e

9.4 Structural models of corporate credit risk

Structural models aim to understand the economics of a company's liabilities and build on the insights of option pricing theory.

  • Structural models are called "structural" because they are based on the structure of a company's balance sheet.

9.4.1 Structural Models Estimations

EXPECTED LOSS Structural models can help estimate expected loss and the present value of expected loss.

Expected loss is equal to the following:

E(loss) KN( ) A ( )N( )

Where:

( ) ( )

( )

sqrt sqrtT t At is the value of assets at time t; K is the face value of debt; N(.) is the cumulative standard normal distribution function with mean 0 and variance 1; T – t is the debt's maturity of debt; u and sigma are the annual expected return and volatility of the company's assets, respectively.

The assumptions of the model are:

  • The company's assets trade in frictionless markets that are arbitrage free,
  • The riskless rate of interest, r, is constant over time, and
  • The time T value of the company's assets has a lognormal distribution with mean uT and variance sigma2T.

PRESENT VALUE OF EXPECTED LOSS Present value of expected loss is calculated as follows:

P(t T) (t T) ( )N( ) A N( )

Where l ( ) r(T t)

(T t)

sqrtT t sqrtT t P(t T) ( )

r is the risk-free rate of interest.

Calculating expected loss Example: Assume a company has the following values: At = $1,000; ut=0.03 per year; r =0.01 per year; K = $700; time to maturity of debt, T – t = 1 year; and sigma = 0.3 per year. Estimate the expected loss and the present value of the expected loss on this debt:

l ( )

( )

sqrt sqrt Using normal distribution table N( ) , N( )

E( oss)

The present value of expected loss is calculated by:

l ( )

( )

sqrt sqrt Using a normal distribution table N( ) , N( )

P(t T) (t T)

  • The $1.50 difference between expected loss and present value of expected loss includes both a discount for the time value of money and the risk premium required by the market to bear the risk of credit loss.
  • In this case, the present value of the expected loss exceeds the expected loss. This means that the risk premium must dominate the difference because the time-value-of- money discount will reduce the present value of the expected loss compared with the expected loss. In other words, in the absence of a risk premium, the present value of the expected loss will be less than the expected loss.

9.4.2 Reasons for equity being viewed as a call option on the company's assets

  • In a structural model, the company's equity can be viewed as a European call option on the assets of the company, with a strike price equal to the debt's face value
  • The link between option pricing theory and structural models comes from the call option analogy for equity
  • The company's owners (equity holders) have limited liability.
  • If the equity holders default on the debt payment at time T, the debtholders' only recourse is to the company's assets. They have no additional claim on the equity holders' personal wealth

Lesson Wrap-Up

This lesson should leave you able to explain the structural models of corporate credit risk in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain structural models estimations in your own words.
  2. Explain reasons for equity being viewed as a call option on the company's assets in your own words.
  3. State one exam-style risk, valuation, or market implication of the structural models of corporate credit risk.