A New Stochastic Duration Based on. the Vasicek and CIR Term Structure Theories

Transcription

1 A New Stochastic Duration Based on the Vasicek and CIR Term Structure Theories XUEPING WU * Fina Version: February 000 Key words: Bond; One-Factor; Term Structure; Stochastic Duration; Immunization JEL Cassification Code: G10; G11; G13 * The author is Assistant Professor in Finance at Department of Economics & Finance, City University of Hong Kong. He is gratefu to Stephen Brown, Piet Sercu, Sanjay Srivastava, and an anonymous referee for hepfu comments and suggestions. He aso thanks the City University of Hong Kong for research funding (project number ). Address for correspondence: Xueping WU, Assistant Professor in Finance, Department of Economics & Finance, City University of Hong Kong, 83 Tat Chee Avenue, Kowoon, Hong Kong, Te: (85) ; Fax: (85) ; emai: efxpwu@cityu.edu.hk.

2 A New Stochastic Duration Based on the Vasicek and CIR Term Structure Theories Abstract The stochastic duration based on the Vasicek and CIR modes is theoreticay superior to Macauay s duration. However, empirica tests on bond immunization performance have so far faied to show its superiority. Within the one-factor framework, this paper proposes to use a onger zero-curve yied instead of the origina instantaneous interest rate as a proxy for the reevant risk source(s). We prove that the new duration becomes arger, increasing with bond maturity, than the origina duration. Bond immunization using Begian data shows that the new duration definitey beats the origina duration and can in some cases outperform Macauay s duration. Key words: Bond; One-Factor; Term Structure; Stochastic Duration; Immunization JEL Cassification Code: G10; G11; G13

3 Introduction Macauay s duration, being the most easiy understandabe measure of exposure to interest rate risk, is widey used by practitioners for the purpose of bond immunization. 1 In its origina form, the mode unreaisticay assumes a fat yied curve and purey parae shifts in the yied curve; but modifications made in Bierwag and Kaufman (1977), Bierwag (1977) and Khang (1979), now aow for non-fat term structures as we as inear and decreasingin-term shifts. However, as pointed out by Ingerso, Sketon and Wei (hereafter, ISW) (1978) and Cox, Ingerso and Ross (hereafter, CIR) (1979), a more fundamenta probem with the mode is that even in its generaized form it cannot be an equiibrium mode, because it vioates the no-arbitrage condition. For these reasons, they suggest a theoreticay sounder measure of yied curve risk that is based on the CIR one-factor term structure mode. Boye (1977) discusses a simiar duration measure that is based on the Vasicek mode. However, ISW (1978) and CIR (1979) did not empiricay test the so-caed stochastic duration (hereafter, ISW/CIR s duration), athough they show numericay that the empiricay observed mean-reverting property of short-term interest rate dynamics is consistent with dampened fuctuations of ong yieds, which are captured by their stochastic duration but missed bady by the traditiona duration measure. Unfortunatey, using coupon bond data, empirica tests of stochastic duration measures based on these (or even more sophisticated) theoretica modes have not demonstrated any 1

4 actua superiority to the simpe Macauay duration. One apparent reason is that both the Vasicek and CIR modes impy near-constant zero-coupon yieds at the ongest maturities; thus, they fai to capture the movement in the ong end of yied curves so bady that even the ad hoc Macauay duration sti turns out to be superior. The practica advantages of Macauay s duration seem to stem from two aspects: (a) interna rates of return refect averages of zero-coupon rates, which means that they can capture a ot of yied-curve information; and (b) interna rates of return are bond- and hence maturity-specific, so that they capture the average zero-coupon rates that are reevant for that particuar bond. In contrast, one-factor modes focus on one specific yied ony, the instantaneous rate, which woud be justified ony if, as predicted by the theoretica term structure modes, a zerocoupon yieds were intimatey inked to this short-term rate and this is manifesty not the case. Given the poor immunization performance caused by the misspecification of the onefactor equiibrium term structure modes, this paper proposes an aternative stochastic duration measure within the one-factor framework. The proposed stochastic duration uses the change in a onger zero-coupon yied rather than the instantaneous rate as a proxy for the reevant risk source of the unexpected changes in interest rates. Empiricay, a onger yied 1 Fisher and Wei (1971). See aso Bierwag, Kaufman and Khang (1978) for a review of appications of Macauay s Duration. See aso Ingerso (1983), Neson and Schaefer (1983) and Brennan and Schwartz (1983). There is an exception. According to Haugen (1993), the unpubished work of Lau (1983) did show that the CIR

