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Technical Notes Interest Rate Volatility and Bond Prices

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Technical Notes Interest Rate Volatility and Bond Prices Technical- Notes Interest Rate Volatility and Bond Prices Francis A. Longstaff, allel shifts. Furthermore, because shifts were always parallel, Associate Professor, Ander• duration and convexity analyses changes in volatility would not be son Graduate School of focus on the risk of changes in the useful in explaining yield move• Management, UClA and level of interest rates, they ignore ments, after controlling for the Eduardo S. Schwartz, other types of relevant risks, in• effects of a change in r, and y cluding changes in the frequency would equal zero. California Chair Professor in Real Estate and of large movements in interest rates. Table I gives the results of this Professor of Finance, regression, estimated using Trea• Anderson Graduate School This paper shows that the price sury yield data, for the 1964-89 of Management, UCLA risk inherent in a default-free period. Note that the parameter {3 bond has two major sources-the is always less than one, and gets - risk of changes in the level of smaller as we go farther out along The risk of a defaultfree interest rates and the risk of the yield curve. Table I implies, bond stems from two major changes in the volatility of inter• for example, that if the one• sources-interest rate shifts est rates. The volatility risk of a month T-bill yield increased by and changes in bond mar• bond can be an important com• 100 basis points, the one-year ponent of its total risk, but few yield would increase by only 52.2 ket volatility. The first type managers currently hedge fixed• basis points, while the five-year of risk is well known. The income portfolios against volatil• yield would increase by only 25.4 second is lessfamilia1~ al• ityrisk. basis points. This is clear evi• though it can represent a dence that shifts in the yield curve major component of the Historical Volatility Risk are dramatically different from To demonstrate that interest rate parallel shifts-a fundamental as• total risk of a fixed-income volatility has a significant effect on sumption of standard duration portfolio. bond prices, we first examine the and convexity analyses. historical relation between vola• - tility and yields. In doing this, we Table I also shows that, even after The past 15 years have included let AYT represent the monthly considering the effects of changes some of the most volatile periods change in the T-maturity Treasury in the level of the yield curve as fixed-income portfolio managers yield; Ar the monthly change in measured by r, there is a signifi• have experienced this century. the one-m,onth Treasury-bill cant relation between yields and One-month Treasury-bill yields yield; and I1V the change in the interest rate volatility for all ma• have been higher than 16% and volatility of interest rates. 1 turities. This is direct evidence m(Y) m lower than 4%. As a result, fixed• that prices of Treasury bonds are income managers have become Now consider the following re• affected by both the level of inter• acutely aware of interest rate risk gression equation: est rates and the volatility of inter• and their exposure to it. est rates. AYT = a + {3Ar+ yAY + E. Managing interest rate risk re• Interestingly, Table I indicates quires measuring it first. Duration In this regression, {3 measures that the relation between yields analysis has become an important how sensitive changes in T-matu• and volatility is negative. This tool, allowing portfolio managers rity Treasury yields are to changes means that an increase in the to measure the sensitivitv of their in the short-term interest rate r. If uncertainty about interest rate t•V> V> portfolios to changes in 'the level shifts in the yield curve were al• changes leads to a decrease in ~ of interest rates. But duration ways parallel, there would be a z« bond yields. This makes sense, « analysis has a number of serious one-to-one relation between AYT because it means that investors «--' drawbacks. Standard duration and Ar, and the slope parameter are willing to pay more for secu• zo analysis, for instance, allows for (3 would equal one. The regres• rities that allow them to lock in a z« only parallel shifts in the term sion parameter y is a measure of long-term, guaranteed rate of re• i:L structure. Thus portfolio manag• how sensitive changes in yields turn when the uncertainty about ers may remain exposed to sub• are to changes in interest rate future money-market yields in• 70 stantial risk arising from nonpar- vOlaLility,2Again) if term structure creases. Table I Regression of Changes in Short-Term Interest Rate and Glossary Changes in Interest Rate Volatility on Changes in Yields to Maturity, 1964 to 1989 ~ Closed-Form Solution: An explicit formula that in• ~ Y T = a + f3~r + y~ V + E volves only simple and easily programmable mathematical 22.5117.2111.9214.6515.3411.16937.5629.40.328.254.666.278f3.377.514.522-1.141-3.01-2.50~to<-2.69-.991-2.29-962-3.09-3.08-.788-.777-.730-.731-2.51.451305.504.239.3433.29.41.422.46.24633663031.000.000a Maturity t /3-3.41-.820ty.34y expressions. 