Bond Portfolio Holding Period Return Decomposition

Transcription

1 Bond Portfolio Holding Period Return Decomposition Robert Brooks Kate Upton January 15, 14 COTACT: Robert Brooks, Department of Finance, The University of Alabama, Alston Hall, Tuscaloosa, AL 35487, (5) The author gratefully acknowledges the helpful comments of James Steeley and Hui-Ju Tsai. Key words: Bonds, Return decomposition, Term structure of interest rates, Duration, Convexity

2 Bond Portfolio Holding Period Return Decomposition ABSTRACT Bond portfolio holding period returns are decomposed into four main components. The horizon component captures the return attributable to the mere passage of time. The spread component captures the return attributable to any change in the spread over the fitted base spot rate curve. The base-rate component captures movement in the fitted base spot rate curve. There is an interaction component. The base-rate component is further decomposed into three components attributable to modified duration, convexity, and cross-convexity. Each of these three base-rate components is further decomposed into three subcomponents tied to level, slope, and curvature components. Illustrations of the decomposition provided. Applying a parsimonious version of this model on the Morningstar universe of bond funds indicates that using level and slope in the duration portion of the base-rate component accounts for approximately 5 percent of unknown returns. Using level, slope, and curvature in the duration base-rate component and adding the spread component explains almost 5 percent of returns. JEL Classification Code: G1, G17, G3

3 Bond Portfolio Holding Period Return Decomposition Bond investment professionals seek to understand the sources of return and risk when managing their bond portfolios. We apply the level, slope, and curvature model (denoted the LSC model) of elson and Siegel (1987) with a general extension along the lines of Svensson (1995) to the problem of decomposing bond portfolio holding period returns (HPRs). The model illustrated here is general in form and can have as many factors as required by the user. Bond portfolio HPR is decomposed into four main components, the non-random horizon component, the spread component, the base-rate component, and an interaction component. The horizon component captures the bond return attributable to the mere passage of time over the holding period horizon based solely on the known spot base-rate curve. The spread component captures the bond return attributable to any change in the spread over the fitted spot base-rate curve. The base-rate component captures movement in the fitted spot rate curve, fitted with the LSC model. The interaction component contains the residual bond return. Figure I a) illustrates the structure of the bond HPR decomposition presented in this paper. The base-rate component of bond returns can be further decomposed into three components attributable to modified duration, convexity, and cross-convexity based on a Taylor series approximation. Thus, the approximation contains an error component. Each of these three rate components can be further decomposed into three subcomponents tied to movement in level, movement in slope, and movement in curvature. Our LSC model described in detail later is general enough to have multiple curvature components. Although not presented here, the LSC model can also be applied to the spread component and a similar decomposition applied to the resultant spread curve as illustrated in Figure I b). 3

4 Figure I. Illustration of Bond Holding Period Return Decomposition a) Spot Rate Decomposition b) Potential Spread Decomposition The remainder of the paper is organized as follows. First, we provide a conceptual illustration without any mathematical details to clarify the approach presented here. Second, we review the historical development of bond return and risk measures. We also briefly survey various methods for fitting the term structure of interest rates with a particular focus on the LSC model used in this paper. Third, we introduce bond HPR measures based on the LSC model for fitting the term structure of interest rates. Fourth, we provide several illustrations of the application of the LSC model to bond return measurement. Fifth, we apply several parsimonious versions of this model to the returns generated by US bond mutual funds from 5 to 14. Finally, we give some summary comments as well as potential areas of future research. 4

5 I. A Conceptual Illustration To illustrate the importance of this decomposition, we sketch the results of applying this methodology based roughly on U.S. Treasury data in February 1. Suppose we have a twoyear corporate bond trading at a 1 basis point spread over the U.S. Treasury-based zero coupon spot rate curve (level = 4%, slope = 3.5%, curvature = 6%, defined in detail later). If the investment horizon is one month, how is this bond s HPR decomposed if it is now trading for basis points over the horizon spot rate curve (level = 3%, slope = 1.5%, curvature = 1%)? ote that these two spot rates will converge to 5% at perpetuity. Figure II illustrates these spot rate curves incorporating spreads (the all-in rate) as well as the difference between them. The applications for the analytics presented in this paper include ex-post decomposition of debt portfolios, bank asset-liability management, pension management, and so forth. These analytics can also be used in scenario analysis, historical simulation, and Monte Carlo simulation related to ex-ante risk management. Hence, the one-month change in the spot rate curve could be anything from the actual historical change to a particular potential scenario. Figure II. Spot Rate Curves Based on LSC Model and Difference 5

6 We briefly consider three par bonds, -year, 1-year, and 3-year non-callable bullet corporate bonds with annual coupon rates of 1.1%, 3.7%, and 4.6%, respectively. We use the same base rate information presented in Figure II and one-month holding period. The analytical tools presented here provide the capacity to decompose the bond HPR into specific components illustrated in Table 1. Cross-convexity involves two of the LSC components, hence in Table 1 level cross-convexity denotes level and slope, slope cross-convexity denotes slope and curvature, and curvature cross-convexity denotes curvature and level. Several observations can be made from the data in Table 1. First, a positive portion of the total HPR is attributable to just the mere passage of time (horizon). The higher the coupon generally results in a higher horizon return, but this portion of HPR is also dependent on the current shape of the spot-base rate curve. Second, the relative portion of the HPR between the spot rate component and the spread component varies significantly by the bond s maturity. Obviously, the longer maturity bond is more influenced by the 1 basis point increase in the spread. What is not immediately obvious is the contribution of the spot rate component as maturity increases due to the unusual behavior of the scenario. Third, the relative contribution of level, slope, and curvature are clearly heavily influenced by the bond s maturity. Thus, the simplistic parallel shift assumption lacks the capacity to provide this degree of decomposition granularity. Finally, in the case of bonds, the decomposition of total duration into level, slope and curvature durations is much more important than the decomposition of the spot rate component into total duration, total convexity and total cross-convexity components. Although we do not focus in this paper on pension surplus management or bank asset-liability management, a topic of another paper, the convexity components and sub-components play a much more important role, especially when duration matching strategies are pursued. 6