5 carries more usefu information on the movement of the term structure than the instantaneous interest rate and, thus, compensates for much of the information oss due to specification errors in the theoretica modes. More precisey, the risk-factor proxy we use in the price sensitivity (or duration measure) of a coupon bond with time to maturity τ is specified as the change in a wτ-period zero-coupon yied, where w is a number beow unity and, in fact, cose to zero. Of course, these risk factors differ across bonds with different maturities; thus, in the ogic of the one-factor modes these wτ-yieds (factors) shoud a have different standard deviations. To make the modified duration work effectivey, however, one has to assume that the differences of these factors are trivia. Athough such an assumption is not iteray compatibe with the underying bond pricing mode, it is not a priori a very unreasonabe assumption if a these wτ-yieds (factors) sti fa in the short end of the maturity spectrum and are, therefore, very simiar across bonds. Nevertheess, the optima choice of w is argey empiricay determined. The advantages of the proposed approach are two-fod. First, ike the bond s theoretica sensitivity to the instantaneous rate and in contrast to the standard Macauay duration, the proposed interest-exposure measure preserves much of the theoretica and empirica tractabiity that one-factor term structure modes enjoy and can be directy obtained from the modes that a financia firm may aready be using for the pricing of bonds. Second, the performance of the proposed measure of interest risk is quite comparabe to Macauay s duration and definitey superior to the performance of a bond s theoretica duration measure is comparabe to Macauay s duration in immunizing a singe iabiity using two monthy rebaanced highest-yied bonds from each side of the duration of the iabiity. 3

6 sensitivity to the instantaneous rate. This caim is substantiated by the resuts of a bondimmunization performance test using data on Begian defaut-free, non-caabe bonds. The test shows unequivocay that the new durations from both the Vasicek and CIR versions are superior to the origina ISW/CIR duration and, for vaues of w between.5% and 5%, often outperform Macauay s duration. The success of the modified stochastic duration becomes possibe because, as proven in Appendix A, the new stochastic duration magnifies the origina ISW/CIR duration, and does so more for ong bonds than for short bonds. Thus, one contribution of this paper to the iterature is that it provides a stochastic duration measure preserving considerabe theoretica and empirica tractabiity, which outperforms Macauay s duration in reasonabe cases and beats definitey the theoretica measures of term structure risk. Most importanty, the proposed approach sheds ight on how practitioners can appy term structure modes which are bound to suffer specification errors. A second contribution is that the paper adds one more dimension in testing the Vasicek and CIR one-factor term structure modes. Whie much work has been done in estimating the competing modes parameters, itte empirica work has been provided as to how we these modes fare in bond immunization. 3 At first sight, immunizing M bonds with M factors may ook simiar to using an M- factor, APT-stye mode (Ross, 1976). However, there is a fundamenta difference: in our appication, the M yieds are mathematicay inked by a term structure theory whie no 3 See Brown and Dybvig (1985), Pearson and Sun (1994), De Munnik and Schotman (1994), and Sercu and Wu (1997), among others. 4

7 functiona structure is imposed among the APT factors. One impication is that, in this paper, the sensitivity of each bond to its own factor is not constant, but varies over time in a way that is determined by the underying bond pricing mode. In contrast, empirica work on APT bond modes assumes constant exposures to pre-specified factors. Because of this fundamenta difference in modeing, our use of the zero-coupon yieds with ony short maturities does not necessariy contradict, for instance, the finding by Eton, Gruber and Michaey (1990) that the four-year spot rate best proxies for the risk source of a one-factor mode of the APT type. The remaining of this paper is organized as foows. Section 1 gives a brief review of the Vasicek and CIR one-factor term structure modes and proposes the new stochastic duration measure in both the Vasicek and CIR versions. Section describes the data and the estimates of mode parameters. Section 3 compares immunization performance among different duration measures. Performance evauation is based on a comparison, across modes, of root mean square errors (RMSE s) of daiy time series of residua returns of individua bonds. The residua return of a bond is defined as the difference between the hoding period return of the bond and that of a duration-and-vaue matched portfoio formed from the rest of the bonds in the sampe. Section 4 concudes the paper.. Term Structure Mode and Stochastic Duration For the sake of carity and continuity, the Vasicek and CIR term structure modes are briefy presented before introducing the new duration measure. In a one-state-variabe mode, the price at t, P(r,t,T), of a zero bond maturing at T, is treated as a contingent caim on the 5

8 instantaneous interest rate, r(t), which usuay foows a mean reverting stochastic process: dr = κ[ m r( t)] + δ( r, t) dz. (1) Mean reversion means that r(t) is pued back toward its ong term mean, m, at rate of κ. The voatiity, δ(r, t), is constant, σ, for the Vasicek mode and σ r( t) for the CIR mode. In (1), dz is a Wiener process, and κ and m are positive constants. Since dz is the ony risk source of both the contingent caim and the state variabe, r(t), a ocay risk-free hedge can be estabished and hence there exists an intertempora noarbitrage condition, known as the fundamenta PDE, P P δ( r, t) P + µ ( r, t ) + rp = 0. () t r r In the Vasicek mode, µ(r,t) equas κ[m-r(t)]-qσ with q being specified as a constant price of risk of changes in the instantaneous rate, r(t), whie in the CIR mode, µ(r,t) equas κ[mr(t)]-q(r,t)σ(r,t)=κm-(κ+λ)r(t) with the risk premium, λr(r,t)=q(r,t)σ(r,t), being endogenousy determined. The other parameters are the same as in (1). Given boundary conditions of a zero-coupon bond, both the Vasicek and CIR modes ead to cosed-form bond pricing formuas. Because of the constant voatiity specification in (1), the Vasicek zero-coupon bond pricing mode takes a simper form, P( r, τ) = A( τ) e r t B ( ) ( τ), (3) with A( τ) = e κτ κτ φ1 (1 κτ e ) φ (1 e ), (4) 6

