Academic journal article Quarterly Journal of Business and Economics

Valuing Convertible Bonds under the Assumption of Stochastic Interest Rates: An Empirical Investigation

Academic journal article Quarterly Journal of Business and Economics

Valuing Convertible Bonds under the Assumption of Stochastic Interest Rates: An Empirical Investigation

Article excerpt


Numerous models for pricing default-free fixed income securities have appeared, but the contingent claims approach to pricing corporate debt has not received a great deal of attention. This paper empirically tests the contingent claims approach to the valuation of corporate convertible bonds under the assumption of stochastic interest rates.

A contingent claims approach to the valuation of convertible bonds first appeared in the late 1970s. Ingersoll (May 1977, September 1977) and Brennan and Schwartz (1977) introduce similar valuation models where the convertible bond price depends on one underlying variable: the firm value. This underlying variable is assumed to follow a diffusion-type stochastic process, and the price of the convertible bond is obtained by solving the resulting partial differential equation under appropriate terminal and boundary conditions. These conditions are determined by the bond's cash flow at maturity, the firm's optimal call policy, and the bondholders' optimal conversion strategy. Brennan and Schwartz (1980) extend theft previous work and allow for uncertainty inherent in interest rates by introducing the short-term risk-free interest rate as an additional stochastic variable. As a result, the model captures the stochastic nature of interest rates and the impact of the correlation that likely exists between firm value and the risk-free interest rate.

An empirical test of a simpler model that uses the value of the firm as the only underlying variable is provided by King (1986). King argues that when market prices are compared with model valuations, the means are not significantly different. Furthermore, he argues that 90 percent of model predictions fall within 10 percent of market values. The model generally generates prices that are slightly higher than actual market prices. A partition of the sample into out-of-the-money bonds (i.e., the bond's conversion value is lower than its straight debt value) and in-the-money bonds (i.e., conversion value exceeds the straight debt value) indicates that the former are slightly overpriced by the model while the latter are slightly underpriced. There is evidence in the paper, however, that the magnitude of the pricing differences can vary significantly even within the sample partitions considered. Further analysis is necessary to assess the accuracy of the model.

This paper takes a closer look at the application of a contingent claims valuation model for pricing corporate convertible bonds and extends the results suggested by King's (1986) study. It empirically tests a model that considers the stochastic nature of interest rates. Prices are generated for 30 of the most commonly traded convertible bonds for a number of term structures observed during the fourth quarter of 1989 and the first three quarters of 1990. These model-generated prices are compared to actual prices observed in the market.


The model is developed under the usual set of assumptions of capital structure irrelevancy and perfect capital markets, i.e., no taxes, no transaction costs, and access to information for all investors. It further is assumed that if the bond is callable, bondholders must immediately surrender their bonds upon the firm's call notice. It is similar to a model developed by Brennan and Schwartz (1980); the only difference is the interest rate process assumed.(1) The short-term interest rate, r, follows the mean-reverting process suggested by Cox, Ingersoll, and Ross (1985):

(1) dr = g(l - r)dt + [[Sigma].sub.1] [square root of r][dz.sub.1],


g, l, [[Sigma].sub.1] = Positive constants; and

[dz.sub.1] = A standard Wiener process with mean zero and variance dt.

The process retains the benefit of mean reversion while it precludes negative interest rates. The existence of a closed form solution for the pricing of a pure discount default-free bond makes empirical estimation of the parameters associated with the process easier. …

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