Exploiting the Conditional Density in Estimating the Term Structure : An Application to the Cox , Ingersoll , and Ross Model

One of the most puzzling features of currency prices is the forward premium anomaly: the tendency for high interest rate currencies to appreciate. We characterize the anomaly in the context of affine models of the term structure of interest rates. In affine models, the anomaly requires either that state variables have asymmetric effects on state prices in different currencies or that nominal interest rates take on negative values with positive probability. We find the quantitative properties of either alternative to have important shortcomings. PERHAPS THE MOST PUZZLING FEATURE of currency prices is the tendency for high interest rate currencies to appreciate when one might guess, instead, that investors would demand higher interest rates on currencies expected to fall in value. This departure from uncovered interest parity, which we term the forward premium anomaly, has been documented in dozens-and possibly hundreds-of studies, and has spawned a second generation of papers attempting to account for it. One of the most influential of these is Fama (1984),who attributes the behavior of forward and spot exchange rates to a time-varying risk premium. Fama shows that the implied risk premium on a currency must (1)be negatively correlated with its expected rate of depreciation and (2) have greater variance. We refer to this feature of the data as an anomaly because asset pricing theory to date has been notably unsuccessful in producing a risk premium with the requisite properties. Attempts include applications of the capital asset pricing model to currency prices (Frankel and Engel (1984),Mark (1988)), statistical models relating risk premiums to changing second moments (Hansen and Hodrick (1983), Domowitz and Hakkio (1985), Cumby (1988)), and consumption-based asset pricing theories, including departures from time* Backus is with the Stern School of Business, New York University and the National Bureau of Economic Research; Foresi is with Goldman Sachs. Telmer is with the Graduate School of Industrial Administration, Carnegie Mellon University. In addition to numerous participants a t seminars and conferences, we thank Ravi Bansal, Geert Bekaert, Wayne Ferson, Burton Hollifield, Andrew Karolyi, Kenneth Singleton, Amir Yaron, Stanley Zin, and especially the editor, Rene Stulz, and two referees of this journal for helpful comments and suggestions. Earlier versions of this paper circulated as "The forward premium anomaly: Three examples in search of a solution" (presented at the NBER Summer Institute, July 1994) and "Affine models of currency pricing" (NBER Working Paper 5623). Backus thanks the National Science Foundation for financial support. 280 The Journal of Finance additive preferences (Backus, Gregory, and Telmer (1993), Bansal et al. (1995), and Bekaert (1996)), from expected utility (Bekaert, Hodrick, and Marshall (1997)), and from frictionless trade in goods (Hollifield and Uppal (1997)). Against this backdrop, we ask whether popular models of the term structure of interest rates are consistent with the anomaly, once the models are adapted to a multicurrency setting. Term structure models across currencies imply a term structure of forward exchange rates. The main empirical regularity associated with forward rates is, arguably, the forward premium anomaly. An important question, therefore, asks whether standard term structure models-models which have proven useful for understanding yields in a single currency-imply forward rates that are consistent with the anomaly. Papers that ask a similar question include Amin and Jarrow (1991), Nielsen and Saa-Requejo (1993), Saa-Requejo (1994), Frachot (1996), Ahn (1997), Bakshi and Chen (1997), and Bansal (1997). Our approach considers a generalization of the models used in related work, an adaptation to currencies of Duffie and Kan's (1996) class of affine yield models. We formulate our models as discrete time processes for currencyspecific pricing kernels-essentially, processes for prices of state-contingent claims-and translate Fama's (1984) conditions for risk premiums into restrictions on pricing kernels. We show that these restrictions have strong implications for the structure and parameter values of affine models, and then consider several specific examples. We find, based on both theory and estimation, that affine models have difficulty accounting for the anomaly. Such models must either allow for some probability of negative interest rates or for asymmetric effects of state prices on interest rates in different currencies. Our estimates suggest important quantitative drawbacks associated with either alternative. We proceed as follows. In Section I we briefly review the properties of currency and interest rate data that our quantitative analysis requires. We then go on to describe the relations among pricing kernels, currency prices, and interest rates that are dictated by general arbitrage-free environments and our specific class of affine models. In Section 11, we develop and estimate several affine models and show how different features and parameter values bear on the forward premium anomaly. Section I11 offers conclusions and suggestions for further research.