9 1 and B( τ) = ( 1 e κτ ). (5) κ where φ κm qσ = κ 1 1 σ κ 3 ; φ 1 σ = ; and τ is time-to-maturity, i.e., τ=t-t. 3 4 κ The CIR mode aso conforms with the genera form in (3). However, both A(τ) and B(τ), which are soey determined by time to maturity, become ess neat, namey, A( τ) = θ θ e 1 θ τ θ1τ ( e 1) + θ 1 θ 3, (6) and B( τ ) = θ e θ τ 1 1 θ 1τ ( e 1) + θ 1, (7) where θ = ( κ + λ) + σ ; θ 1 κ λ θ1 = + + ; and θ = κm. σ 3 In cross-sectiona estimation, the unobservabe state variabe, r(t), can be estimated as an impied instantaneous rate. So, there are four parameters in each of the modes. Since a zero coupon bond with a maturity more than one year is not aways avaiabe in a countries, estimation is usuay done using coupon bond data. In fact, the price of a coupon bond at t, P(r,c,t,T), with coupon rate, c, maturing at T, is simpy a portfoio of zero coupon bonds, whose prices are determined directy by the zero-coupon bond pricing modes in (3), (4) and (5) or (3), (6) and (7). These modes can be easiy expressed in the form of the continuousy compounded zero-coupon yied, namey, R( r, τ) = 1 n P( r, τ). (8) τ 7

10 With the above estimated term structures, bond price sensitivity to interest rate changes can be expressed in a more eegant fashion. Duration in genera is a measure of bond return risk caused by unanticipated changes in interest rates. In the traditiona Macauay duration measure, the unanticipated changes are assumed to come from the parae shifts in a fat term structure of interest rates; that is, a yieds are assumed to have the same variance and to be perfecty correated. Theoreticay as we as empiricay, the assumption of pure parae shifts and fat term structures is untenabe. In contrast, the onefactor Vasicek and CIR framework identifies a specific risk source of the unanticipated changes in interest rates because the whoe term structure movement is assumed to hinge on the instantaneous rate, r(t). Sti, it woud be naive to beieve that the unexpected changes in interest rates can be satisfactoriy expained by just the instantaneous rate. To sove this probem, one coud adopt two- or mutipe-factor modes; however, these are often hard to estimate. Aternativey, one coud ook for a singe risk-factor that (a) is better at capturing twists and shifts of the term structure of interest rates than the instantaneous interest rate, and (b) is actuay observabe. The atter aternative is the route adopted in this paper. Suppose, initiay, that the unexpected changes in interest rates can be tracked by one state variabe, the instantaneous interest rate. Even though one-week (and, in many markets, overnight) interest rates are observabe, the instantaneous rate itsef is not. However, one can aways construct an intermediary and observabe factor, F, that, in turn, is driven by the instantaneous rate. Then a genera duration measure takes the foowing form, 8

11 Duration = 1 dp( r, c, t, T ) P( r, c, t, T ) df N 1 dp( r, τ ) = CF( τ ), (9) P( r, c, t, T ) df = 1 N where P( r, c, t, T ) = CF( τ ) P( r, τ ) is the price of a bond with N coupons (pus face = 1 vaue at T) to be due and CF(τ ) is the -th component cash fow with time to due date, τ (τ τ=τ-t ; ). As the intermediary risk factor, F, we propose one of the zero-coupon yieds in (8), and we assume it is abe to capture, with the hep of a term structure mode, the unanticipated changes in a interest rates. Thus, one can easiy get around the probem of unobservabiity of the instantaneous rate. More probematic is the fundamenta assumption that one singe factor, such as F(r(t)) or r(t) itsef, drives a bond prices. In reaity, even the best-chosen singe factor can ony capture the systematic risk for a bonds, that is, the risk caused by the average component of shifts in the term structure. But the term structure movements often cause more unanticipated price changes for one group of bonds than for another, as iustrated by, for instance, Eton, Gruber and Michaey (1990). In other words, one factor does not seem to be abe to take care, at the same time, of average (systematic) risk as we as maturityspecific risk. Nor can we reasonaby assume, as we often do in modes for equity markets, that the maturity-specific risk can be satisfactoriy diversified away: by definition, that specific risk equay affects a bonds with simiar terms to maturity. To remedy this, we et the risk factor be party determined by the time-to-maturity of the bond. More precisey, the 9

12 zero-coupon yied that we choose as our intermediary factor is bond-specific, in the sense that we seect the yied that corresponds to a fixed fraction, w, of the time-to-maturity of the coupon bond in question, τ. 4 Using (3), (8) and defining F = R(r, wτ), we find, N 1 dp( r, τ ) Duration = CF( τ ) (10) P( r, c, t, T ) dr( r, w ) = 1 τ N 1 = CF( τ P( r, c, t, T) = 1 B( τ ) P( r, τ ) dr ) B( wτ) dr wτ (11) N CF( τ ) P( r, τ ) wτ = B( τ ). (1) = 1 P( r, c, t, T) B( wτ) Equation (10) specifies the risk-factor proxy to be the change in a zero-coupon yied for time to maturity wτ. The time to maturity of the yied factor is tied to the maturity of the bond in question. As a resut, the factor for a ong bond is a onger zero-coupon yied whie the factor for a short bond is a shorter one. For exampe, if we set w at 0.05, the factor for a 10-year bond is the six-month interest rate, whie for a 5-year bond the factor is the three-month rate. Equation (11) says that both zero-coupon bond prices, P r, τ ), and ( the risk factor, R( r, wτ ), for the specific coupon bond in turn depend on the instantaneous rate, r(t), with B(τ) being defined in (3); and equation (1) simpy resuts from rearranging (11). Notice that the first fraction after the summation operator is the reative present vaue 4 If there are M coupon bonds with different maturities, we wi have M factors, τ(k), k=1,, M, just ike M interna yieds in the case of Macauay s duration. However, the M wτ-yieds are 10