3942561 YearsMonthsYear 3 Months ~ Convexity: A measure of the nonlinear effect on bond prices of changes in the level of the yield curve . ~ Duration: A measure of the percentage senSitivity of a bond price to This negative relationship, how• sions-the risk of changes in in• changes in the level of the ever, does not grow in line with terest rates and the risk of yield curve. the maturity of the bond. In par• changes in interest rate volatility. ticular, y is -0.730 for three• To give a historical perspective of ~ Interest Rate Volatility: month yields, increases to a max• how variable these two sources of The variance of changes in the imum of -1.141 for one-year risk are, Figure A plots the one• level of the yield curve. yields, and then decreases for month Treasury-bill yield during longer maturities. Table I shows the 1964-89 period, and Figure B ~ Macaulay Duration: that three-month and four-year plots the volatility of interest rates The simplest and most com• yields are almost equally sensitive over the same period. Note that mon approach to measuring to changes in volatility. the one-month rate and the vola• bond duration, the Macaulay tility measure move together but duration measure assumes These results demonstrate that are not perfectly correlated. Fur• that all yield curve shifts are the price risk of default-free thermore, while interest rates ex• parallel. bonds has two important dimen- perience dramatic changes, the ~ Volatility Risk: The relation between bond Figure A One-Month Treasury-Bill Rate prices and interest rate volatil• ity makes fixed-income portfo• lios susceptible to changes in 0.120.18 1970197519801985 volatility. ~" 0::0E 0.080.1 0) =?>="" 1965 i:ii 0.06 ~ Yield Sensitivity: ." 0.140.040.160.02 The relation between yields and the level of interest rate volatility. - level of interest rate volatility is even more variable. The Longstaff-Schwartz Model To manage the price risk of a t•V'> fixed-income portfolio, a man• V'> ager clearly needs to be able to ~ z<!' estimate the sensitivity of the <!' --' portfolio to changes in interest <!' rate volatility. In developing tools zo for measuring volatility risk, we z<!' draw upon a recent paper by u:: Longstaffand Schwartz that devel• ops a simple general-equilibrium, 71 Figure B Variance of Changes in One-Month Treasury-Bill Rate basis-point change in the stan• dard deviation of interest rate changes for bonds with varying 0.014 maturities and coupon rates, holding r constant. In computing these price changes, we used pa• 0.012 rameter values for the Longstaff• Schwartz model that allowed the model to match the one-month, one-year, two-year, three-year, 0.0060.0080.01 1975197019801985 ~ 0;0::., I@'" four-year and 30-year Treasury ~ 19650;., 0.004 yields as of January 15th, 1992, as .q well as imply long-run average 0.0020 values for r and the standard de• viation of changes in r of 0.05 and 0.0225, respectively.5 As Table II shows, percentage changes in bond values increase with maturity. For example, a 25• basis-point increase in volatility increases a six-month, zero• coupon bond price by 0.19%, a one-year, zero-coupon bond price by 0.49%, and a 30-year, zero-coupon bond price by 1.31%. Similar results hold for the other coupon bonds. The table also shows that the percentage change in bond price is smaller, the larger the coupon rate of the term-structure model that explic• Measuring Volatility Risk bond. Observe that the percent• itly captures the effect on bond Since the Longstaff-Schwartz age changes increase rapidly out prices of changes in interest rate model results in simple, closed• to about five years, and then level volatility.3 form solutions for the value of off. bonds in terms of the variables r The Longstaff and Schwartz arid V, it is easy to use the model Table III shows the effects of a model starts from fundamental to compute the price effect of a 25-basis-point increase in the economic considerations' about change in interest rate volatility. standard deviation of interest rate investment opportunities, inves• Table II shows the percentage volatility on bond prices. For the tors' risk preferences, technolog• price change resulting from a 25- zero-coupon bond, the price ef- ical change, the nature of finan- cial .markets, arid investment uncertainty in the economy.
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