Term Structure Estimation for U.S. Corporate Bond Yields Jerry Yi Xiao RiskMetrics Group jerry.xiao@riskmetrics.com RiskMetrics has developed a methodology for estimating the term structures of U.S. corporate bond yields. This paper describes the domestic U.S. corporate bond market and presents the RiskMetrics approach to yield curve estimation for this market. We also review some salient features and applications of our yield curve estimates for risk managers. 1 Introduction The term structure of corporate bond yields is essential to the calibration and testing of credit models and the pricing of credit sensitive instruments. For risk management, the yield spread over the default-risk-free or government curve is a crucial measure for credit risk analysis, while the historical information of the term structure is essential for the calculation of the volatilities and correlations of defaultable future cash flows. The evolution of the credit term structure and of credit spreads is important when bonds of different credit ratings are actively traded and hedged. For regulators, the credit term structure provides information on the characteristics of credit risk and the correlation of credit risk with interest rate risk, which must be considered in setting regulatory standards for banking. The term structure of corporate credit yields is defined here as the relationship between the yields on corporate discount bonds, which generate only one cash flow at maturity, and the maturity dates of these instruments. For the US corporate bond market, as in any other bond market, the prices of discount bonds for a continuum of maturity dates are not available. Therefore, the term structure has to be estimated from the observed prices of a set of coupon bonds issued by corporations. Considerable effort has been made in the estimation of term structure of risk-free interest rates from the coupon bonds issued by governments. 1 These techniques have been widely adopted to empirically describe and forecast the term structure of interest rates. However, when it comes to the estimation of the term structure of corporate credit yields, accurate and robust techniques are still lacking. Indexes for corporate bond yields are typically averages of yields on a bucket of bonds that are chosen based on the characteristics of the bond such as remaining maturities and face values. These indexes, including Moody s Aaa and Baa yield indexes for non-financial seasoned corporate bonds, are defined for maturity buckets rather than specific times to maturity. They are generally 1 For example, see the survey by Zangari (1)
RiskMetrics Journal, Volume 2(1) 20 employed as benchmarks for performance and investment, i.e., to determine risk/return characteristics. In this document, we present RiskMetrics methodology for estimating term structures of U.S. corporate bond yields using exponential polynomials curve fitting. The characteristics and applications of these term structures will also be discussed. 2 Overview of U.S. Corporate Bond Market We begin by discussing some unique features of the U.S. corporate bond market. Generally, a corporate bond issue is characterized by its coupon rate, coupon frequency, maturity date, outstanding par value, seniority and security type, agency credit ratings (if rated), and possibly, provisions for paying off. The coupon rate of the corporate bond can be either fixed or floating according to some index rate. Although there are some exceptions, most of the U.S. corporate bonds pay coupons semiannually. In contrast to a Treasury bond which is backed by the full faith of the U.S. government, a corporate bond has a non-zero default probability. This default risk is gauged by credit ratings assigned by generally recognized rating agencies. Moody s and Standard & Poor s rating systems are the two most often used systems for corporate bonds. A one-to-one mapping of the bond ratings between the two rating systems is also widely accepted. Bonds with a top four credit rating ( Moody s Aaa to Baa, Standard & Poor s AAA to BBB) are referred to as investment-grade bonds while those with lower ratings are referred to as high-yield or junk bonds. Seniority determines the priority order in which investors are compensated if default occurs. Junior bond holders will be compensated only after all the senior bond holders s claims are fulfilled. The security of a bond indicates how a bond issue is guaranteed by the issuer s property. A bond may be classified into mortgage bond, collateral trust bonds, equipment trust certificates, debenture bond, etc. These features will affect the risks associated with the investment in a bond. Embedded options are a feature of some corporate bonds. For example, a call provision allows the issuer to redeem the outstanding bond issue prior to maturity, while a put provision allows the investor to demand an early return of principal. Other bonds with embedded options include convertible bonds and bonds with sinking fund provision. The yields of these bonds are adjusted by the market to reflect the value of the embedded options. The prices reported by traders are clean prices as a percentage of par, that is, excluding the accrued interest since last coupon payment. The value of a bond investment is typically measured by yield to maturity, which is the internal rate of return paid on a bond if the investor buys and holds it to its maturity date with all coupons reinvested in the same bond. The yield to maturity of a bond can be calculated from its market price and cash flows. The spread between corporate bond yield and Treasury rate for