13 weights of component cash fows of the coupon bond. The term in the square brackets is the price sensitivity of a zero-coupon bond with a time-to-maturity of τ for the -th coupon to be due. It is worth mentioning that, unike Macauay s duration, the dimension of the price sensitivity in (1) is not a number of units of time. 5 Since the change in the wτ-period yied is taken as the risk factor instead of the change in the instantaneous interest rate, the price sensitivity is modified by wτ. It wi be argued beow that such an adjustment is B( wτ) appropriate. It foows that the genera form of duration in (1) can be specified according to the term structure mode either in (3), (4) and (5) or in (3), (6) and (7). That is, N CF( τ ) P( r, τ ) 1 κτ κ ( wτ) DVasicek = (1 e ) κ ( τ ), (13) w = 1 P( r, c, t, T ) κ 1 e and D CIR θ1 ( wτ ) { θ [ e 1] + θ } N θ1τ CF( τ ) P( r, τ ) e 1 = θ1τ θ1 ( wτ ) = 1 P( r, c, t, T) θ ( e 1) + θ1 e 1 1 wτ (14) fundamentay different from the M interna yieds because the reation among the formers is governed by a bond pricing mode. 5 The stochastic duration measure in (10) stands for the price sensitivity and is consistent with that in Ingerso, Sketon and Wei (1978). However, without the concern on duration matching, Cox, Ingerso and Ross (1979) further propose to convert, non-ineary, the dimensioness stochastic duration measure into one in units of time. It is straightforward to make duration matching with the measure in (10) because price sensitivities shoud be additive. Therefore, this paper uses the dimensioness stochastic duration measure. 11

14 where a parameters were defined before. The first part of the bracketed terms in (13), 1 (1 e κτ ) =B(τ) (for the Vasicek mode from (5)), and in (14), κ θ e θ1τ θ1τ ( e 1) 1 + θ 1 = B(τ) (for the CIR mode, from (7)), represent the price sensitivity using the singe instantaneous rate as the risk factor. This part becomes constant as τ. It is we known that shortterm interest rates fuctuate much more than ong rates. Thus, prices of a ong zero-coupon bond tend to be much ess sensitive to changes in short-term interest rates than those of a short one, as reasonaby described by both the Vasicek and CIR modes. The second part, κ( wτ) { θ } 1 ( wτ) θ [ e 1] + θ1 wτ in (13) and in (14), hep to recoup part of the κ ( wτ ) 1 e θ1 ( wτ) e 1 maturity-specific risk ost due to specification errors. This part is greater than unity and approaches unity as w shrinks to zero (see proofs in Appendix A). Therefore, the use of the wτ-period yied as a risk factor for a specific coupon bond does not simpy modify but magnify, more for ong bonds than for short bonds, the price sensitivity with itte oss of both theoretica and empirica tractabiity. At the same time, the origina ISW/CIR stochastic duration measure (w=0) is nested in the new duration measure (w>0). According to a one-factor equiibrium mode, of course, any given bond has a unique theoretica price variabiity. Thus, if we re-specify the factor in a way that the bond s duration increases, there woud be an offsetting drop in the variabiity of the factor which brings us back to square one. To get out of this circe, an ad hoc assumption is needed that is simiar to the hypothesis of parae shifts of interna yieds in Macauay s duration. We assume that the wτ(i)- and wτ(j)-yieds of two different bonds i and j, have the same 1

15 variabiity. This obviousy vioates the spirit of the theoretica modes. However, if w is sma, say, 0.05 or 0.05, then a bond-specific factors are reay short-term interest rates, which, athough they are of sighty different maturities, sti do not appear to have overy different voatiities in practice. Our second defense of the ad hoc assumption is eminenty pragmatic. If, in practice, this approach does better than either the theoreticay correct approach (which fais to capture term structure movements) or the generaized Macauay mode (that has a totay unstructured approach to the term structure), then the assumption of equa voatiities is not a bad one after a. To verify whether the assumption hods we, one can immunize bonds by going short in a portfoio with a matched duration measure. Thus, we test whether that immunization strategy outperforms strategies based on the theoreticay correct approach or on the purey ad-hoc Macauay duration. The potentia gains with the wτ-period yied for a specific coupon bond in capturing the reevant risk come from two aspects. First, any (finite) positive vaue of w means that we are choosing, as the factor, a zero-coupon yied with a finite maturity rather than the instantaneous interest rate. Thus, the risk factor carries some yied-curve information that woud otherwise have been missed out due to specification errors in the mode. Second, wτ is tied to the time-to-maturity of the (coupon) bond in question. This is important because it is known that, the onger the time-to-maturity of the bond, the ess the instantaneous rate tends to matter, and hence the more information on onger yieds is needed. The ony remaining issue is the choice of w for the wτ-period yied. For simpicity, unike τ, we prefer w not to be bond-specific. And w shoud be sma if the equa variabiity for yieds R(r,wτ(i)) and R(r,wτ(j)) is abe to hod reasonaby. Apart from these a priori 13

