Introduction

Investors have long understood the need to measure how changes in interest rates will affect the value of fixed-income portfolios. Duration and convexity, used to measure these effects, belong in every portfolio manager's tool kit. But these alone do not give a complete picture of the risk in a portfolio. Changes in interest rates are not the only source of risk in fixed-income investment. What about changes in interest rate volatility?

Nearly all fixed-income instruments contain embedded options. The price of a callable bond, for example, depends on the value of the call option; this, in turn, depends on the volatility of interest rates. Measuring an instrument's sensitivity to interest rate volatility is thus central to valuing the instrument as a whole.

The Black-Scholes formula shows that options' sensitivity to volatility, and the value of callable bonds, pass-throughs, futures, and other instruments with option-like features also depends on market volatility. Even noncallable bonds are volatility-dependent. The published results from Vasicek (1977) (Chapter 6 of this volume), Cox, Ingersoll, and Ross (1985), and others on the behavior of the term structure of interest rates show the presence of the volatility parameter in the bond pricing formula.

Just as the fixed-income investor needs to know how changing interest rates affect portfolio value, he or she should be concerned about the effects of random (stochastic) changes in volatility. This article outlines a new two-factor term structure model that explicitly incorporates volatility as a stochastic factor and produces a new risk measure—volatility exposure.

Stochastic Volatility Term Structure

Term structure theory attempts to define the behavior of interest rates. Its starting point is to identify the stochastic factors that explain the movement of interest rates. The stochastic processes that govern the behavior of the factors are then specified.

The next step is to derive an equilibrium condition that precludes riskless arbitrage, and to define the risk premia associated with the factors. This results in a partial differential equation for the bond price. For the theory to be practicable, a closed-form solution should be achievable. The exposure of the bond price to the stochastic factors can then be evaluated, and the risk measures quantified. Finally, the pricing can be extended to more complex instruments such as interest rate–contingent claims.

The stochastic volatility term structure (SVTS) describes the behavior of the short rate r by a diffusion process:

1

where

dr	=	change in the short rate,
α	=	speed of reversion to the mean ,
	=	long-term mean of the short rate,
dt	=	change in time,
v	=	instantaneous variance (volatility), and
dx	=	random element.

Eq. (1) describes the short rate as a continuous process with a tendency to revert to a long-term mean value. The strength of this tendency is proportional to its current deviation from the mean. Thus, high rates have a tendency to come down, while low rates tend to go up. In all cases, however, there is a random component associated with the change in interest rates, which can make high rates go higher or low rates go lower. The magnitude of this random component is described by its variance .

If the variance v is a constant, as previous models have assumed, a one-factor description of the term structure can be derived. In the SVTS specification, the variance (volatility) v is a second stochastic factor, described by the following equation:

2

where

dv	=	change in volatility,
γ	=	speed of reversion to the mean ,
	=	long-term average volatility,
dt	=	change in time,
ξ2v	=	instantaneous variance, and
dy	=	random element.

Similar in form to the short-rate equation, the volatility equation (2) also has a mean reverting tendency with strength proportional to the current deviation from the mean level. Unlike the equation for the short rate, however, the random component has a variance proportional to the current level of volatility. This means that volatility changes less abruptly in very quiet markets than in very unstable markets.

In addition, the random element dx of the short rate and the random element dy of the volatility can be correlated with a correlation coefficient ρ. Thus, increasing levels of rates are typically accompanied by an increase in their volatility and vice versa, as indeed happens in reality.

Under this description of the term structure, the price P = P(t, r, v) of a zero coupon bond with term t depends on the values of the two stochastic factors, r and v. From Ito's lemma, the price change is then governed by the factor changes dr, dv according to the equation

3

Given the nature of the price changes specified by Eq. (3), it is possible to form a portfolio of three bonds of different maturities in such proportions that the dependence on the risk factors dr, dv is eliminated. Since this portfolio is riskless, its rate of return must be equal to the riskless rate r.

This is the arbitrage argument first invoked by Black and Scholes in their 1973 article. Indeed, the impossibility of a riskless arbitrage is a necessary condition in an efficient market. If excess profits are to be achieved, then risk must be assumed.

Formalizing the argument results in the partial differential equation

4

that must be satisfied by the bond prices P(t, r, v) in order for the market to be efficient.

Eq. (4) contains terms that capture the pricing of risk in a risk-averse market. We have assumed that the market price of risk corresponding to each of the stochastic factors is proportional to the level of risk in the market, with the proportionality constants λ and η.

The solution of Eq. (4) subject to the boundary condition P(0, r, v) = 1 is given by the expression

5

The quantities in Eq. (5) are functions of the term t alone. They are obtained as the solutions of ordinary differential equations to which the partial differential equation reduces. In particular, the function D(t) is given by

6

The functions and are given by more complicated (but closed-form) expressions, involving the confluent hypergeometric function. For the exact formulas, refer to Fong and Vasicek (1991).

We may point out that the form of the bond pricing equation (5) and the specifications of the functions D, F, and G are deduced from the condition of market efficiency, rather than simply declared. This provides a rigorous framework that goes beyond the intuitive description that is commonly the first and only step in many term structure formulations.

