When an enterprise has a low level of equity capital compared to its level of debt capital there is a substantial chance that the equity will be wiped out by a small adverse development for the firm. The effect of leverage on the risk faced by the stockholders is relatively easy to express analytically. For a firm with a rate of return on total capital of r_a and an interest rate on debt of r_d the rate of return on equity r_eq is:

r_eq = r_a + L(r_a - r_d)
 

where L is the debt-equity ratio. This relationship holds not only for intantaneous rates but also for the expected rates of return. Thus,

E(r_eq) = E(r_a) + L(E(r_a) - r_d)
 

and hence for the deviations from the expected values; i.e.,

Δr_eq = Δr_a + L(Δr_a)
or
Δr_eq = (L+1)Δr_a
 

This above relation leads to the risk for equity holders being (L+1) times the risk for the unleveraged corporation.

Risk_lev = (1+L)Risk_unlev
 

This hold true whether risk is defined as the standard deviation of the rate of return as in portfolio analysis or the beta coefficient as in the capital asset pricing model; i.e.,

σ_lev = (1+L)σ_unlev
β_lev = (1+L)β_unlev
 

This analysis assumes that the interest on the debt will be paid with certainty.

But when there is little cushion of equity to shield the debt holders from adverse developments for the firm the debt securities become risky. This is debt-holder risk can be analyzed using the Contingent Commodities approach of options theory. First note the profile of payoffs to equity and debt holders as a function of the value of the enterprise, as shown below.

From the shape of the profile it is clear that the equity holder have a claim that is in the nature of a call option. On the other hand the debt holders are like someone who owns a share of a stock but has sold a call option on it.

Contingent claims analysis applies the methods of options valuation but the variables are different. Below are given the correspondences between option valuation and ordinary option theory.

This analysis presumes there is only one debt issue. Furthermore, for the Black-Scholes option valuation formula to be used there would have to be no dividend payment before the payoff of the debt and no interest paid on the debt until the payoff.

For a calculator which computes the value of equity and debt using the above model and the Black-Scholes formula for call option value click here