7 Table 1. Illustration of Bond Holding Period Return Decomposition -Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.1174% Duration % 1.96% -6.15% % Spot Rates % Convexity.1%.1%.%.% Spread -1.93% Cross-.1% -.3%.6% -.% Convexity Interaction.8% Estimation.% Error Total % % 1.94% % % 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.344% Duration % % % % Spot Rates % Convexity.355%.34%.4%.1% Spread % Cross- -.34% -.16%.1% -.388% Convexity Interaction.478% Estimation.673% Error Total -1.1% % % % % 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3433% Duration % % % -1.37% Spot Rates % Convexity.568%.561%.5%.13% Spread % Cross % -.483%.61% % Convexity Interaction % Estimation % Error Total -.755% % % % % II. Bond Risk Measures, Bond Returns and the Term Structure In this section, various historical bond return and risk measures are reviewed to provide context for the approach taken in this paper. We also briefly review fitting the term structure of interest rates as our approach relies on a fitted maturity time term structure model. The mathematical details are presented consistently with continuous compounding used here, not as they were originally published. Continuously compounded rates are much easier to manipulate mathematically for the goals of this paper. Let r i : t k denote the annualized, continuously compounded spot interest rates observed at calendar time t k for maturity time i. CF i denotes the i th cash flow, Coupon Bl denotes the annual fixed coupon percentage on bond B l with par value Par Bl, and m denotes the number of 7

8 payments per year. Thus, the market value of fixed coupon bond ( B l ) observed at calendar time t k with years to maturity can be expressed as (notation is suppressed where possible): V Bl,,t k i1 i1 Coupon Bl Par Bl e r i :t k m Coupon Par e r i i m i Par Bl e r :t k Par e r CF i e r i i i1 (1) This framework could easily be applied to a bond portfolio, but would require yet more precise notation. Bond Risk Measures Macaulay (1938) is usually credited with providing the first mathematically detailed definition of an adjusted bond term to maturity, a definition Macaulay gave as duration. Thus, using our notation and continuous compounding, Macaulay duration is expressed as CF Duration B i e r i i w i i () i1 V B i1 Macaulay assumed a constant yield for all cash flows. We provide a new perspective on appraising the importance of the constant yield assumption for bond risk management. Hicks (1939) identified Macaulay duration as the elasticity of the bond value with respect to the discount factor. Let denote e r, a single period discount factor. Hicks elasticity measure can be expressed as HE B i1 dv B VB d CF i e r i V B dv B d i V B i1 i1 i CF i i 1 V B w i i Duration B (3) and Hicks assigned the term average period to this measure. In our context of bonds, Hicks insight is that if the average period of a bond is greater (less) than some benchmark, then a 8

9 decline (rise) in interest rates will result in the change in the bond being greater (less) than the change in the benchmark. Thus, Hicks provides the tools for future asset-liability management. Bond Return Measures Redington (195) introduced the concept of immunization by focusing on the value of assets less the value of liabilities. For consistency, we present just a single bond as an asset. Also, this expression is assuming discretely compounded bond returns. Thus, using our notation, Redington s Taylor series approach to duration and convexity is based on V B i V B i r r r i! i i V B i r r i V B V B i! r r 1 V B r (4) r and solving for the discretely compounded rate of return on the bond, we have i1 i i R B,dc V B V B 1 V B r r r i 1 V B r r i V B V B i! V B i! 1 V B V B r r 1 1 V B V B r i1 (5) r Duration B r 1 Convexity B r where Duration B 1 V B V B r CF i e r i i w i i i1 V B i1 (6) Convexity B 1 V B CF i e r i V B r i w i i i1 V B i1 (7) This perspective of duration became known as modified duration as opposed to Macaulay duration. If continuous compounding for computing bond value is used, then these two perspectives of duration are identical. This equality is not true with discrete compounding for computing bond values. Both bond valuation and bond returns influence bond risk measurement. Assuming continuously compounded bond returns, we have 9

10 lnv B ln i V B r i! i r r i i ln i V B r i! r i lnv B ln V B r r r 1 lnv B r r (8) and thus R B,cc ln V B V B i1 1 V B V B r r 1 1 ln V B i V B r i! r r i i1 ln i V B r i! r i V B 1 r V B V B r r Duration B r 1 Convexity B Duration B r (9) ote that the second derivative produces an additional term with continuous compounding (see Barber (1995)). Several other authors have examined duration and other bond risk measures and determined certain modifications are beneficial. Though not exhaustive, major contributions were made by Durand (1957), Fisher (1966), Fisher and Weil (1971), Granito (1984), Fong and Vasicek (1983, 1984), Fong and Fabozzi (1985), Chambers, Carleton, and McEnally (1988), Prisman and Shores (1988), Bierwag, Kaufman, and Latta (1988), Ho (199), Barber (1995), awalkha and Chambers (1996), Soto (1) and awalkha, Soto and Beliaeva (5). The numerous extensions to the original duration and convexity formulas indicate that a richer framework is needed to implement these measures in practice. We pursue a unique approach to provide more rigorous bond return measurement and bond risk management tools through a parsimonious yield curve model that we now introduce. Fitted Term Structure Models Fitted term structure models can be viewed from two perspectives, calendar time and maturity time. The calendar time perspective is focused on the stochastic behavior of the term structure over time. The maturity time perspective is focused on the non-stochastic shape of the term structure at a particular point in calendar time. Prior empirical studies of the calendar time 1

