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Structural models focus on a firm’s assets and liabilities and define a mechanism for default. The probability of default is endogenous as default normally occurs when the value of the firm’s assets hits a barrier representing default. They are called structural models as default is based on the company’s balance sheet structure.
Structural models allow the interpretation of debt and equity values in terms of options. The probability of default is then modeled using option pricing theory, for example via the Black Scholes-Merton option pricing model.
Let \(F_t\) be the value of a firm, which is the sum of debt \(B_t\) and equity \(E_t\):
$$ F_t=B_t+E_t $$
Since debt is senior to equity, the value of equity at maturity is given by:
$$ E_T = Max(F_T-K,0) $$
Notice that \({Max}(F_T-K,0)\) is the same as the payoff of a European call on a company’s assets with a strike price of \(K\) and a maturity of \(T\).
According to put-call parity, a firm’s equity can also be interpreted as a long position in the assets, long put option, and short bond:
$$ E_T=F_T-K+Max(K-F_T,0) $$
Owning a firm’s debt is economically equivalent to:
Let \(K\) be the face value of a single risk-free zero-coupon bond that matures at time \(T\).
The value of debt, \(B_T\) is given by:
$$ B_T=F_T-Max(F_T-K,0) $$
According to the put-call parity, debt can be viewed as a long bond and a short put option:
$$ B_T=K- \left\{\begin{matrix} {0\ if\ F_T\geq K} \\ {{K-F}_T\ if\ F_T < K } \end{matrix}\right.=K-Max(K-F_T,0) $$
Contrary to structural models, default is no longer tied to the firm’s assets falling below a threshold level under reduced-form models. Instead, it occurs according to some exogenous hazard rate process.
Question
Which of the following is most accurate about reduced-form models?
- Reduced form models provide an economic explanation of the occurrence of default.
- Default occurs according to some endogenous hazard rate process.
- The probability of default and recovery rate vary with the business cycle.
Solution
The correct answer is C.
Reduced form models allow the default intensity to change as the firm’s fundamentals and economy changes.
A is incorrect. Reduced form models do not provide an economic rationale behind a default
B is incorrect. The reduced form approach does not consider the endogenous cause of defaults; rather, they depend on exogenous specifications for credit default and debt recovery.
Generally speaking, reduced-form models assume an exogenous recovery rate independent of the default probability and the dynamics of a firm’s assets and take as basics the behavior of default-free interest rates, the recovery rate of defaultable bonds default, as well as a stochastic process for default intensity.
Reading 31: Credit Analysis Models
LOS 31 (d) Explain structural and reduced-form models of corporate credit risk, including assumptions, strengths, and weaknesses.