Value at Risk for Interest Rate-Dependent Securities

MARCH 2003 THE JOURNAL OF FIXED INCOME 81 V alue at risk modelers face difficult trade-offs in choosing between the two major VaR methodologies: historical/ Monte Carlo simulations or parametric models. The historical approach uses the recent history of the asset price (or, for derivatives, the price of the underlying), while the parametric approach imposes functional form assumptions upon the expected returns and uses history only to estimate the few governing parameters. Each model has its limitations. The parametric model gains computational ease at the cost of unrealistic assumptions for the forward distribution, often even assuming normal errors despite a long history of demonstrated abnormal tail behavior (Mandelbrot [1963]). The historical model suffers the “rewind” problem; there is no guarantee that the future will replay the past in the same way. Randomization (Monte Carlo) experiments mitigate this problem but at computational cost. Duffie and Pan [1997] and Linsmeier and Pearson [2000] give excellent overviews.1 Interest rate modeling has allowed for a wide degree of non-parametric flexibility in how the forward distribution of rate changes depends on the level. Whether one uses the classic Cox, Ingersoll, and Ross (CIR [1985]) specification that volatility increases with the square root of the level, or the non-parametric Aït-Sahalia [1996a] and Stanton [1997] models, it is widely believed that this relationship is important in accurately representing interest rate changes. A VaR calculated by ignoring this dependence of volatility on the level of rates would miss this essential feature. Parametric methods that explicitly model this dependence, such as CIR, assume normal errors (actually, non-central chi-squared in the discrete case, although this is built from Wiener processes), and so understate the actual kurtosis of the distribution, leading to inaccurate tail probability and VaR estimates. Contemporary non-parametric methods for valuing derivatives, such as Stutzer [1996] and Stanton [1997], are increasingly adopted in option pricing, but are only just beginning to find prominence in VaR analyses. They can accurately reflect the thick tails, pronounced skewness, and excess kurtosis of financial asset price returns. Kernel density estimation allows modeling of this tail behavior in flexible yet workable ways. Stutzer [1996] provides a way to construct a risk-neutral probability distribution using the maximum amount of information from the historical record of the underlying security. Replacement of the conventional lognormal assumption with this risk-neutral historical distribution allows a rich variety of effects on option values. Rather than squeezing the empirical variation into a single volatility parameter, Stutzer allows an entire dimension of variation to enter non-parametrically, but only a single dimension, while interest ratedependent securities need more. Value at Risk for Interest Rate-Dependent Securities