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

Abstract

We propose an empirical method that utilizes the conditional density of the state variables to estimate and test a term structure model with known price formulae, using data on both discount and coupon bonds. The method is applied to an extension of a two‐factor model due to Cox, Ingersoll, and Ross (1985; CIR). Our results show that estimates based on only bills imply unreasonably large price errors for longer maturities. We reject the original CIR model using a likelihood ratio test, and conclude that the extended CIR model also fails to provide a good description of the Treasury market.

An empirical analysis of alternative recovery risk models and implied recovery rates

“…These estimates imply a long-run interest rate mean of 9.4% and a long-run interest rate volatility of about 9.5%. The parameter values are in-line with such well-known counterparts as Chan et al (1992) and Pearson and Sun (1994). Suggestive of parameter stability, the coefficients of variation are each estimated to be lower than 0.5.…”

Heterogeneous Beliefs, the Term Structure and Time-varying Risk Premia

“…The consumption-based CAPM developed by Lucas (1978) and Breeden (1979), and later applied to the term structure in Cox, Ingersoll and Ross (1985) and Backus et al (1989) implies the expectations hypothesis that cannot account for the observed fluctuations in bond returns (e.g see results of Fama and Bliss 1987;Shiller and McCulloch 1990;Campbell and Shiller 1991). Arbitrage-free dynamic affine factor models, as in Litterman and Scheinkman (1991), Pearson and Sun (1994), and Dai and Singleton (2000), do not provide economic interpretations of these latent factors or the formation of time-varying risk premia. As shown in Ang and Piazzesi (2003), the latent yield factors, which depend upon observed yields and have no connection with macroeconomic fundamentals, account for most of the movements of the yield curve.…”

Bond pricing in a hidden Markov model of the short rate

“…Then the model has a semi-affine term structure and A and B are the solutions of (17) and (16), respectively. Remark 3.3 Notice that B does not depend on x , but remember that the conditions in Proposition 3.2 are only sufficient for the existence of a semi-affine term structure.…”

“…Proof It is easily verified that F T (t, r, x ) = e A(t,x ,T )−B (t,T )r , with A the solution of (17) and B the solution of (16), satisfies the term structure equation (10) with the boundary condition F T (T , r, x ) = 1. Since the term structure is uniquely determined by the term structure equation we must have F T (t, r, x ) = p(t, T ) and the proof is complete.…”

“…A multi-factor model does not assume perfect correlation between different points on the yield curve. Furthermore, empirical studies of multi-factor models indicate that the introduction of additional factors improves the fit, see for instance Stambaugh [20], Longstaff and Schwartz [14] and Pearson and Sun [17]. The drawbacks of the multi-factor models are that they are more difficult to analyse and that the additional factors are not easily identified.…”