The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 686 Wednesday, February 4, 2004 1:12 PM

686 The Mathematics of Financial Modeling and Investment Management

porate debt get paid either the face value, K, under no default or take

over the firm, A, under default. Hence the value of the debt on the

maturity date is given by

DTT ( , )= min{AT(),K}

= AT()– max{AT()– K, 0 } (22.1)

= K– max{KAT– (), 0 } (22.2)

The equations provide two interpretations. Equation (22.1) decom-

poses the risky debt into the asset and a short call. This interpretation

was first given by Black and Scholes that equity owners essentially own

a call option of the company. If the company performs well, then the

equity owners should call the company; or otherwise, the equity owners

let the debt owners own the company. Equation (22.2) decomposes the

risky debt into a risk-free debt and a short put. This interpretation

explains the default risk of the corporate debt. The issuer (equity own-

ers) can put the company back to the debt owner when the performance

is bad.^7 The default risk hence is the put option. These relationships are

shown in Exhibit 22.1. Exhibits 22.1(a) and 22.1(b) explain the rela-

tionship between equity and risky debt and Exhibits 22.1(b) and 22.1(c)

explain the relationship between risky and risk-free debts.

Note that the value of the equity and debt when added together must

equal the assets of the firm at all times, that is, A(t) = E(t) + D(t,T). Clearly,

at maturity, this is true as we have

EXHIBIT 22.1 Payoff Diagrams at Maturity for Equity, Risky Debt, and

Risk-Free Debt

(^7) A covered call is a combination of a selling call option and owning the same face
value of the shares which might have to be delivered should the option expire in the
money. If the option expires in the money, a net profit equal to the strike is made. If
the option expires worthless, then the position is worth the stock price.