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An Analytical Model of Bond Risk Differentials

Published online by Cambridge University Press:  19 October 2009

Harold Bierman Jr.  and
Jerome E. Hass
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Extract

There is broad consensus that three types of risk confront the potential bond purchaser: the risk of default (possible interest and/or principal loss), the risk of interest rate changes (possible principal loss or gain if the bonds are sold before maturity), and price level risk (loss of purchasing power). The analysis in this paper is directed toward the first of these risks, the risk of default. By assuming that investors require interest rate adjustments on debt subject to default sufficient to give them an expected present value equal to the present value associated with the investment of their funds in default-free securities, we examine the process that determines the risk-adjusted equilibrium interest rate and the factors affecting that rate. We also examine the implications of the model for the cost of debt and a firm's debt capacity.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 10 , Issue 5 , December 1975 , pp. 757 - 773
Copyright
Copyright © School of Business Administration, University of Washington 1975

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References

1 Silvers, J. B. “An Alternative to the Yield Spread as a Measure of Risk,” The Journal of Finance. (September 1973), pp. 933–955.CrossRefGoogle Scholar

2 Fisher, Lawrence. “Determinants of Risk Premiums on Corporate Bonds,” The Journal of Political Economy (June 1959), pp. 217–237CrossRefGoogle Scholar. Since the capital asset pricing literature uses the term “risk premium” to describe the difference between the expected rate of return and the risk-free rate, we use the term “risk differential” to describe the difference between the contractual rate and the risk-free rate.

3 Both of these assumptions will be relaxed later in the paper.

4 Introducing risk-aversion behavior by the investor would intensify the inverse relationship between the risk of survival and the investor-required return.

5 Should MM and QQ cross more than once, the lower or lowest r is, of course, the correct interest rate since the firm would have to pay higher interest costs at all other r's.

6 Referring to Figure 3, as B increases and QQ shifts inward, the relevant intersection of QQ and MM slides down MM, so that r*(B) increases as B increases.

7 For a general discussion of corporate debt capacity see G. Donaldson, Corporate Debt Capacity (Harvard University, 1961).

8 Richard Holman suggested this method of analysis.

9 Since r*(Bo) < ro, the firm will issue bonds at r*(Bo) rather than r o. See footnote 5.

10 If the bondholders were risk averters, the MM curve in quadrant I of Figure 4 would be flatter; its corresponding RR curve would fall below the existing one in quadrant IV and the consequent maximum level of debt and the promised interest payment (I) would be less than it would in the absence of risk aversion and the expected equity dividend greater.

11 Suppose the maturity date is put off from N to N + M. Then the change (loss) in the net expected present value of the principal repayment is

while the change (gain) in the net expected present value of the interest payment is

Equating these two differences we find that

or

(4) as we anticipated.

12 The calculations for all the tables in this section were performed by David Downes.

13 We could also interpret the situation as one where we do not know the probability of survival, make an estimate of the prior probability, and revise the probability as new information is obtained.