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This blog is an extension of our blog on Credit Risk and Credit Derivatives.

The Merton Model

A Merton model is an example of a value-based model. These structural models assume that credit risk is a function of the value of the firm and we can calculate the value of debt and equity from that information.

The value of a firm’s debt can serve as an indicator of the firm’s default risk.

The basic equation is : V = D + E

As E & D are contingent claims, option pricing can be used to determine their values.

Debt is considered as a single ZCB bond which matures at time M and has a face value of DM

The payoffs at maturity are as follows:

  • Payment to debt-holders: DM – max(DM – VM, 0)
  • Payment to stock-holders: max(VM – DM,0)

The Black-Scholes Merton (BSM) option pricing model can be used to calculate the value of the stockholder & bondholder’s claim.

The payoff to the stock holder is similar to a long call position

While the payoff to the bond holder is similar to a short-put and a risk-free bond.

Distance to Default (1/2)

The KMV model assumes that debt is issued twice.

The first matures before the chosen horizon and the second matures after that horizon.

The maturity or default threshold is a linear combination of short-term and long-term liabilities. A practical rule is given as

If LT liabilities-to-S.T liabilities < 1.5 Then
Short Term Liabilities + 0.5*Long-term Liabilities
Otherwise

 

Distance to Default (2/2)

For calculating the distance to default (DD) we can use the following formula

 

 

A more precise formula is given below

 

 

E(RoA): expected return on assets

V : Value of the firm assets

Σv : std. deviation of firms assets

 

Decision Rules in Credit Analysis

Decision rules are used in credit analysis to place an observation/firm into a particular sub-group

Minimum Error

The decision rule uses Baye’s Theorem to determine a probability.

P(C given default) * p(default) > or < p(C given not default)* p(not default)

If the left is greater than the right then this will indicate that the firm should be in the likely-to-default group

If the inequality is reversed then the characteristic would indicate that the firm should be in the not-likely-to-default group.

Minimum Risk

It refers to a class of rules that try to either minimize the probability of misclassification or minimize the loss associated with that error

Neyman-Pearson

This uses the statistical concept of Type I and Type II errors.

Type I occurs when a bank lends to a risky firm because it was incorrectly accepted as a non-risky firm

Type II occurs when a bank refuses to lend to a non-risky firm because it was incorrectly rejected as being risky.

Minimax

This rule tries to minimize the maximum risk or error.

The decision is based on calculated probabilities.

Measure of Performance (1/2)

The Risk Operating Characteristic (ROC) is calculated by computing the following

 

Y and X are then graphically represented with the maximum value being unity for both the axis.

Ideally the ray should have infinite slope indicating that all defaults were correctly predicted. If the ray has a 45 degree slope then there are equal proportions of both mistakes.

The cumulative accuracy profile (CAP) compares the probability of default computed by the classification system to the ranking of observed defaults.

The vertical axis represents the fraction of firms that defaulted while the horizontal axis represents the probabilities computed by the classification system.

In both cases we assess the performance by plotting the results on a graph and interpreting the resulting pattern.