Sanjay K. Nawalkha

The M-Vector Model: Derivation and Testing of Extensions to M-Square

This article derives and tests a multiple-factor extension of the M-square model (see Fong and Vasicek [1984] and Fong and Fabozzi [1985]), termed as the M-vector model. Tests of the M-square model indicate that the model reduces the interest rate risk inherent in the traditional duration model by more than half. The M-vector model demonstrates near-perfect hedging performance, eliminating more than 95% of interest rate risk inherent in the traditional duration model.

Generalized M-Vector Models for Hedging Interest Rate Risk

This paper generalizes the M-square and M-vector models (Fong and Fabozzi [1985] and Nawalkha and Chambers [1997]) by using a Taylor series expansion of the bond return function with respect to simple polynomial functions of the cash flow maturities. The classic M-vector computes the weighted averages of the distance between the maturity of each cash flow and the portfolio horizon, raised to integer powers (e.g., (t - H)^1, (t - H)^2, (t - H)^3, etc.). Implementation of the new approach involves computing the weighted averages of the distance between some polynomial function of the maturity of each cash flow and that of the portfolio horizon, raised to integer powers (e.g., (t^0.5 - H^0.5)^1, (t^0.5 - H^0.5)^2, (t^0.5 - H^0.5)^3, etc.). We test six different generalized M-vector models corresponding to six different polynomial functions. It is shown that polynomial functions of lower power (i.e., 0.25 or 0.5) provide significantly enhanced protection from interest rate risk, when higher-order generalized M-vector models are used.

Convexity, Risk, and Returns

This paper tests empirically whether convexity is return enhancing (the traditional view based upon parallel term structure shifts), or return diminishing (the equilibrium view suggesting convexity is priced). Results of empirical tests over different time periods show bond convexity to be either insignificantly or negatively related to ex-ante bond returns. These results are consistent with the critique of the traditional duration model by Ingersoll, Skelton, and Weil [1978] and suggest that bond convexity may be priced. Further, the magnitude of bond convexity is shown to be related directly to the immunization risk inherent in a bond portfolio, consistent with the implications of Fong and Vasiceks [1983, 1984] M-Square model.

An Improved Approach to Computing Implied Volatility

A well-known problem in finance is the absence of a closed form solution for volatility in common option pricing models. Several approaches have been developed to provide closed form approximations to volatility. This paper examines Chances (1993, 1996) model, Corrado and Millers (1996) model and Bharadia, Christofides and Salkins (1996) model for approximating implied volatility. We develop a simplified extension of Chances model that has greater accuracy than previous models. Our tests indicate dramatically improved results.

A Simple Approach to Pricing American Options Under the Heston Stochastic Volatility Model

Lattice models are workhorses of practical option pricing, especially for American options, but an important design criterion is that they need to be set up so that the interior branches recombine, otherwise, they become computationally intractable; the underlying state variable becomes path-dependent, the tree “splinters,” and the number of distinct nodes goes up exponentially rather than arithmetically as the number of time steps grows. Unfortunately, we have accumulated a great deal of evidence that volatility is time varying, which is a factor that splinters the tree. In this article, Beliaeva and Nawalkha show how to get around the problem by transforming the returns process to create two uncorrelated path-independent trees for returns and variance that can be put together into a single recombining two-dimensional lattice. The procedure is general; here they illustrate its use with the popular Heston model.