The term structure of interest rates is determined from the pricing Eq. (5). If we define as the spot rate of term t, then

7

Eq. (7) describes the behavior of interest rates as a function of the term and the development in time of the two stochastic factors r and v. The resulting spot rate curves can be monotone or have one or two humps. Figures 27.1 to 27.3 show the shapes of the spot rate curves for several values of the parameters. Note in particular in Figure 27.3 that the SVTS allows for the possibility of different yield curves when both the short and the long end of the curves are fixed, which cannot happen in a single-factor model. From the form of the solution for the bond price in Eq. (5), we note that

8

9

The quantities D and F are thus, respectively, the rate exposure (i.e., duration) and the exposure to volatility. Together, duration and volatility exposure constitute the risk parameters of a bond. Moreover, the expected rate of return is also fully determined by the two measures. Two securities or portfolios will have the same returns over a given period if their durations and their volatility exposures are kept matched during that period.

Term Structures of Interest Rates

Volatility Exposure

Figure 27.4 depicts the shape of the function F(t) that constitutes the measure of volatility exposure for a zero coupon bond. We note that in most of its range, it is a concave function, unlike, for instance, Macauley duration (linear) or convexity (convex quadratic).

Simple calculation shows that duration or volatility exposure for a coupon bond is each weighted averages of the duration or volatility exposures of its individual cash flows. The same principle applies to portfolios of fixed-income instruments: Both risk measures combine linearly as a function of the market value of the portfolio components.

Index Tracking

One possible application of volatility management is in index tracking. The approach is to apply volatility analysis and control using a strategic optimization system called Stratos. The effectiveness of this system can be measured by comparing the tracking error produced by a Stratos portfolio with that of a more traditional approach to bond tracking, Bondtrac, as shown in Table 27.1. (Both systems come from Gifford Fong Associates.)

We chose the period October 1990 to March 1991 for the simulations, since we wished to select a six-month time period in which the implied volatilities changed significantly. We used the spot rate on one-month T-bill for the risk-free short rate. The target index was the widely used Shearson Lehman Treasury Index. Since the SVTS theory asserts that even the prices of noncallable bonds are affected by volatility, we restricted ourselves to such securities.

To compare the effectiveness of Bondtrac versus Stratos, both systems optimized a portfolio containing the same assets and indexed against the same target index. To replicate the index using the traditional approach (Bondtrac), we separated the bonds into 10 cells; these cells were defined by coupon and maturity break points.

The index was first partitioned into two groups by coupon (0–10% and 10–30%), then both coupon groups were partitioned into five maturity groups (1 to 2 years, 2 to 5 years, 5 to 10 years, 10 to 20 years, and 20 to 30 years). These 10 cells gave a fair representation of the characteristics of the target index, which was made up of high-coupon and low-coupon bonds with short to long maturities.

After randomly selecting one bond from each cell to be included in the portfolio, we ran Bondtrac to calculate the optimal composition of the 10 bonds. In this optimization procedure, Bondtrac tries to match the duration, convexity, and cell representation of the portfolio with those of the index.

We then ran Stratos with the same bonds used by Bondtrac to get the optimal composition for the Stratos portfolio. Unlike Bondtrac, Stratos tries to match duration, convexity, and volatility exposure without aiming to match the index cell representation.

Once we had the optimal portfolio compositions, we calculated the actual returns from each portfolio strategy. The tracking error was defined as the deviation of the portfolio return from the index return ( and , where is the return on the Bondtrac portfolio, is the return on the Stratos portfolio, and denotes the return on the index). The tracking errors found are reported in Table 27.1.

With the randomly selected bond, Stratos generated lower tracking error in terms of absolute value in four out of six months. The average tracking error was 2.8 basis points (bp) for the Bondtrac portfolio and 1.9 bp for Stratos; theoretically, it should be zero for both. We tested the null hypothesis H₀: μ_B = μ_S = 0. The values of the t statistic were 0.43 and 0.76, too low to reject the null hypothesis.

The monthly standard deviations of the tracking errors were 6.5 bp and 2.5 bp, respectively. On an annualized basis, the corresponding figures are 23 bp and 9 bp—a substantial difference.

To test whether this difference in standard deviations in tracking error was statistically significant, we constructed another null hypothesis against the alternative . To test these hypotheses, we computed the ratio of sample variances, which has F distribution if H₀ is true. The computed F value was 6.62 > 5.05 = F (5 percent). We therefore rejected the null hypothesis in favor of the alternative at the 5 percent significance level. The evidence proves that the variance of the tracking error of the Stratos portfolio was smaller than that of the Bondtrac portfolio.

Stratos generally tracked the index better than Bondtrac during the period covered in this study. In particular, it tended to work better when the volatility of the short rate changed significantly, as happened from November 1990 to January 1991.

Applying the concept of volatility exposure to an indexed Treasury portfolio can significantly reduce risk.

References

1. Cox, J.C., J.E. IngersollJr., and S.A. Ross. (1985). “A Theory of the Term Structure of Interest Rates.” Econometrica, 53, 385–407.
2. Fong, H.G., and O.A. Vasicek. (1991). Interest Rate Volatility as a Stochastic Factor, unpublished, Gifford Fong Associates.
3. Fong, H.G., and O.A. Vasicek. (1991). “Fixed Income Volatility Management.” Journal of Portfolio Management (Summer), 41–46.
4. Vasicek, O.A. (1977). “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics, 5, 177–188.