21 Term Structure Estimation for U.S. Corporate Bond Yields the same expiry is the extra yield required to offset potential credit loss, liquidity risk, and security-specific risk and to provide extra reward for risk-taking. U.S. corporate markets use a 30/30 day count basis, which means interests on corporate bonds are based on a year of 30 days made up of 12 30-day months. 3 Term structure estimation for corporate bonds 3.1 Parsimonious approach for government term structure estimation A number of central banks in developed countries have adopted the parsimonious approach to estimation of government bond term structure first proposed by Nelson and Siegel (1). This approach uses a specific type of exponential polynomial function for the instantaneous forward rate f (m) at time m. f (m) = β 0 + β 1 e m/τ 1 + β 2 ( m τ 1 )e m/τ 1, (1) where β 0, β 1, β 2 and τ 1 are four parameters to be determined from the data. The function is characterized by convergence towards a constant interest rate level in the long term. By varying these four parameters, this function is capable of describing the shapes of a wide range of forward rate curves. The spot or zero-coupon rate as a function of maturity is derived by integrating the forward rate (1) from 0 to m and dividing by m: z(m) = β 0 + (β 1 + β 2 ) (1 e m/τ 1 ) m/τ 1 β 2 e m/τ 1 (2) Svensson (14) proposed enhancing the function s flexibility by adding a fourth term to Nelson and Siegel s original forward rate function in (1), f (m) = β 0 + β 1 e m/τ 1 + β 2 ( m τ 1 )e m/τ 1 + β 3 ( m τ 2 )e m/τ 2. (3) The parameters in (3) are estimated via a non-linear optimization procedure. For a given set of parameters, Svensson first uses the spot rates given by (3) to derive the zero-coupon rate for any future time m. For a given bond, he then uses these zero rates to determine its theoretical price, from which its yield to maturity can be calculated. These yields are calculated for all bonds using these parameters and compared to their actual market values. The final parameters are chosen so the sum of squared yield differences are minimized. The Nelson-Siegel or Svensson approach to term-structure estimation is not a theory of the term structure. It does not attempt to explain typical features of the term structure. Nor is it a model for the evolution of the term structure through time. It is a phenomenological approach providing a close representation of the term structure at a given point in time.
RiskMetrics Journal, Volume 2(1) 22 3.2 An empirical curve-fitting approach for corporate bond term structure estimation Corporate and government bond curves The methodology described in Section 3.1 has been applied primarily to government bond yield curves. 2 In applying it to corporate bonds, it is important for us to understand the differences between corporate bonds and Treasury bonds and the difficulties they give rise to. Figure 1 U.S. Treasury yields.5 Off the run On the run.5 Yield (%) 5.5 5 4.5 0 5 15 20 25 30 Year to Maturity Yields to maturity for U.S. Treasury securities with more than 30 days to maturity and more than three months of term at issue, excluding callable bonds and inflation-indexed securities. The most important difference between Treasury and corporate bonds is the credit spread. There are no credit spreads among the Treasury instruments as they are all default-free. As shown in Figure 1, the yields to maturity of seasoned Treasury securities (except for callable or indexed) are very close to a single curve. The largest yield differences are between on-the-run and off-the-run bonds, and like the smaller differences among seasoned bonds, are regarded as mainly due to differences in liquidity. 3 However, for corporate bonds, credit spreads exist not only between two different credit rating categories, as indicated by the overall yield differences for bonds with Aa and A ratings in Figure 2, but also between bond issues in different industry categories within the same credit rating, as will be shown later. Furthermore, as seen in Figure 2, bond issues from different firms within the same industry and credit rating are traded at a wide range of yields. This dispersion generally results from the following causes: 2 Malz (1) further applied it to term structure estimation based on interbank interest rates. 3 For a discussion of the impact of recent liquidity changes in the Treasury market, see Fleming (2000).