11 term structure of interest rates have documented several well-known observations. Crack and awalkha () summarize that... (u)p to 95 percent of the returns to U. S. Treasury security portfolios are explained by term-structure level shifts, slope shifts, and curvature shifts (Letterman and Steinman 1991; Jones 1991; Willner 1996; Jamshidian and Zhu 1997) (34). Maturity time term structure models seek to represent the term structure by some mathematical function that has desirable properties. As quoted in elson and Siegel (1987), Milton Friedman recognized the benefits of a parsimonious term structure model when he states, Students of statistical demand functions might find it more productive to examine how the whole term structure of yields can be described more compactly by a few parameters (474). There is a large literature on fitting the term structure dating at least back to Durand (194) and includes piecewise polynomial splines (McCulloch (1971, 1975)), various parametric models (Fisher (1966), Echols and Elliott (1976), Cooper (1977), Dobson (1978), and Chambers, Carleton and Waldman (1984)), and exponential splines (Vasicek and Fong (198)). Several authors offer subjectively drawn curves, including Woods (1983), Malkiel (1966), and Durand (194). Willner (1996) posits that the desirable properties of a curve fitting routine must address the bond portfolio manager s need for intuitive, descriptive, and comprehensive risk exposure information. (p. 49, italics in original) elson and Siegel (1987) provide one such model and appeared to be motivated by the mathematical relationship between spot rates and forward rates. They put forward a parsimonious model that was... solved from differential equations describing rational interest rate behavior (see page 5, Willner (1996)). We use an expanded version of this model below. Svensson (1995) offers an accurate methodology for the LSC model 11

12 based on the work of elson and Siegel (1987). We call this approach the LSC model for level, slope, and curvature. We use a general form that can be expressed as r i :t k C i,n i ;s n1 (1) n b n,t k where C i, i ;s 1 1 (11) C i,1 i ;s s 1e i s i (1) C i,n i ;s n 1 s n 1 1e i s n1 i e i s n1 ;n 1 (13) where s n denotes scalars that applies various weights to different locations on the term structure (termed the time constant by elson and Siegel (1987) that determines the rate at which the regressor variables decay to zero (478) and the hump position parameter by Willner (1996)), C i,n i ;s n1 denotes LSC maturity coefficients, a parameter that depends solely on maturity time and selected scalars, and b n,t denotes the LSC spot rate factor, a parameter that is typically found using ordinary least squares regression applied to maturity time spot rates. In our later analysis of bond mutual funds return decomposition, we derive the LSC spot factors using this methodology and use the fund s reported effective duration to derive the LSC maturity coefficients. ote that as maturity goes to infinity, i, then r i :t k b,t k. Thus, b,t k is interpreted as the level of spot rates. As maturity goes to zero ( i ), then r i :t k b,t k b 1,t k. Thus, b 1,t k is interpreted as the slope of spot rates. ote that if the interest rate term structure is upward sloping then b 1,t k is negative. The spot rate factors greater than one, b n,t k (n>1), measure the rate of curvature. Higher values lead to flatter slopes and lower values lead to steeper slopes for shorter maturities. Barrett, Gosnell and Heuson (1995) and 1

13 Willner (1996) both report that fitted yield curve functions are not that sensitive to the choice of the scalar. Steeley (8) used daily UK government bond coupon STRIPS from December 8, 1997 to May 15, and thoroughly examines a variety of curve fitting methodologies. Spot yield curve fitting methodologies include cubic spline, polynomial, Vasicek as well as the Level, Slope, and Curvature (LSC) model discussed below (referred by Steeley as the extended Svensson model). Based on a three-factor model as used below, Steeley documents that the LSC model has the lowest average (across the sample) mean (across the curve) absolute yield error (15). With six factors, Steeley reports that the cubic spline has the best fit, but the LSC model is a close second. We now present our approach to bond HPR decomposition. III. LSC Bond Return Decomposition We now turn to decomposing bond returns based on the LSC model. The goal is to provide useful metrics for appraising various bond investment strategies. We then apply this decomposition technique to the universe of US Bond mutual funds. Bond return decomposition The LSC model permits a parsimonious bond HPR decomposition that is particularly useful in appraising various attributes related to bond investments. Based on the LSC model defined above, we express the value of a bond at time t k (denoted in brief as Vr,s, highlighting the dependence upon spot rates, r, and spreads, s, as F C i,n i ;s n1 b n,t sp k B l,i,tk n V i r,s V Bl,,t k CF i e CF i e r LS i,tk C sp B l,tk i (14) i1 If we assume a constant spread over the LSC spot rate, decompose the yield to maturity, i1 y Bl,,t k, into LSC rates, sp Bl, i,t k sp Bl,t k, we can r LSC i,t k, and a constant spread, sp Bl,t k. 13

14 Recall the framework presented here can be extended to include a fitted spread in the same fashion as the base rate. ote that with this approach any error in fitting the LSC model to some base spot rate data is completely captured in the spread. Thus, this spread captures both unique aspects of the particular bond as well as an imprecise fit of the LSC model. Up to this point, there is no uncertainty. In units of time (the horizon), however, the LSC model parameters may change, the spread may change, and the bond may have paid a coupon. Thus, the value of the bond position after units of time can be expressed as (~ denotes uncertainty at current time t k ), V r, s V CF r LSC Bl,,t k i e i,t k s p Bl, i,t k i Coupon Bl, CF r LSC i e i,t k s p Bl,t k i (15) i1 i The coupon paid over the horizon is attributed to counter. ote in the equation above, the last summation is from to, not 1 to. Although omitted from our illustrations below, the accrued interest on the coupon payment could easily be incorporated in the coupon amount. With these expressions for bond value, the bond holding period rate of return can be decomposed in the following manner. R V r, s ln Vr,s ln V r,s V r, s V r,s V r, s V r,s Vr,s V r,s V r,s V r, s V r,s ln V r,s Vr,s ln V r, s V r,s ln V r,s V r,s ln V r, s (16) V r,s V r, s V r,s R h R sp R r I Thus, the bond HPR s four components can be identified as the horizon return, the spread return, the base-rate return and an interaction term, respectively. ote that the return attributable to the horizon return is known at the beginning of the holding period, whereas the return attributable to the spread, the return attributable to the base rate and the interaction term, are all three unknown. Thus, the unknown bond return component can be expressed as the bond position s HPR less the known horizon component. When examining bond mutual funds, we consider this unknown 14

