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Figure 12.2: Annual Average Corporate Default Rates for Speculative Grade Firms, 1983-2010

Figure 12.2: Annual Average Corporate Default Rates for Speculative Grade Firms, 1983-2010

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Figure 12.2: Annual Average Corporate Default

Rates for Speculative Grade Firms, 1983-2010



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



9



Modeling Corporate Default



10



• The Merton model of corporate default provides important

insights into the valuation of equity and debt when the

probability of default is nontrivial

• The model also helps us understand which factors affect the

default probability

• Consider the situation where we are exposed to the risk that

a particular firm defaults

• This risk could arise from the fact that we own stock in the

firm, or it could be that we have lent the firm cash, or it

could be because the firm is a counterparty in a derivative

transaction with us

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Modeling Corporate Default



11



• We would like to use observed stock price on the firm to

assess the probability of the firm defaulting

• Assume that the balance sheet of the company in question

is of a particularly simple form

• The firm is financed with debt and equity and all the debt

expires at time t+T

• The face value of the debt is D and it is fixed.

• The future asset value of the firm, At+T , is uncertain



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Equity is a Call Option on the Assets

of the Firm



12



• At time t+T when the company’s debt comes due the firm

will continue to operate if At+T > D but the firm’s debt

holders will declare the firm bankrupt if At+T < D and the

firm will go into default

• The stock holders of the firm are the residual claimants on

the firm and to the stock holders the firm is therefore worth



• when the debt comes due

• This is exactly the payoff function of a call option with

strike D that matures on day t+T

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Equity is a Call Option on the Assets

of the Firm



13



• Figure 12.4 shows the value of firm equity Et+T as a

function of the asset value At+T at maturity of the debt when

the face value of debt D is $50

• The equity holder of a company can therefore be viewed as

holding a call option on the asset value of the firm

• It is important to note that in the case of stock options the

stock price was the risky variable

• In the present model of default, the asset value of the firm

is the risky variable but the risky equity value can be

derived as an option on the risky asset value

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Figure 12.4: Equity Value as Function of Asset Value

when Face Value of Debt is $50



Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



14



Equity is a Call Option on the Assets of

the Firm



15



• The BSM formula can be used to value the equity in the firm

in the Merton model

• Assuming that asset volatility, σA, and the risk-free rate, rf ,

are constant, and assuming that the log asset value is

normally distributed we get the current value of the equity to

be



where



Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



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Figure 12.2: Annual Average Corporate Default Rates for Speculative Grade Firms, 1983-2010

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