23 Term Structure Estimation for U.S. Corporate Bond Yields Figure 2 Samples of market data Industrial Aa Industrial A.5.5.5.5 Yield(%) Yield (%).5.5.5.5 0 5 15 20 25 30 Time to maturity (year) 0 5 15 20 25 30 Time to maturity (year) Market data from Bridge Information System: observed yields to maturity for plain vanilla U.S. industrial corporate bonds with Moody s Aa and A ratings on August 24, 2000. 1. One reason for the existence of credit spreads within the same credit rating and industry category is that rating agencies allow for some heterogeneity in each credit rating class. However, as indicated by the outliers in Figure 2, sometimes the market may perceive certain issues so differently from their credit ratings that they are traded at a much higher yield, or in other words, traded as if they have a lower credit rating even though the rating agencies have not downgraded them yet. 2. Liquidity is another source of difference. While the Treasury securities, especially onthe-run Treasury instruments, are among the most liquid, corporate bonds are generally less liquid. Liquidity varies tremendously across different issues and contributes to the dispersion of the corporate bond yield. 3. Coupon effects introduced by both the slope of Treasury term structure and tax consideration also have impacts on yield. Since the yield variations due to coupon differences are generally small (usually a few basis points) compared to the first two effects, we ignore the coupon effect in our current term structure estimation. 4. More subtle factors for yields variation include seniority and security types of bonds. Although they are already taken into consideration when rating agencies assign credit ratings, they may still have some residual effects on the yields. 4 Again, we ignore this effect, as the resulting difference is usually small. 4 See, for example, Fridson and Garman (1).
RiskMetrics Journal, Volume 2(1) 24 The dispersion of the yields within a given rating category increases as the credit rating declines, suggesting an increasing disparity of credit quality and liquidity among bonds of lower credit ratings. In view of these features of corporate bond prices and yields, we need to properly group the bonds according to their credit ratings and industrial categories in estimating the corporate bond term structure. Criteria must also be introduced to exclude outliers in each group from the estimation process. Curve fitting for the corporate yield curves Although one would like to adopt the Svensson formula for the instantaneous forward rate and get the spot rates directly from the estimated parameters, the process is too computationally expensive to be practical given the number of curves to fit and the number of bonds involved. Instead, we fit a yield curve directly from the yields to maturity and then use a bootstrapping technique to arrive at the zero-coupon rates. Following the parsimonious approach, an exponential polynomial function is used as an empirical description of the yield of corporate bonds: y(m) = β 0 + β 1 e m/τ 1 + β 2 e m/τ 2, (4) where y(m) is the yield of bonds which mature in m years, τ 1, τ 2, β 0, β 1, and β 2 are the five unknown parameters which will be estimated in the least squares curve fitting process. Like the Nelson-Siegel formula, (4) is flexible enough to incorporate different shapes of term structures. Since coupon effects are ignored in our estimation procedure, we may treat the yield in (4) as a par yield. Par yields are the yields of bonds trading at par and are thus equal to the assumed coupon rates. To account for the variation in liquidity of different bond issues, we use outstanding par values of each bond as one of the weighting factors for our curve fitting so that, in general, more liquid bonds carry higher weights. The curve fitting procedure minimizes the yield errors rather than price errors. However, it is also critical for the model to consistently fit the prices of the bonds used in the estimation. Therefore, a weighting of the yield errors in the minimization is introduced to correct for the variation in sensitivity of the price errors to yield errors. This can be best understood with the concept of duration, which relates the yield to maturity change of a bond to its price change, B B = D y, (5) where B is the bond price, and D is the modified duration. For a given yield change, the price change for a long term, hence a long duration, bond is much larger than that of a short term note with a small duration. Therefore, without proper weighting, a curve