15 return as the excess return generated by the bond manager, or the return not attributable to the mere passage of time. R Unknown R R h R sp R r I I (17) This important insight will result in different measures of historical bond risk. Since the horizon return varies over time, a portion of the measured variance of a bond position s HPR is actually just the time series variation of known horizon. Perhaps this component of the bond position s HPR should not be included in time series measures of bond risk. Bond holding period return decomposition We now turn to decomposing the bond position s HPR based on the LSC model into its components based on modified duration, convexity and cross-convexity. Specifically, we focus solely on the return attributable to the fitted LSC model spot base-rates. Recall the bond value, based on the LSC model adjusted for horizon and the coupon (note i= for the coupon payment), can be expressed as V r,s CF i e r LSC sp i,tk B l,tk i (18) i Let the present value cash flow weights for a given bond cash flow be defined as w i,i,t k CF ie LSC r i,tk sp t k V r,s i (19) and define the change in LSC spot base-rate factors over the next time period (assumed to be units of time) as b n,t k, b n,t k b n,t k () Therefore, the continuously compounded rate of return can be approximated, based on Taylor series applied to the natural log of the bond price (detailed proof available from the author), as 15

16 R r Bl,,t k, ln V Bl,,t k V Bl,,t k F FD b n,b l,,t k b n,t k, 1 n FCC bn,b n ',B l,,t k FD bn,b l,,t k FD bn ',B l,,t k n n'n 1 and R Bl,,t k, n FC bn,b l,,t k b n,t k, b n',t k, FD bn,b l,,t k R Bl,,t k, b n,t k, denotes the estimation error from the Taylor series approximation procedure. We now examine factor durations, factor convexities and factor cross-convexities. LSC factor durations. LSC factor durations are formally defined in a manner similar to modified duration, except LSC factors are used rather than yield to maturity. That is, FD bn,b l,,t k V B l,,t k b t k 1 b n,t k V Bl,,t k b t k i level changes, then LSC factor durations reduce to modified duration. i C i,n i ;s n1 () w i,bl,,t k The LSC coefficients are an important variable in computing LSC factor durations. If only the LSC factor convexities. LSC factor convexities are formally defined in a manner similar to standard convexity, except LSC factors are used rather than yield to maturity. That is, FC bn,b l,,t k V Bl,,t k b t k b n,t k only the level changes, then LSC factor convexities reduce to standard convexity. 1 V Bl,,t k b t k i i C i,n i ;s n1 w i,bl,,t k (3) The LSC coefficients are an important variable in computing LSC factor convexity. Again, if LSC factor cross-convexities. LSC factor cross-convexities are unique to the LSC (1) model. Again, LSC factors are used rather than yield to maturity and the focus is on the crosspartial derivative. That is, FCC bn,b n ',B l,,t k V Bl,,t k b t k 1 b n,t k b n',t k i i C i,n i ;s n 1 C i,n' i ;s n'1 V Bl,,t k b t k w i,bl,,t k (4) 16

17 Once again the LSC coefficients are an important variable in computing LSC factor crossconvexities. If only the level changes, then LSC factor cross-convexities are all zero. Base-Rate Return Decomposition. Therefore, the base-rate return can be further decomposed into the modified duration, convexity, and cross-convexity components, R r Bl,,t k, R FD Bl,,t k, R FC Bl,,t k, R FCC Bl,,t k, R Bl,,t k, (5) Each of the base-rate subcomponent returns can be further decomposed into level, slope, and curvature returns. The factor duration returns can be expressed as R FD Bl,,t k, R FD,Level Bl,,t k, R FD,Slope Bl,,t k, R FD,Curvature Bl,,t k, (6) where R FD Bl,,t k, FD bn,b l,,t k (Factor Duration) (7) n b n,t k, R FD,Level Bl,,t k, FD b,b l,,t k b,t k, (Factor Duration Level) (8) R FD,Slope Bl,,t k, FD b1,b l,,t k b 1,t k, (Factor Duration Slope) (9) R FD,Curvature Bl,,t k, FD b,b l,,t k b,t k, (Factor Duration Curvature) (3) The factor convexity components can be expressed as R FC Bl,,t k, R FC,Level Bl,,t k, R FC,Slope Bl,,t k, R FC,Curvature Bl,,t k, (31) Thus, the factor convexity returns can be expressed as R FC Bl,,t k, 1 FC bn,b l,,t k FD bn,b l,,t k b n,t k, (Factor Convexity) (3) n R FC,Level Bl,,t k, R FC,Slope Bl,,t k, 1 FC b,b l,,t k FD b,b l,,t k 1 FC b 1,B l,,t k FD b1,b l,,t k R FC,Curvature Bl,,t k, 1 FC b,b l,,t k FD b,b l,,t k 17 b,t k, b 1,t k, b,t k, (Factor Convexity Level) (33) (Factor Convexity Slope) (34) (Factor Convexity Curvature) (35)