16 considerations, we et the choice of w be setted as a purey empirica matter.. Data and Term Structure Parameters Data are obtained from the data service of the Financiee Economische Tijd (a major financia newspaper in Begium). The data consist of prices of OLOs (Obigations Linéaires/Lineaire Obigaties, a cass of non-caabe Begian government bonds first introduced around 1990) and short-term discount bonds constructed from Brusses interbank offer rates in Begian Franc (BIBORs). There are 351 daiy cross sections, after deeting non-trading and thin-trading days from March 7, 1991 through September 16, 199. Parameter estimates of the Vasicek and CIR term structure modes are directy taken from Sercu and Wu (1997), who provide (time-varying) daiy cross-sectiona estimations. Over time, the number of traded OLOs increases from six to eeven because of introduction of new issues. At the beginning of the sampe period, times to maturity of OLOs range from three to 1 years whie near the end, from 1.5 to 15 years. The OLO prices are ast-trade quotes from the continuous on-screen trading in the CATS (Computer Aided Trading System). Bond prices (invoice or trade prices) were computed from cosing quotes (fat prices) pus accrued interest according to the 360-day year rue that hods in the bond market. The one-week settement rue means that the invoice prices are actuay one-week forward prices. This effect is corrected for by computing the impied spot prices from the forward prices. Our use of BIBORs rather than T-bi yieds is dictated by the fact that, because of poor iquidity, the yieds on the T-bis were often higher than comparabe BIBORs by at east ten basis points. Therefore, we prefer BIBORs to fi in the gap in the 14

17 short end of the fu yied curve. The bid-ask spread is 1.5 basis point usuay, and we used the midpoint rates. Five BIBORs, namey, 1-, -, 3-, 6- and 1-month, are avaiabe throughout the whoe sampe period. As shown in Fig. 1 in Sercu and Wu (1997), estimated zero-coupon-yied curves for both the Vasicek and CIR modes are sharpy humped around five months, and the zerocoupon-yied curves tend to shift downward over the sampe period indicating a bu market. With time to maturity in units of days, the mean cross-sectiona estimates of the parameters are φ 1 =0.040, φ =0.0048, κ= and r=8.76% per annum (Vasicek) and θ 1 =0.0103, θ =0.0079, θ 3 =0.061, and r=8.90% per annum (CIR). The mean RMSEs of cross-sectiona regressions are 13.5 (Vasicek) and 1.5 basis points (CIR), respectivey. 3. Comparison of Immunization Performance Tabe 1 reports, for each asset, the time-series averages of the Macauay, Vasicek and CIR duration measures during the sampe period, as we as the modified stochastic durations for various finite vaues of w. The origina ISW/CIR stochastic duration in both the Vasicek and CIR versions, which takes the instantaneous rate as the risk factor, is tabuated in the coumns under w=0%. To interpret the numbers, Macauay s duration of five years means that a 1% change in the continuousy compounded interna yied eads to a 5% change in bond prices. 6 Likewise, a stochastic duration of, say, 0.3 indicates that a 1% change in the 6 The mo dified version of Macauay s duration (Macauay s duration divided by one pus the interna yied) is used. 15

18 wτ-zero-coupon yied eads to a 0.3% change in the prices of a bond with time to maturity, τ. From the tabe we see that when time-to-maturity is very short, the Macauay, Vasicek and CIR duration measures are cose. 7 The stochastic duration measures initiay increase with the bond s maturity (but more sowy than their Macauay counterpart), then remain amost constant for a wide range of coupon bonds in the midde, and decrease for the ong bonds, a pattern that refects the modes prediction that very ong-term yieds are constant. When the Macauay duration reaches the vaue of 4.785, the Vasicek and CIR duration measures peak at 0.6 and 0.35 respectivey. Different duration definitions take different proxies for the unobservabe risk sources, so one cannot draw any concusions from the different eves of the durations. Specificay, if the instantaneous interest rate has a sufficienty high variabiity and reasonaby succeeds in capturing movements in the entire term structure (via a bond pricing mode), then a ow ISW/CIR duration can sti expain a substantia part of bond price changes. Thus, even though in Tabe 1 the price sensitivities of ong (coupon) bonds according to the origina ISW/CIR duration (w=0%) are much ower than those according to the Macauay counterpart and the difference between the two tend to increase with τ, one cannot concude from this that the Macauay duration woud overstate the true price sensitivity of 7 For a very short τ, the dimensioness zero bond price sensitivity, τdr(τ)/dr, approaches τ. Therefore, as ong as the unit of interest rates is consistent (annuaized), Macauay s duration and stochastic duration measures are comparabe at the very short end. 16