Interest Rate Sensitivities of Bond Risk Measures

We present a simple expression for the sensitivity of duration, convexity, and higher-order bond risk measures to changes in term-structure shape parameters. Our analysis enables fixed-income portfolio managers to capture the combined effects of shifts in term-structure level, slope, and curvature on any specific bond risk measure. These results are particularly important in environments characterized by volatile interest rates. We provide simple numerical examples. Building on previous research into the sensitivities of bond risk measures, we present a simple expression for the sensitivity of duration, convexity, and higher-order bond risk measures to nonparallel changes in the shape of the yield curve. Although researchers have analyzed the sensitivity of a bonds duration to changes in the bonds yield, little is known about the interest rate sensitivity of duration, convexity, and so on, to changes in level, slope, and curvature of the term structure. The subject is important because up to 95 percent of returns to portfolios of U.S. Treasury securities are explained by term-structure level, slope, and curvature shifts—and these shifts can be quite extreme in volatile interest rate environments. We captured these parameters of term-structure shape by using a simple polynomial representation of the continuously compounded spot yield curve. Given a noninfinitesimal, nonparallel shift in the yield curve, we were able to derive closed-form expressions for the resulting changes in bond risk measures as a function of changes in the level, slope, and curvature of term structure and as a function of the bond risk measures themselves. Our framework enabled us to answer questions that are relevant to the work of managers who are required to maintain target durations for their bond portfolios and who wish to know how sensitive their bond risk positions are to general interest rate changes: How does the duration of a bond change with respect to a change in the slope of the term structure? How does the convexity of a bond change with respect to a change in the level of the term structure? Do the duration and convexity of a barbell portfolio change more rapidly than those of a bullet portfolio? These questions are relevant to managers of fixed-income portfolios and managers of financial institutions. Shifts in term-structure level, slope, and curvature are not independent. For example, increases in level tend to be associated with decreases in slope. We used such interrelationships to derive a simple but realistic numerical example of the effect of a noninfinitesimal, nonparallel term structure shift on a bullet bond and two barbell bonds. We found that if we ignored the slope and curvature shifts and accounted only for the level shift, we seriously misestimated the effect of the full term-structure shift on bond duration measures for the barbell bonds. The percentage error we made became larger as the cash flow spacing of the barbell became wider. When we added a term (i.e., slope to level) and then two terms (i.e., curvature and slope to level), the magnitude of our estimation errors decreased substantially. Therefore, accounting for the impact of level shifts alone (i.e., parallel shifts) is not sufficient when estimating the effect of changes in term-structure shape on bond risk measures. We also note one simple result: Although the bonds in our numerical example all had the same initial price and duration, the effect of the nonparallel shift in term structure on their prices was quite different. This outcome is a simple reminder that practitioners must look beyond parallel term-structure shifts when analyzing bonds.

Pricing American Interest Rate Options under the Jump-Extended Vasicek Model

This paper shows how to price American interest rate options under the exponential jumps-extended Vasicek model. We modify the Gaussian jump-diffusion tree of Amin [1993] and apply to the exponential jumps-based short rate process under the Vasicek-EJ model. The tree is truncated at both ends to allow fast computation of option prices. We also consider the time-inhomogeneous version of this model, denoted as the Vasicek-EJ++ model that allows exact calibration to the initially observable bond prices. Our simulations show fast convergence of European option prices obtained using the jump-diffusion tree, to those obtained using the Fourier inversion method and the cumulant expansion method.

Managing Interest Rate Risk: The Next Challenge?

Are the managers of financial institutions ready for the small but increasingly significant risk of inflation in the near future, due to the unprecedented fiscal and monetary responses of the U.S. government to prevent an economic collapse? This paper addresses this important issue by reviewing important findings in the area of interest rate risk management. We discuss five classes of models in the fixed income literature that deal with hedging the risk of large, non-parallel yield curve shifts. These models are given as M-Absolute/M-Square models, duration vector models, key rate duration models, principal component duration models, and extensions of these models for fixed income derivatives, for valuing and hedging bonds, loans, demand deposits, and other fixed income instruments. These models can be used for designing various hedging strategies such as portfolio immunization, bond index replication, duration gap management, and contingent immunization, to protect against changes in the height, slope, and curvature of the yield curve. We argue that the current regulatory models proposed by the U.S. Federal Reserve, the Office of Thrift Supervision, and the Bank of International Settlements, may understate the true interest rate risk exposure of financial institutions, if sharp increases in interest rates lead to higher default risk and quickening of the pace of deposit withdrawals.

Term Structure Estimation

The term structure of interest rates gives the relationship between the yield on an investment and the term to maturity of the investment. Since the term structure is typically measured using default-free, continuously-compounded, annualized zero-coupon yields, it is not directly observable from the published coupon bond prices and yields. This paper focuses on how to estimate the default-free term structure of interest rates from bond data using three methods: the bootstrapping method, the McCulloch cubic-spline method, and the Nelson and Siegel method. Nelson and Siegel method is shown to be more robust than the other two methods. The results of this paper can be implemented using user-friendly Excel spreadsheets.

Closed-Form Solutions of Convexity and M-Square

Closed-form formulas for Macaulay duration, as given by Babcock and Chua, provide the user with a less cumbersome and more efficient procedure for calculating duration. Recent developments, however, have suggested alternative measures of bond portfolio immunization designed to overcome the severe restrictions that Macaulay duration places on permitted interest rate behavior. This note presents closed-form formulas for two such alternative measures - convexity and M-square and demonstrates how these measures can be used in an immunization strategy.