18 The factor cross-convexity components can be expressed as R FCC Bl,,t k, FCC,L,S R Bl,,t k, R FCC,L,C Bl,,t k, R FCC,S,C Bl,,t k, (36) R FCC Bl,,t k, R FCC,L,S Bl,,t k, R FCC,L,C Bl,,t k, R FCC,S,C Bl,,t k, FCC bn,b n ',B l,,t k FD bn,b l,,t k FD bn ',B l,,t k n n'n 1 FCC b,b 1,B l,,t k FD b,b l,,t k FD b1,b l,,t k FCC b,b,b l,,t k FD b,b l,,t k FD b,b l,,t k FCC b1,b,b l,,t k FD b1,b l,,t k FD b,b l,,t k b n,t k, b n',t k, (Factor Cross-Convexity) (37) b,t k, b 1,t k, (Factor Cross-Convexity Level-Slope) (38) b,t k, b,t k, (Factor Cross-Convexity Level-Curvature) (39) b 1,t k, b,t k, (Factor Cross-Convexity Slope-Curvature) (4) IV. Illustrations of the LSC Model Decomposition The bond HPR decomposition presented here is illustrated in two ways. First, we illustrate the decomposition with simple examples to highlight selected issues. Second, we illustrate the decomposition with actual fitted spot rates based on constant maturity treasuries from the U.S. Treasury market. For both ways, we ignore the spread component to simplify the presentation. For the simple examples we use four semiannual coupon bonds with maturities of 1-, 5-, 1-, and 3-year maturities. Each bond is assumed to carry a 5 percent coupon and $1 par. The bond holding period is assumed to be the one-month just after a coupon payment. These four bonds will be subjected to four alternative spot rate curve shifts, flat to flat (Level = 5% to Level = 4%), flat to upward (Level = 5% to Level = 5% and Slope = 1%), flat to downward (Level = 5% to Level = 5% and Slope = +1%), and downward to upward (Level = 5% and Slope = +1% to Level = 5% and Slope = 1%). 18

19 Table present the detailed return decomposition. For brevity, we do not report the spread, interaction, cross-convexity and estimation error terms as they are not significant at the first two decimal places. For a parallel shift in the spot rate curve, Table a confirms that the slope and curvature components are essentially zero. Clearly, the longer maturity bonds are more rate sensitive and convexity has an increasing role. If the slope shifts as presented in Table b from flat to upward sloping, then level and curvature components are zero and only the slope components contribute. ote that convexity does not contribute because only the short end of the spot rate curve shifted down. Table c presents similar results except the spot rate curve slope inverted. otice that the total bond HPRs are different when compared to Table b, but the components are nearly identical except for a sign change. Thus, HPRs are roughly symmetrical around the horizon return. Table d illustrates the combination of the two slope changes moving from an inverted curve to an upward sloping curve. 19

20 Table a. Bond HPR Decomposition Change in Level 5% Level to 4% Level (no change in other parameters) 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration.981%.981%.%.% Spot Rates.98% Convexity.%.%.%.% Total 1.316%.981%.981%.%.% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration 4.458% 4.458%.%.% Spot Rates 4.417% Convexity.81%.7%.5%.6% Total 4.861% 4.417% 4.418%.5%.6% 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration 7.998% 7.998%.%.% Spot Rates % Convexity.497%.477%.6%.14% Total 8.353% % %.6%.14% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration % %.%.% Spot Rates % Convexity.5575%.5553%.6%.16% Total % % %.6%.16% Table b. Bond HPR Decomposition Change in Slope (To Upward) 5% Level, % Slope to 5% Level, -1% Slope (no change in other parameters) 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration.789%.%.789%.% Spot Rates.789% Convexity.%.%.%.% Total 1.14%.789%.%.789%.% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration 1.715%.% 1.715%.% Spot Rates 1.713% Convexity.%.%.%.% Total.164% 1.713%.% 1.715%.% 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration 1.875%.% 1.875%.% Spot Rates 1.88% Convexity.%.%.%.% Total.16% 1.881%.% 1.875%.% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration %.% %.% Spot Rates % Convexity.%.%.%.% Total.95% 1.836%.% %.%

21 Table c. Bond HPR Decomposition Change in Slope (To Downward) 5% Level, % Slope to 5% Level, +1% Slope (no change in other parameters) 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration -.789%.% -.789%.% Spot Rates -.789% Convexity.%.%.%.% Total % -.789%.% -.789%.% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration %.% %.% Spot Rates -1.71% Convexity.%.%.%.% Total % %.% %.% 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration %.% %.% Spot Rates % Convexity.%.%.%.% Total % %.% %.% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3935% Duration %.% %.% Spot Rates % Convexity.%.%.%.% Total % %.% %.% Table d. Bond HPR Decomposition Change in Slope (To Downward) 5% Level, +1% Slope to 5% Level, -1% Slope 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.4431% Duration %.% %.% Spot Rates % Convexity.%.%.%.% Total 1.99% %.% %.% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.447% Duration 3.431%.% 3.431%.% Spot Rates 3.45% Convexity.%.%.%.% Total 3.897% 3.45%.% 3.431%.% 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.4% Duration 3.655%.% 3.655%.% Spot Rates 3.655% Convexity.%.%.%.% Total 4.553% %.% 3.655%.% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3999% Duration 3.668%.% 3.668%.% Spot Rates 3.678% Convexity.%.%.%.% Total 4.77% 3.678%.% 3.668%.% Table 3 reports selected one-month HPRs for four points in time. The spot rate curve was fitted based on the constant maturity treasury yield reported in the H.15 file of the Federal Reserve Bank. We assume fictitious bonds exist with the coupon rate closest to par and mature in exactly 1-, 5-, 1- and 3-years. We also assume no spread over the spot rate curve. In practice, off-the-run treasuries do trade for a non-constant spread over the on-the-run treasury yields. ote 1