19 ong bonds or that the stochastic duration measure is ikey to understate it. 8 Even more importanty, it is not sufficient that a factor has the abiity to predict the variance of bond prices; the factor s predicted bond price shoud aso be highy correated with the actua bond price. Thus, what counts is not the durations individua magnitudes, but their crosssectiona patterns across bonds, the variabiity of the chosen source of risk, and the factor s abiity to capture overa movements of the term structure. In practice, the variabiity of the short-term interest rate turns out to be too ow to expain even the variance of price changes of onger-term coupon bonds. Improvement may be possibe with a magnified stochastic duration measure, that is, a duration measure with w > 0. If the zero-coupon yieds for maturity wτ are taken as the reevant risk factors, the Vasicek and CIR duration measures become arger, as expected, and tend, reasonaby, to increase with time-of-maturity (see coumns under w=5% and w=10%, respectivey). This confirms the price-sensitivity-intensifying effect of taking zero-coupon yieds onger than the instantaneous rate as the risk sources. However, no concusion on the effectiveness of such a modification can be drawn unti it is verified that bond portfoios with equa modified durations aso have highy correated price changes. Thus, we need to evauate the immunization performance. 8 With numerica exampes of duration measures, Cox, Ingerso and Ross (1979) point out that, for coupon bonds, the stochastic duration measure converted into units of years peaks at 10 years, and they contend that this is reaistic compared to the remote peak at 50 years with the Macauay counterpart. However, which duration measure better captures the risk in bond returns requires an empirica check on immunization performance. And the remaining of the paper wi carry out such a task. 17

20 Instead of immunizing a singe iabiity for one specific date, we immunize individua coupon bonds using each of the aternative duration measure. The procedure is as foows. On each trading day, t, we obtain, from previous trading day, duration information about three assets: a duration for a specific coupon bond, m, the duration for the equa-weight short maturity portfoio consisting of five zero-coupon bonds constructed from BIBORs, Sd, and the duration for the equa-weight ong maturity portfoio consisting of a avaiabe OLOs excuding the one to be matched, Ld. Since the specification of duration (either a stochastic one or Macauay s) has not been made here, what foows is very genera. The equa-weight matched portfoio is determined by finding out X and Y (weights on the short portfoio and ong portfoio, respectivey) such that m=x Sd+Y Ld (duration matching) and 1=X+Y (vaue matching). Thus, one can cacuate the matched portfoio effective return as mr=x Sr(short portfoio return)+y Lr(ong portfoio return), and hence the abnorma (residua) return using this matched portfoio as benchmark. Of course, using different duration measures wi generate different matched portfoios (X and Y) and hence resut in different hedging performances. Besides the residua return for the 1-day horizon, five other residua returns for the -, 3-,..., and the 6-day horizon are cacuated and measured by j-day averages. 9 The 9 If the RMSE of the cumuative return is wanted rather than the average, it suffices to mutipy by the number of days. Whether one uses averages or sums, the RMSEs for different horizons obviousy cannot be compared across horizons; however, either RMSE can aways be compared across immunization methods. Note aso that the j-day cumuative return starting from date t substantiay overaps with the j-day cumuative return starting from date t+1 etc., which provides one more reason not to compare the RMSEs across horizons. However, there is no reason to beieve the overap in the 18

21 motivation for ooking at a onger hoding period is that a stochastic duration measure may not be abe to show its potentia within a short horizon. The reason is that bond prices tend to fuctuate around their fundamenta vaues, which are impied by the estimated term structures using cross-sectiona bond prices. 10 Therefore, within a short horizon, such a temporary departure wi introduce noise into the performance of the stochastic duration measures that argey rey on the estimated term structures. Average RMSE s for different hoding periods and for different vaues of w (ony reevant for the Vasicek and CIR duration measures) are reported in Tabe. The concusions are as foows. Comparing across methods we see that for a horizons, the Macauay duration measure resoundingy beats the origina ISW/CIR duration in both the Vasicek and CIR versions (w=0%). For exampe, at the one-day horizon, Macauay immunization has a RMSE of ess than 6.1 basis points for the 1-day horizon, whie the ISW/CIR immunization in both the Vasicek and CIR versions has a RMSE cose to 8.6 basis. However, the performance of stochastic duration measures does improve across the board when w is set greater than zero, confirming that onger zero-coupon yieds better capture the unexpected changes in interest rates than the instantaneous interest rate. Moreover, in some cases (marked with a singe underine) the performance of the modified stochastic duration measures is comparabe to, or better than, the performance of observations woud be in favor of a particuar duration measure; that is, it is unikey to undermine the cross-sectiona comparison. 10 There are aways bond pricing mode residuas. Nevertheess, Sercu and Wu (1997) find that these residuas tend to revert to the mean over time. 19