25 Term Structure Estimation for U.S. Corporate Bond Yields fitting process that uniformly minimizes yield errors across the terms will tend to overfit the short term bond prices and underfit the long term bond prices. To correct for this effect, another weighting factor equal to the duration of a bond could be introduced for each yield error. To simplify the calculation, we use time to maturity as a weighting factor, as the differences are minimal. For a given set of bonds, the curve fitting process is accomplished into two steps: nonlinear estimation of (τ 1,τ 2 ) and linear regression to estimate β 0,β 1,β 2 for a given pair (τ 1,τ 2 ). We optimize (τ 1,τ 2 ) by minimizing the following weighted sum of squared yield errors F(τ 1,τ 2 ) = i [y i y(m i,τ 1,τ 2 )] 2 W i, () where W i is the weighting factor for ith bond, and y(m i,τ 1,τ 2 ) = β 0 (τ 1,τ 2 ) + β 1 (τ 1,τ 2 )e m i/τ 1 + β 2 (τ 1,τ 2 )e m i/τ 2, () in which β 0 (τ 1,τ 2 ), β 1 (τ 1,τ 2 ) and β 2 (τ 1,τ 2 ) represent parameters obtained from linear regression for given (τ 1,τ 2 ). Notice that the linear regression is also weighted with factor W i. For the estimation of τ 1 and τ 2, we first limit their ranges to an empirically plausible range [0.2, 50.0]. Then we construct a two-dimensional grid for the whole range of τ 1 and τ 2, and evaluate F(τ 1,τ 2 ) on each of the nodes to get a rough estimate of (τ 1,τ 2 ). A finer grid is constructed around the first estimate for the final estimate of (τ 1,τ 2 ). This procedure increases the likelihood that we obtain a global optimum for (τ 1,τ 2 ). In the estimation procedure, there are outliers whose yields are far from those of the bulk of the group, indicating that the market does not perceive them to be in the same rating category or to have poor liquidity. To eliminate those bonds from the pool, we use the following iteration process: we first use all the bonds to fit a curve and measure the yield spread of each bond to the curve. We then select the bonds whose spreads to the fitted curve are within a certain region to do the next iteration of curve fitting. We repeat the process until the change of the curves for two consecutive steps is small. Note that for each step, we always select from the whole pool of bonds in the rating/industry category. To define the proper inclusion region for the yield differences, we specify the lower and upper cutoff levels and determine the two corresponding cutoff thresholds based on the distribution of these differences. Experience suggests a lower cutoff of 2% and upper cutoff of 5%. We illustrate in Figure 3 with the outliers excluded from the final fitting process marked by open squares.
RiskMetrics Journal, Volume 2(1) 2 Figure 3 Curve fitting and outlier elimination: investment grade bonds Industrial Aa Industrial A.5.5.5.5 Yield(%) Yield (%).5.5.5.5 0 5 15 20 25 30 Time to maturity (year) 0 5 15 20 25 30 Time to maturity (year) U.S. industrial corporate bonds with Moody s Aa and A ratings on August 24, 2000. Although the above procedure works well in most cases, the least-square procedure curve becomes unstable when τ 1 and τ 2 are very close to each other, that is, when (4) is an overfit for the yield. When this occurs, we reduce the number of exponential terms and use the following formula instead: Y (m) = β 0 + β 1 e m/τ 1. () For high-yield categories, because of their extremely wide range of yields, as shown in Figue 4, we assume a flat term structure and carry out the same outlier elimination process. Once estimates of the parameters in (4) are obtained, we can compute the par yield for bonds with any time left to maturity, for example, 0.5 year, 1 year, 1.5 years, 2 years, etc. We then work from the short end of the yield curve to compute the zero rates via bootstrapping. 3.3 Bond data and results The raw data of some,000 actively traded fixed-coupon U.S. corporate bonds are provided daily to RiskMetrics by Bridge Information Systems. For our term structure estimates, we first filter out bonds with embedded options and bonds with coupon payment frequencies other than semiannual. We also exclude illiquid bonds with less than USD 0 million of outstanding par values. The total number of qualified bonds is about 5,000 each day.