22 cross-convexity involves two of the LSC components, hence level cross-convexity denotes level and slope, slope cross-convexity denotes slope and curvature, and curvature cross-convexity denotes curvature and level. Rows that do not contain values at the second decimal place are omitted. Several interesting observations can be made based on Table 3. First, the bond HPR decomposition presented here provides visibility of the various sources of return. For example, in Table 3a from 1/31/1979, the high coupons result in over 8 basis points of return attributable to just the horizon component and in Table 3d from 4/3/1, the low coupons result in less than 43 basis points of return attributable to just the horizon component. Second, the spot rate component is not always the same as it is the combination of level, slope, and curvature movements. For example, in Table 3c we see a positive contribution of the spot rate component for the 1-year bond, but a negative contribution for the other three bonds. Third, the magnitude of the contribution of level, slope, and curvature varies widely across time and across maturities. For longer maturities, level tends to dominate and for shorter maturities slope tends to dominate. Fourth, the duration components tend to be orders of magnitude more significant than convexity and cross-convexity components. Thus, when managing bond risk based on this decomposition, it appears to be more important to focus on managing the individual duration components as opposed to convexity components. Intuitively, when analyzing an asset-liability management or pension surplus problem, the convexity terms will become much more significant, especially if the asset portfolios were constructed to immunize the liabilities based on duration measures.

23 Table 3a. Bond HPR Decomposition for 1/31/1979 to 1/31/ % Level,.% Slope, -1.38% Curvature, % Spread to 1.94% Level, 1.46% Slope, -.68% Curvature, % Spread Coupons: % 1-year, 1.375% 5-year, 1.15% 1-year, and 1.% 3-year 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.8917% Duration % %.386%.1937% Spot Rates % Convexity.%.1%.1%.% Total.343% % -1.18%.3863%.1937% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.835% Duration -.696% -5.13%.8668% % Spot Rates % Convexity.11%.181%.13%.17% Total % % %.8687% % 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.868% Duration % %.9155% % Spot Rates -5.48% Convexity.131%.981%.17%.33% Spread.% Cross- -.16% -.47%.1% -.179% Convexity Interaction.% Estimation -.6% Error Total % % %.918% 1.88% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.876% Duration % %.9193% % Spot Rates % Convexity.5847%.5794%.17%.36% Spread.% Cross- -.4% -.8%.1% -.348% Convexity Interaction.% Estimation -.65% Error Total % % 1.918%.9% 1.841% Table 3b. Bond HPR Decomposition for 9/8/1984 to 1/31/ % Level, -.99% Slope, 4.98% Curvature, % Spread to 1.81% Level, -1.83% Slope, 5.8% Curvature, % Spread Coupons: 11.65% 1-year, 1.5% 5-year, 1.35% 1-year, and 1.15% 3-year 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon 1.965% Duration.9347%.355%.5968% -.147% Spot Rates.9347% Convexity.%.%.%.% Total.313%.9347%.355%.5968% -.147% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon % Duration.715% 1.558% 1.311% -.165% Spot Rates.7149% Convexity.3%.%.%.% Total 3.896%.715% 1.589% % -.165% 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon 1.8% Duration 3.69%.363% % -.134% Spot Rates 3.637% Convexity.14%.99%.%.4% Total 4.757% 3.637%.3754% % -.135% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon 1.773% Duration 4.671% 3.45% % % Spot Rates 4.771% Convexity.481%.475%.%.4% Total 5.844% 4.77% 3.47% % % 3

24 Table 3c. Bond HPR Decomposition for 9/3/8 to 1/31/8 5.% Level, -4.9% Slope,.38% Curvature, % Spread to 5.33% Level, -5.43% Slope,.93% Curvature, % Spread Coupons: 1.75% 1-year, 3.5% 5-year, 4.15% 1-year, and 4.65% 3-year 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.113% Duration.466% -.81%.8331% -.847% Spot Rates.466% Convexity.%.%.%.% Total.6775%.466% -.81%.8331% -.847% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.391% Duration -.131% % % -.719% Spot Rates % Convexity.6%.5%.%.% Total.591% % % % -.716% 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.457% Duration % -.593%.176% -.914% Spot Rates -1.33% Convexity.44%.4%.1%.1% Total -.964% % -.566%.17% -.919% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.439% Duration % -4.96%.985% -.938% Spot Rates % Convexity.54%.538%.1%.1% Total % % %.98% -.91% Table 3d. Bond HPR Decomposition for 4/3/1 to 5/31/1 5.5% Level, -6.% Slope, -.33% Curvature, % Spread to 5.7% Level, -5.66% Slope, -1.4% Curvature, % Spread Coupons:.75% 1-year, 3.15% 5-year, 4.15% 1-year, and 4.65% 3-year 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.143% Duration.153%.84% -.639%.149% Spot Rates.153% Convexity.%.%.%.% Total.484%.153%.84% -.639%.149% 5-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.3994% Duration 1.75% % -.637% % Spot Rates 1.735% Convexity.4%.3%.%.% Total.19% 1.735% 1.144% -.638% % 1-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.469% Duration.885%.353% -.665% 1.513% Spot Rates.8863% Convexity.9%.7%.%.1% Total 3.313%.8863%.376% % 1.511% 3-Year Bond Decomposition Spot Rates Total Level Slope Curvature Horizon.447% Duration 4.644% 3.36% % 1.54% Spot Rates 4.866% Convexity.158%.157%.%.1% Total % 4.865% 3.185% % 1.56% V. Application of the LSC Model Decomposition An application of the model presented here can also be employed to decompose the returns of bond mutual funds. We utilize a parsimonious version of the model and find that on average the LSC spot factors interacted with the LSC maturity coefficients and the spread factor 4