22 Macauay s duration. For exampe, this is true for the modified duration of the Vasicek mode when w=.5% for the 4-, 5- and 6-day horizons, and for the CIR mode when w=.5% for the 6-day horizon and when w=5% for the 4-, 5- and 6-day horizons. Tabe 3 and 4 further show more detaied immunization performance resuts by asset, which are in genera consistent with the resuts in Tabe. These resuts confirm our eary conjecture that the new duration may ony work effectivey if w is smaish. In a nutshe, a stochastic duration becomes effective if one considers changes in onger zero-coupon yieds rather than the singe instantaneous rate, as the reevant risk sources of the unexpected changes in interest rates. The proposed stochastic duration measure undoubtedy beats the origina ISW/CIR duration. The bond immunization performance race aso shows that the new duration outperforms the popuar Macauay duration in some cases when the vaue of w is in a range between.5% and 5%. The optimum vaue of w tends to be sma, at east in this particuar sampe. It is ikey that, in different sampes, the optima vaue for w woud be different. However, even in different databases one woud not expect substantiay arger vaues for the optima w. The reason is that, if w is very arge, say, 0.5, the factors woud become very different across bonds, which woud vioate the assumption that they are a driven by the same singe factor and have comparabe voatiities. 4. Concusion In this paper, we argue that changes in zero-coupon yieds, which are sighty onger than the instantaneous rate, can be a better proxy for the reevant risk sources of unexpected 0

23 changes in interest rates. We prove that, using such a zero-coupon yied as a risk factor for a specific coupon bond, the stochastic duration derived from the Vasicek and CIR modes is arger, increasing with bond maturity, than the origina ISW/CIR duration. The immunization performance test shows that the proposed stochastic duration definitey beats ISW/CIR s duration and can in some cases outperform Macauay s duration. Appendix A First, using L Hopita s rue, it is trivia to prove that the modified part of the price sensitivity, κ( wτ) { θ } 1 ( wτ) θ [ e 1] + θ1 wτ, (Vasicek) or (CIR) approaches unity when w goes κ ( wτ ) 1 e θ1 ( wτ) e 1 to zero. Second, to prove that the modified part is greater than unity, et us first ook at the case of the Vasicek duration measure. Let f ( x ) = x 1 + e x, ( x = κwτ 0 ), then, the probem is to prove f ( x ) = x 1+ e x 0. A.1 We have f '( x) = 1 e x and f ''( x) = e x. Because f '' ( x ) = e x > 0, there exists a goba minimum for x 0. For f '( x) = 1 e x = 0, there is ony one soution, x = 0. So, f ( x) has a minima vaue at x = 0. Therefore, (A.1) hods. Next, et us ook at the case of the CIR duration measure. Let θ x [ θ θ1] 1 θ1x f ( x ) = ( e 1) + x ( e 1 ), ( x = wτ 0 ), ikewise, the probem becomes to prove 1

24 θ x [ θ θ1] 1 θ1x f ( x ) = ( e 1) + x ( e 1) 0. A. And we have [ x ] [ 1 ] f '( x) = θ θ xe + θ ( e ) + θ θ e θ1x θ1x θ1x θ1x = θ θ ( θ θ ) e + ( θ θ ) A.3 and [ ] θ x f '' ( x) = θ θ e + θ θ θ x ( θ θ ) e 1 θ1x [ θ1 θ θ1 xθ1 θ ] θ1x = e ( ) +. A.4 Note that, when the current interest rate is above κ m, the term structure is faing; κ + λ and when the rate is beow it, the term structure is humped or rising. See Cox, Ingerso and Ross (1985), page 394, foowing equation (6). It foows that κ+λ>0 because κm>0 and an interest rate is positive, and hence that θ >0. See the definition of the parameter θ beow equation (7), and aso notice that θ θ 1 =κ+λ>0. Therefore, f hence there exists a goba minimum for x 0. For f ''( x) > 0, (from A.4), and '( x) = 0, there is ony one soution, x = 0, (from A.3). So, f ( x) has a minima vaue at x = 0. Therefore, (A.) hods. Note that in both cases, we have f ''( x) > 0. Thus, f ( x) is an increasingy monotonic function in τ, indicating that the price sensitivity of ong bonds is more magnified than that of short bonds.

25 REFERENCES Bierwag, G. (1977), Immunization, Duration and the Term Structure of Interest Rates, Journa of Financia and Quantitative Anaysis, 1 (December), pp Bierwag, G. and G. Kaufman (1977), Coping with the Risk of Interest Rate Fuctuations: A Note, Journa of Business, 50, No. 3 (Juy), pp Bierwag, G., G. Kaufman and C. Khang (1978), Duration and Bond Portfoio Anaysis: An Overview, Journa of Financia and Quantitative Anaysis, 13 (November), pp Boye, P. (1977), Immunization under Stochastic Modes of the Term Structure, Working paper (University of British Coumbia). Brennan, M. and E. Schwartz (1983), Duration, Bond Pricing, and Portfoio management, in G. Bierwag, G. Kaufman, and A. Toevs, eds.: Innovations in Bond Portfoio Management: Duration Anaysis and Immunization (JAI Press, Greenwich, CT), pp Brown, S. and P. Dybvig (1986), The Empirica Impications of the Cox, Ingerso, Ross Theory of the Term Structure of Interest Rates, Journa of Finance, 41, No. 3 (Juy), pp Cox, J., J. Ingerso and S. Ross (1979), Duration and the Measurement of Basis Risk, Journa of Business, 5, pp Cox, J., J. Ingerso and S. Ross (1985), A Theory of the Term Structure of Interest Rates, Econometrica, 53 (March), pp

