests and the following more qualitative considerations: • Longer half-lives give more stable forecasts with smaller statistical uncertainty and random fluctuations. • The risks of a portfolio with primarily investment grade securities are likely to be determined largely by the interest rate factors, for which the optimal half-life is around two years. • The risks of a portfolio with heavy exposure to lower grade securities, for which spread risk may be dominant, are optimally forecast with the longest half-life. • The Barra Global Fixed Income risk model is already in production using a half-life of two years (based upon similar tests across many fixed income markets). • Finally (as described later), the two-year half-life worked quite well in the aftermath of the liquidity crisis of late 1998, which saw large short-term jumps in spreads, and where volatilities were as much as 10 times higher than previous levels. The average spread volatility has since settled down to a level about twice as high as it was prior to the 1998 events, and the model has tracked this behavior with remarkable fidelity. Figure 3. Negative log-likelihood of observed spread and muni factor changes as a function of half-life in the covariance calculation. The best fit is a few months for high-grade corporates, one year for MBS, and too long to measure with the sample data for the others. 0 10 20 30 40 0 1 2 3 4 5 6 7 8 Halflife -lo g( lik el ih oo d) AAA-A BBB-CCC MBS MuniSpot MuniSpd Swap The Barra U.S. Fixed Income Risk Model 10 Incomplete data methodology When working with complete time series (that is, where there are neither missing values nor incomplete series), it is easy to calculate the optimal estimator of the covariance of the ith and jth factors; it is given by the usual pairwise formula from basic statistics: This formula cannot be used, however, when the factor series are of different lengths N or have missing values. Calculated in this naive fashion for such series, the resulting covariance matrices may imply that certain portfolios have negative return variance, which is mathematical nonsense. In other words, this simple formula does not give a good estimate (or even a valid one) for the covariance matrix when the factor series have gaps or are otherwise incomplete. There are two sources of incompleteness in the factor return series. One is that for each series there is no information prior to some date. For example, the taxable spot rate and spread series begin August 1992, while the municipal data begin January 1993. The second source of incompleteness is that many of the series contain gaps. These gaps usually arise because of the absence of primary data (for instance, when there are too few bonds to estimate a sector-by-rating category’s spread change). Some of these gaps can be remedied by proxy. For example, if there are too few bonds to estimate the Utility-B spread change, the Utility-B spread change can be proxied with the average spread chang