Technical- Notes and 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 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 , 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 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 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 22.5117.2114.6515.3411.9211.16.5629.40.254.666f3937.328.278.514.522.377-1.141-3.01-2.50~to<-.991-3.08-2.29-.788-962-3.09-2.69-.777-2.51-.730-.731.451.46.246.3433.504.239.41.29.42230533663031.000.000a Maturity t /3-3.41-.820tyy.34 expressions. 9346251 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: ." 0.040.020.140.16 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

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. of interest rate 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. It Table II Percentage Change in Price for Bonds with Par Value of solves for the market price of 100 Resulting from 25-Basis-Point Change in Annualized interest rate risk that determines Standard Deviation of Changes in Short-Term Rate (from the required expected rate of re• 0.0275 to 0.0300) turn Qn securities with interest• rate-sensitive prices. Once the re• 5%1211.200190490.901091.17]0%U9U51.17U80190.480.87104U2US1.110.190470.851011.081.17u6]5% 0491310.190931.28U41.241.311310% Coupon Rate 305420251015321YearsYearMonthsYears 6 Maturityquired expected rate of return is known, the equilibrium prices for default-free bonds can be deter• f•l/) l/) mined simply by discounting >- -.J their cash flows at the appropriate z

2.5 volatility sensitivity of a zero• coupon bond levels off rapidly with increasing maturity. Further• -'---" more, the volatility sensitivity of a 2 bond is much less affected by the coupon rate than is the duration / of the bond. / .~.c:Uc:'"00 1.5 p.':" ---.- . -" Finally, it is easy to show that the " /' volatility sensitivity of a bond is 0.5 quite different from its convexity. /:.(.::: ~ ' . /.:-./~ This is because convexity in• .~...•....------creases gradually for shorter du• I'~ -;::...---- " -- -- rations but more rapidly for ------longer durations.6 The opposite #:~/ is true for volatility sensitivity. Conclusion o We have shown that changes in 2 3 4 5 10 15 20 25 30 the volatility of interest rates can have large effects on the prices ~- Zero-Coupon - - - 5% ..... 10% - . -15% and yields of bonds. These effects 73 Table N Change in Yield to Maturity (measured in basis points) 53 (1985), pp. 363-84. In the Long• for Bonds with Par Value of 100 Resulting from a 25• Skiff and Scbwartz model, the two Basis-Point Change in Annualized Standard Deviation of factors are tbe short-term riskless in• terest rate and the volatility of the Changes in Short-Term Rate (from 0.0275 to 0.0300) short-term interest rate. See Longstaff and Schwartz, ''Interest Rate Volatil• Coupon Rate ity and the Term Structure, " op. cit. Maturity 0% 5% 10% 15% 4. In the Longstaff and Schwartz model, -38.6 the present value of one dollar to be 6 Months -38.6 -386 -38.6 received T periods in thefuture can 1 Year -49.3 -50.3 -50.1 -50.2 be expressed as A(T)exp[B(T)r + 2 Years -46.3 -47.7 -47.7 -47.7 C(T)V},where A(T), B(T) and C(T) 3 Years -37.9 -39.5 -39.9 -40.2 4 Years· -30.8 -32.7 -334 -34.0 are functions of T as well as six pa• 5 Years -25.5 -27.6 -28.6 -29.4 rameters a, {3, y, 0, 1] and v that de• 10 Years -13.1 -16.1 -17.6 -18.6 scribe the dynamic evolution of the short-term interest rate over time. 15 Years -8.7 -12.4 -13.9 -14.8 20 Years -6.5 -10.7 -12.1 -12.8 This means that yields to maturity 25 Years -5.2 -9.9 -111 -11.6 are linear functions of r and Vfor a 30 Years -4.4 -9.4 -10.4 -10.7 given T 5. There are six parameters. in the Long• staff and Schwartz model. By specify• ing the long-run average values of r and V, as well as the average yield bear littie or no relation to the sures to shifts in interest-rate on a consol bond, two of the six pa• duration or convexity of the volatility. rameters can be solvedfor analyti• bond. In fact, these effects have cally. The remaining four are deter• the greatest impact on the prices Footnotes mined by setting actual yields for and yields of intermediate-term 1. The one-month Treasu7J!bill yields four different maturities equal to the bonds. used are from the data maintained closedform expression implied by the by the Centerfor Research in Security model and then inverting the system The sensitivity of bond prices to Prices (CRSP)at the University of Chi• of four equations to solve for the four parameters. This is similar to the changes in volatility has many im• cago. Theseyields are based on the average of bid and ask prices for procedure of inverting Black-Scholes poi-tant implications for fixed• Treasu7J'bills and are normalized to option prices to solve for the implied income portfolio managers. reflect a standard month of 30.4 volatility parameter. We solve the Clearly, if volatility risk is not days. The longer-maturity yields are system offour equations using a nu• recognized and hedged, the port• from the data set used in E. F. Fama merical gridsearch algorithm that is folio manager may be exposed to and R. Bliss, "The Information in easily implemented on a Pc. significant losses in the event of a Long-Maturity Forward Rates," Amer• G This is illustrated on page 78 of F.J Fabozzi and T D. Fabo=~ Bond sudden change in the level of ican Economic Review 77 (1987), pp. market uncertainty-an event that 680-92. Markers, Analysis and Strategies (En• 2. The volatility of changes in the short• glewood Cliffs,N}: Prentice Hall, has occurred with increasing fre• 1989). quency in rec~nt years. The Long• term interest rate is estimated using staff-Schwartz model discussed a simple GARCH(1,1) model. In this model, the volatility of changes in here gives fixed-income manag- the short-term interest rate is as• ers an important tool for quanti• sumed to be a linear function of its fying and hedging their expo- lagged value, the short-term interest rate, and the square of the last unex• pected change in the short-term inter• Table V Macaulay Durations est rate. See F. A. Longstaff and E. S. Schwartz, ''Interest Rate Volatility and the Term Structure: A Two-Fac• Coupon Rate tor General Equilibrium Model, " Maturity 0% 5% 10% 15% Journal of Finance 47 (1992), 1259• 82. 6 Months 0.50 0.50 050 0.50 3. The Longstaff and Scbwartz model is V) 1 Year 1.00 0.99 0.98 0.97 t• a twofactor extension of tbe Cox, V) 2 Years 2.00 1.93 1.87 1.82 ~ 3 Years 3.00 2.82 2.68 Ingersoll and Ross term structure <1: 258 z 4 Years 4.00 3.66 3.43 3.26 model in which production retums -'<1: 5 Years 5.00 4.46 4.12 3.88 in the economy are random and o<1: 10 Years 10.00 7.75 6.84 6.35 technological change occurs via z 15 Years 15.00 10.03 8.71 8.10 changes in the mean and variance z<1: 20 Years 20.00 1152 1003 941 of production returns. SeeJ c. Cox, u:: 25 Years 25.00 12.44 10.97 10.39 J E. Ingersoll and S. A. Ross, "An 30 Years 30.00 1298 11.64 1114 Intel-temporal General Equilibrium 74 Model of Asset Pn'ces," Econometrica