26 De Munnik, J. and P. Schotman (1994), Cross Sectiona versus Time Series Estimation of Term Structure Modes: Empirica Resuts for the Dutch Bond Market, Journa of Banking and Finance, 18, pp Eton, E., M. Gruber and R. Michaey (1990), The Structure of Spot Rates and Immunization, Journa of Finance 45 (June), pp Fisher, L. and R. Wei (1971), Coping with the Risk of Interest-Rate Fuctuations: Returns to Bondhoders from Naive and Optima Strategies, Journa of Business 44 (October), pp Haugen, R. (1993), Modern Investment Theory, 3rd Ed. Prentice Ha. Ingerso, J. (1983), Is Immunization Feasibe? Evidence from the CRSP Data, in G. Bierwag, G. Kaufman, and A. Toevs, eds.: Innovations in Bond Portfoio Management: Duration Anaysis and Immunization (JAI Press, Greenwich, CT), pp Ingerso, J., J. Sketon and R. Wei (1978), Duration Forty Years Later, Journa of Financia and Quantitative Anaysis (Proceedings Issue, November), pp Khang, C. (1979), Bond Immunization When Short-Term Rates Fuctuate More Than Long-Term Rates, Journa of Financia and Quantitative Anaysis, Vo. 14, No. 5 (December), pp Lau, P. (1983), An Empirica Examination of Aternative Interest Rate Risk Immunization Strategies, Unpubished Ph.D. dissertation (University of Wisconsin). 4

27 Macauay, F. (1938), Some Theoretica Probems Suggested by the Movement of Interest Rates, Bond Yieds and Stock Prices in the United States since New York: Coumbia University Press. Neson, J. and S. Schaefer (1983), The Dynamics of the Term Structure and Aternative Portfoio Immunization Strategies, in G. Bierwag, G. Kaufman, and A. Toevs, eds.: Innovations in Bond Portfoio Management: Duration Anaysis and Immunization (JAI Press, Greenwich, CT), pp Pearson, N. and T. Sun (1994), Expoiting the Conditiona Density in Estimating the Term Structure: An Appication to the Cox, Ingerso and Ross Mode, Journa of Finance 49 (September), pp Ross, S. (1976), The Arbitrage Theory of Capita Asset Pricing, Journa of Economic Theory 13, pp Sercu, P. and X.P. Wu (1997), The Information Content in Bond Mode Residuas: An Empirica Study on the Begian Bond Market, Journa of Banking and Finance 1, pp Vasicek, O. (1977), An Equiibrium Characterization of the Term Structure, Journa of Financia Economics 5 (November), pp

28 Tabe 1: Duration Measures Maturity w=0% w=5% w=10% Macauay Vasicek CIR Vasicek CIR Vasicek CIR Bibor-1m Bibor-m Bibor-3m Bibor-6m Bibor-1m Bond05 8-Feb Bond0 5-Apr Bond08 9-Aug Bond04 1-Jan Bond11 30-Ju Bond01 1-Jun Bond03 1-Aug Bond07 7-Jun Bond10 5-Jun Bond06 1-Mar Bond09 1-Oct otes: N The average duration measures over the sampe period (from March 7, 1991, or the first issue date, through September 16, 199) are reported. Macauay s duration is in units of time (years) but the stochastic duration measures have no dimension. For the Vasicek and the CIR duration measures, w is the fraction of the time to maturity (τ) of each bond, and the change in the wτ-period yied (annuaized) is used as a risk factor proxy for the unexpected changes in interest-rates. Bonds are ascendingy tabuated from top to bottom by maturity. 6

29 Tabe : Bond Immunization Performance Comparison Duration Horizon (Trading Days) w Measure Macauay % Vasicek CIR % Vasicek CIR % Vasicek CIR % Vasicek CIR % Vasicek CIR Notes: The performance is gauged by the average of RMSE s of the daiy time-series of residua returns (in basis points) over a OLO bonds. The residua (abnorma) return is the difference between a hoding period effective return of individua bonds and the benchmark return of the duration-and-vaue matched portfoio. The residua return of the j-day horizon at a cross section is defined as the cumuative daiy residua returns up to the j-day horizon divided by j days. The average RMSE s no greater than the Macauay counterparts are marked with a singe underine. w is the fraction of the time-to-maturity (τ) of a bond, and the bond-specific wτ-period yieds (annuaized) are used to proxy for the reevant risk sources of the unexpected changes in interest-rates. 7

30 Notes: Tabe 3: Performance of Macauay s and ISW/CIR s Duration Measures by Asset Horizon (Trading Days) Asset Obs Pane A: Macauay's Duration Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Pane B: Vasicek Duration (w =0%) Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Pane C: CIR Duration (w=0%) Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond RMSE s no greater than the Macauay counterparts are marked with a singe underine. See aso the detaied expanatory notes in Tabe. 8

31 Tabe 4: Performance of the New Stochastic Duration Measures (w=.5%) by Asset Horizon (Trading Days) Asset Obs Pane A: Vasicek Duration (w=.5%) Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Pane B: CIR Duration (w =.5%) Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Bond Notes: RMSE s no greater than the Macauay counterparts are marked with a singe underine. See aso the detaied expanatory notes in Tabe. 9