Recent Advances in Estimating Term-Structure Models

In the past 10 years, increasingly sophisticated statistical techniques have been applied to the estimation of increasingly complex models of the term structure of interest rates. In reviewing this literature, we highlight the facts that have been established and the key unresolved issues. The data indicate that within a wide range of interest rates, mean reversion in rates is, at best, weak. Whether mean reversion is stronger for very high or very low levels of rates is an unresolved issue. The absolute volatility of rates increases as the level of rates increases, but the strength of this effect and the role and nature of either stochastic-volatility or regime-switching components in rates are still unclear. Unfortunately, these unresolved issues have important implications for fixed-income option pricing and risk measurement, including value-at-risk calculations. Models of the term structure of interest rates are widely used in pricing interest rate derivatives and instruments with embedded options, such as callable bonds and mortgage-backed securities. Many such models are based on the simplifying assumption that changes in interest rates of all maturities are driven by changes in a single underlying random factor, often taken to be the “short” or “instantaneous” rate of interest. Both the decision to use a one-factor model and the choices made within that framework are crucial. The evolution of interest rates over time, and thus the prices of options and other derivatives, is determined entirely by these choices. Models of interest rate volatility (sometimes implicit) also play key roles in risk measurement—for example, in value-at-risk calculations. Unfortunately, theory provides little guidance about the modeling choices. As a result, the last decade has seen the development of a large and growing academic literature devoted to estimating how expected changes and volatilities of interest rates are related to their levels and, sometimes, other variables. This literature is scattered in different places and often emphasizes the statistical and econometric techniques used rather than the implications of the analysis for interest rate models. Therefore, it is not easily accessible to many practitioners. This article is the first step in remedying this problem. We summarize in one place the substantive implications of the recent academic literature for dynamic models of the evolution of interest rates. Much of the recent academic literature we review focuses on one-factor models, but researchers have known for at least 10 years that at least three factors are needed to fully capture the variability of interest rates. Why then consider one-factor models? The answer is that research has shown that roughly 90 percent of the variation in U.S. Treasury rates can be explained by the first factor, which can be interpreted as corresponding to changes in the general level of interest rates. Thus, any relationship between the level of interest rates and their expected changes and volatilities will be dominated by the influence of this first factor. In one-factor models, this factor is typically identified with the instantaneous or short rate of interest. The recent academic literature studying the behavior of the yields on short-term bonds or deposits can be interpreted as a detailed explanation of the first factor by using the yield on a particular instrument (e.g., one-month LIBOR) as a proxy for the short rate. According to the recent literature, what model features appear to be essential in describing the fundamental properties of interest rates? First, the new literature does not provide conclusive evidence based solely on the data about whether interest rate levels tend to return to a constant long-run level and, if they do, whether this tendency is stronger for extreme levels of interest rates. Second, with respect to interest rate volatility, the “absolute” volatility of the short rate, defined as the standard deviation of rate changes scaled by the square root of the time between changes, clearly increases as the level of interest rates increases. Inferences about the relationship between the level and volatility of the short rate are sensitive, however, to the treatment of the years between 1979 and 1982, the years of the U.S. Federal Reserve's so-called experiment in targeting monetary aggregates rather than targeting interest rate levels. In particular, the data from this period suggest a strong relationship between volatility and the level of interest rates; when this period is excluded or treated as a distinct regime with a lower probability of occurring, the data suggest a much weaker relationship between interest rate level and volatility. Finally, modeling the volatility of interest rates requires more than a simple “level effect”; that is, some sort of stochastic-volatility effect seems to be in play. But the additional volatility component can be described adequately (in a statistical sense) in a variety of competing ways.