25 explain 49.15%, according to adjusted R-squared, of the returns generated by the bond mutual fund universe between 5 and 14. Further, certain mutual fund bond categories had almost 85% of their returns explained by the LSC decomposition model. Data To generate the LSC spot factors, we obtain maturity time spot rates of Treasury securities from the H.15 file and use an ordinary least squares regression. Specifically the regression is: (41) The α t from this regression is the Level factor (L t ), the β 1,t coefficient is the slope factor (S t ), and the β,t coefficient is the curvature factor (C t ). The fund specific data, effective duration and average credit quality, are available via Morningstar Direct. Missing data are interpolated since many funds only report this information quarterly, however a fund must have reported within the last 6 months for a missing value to be interpolated. The funds returns are from the CRSP mutual fund survivorship-free database. The universe of bond funds is defined as all open-ended funds in the fixed income broad category group from Morningstar which reported between 1/1/5 and 9/3/14; only Class A shares are used, but results are robust to using weighted averages of the share classes. The initial universe is 767 funds, and data is available for the models for 567 of these funds, resulting in 41,968 fund-month observations. For each fund we use the effective duration to form fund specific C 1 and C variables for each month using equations (1) and (13). The fund s average credit quality is used to create the credit spread (CS) variable as the difference between the funds quality index return, according to their average credit quality and the 1 year CMT. 5

26 Horizon Return The horizon return captures the return attributable to the mere passage of time, and this return is known at the beginning of the holding period. We generate horizon return in the following manner. The C 1 and C are generated using the effective duration for each fund, and the LSC spot factors (L t, S t, and C t ) are used for each time period t. These factors are then used to estimate the fund s yield as: The bond fund s value based on this yield is computed as: (4) We then decrease the effective duration by Δt =1/1 and regenerate C 1 and C. Decreasing (43) effective duration by one month simulates the passage of time. We then utilize the new Ć 1 and Ć into equation (4) to generate. This new yield is employed in the following equation to find the bond s theoretical value one month ago, with no changes to the yield curve or credit spread. The horizon return can then be estimated as: (44) (45) For the purposes of the bond return decomposition model, the return analyzed should be the unknown portion or excess return. We compute the excess return of a fund as the reported return less the horizon return component. 6

27 Results The model implies a bond s holding period return is a function of the horizon component, the spread component, and the base-rate components. HPR= f(δl t, ΔS t *C 1i,t, ΔC t *C i,t, Credit Spd i,t ) (46) where ΔL t, ΔS t, and ΔC t are factors derived from the LSC model and C 1 and C are fund specific factors generated from the fund s reported effective duration. We test three models from the theoretical model presented in this paper. Each model effectively adds another layer of the decomposition components to the previous model, i.e. Model 1 is nested in Model, Model is nested in Model 3. The results from these models are presented in Table 4a and 4b. The first model includes only level and slope as factors. Model 1: (47) As illustrated in Table 4a and 4b, Model 1 explains 6.85% of the bond fund universe excess return, according to the adjusted R-squared for the regression. The Emerging Markets Fixed Income and Short Government categories have the largest amounts of returns accounted for by the model with 45.88% and 46.5%, respectively. Bond fund categories with a very short duration and corporate focus, namely Short-Term Bond and Ultrashort Bond, have the lowest R- squared values. Funds with a low duration tend to have a larger proportion of their return attributable to horizon return, so it is not surprising these funds are not as well explained by Model 1. The second model incorporates curvature. Model : (48) 7

28 The results show Model accounts for 47.3% of the bond return variability for the universe of funds. Excluding Morningstar categories with a small number of observations, the category with the highest R-squared for Model is the Intermediate-Term Bond category with 7.59%. High Yield Bond funds and short duration bond funds have the least amount of returns explained by Model. The final model includes the spread component. Model 3: (49) Table 4a indicates that Model 3 explains almost 5% of the returns generated by the bond mutual fund universe. Multisector bond funds and World bond funds have some of the least variation explained with 39.85% and 41.%, respectively. Bond fund investments in these categories are spread across various types of fixed income investments and potentially have more investments whose returns are not explained by a curve fitting model with credit spread incorporated. These type of investments could be contributing to the poor fit of the model. Table 4a: Bond Fund Return Decomposition-R--Squared Values for Bond Mutual Fund Universe & by Global Category Model 1 Model Model 3 # of Funds umber of Obs Entire Universe 6.85% 47.3% 49.15% ,968 by Morningstar Global Category US Fixed Income 5.1% 48.7% 5.49% 341 7,818 Other Fixed Income 37.1% 53.5% 55.55% 34 1,433 Inflation Linked 5.6% 48.99% 51.66% 5 1,91 High Yield Fixed Income 3.69% 38.6% 4.17% 89 6,494 Global Fixed Income 5.56% 39.53% 41.% 45,95 Emerging Markets Fixed Income 45.88% 6.3% 61.44% 33 1,35 8

29 Table 4b: Bond Fund Return Decomposition-R--Squared Values for Bond Mutual Fund Universe & by Morningstar Category Model 1 Model Model 3 # of Funds umber of Obs Entire Universe 6.85% 47.3% 49.15% ,968 by Morningstar Global Category Bank Loan 37.3% 51.41% 5.68% 8 1,568 Corporate Bond 16.53% 38.6% 4.57% 3 1,939 Emerging Markets 45.88% 6.3% 61.44% 33 1,35 High Yield Bond 3.69% 38.6% 4.18% 89 6,494 Inflation Protected Bond 5.6% 48.99% 51.66% 5 1,91 Intermediate Government 8.36% 7.59% 73.98% 41 3,94 Intermediate-Term Bond.64% 48.84% 5.43% 119 9,764 Long Government 1.7% 84.6% 84.56% 14 Long-Term Bond 13.93% 8.95% 85.4% 3 6 Multisector Bond.55% 36.87% 39.85% 35,39 ontraditional Bond 37.1% 53.5% 55.55% 34 1,433 Preferred Stock 31.99% 45.76% 5.5% 137 Short Government 46.5% 6.1% 6.56% 13 1,399 Short-Term Bond 8.37% 4.37% 41.97% 61 5,554 Ultrashort Bond.31% 3.14% 35.51% 14 1,4 World Bond 5.57% 39.53% 41.% 45,95 VI. Summary The problem of decomposing bond portfolio HPRs was addressed in this paper. We applied the LSC model of elson and Siegel (1987) with extensions along the lines of Svensson (1995). Based on the LSC model, bond HPRs are decomposed into four main components, the non-random horizon component, the spread component, the coupon component, and the spot rate component. The spot rate component of returns can be further decomposed into three components attributable to modified duration, convexity, and cross-convexity based on a Taylor series approximation. Each of these three spot rate components can be further decomposed into three 9