2 Term Structure Estimation for U.S. Corporate Bond Yields Figure 4 Curve fitting and outlier elimination: high-yield bonds 15 14 13 12 Yield (%) 11 0 5 15 20 25 30 Time to maturity (year) Industrial Ba bond yields on August 24, 2000 The qualified bonds are categorized into three industry groups, financial, industrial and utility, according to the sectors of issuers. The bonds are then further grouped by their credit ratings. A typical data example is shown in Table 1 with industry and credit rating breakdown. We use Moody s symbols for credit ratings for convenience. For each rating category, we pick only bonds whose unmodified ratings from Moody s and S&P agree. Table 1 Number of qualified Vanilla Bonds on 0/14/2000 Financial Industrial Utility Total Aaa 125 1 21 Aa 52 111 33 A 0 2 15 Baa 353 1 24 152 Ba 4 14 52 24 B 22 4 1 124 Caa 2 2 2 Total 204 13 42 Based on available data, we group bonds into 1 categories (see Table 2) and estimate the term structure for each pool. One may notice that there is no term structure estimation for industrial Aaa bonds. This is due to the small number of bonds in this group which makes a yield curve estimate unreliable. For utilities, since the Aaa, Aa, and A bonds
RiskMetrics Journal, Volume 2(1) 2 Table 2 The categories for which term structures are estimated Financial: Aaa Aa A Baa Ba Industrial: Aa A Baa Ba Utilities: Aaa+Aa+A Baa Ba All Industries: Aaa Aa A Baa Ba B+Caa+Ca+C Constant par yield, flat term structure. appear to trade on the same curve, a single yield curve is estimated for these three credit ratings. In reporting term structures, we choose a set of terms to maturity or vertexes and report daily yields and zero rates corresponding to those vertexes. If, for a certain group on a certain day, there are not enough bonds in the long or short end, the term structure vertexes in those ranges will not be reported. For example, if there are no financial Aaa bonds with time to maturity more than years, financial Aaa yields on that day for terms longer than years will not be reported. A snapshot of all the corporate yield curves on August 24, 2000 is displayed in Figure 5. For each credit rating, yields are generally different for different industry groups; bonds of financial firms, for example, have higher yields. The differences among industries increase as the credit rating drops. We also note a large rise in yield from Baa to Ba across all industries as the bonds cross from investment grade to high yield. An important result of the term structure estimation for credit sensitive derivatives is credit spread over the Treasury curve. For each vertex of an industry-rating group, we may get a rate estimate for each business day using our methodology, resulting in a time series of the corresponding rates. We will call each of these rate estimates for a given vertex and industry-rating group an index. Figure displays the time series of the -year yield indexes and their spreads over Treasury yields for all industries. Yield spreads for these indexes were negatively correlated with Treasury rates in the year 2000. This result is in agreement with the study using monthly changes of Moody s seasoned bond index by Duffie (1). 4 Volatilities of the indexes 4.1 Effects of pool size Each corporate yield curve is estimated from a different bond pool, which is refreshed every business day. As seen in Table 1, the number of bonds in each pool varies widely. A natural question to ask is whether the difference in the number of bonds affects the volatility of the indexes.
2 Term Structure Estimation for U.S. Corporate Bond Yields Figure 5 Example of term structure output 11 Financial 0/24/2000 11 Industrial 0/24/2000 yield Baa A Aa yield Ba Baa A Aaa Aa 0 20 30 maturity 0 20 30 maturity 11 Utility 0/24/2000 11 General 0/24/2000 yield Ba Baa yield Ba Baa Aaa+Aa+A Aaa 0 20 30 maturity Snapshot of term structures on August 24, 2000 0 20 30 maturity Figure Curve time series 11 Ba 5 Ba 4 Baa Yield (%) A Aaa Spread (%) 3 Baa 2 A 5 Treasury 1 Aaa 4 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year 2000 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Year 2000 Time series of U.S. corporate -year yields and spreads over Treasury yields. To answer this question, we first choose a group of bonds, e.g. financial Aa, and randomly select, for each day, a certain number of bonds to construct an index. Since each daily
RiskMetrics Journal, Volume 2(1) 30 set of bonds is selected randomly, the volatility of this pseudo-index is higher than the real index. As we increase the number of bonds in building these pseudo-indexes, the volatility decreases due to diversification effects. Figure shows this relationship for -year indexes of different groups. It is clear from Figure that for all the different groups, as long as we have about 50 bonds in the pool, the diversification effects are high enough to yield an index whose volatility level will be relatively stable for further increases of the pool size. On one hand, this lower bound for bond numbers restricts the range of indexes we can reliably construct. On the other hand, this result provides assurance that volatilities of the indexes are accurate measures of market volatilities. Figure Effects of pool size on index volatility 50 40 Volatility (%) 30 20 0 0 50 0 150 200 250 300 Number of Bonds Used for Calculation The horizontal axis shows the number of bonds randomly picked within the corresponding groups to calculate -year pseudo-indexes. 4.2 Index volatility and credit rating As we measure the volatility of each index, we find that high yield indexes usually have a lower volatility than investment grade indexes. Within the investment-grade universe, the volatility between different rating categories is smaller, but can still be observed, with Baa curves having lower volatility than Aaa curves. Since an index can be roughly regarded as an average of a pool of bonds, this apparently counterintuitive result can be simply understood in terms of diversification. It is known that diversification effect is greater when there is lower correlation between individual instruments in the portfolio. As it turns out, the lower the credit ratings, the lower the correlation between individual bonds. Therefore, this reverse volatility order is caused by the difference in correlation as the credit rating drops.
