4. Credit Analysis Models

Question 1

Explain ‘Expected Loss’ and ‘Present Value of Expected Loss’.

Answer 1

• Expected Loss: is equal to the probability of default multiplied by the loss given default. Expected loss can be stated as a monetary value or as a percentage of a bond’s value. Expected loss depends on the state of the economy: during boom times, the probability of default as well as loss given default will be lower
• Present Value of Expected Loss: is the highest price a hypothetical investor would be willing to pay to an insurer to bear the credit risk of the investment. The present value of expected loss makes two adjustments to the expected loss measure:
  1) Time value adjustment: future expected losses are discounted to their present value, reflecting that a loss of principal far in the future less of a concern than a near-term loss
  2) Risk-neutral probabilities: adjustment to the probabilities of default is a more complex adjustment that accounts for the risk of the cash flows and is called the risk premium. Instead of using the probability of default, we use the risk-neutral probability of default. The difference between actual probabilities of default and risk-neutral probabilities is the adjustment for risk

The present value of expected loss is the difference between the value of a credit-risky bond and an otherwise identical risk-free bond.

PVEL = (V_risk-free bond) - (V_Credit-risky bond)

PVEL = Expected Loss + Risk Premium - Time Value Discount

PVEL - Expected Loss = Risk Premium - Time Value Discount

PVEL = FV * exp(-rT)

Question 2

Explain ‘Structural Models of Corporate Credit Risk’.

Answer 2

Structural models of corporate credit risk are based on the structure of a company’s balance sheet and rely on insights provided by option pricing theory.

• Option Analogy: consider a hypothetical company with assets that are financed by equity and a single issue of zero-coupon debt. The value of the assets at any point in time is the sum of the value of equity and value of debt. Due to the limited liability nature of corporate equity, the shareholders effectively have a call option on the company’s assets with a strike price equal to the face value of debt. If at the maturity of the debt, the value of the company’s assets is higher than the face value of debt, the shareholders will exercise their call option to acquire the assets (and then pay off the debt and keep the residual). On the other hand, if the value of the company’s assets is less than the face value of debt, the shareholders will let the option expire worthless (i.e. default on the debt), leaving the company’s assets to the debt holders.

Hence, at time T (maturity of debt):

Value of Stock = MAX (0, Assets(t) - Face Value of Debt)

Value of Debt(t) = MIN (Assets(t);Value of Debt(t))

Based on above analysis, we can also say that owning risky debt with a face value of K is equivalent to owning a risk-free bond with the same face value (K) and writing a European put option on the assets of the company with a strike price of K. We can express as:

Value of Risky Debt = Value of Risk-Free Debt - Value of Put Option On Company’s Assets