30 subcomponents tied to movement in level, movement in slope, and movements in a set of curvature components. We illustrate our results with several simple numerical examples as well as selected illustrations based on U.S. Treasury data. From these illustrations, we saw that the horizon component contribution has varied widely over time. Also, when spot rates are decomposed, the duration components dominate in magnitude over convexity and cross-convexity components. This will not be the case with asset-liability management and pension surplus issues. We apply the model to decompose the unknown return, the holding period return less the horizon return, for the Morningstar universe of bond funds between 5 and 14. We test three models generated from the theoretical model presented in this paper, with each model effectively adding another layer of the decomposition components to the previous model. The first model includes only level and slope as factors and explains 6.85% of the universe returns. The second model adds curvature factors and accounts for 47.3% of the returns. Finally incorporating the credit spread factor results in 49.15% of returns explained. The ability to decompose bond returns into increasingly refined components will provide analytical tools to improve bond portfolio management. 3

31 REFERECES Barber, J., 1995, A note on approximating bond price sensitivity using duration and convexity, Journal of Fixed Income 5, Barrett, W. R., Gosnell, Jr., T., and A. Heuson, 1995, Yield curve shifts and the selection of immunization strategies, Journal of Fixed Income 5, Bierwag, G., Kaufman, G., and C. Latta, 1988, Duration models: A taxonomy, Journal of Portfolio Management 14, Brooks, R. and M. Livingston, 199, Relative impact of duration and convexity on bond price changes, Financial Practice and Education, Chambers, D., Carleton, W., and D. Waldman, 1984, A new approach to estimation of the term structure of interest rates, Journal of Financial and Quantitative Analysis 19, Chambers, D., Carleton, W., and R. McEnally, 1988, Immunizing default-free bond portfolios with a duration vector, Journal of Financial and Quantitative Analysis 3, Cooper, I., 1977, Asset values, interest-rate changes, and duration, Journal of Financial and Quantitative Analysis 1, Crack, T. and S. awalkha,, Interest rate sensitivities of bond risk measures, Financial Analysts Journal 56, Dobson, S., 1978, Estimating term structure equations with individual bond data, Journal of Finance 33, Durand, D., 194, Basic yields of corporate bonds, , ational Bureau of Economic Research, Technical Paper no. 3, Cambridge, MA. Durand, D., 1957, Growth stocks and the Petersburg paradox, Journal of Finance 1, Echols, M. and J. Elliott, 1976, A quantitative yield curve model for estimating the term structure of interest rates, Journal of Finance and Quantitative Analysis 11, Fisher, L., 1966, An algorithm for finding exact rates of return, Journal of Business 39, Fisher, L. and R. Weil, 1971, Coping with the risk of interest-rate fluctuations: Returns to bondholders from a naïve and optimal strategies, Journal of Business 44, Fong, H. and F. Fabozzi, 1985, Appendix E: Derivation of risk immunization measures, Fixed income portfolio management (Dow Jones-Irwin, Homewood, IL). 31

32 Fong, H. and O. Vasicek, 1983, Return maximization for immunized portfolios, in Kaufman, G., G. Bierwag, and A. Toevs, eds.: Innovations in bond portfolio management: Duration analysis and immunization (JAI Press, Inc., Greenwich, CT). Fong, H. and O. Vasicek, 1984, A risk minimizing strategy for portfolio immunization, Journal of Finance 39, Granito, M., 1984, Bond Portfolio Immunization (LexingtonBooks, Lexington, MA). Hawawini, G., ed., 198, Bond duration and immunization: early developments and recent contributions (Garland Publishing, ew York). Hicks, J., 1939, Value and capital (Clarendon Press, Oxford, UK). Ho, T., 199, Key rate durations: Measures of interest rate risk, Journal of Fixed Income, Jamshidian, F. and Y. Zhu, 1997, Scenario simulation: Theory and methodology, Finance and Stochastics 1, Jones, F., 1991, Yield curve strategies, Journal of Fixed Income 1, Knez, P., Litterman, R., and J. Scheinkman, 1994, Explorations into factors explaining money market returns, Journal of Finance 49, Lidstone, G., 1895, On the approximate calculation of the values of increasing annuities and assurances, Journal of the Institute of Actuaries 31, pp (Alternative page numbers pp ) Quoted in Hawawini (198). Litterman, R. and J. Scheinkman, 1991, Common factors affecting bond returns, Journal of Fixed Income 1, Macaulay, F., 1938, Some theoretical problems suggested by the movement of interest rates, bonds, yields, and stock prices in the United States since 1856 (Columbia University Press, ew York, Y). Malkiel, B., 1966, The term structure of interest rates (Princeton University Press, Princeton, J). McCulloch, J., 1971, Measuring the term structure of interest rates, Journal of Business 34, McCulloch, J., 1975, The tax-adjusted yield curve, Journal of Finance 3, awalkha, S. and D. Chambers, 1996, An improved immunization strategy: M-Absolute. Financial Analysts Journal 5,