31 Term Structure Estimation for U.S. Corporate Bond Yields 5 Volatility of individual bonds 5.1 Characterizing bond volatility relative to an index To characterize the volatility of individual bonds, we can perform an analysis for individual bonds and the indexes based on the theme that is analogous to the definition of beta for equity market based on the Capital Asset Pricing Model. To do this, we first transform the daily change of a par yield index to daily change of prices, which in turn give us the returns based on the index. In Figure, we plot the daily returns for the -year Baa index against returns for three Baa rated individual bonds with time to maturity close to years. One important observation is that, except for some large jumps, the individual returns are close to the market return. In other words, the beta is close to one. 5 Figure Index and individual bond returns IIN.250 04/15/0 IGR LMT.200 12/01/0 IGR CTX.50 03/01/0 IGR 4 4 4 2 2 2 r IIN 0 r LMT 0 r CTX 0 2 2 2 4 4 4 2 0 2 2 0 2 2 0 2 r Baa r Baa r Baa Daily returns for index plotted against returns for individual bond in 2000 For a more precise analysis, we use a one-factor model to analyze a pool of bonds with maturity close to the term of an index, 5 If we carefully study the time series of individual bonds of a given group, we may notice that most of their yields usually follow a general trend (the index) and maintain constant spreads. Individual spreads may jump, from time to time, to other values and then keep constant again. This leads to the behavior shown by the returns in Figure. The phenomenon shows the pricing practices of a lot of bond traders.
RiskMetrics Journal, Volume 2(1) 32 r i,t µ i σ i = ρ i ( rm,t µ m σ m ) + 1 ρi 2 ɛ i,t, () where r i,t is the daily return from the ith bond, r m,t is the daily returns from the corresponding index, ρ i is the correlation between the bond and index returns, and ɛ i,t is the idiosyncratic term of the bond. Individual bonds have a mean return µ i and a variance of σi 2, while the index has a mean return µ m and a variance σm 2. The return for an individual bond can then be expressed as σ i σ i r i,t = (µ i ρ i µ m ) + ρ i r m,t + 1 ρi 2 σ i ɛ i,t, () σ m σ m from which we may estimate the parameters β i = ρ i σ i σ m, (11) α i = µ i ρ i σ i σ m µ m. (12) As suggested by Figure and confirmed again by the results in Table 3, α i is essentially zero and beta i is close to one for all the groups considered. Therefore, () can be simplified to r i,t = r m,t + ε i,t. (13) To capture the individual bond volatility relative to an index, instead of introducing beta as equity market models, we may introduce a new factor R i = σ i σ m, (14) which is simply the ratio of the bond volatility to the index. For simplicity we may compute the parameter for each industry-rating category which is the average of R i over all the bonds within the category. Some results are shown in Table 3. With the relationship in (11) and β 1, R should be equal to 1/ρ, where ρ is the average correlation between individual bond and the index of a certain category. As reflected in the last column of Table 3, the average correlation of individual bonds to the index is higher for higher-rated bonds. Therefore, lower correlation is associated with higher volatility of individual bonds relative to the volatility of the index. This is consistent with the finding of section 4 that the average correlation of individual bonds with one another is higher for higher-rated pools.
33 Term Structure Estimation for U.S. Corporate Bond Yields Table 3 Statistical characteristics for industrial bonds α β R ρ Aaa 0.003 ± 0.02 1.02 ± 0.04 1.15 ± 0. 0.0 Aa 0.00 ± 0.02 1.00 ± 0.03 1.12 ± 0.0 0.0 A 0.003 ± 0.03 1.05 ± 0.0 1.1 ± 0.20 0.1 Baa 0.002 ± 0.0 0. ± 0.21 1.2 ± 0.2 0. An important implication of these observations is that a bond index should not be treated as the market portfolio. As indicated by the high correlations shown in Table 3, systematic risk plays a dominant role in the risk of corporate bond portfolios, and diversifiable risk is insignificant. This is especially the case for bonds with high credit ratings due to their higher correlations with one other. Portfolio theory, which is focused mainly on the notion of diversification, is not very applicable in analyzing bond portfolios. It thus does not justify the use of bond indexes as an optimal market portfolio. Therefore, systematic risk analysis based on the shape and level of the term structure, instead of analysis based on classic portfolio theory, should be the focus of bond investment. 5.2 Decomposition of bond volatility Another interesting result arises from the decomposition of individual bond volatility into the contribution from Treasury volatility, index spread volatility, and residual spread volatility. The spread volatility itself can also be used in valuation of some credit derivatives. To a first approximation, the daily return of a bond can be related to the daily change of bond yield through the duration relationship in (5), in which y should be taken as the yield change between two consecutive business days. Since we have the times series of the indexes, we may decompose the times series of individual bond yield into three parts: Bond yield = Treasury par yield + Corporate index spread + Residual spread, (15) in which the term for the Treasury and corporate index equals to the time to maturity of the bond. Now (5) can be written as B B = D ( T + S + ɛ), (1) where T, S, and ɛ are the daily changes in Treasury par yield, index spread, and residual spread respectively. With the time series of T, S, and ɛ, we may calculate the volatility of each component and the correlation between one another, and relate them to the price volatility of the bond by assuming that the modified duration is reasonably stable within the time frame under consideration.
RiskMetrics Journal, Volume 2(1) 34 This simple volatility decomposition gives us a general idea of the relative contributions to the total risk of a bond. The term structure of Treasury yield and corporate index enters into the equation indirectly through the modified duration D. A more elaborate approach to volatility decomposition, which uses zero coupon rates and calibrated residual spread, incorporates term structure through the notion of partial duration. A detailed description of this approach and its application to risk budgeting for corporate bond portfolios is presented by J. Mina in this issue of the Journal. Conclusion In this article, we present our methodology for the estimation of U.S. corporate bond term structures based on market data. The technique is based on a parsimonious approach which properly balances accuracy and computational costs. The estimation procedure can successfully handle the wide dispersion of the yields within each credit rating/industry group, while capturing the liquidity and duration effects as well. These corporate curve time series data can provide important yield, spread, volatility and correlation information for U.S. corporate bonds, which can be useful for risk managers, traders, and regulators. References Duffie, G. R. (1). The relation between Treasury yields and corporate bond yield spreads, Journal of Finance 53(): 2225 2242. Fleming, M. J. (2000). The benchmark U.S. Treasury market: recent performance and possible alternatives, FRBNY Economics Ploicy Review (April): 12 145. Fridson, M. S. and Garman, M. C. (1). Valuing Like-Rated Senior and Subordinated Debt, Journal of Fixed Income (3): 3 3. Malz, A. M. (1). Interbank Interest Rates as Term Structure Indicators, Federal Reserve Bank of New York, mimeo. Nelson, C. R. and Siegel, A. F. (1). Parsimonious modeling of yield curves, Journal of Business 0(4): 43 4. Svensson, L. E. (14). Estimating and interpreting forward interest rates: Sweden 12-4, Discussion Paper 51, Center for Economic Policy Research. Zangari, P. (1). An investigation into term structure estimation methods for RiskMetrics, RiskMetrics Monitor pp. 3 31.