Elements of financial risk management chapter 12

• Credit risk can be defined as the risk of loss due to a counterparty’s failure to honor an obligation in part or in full • Credit risk can take several forms • For banks credit risk ar

Credit Risk Management

Elements of Financial Risk Management

Chapter 12Peter Christoffersen

• Credit risk can be defined as the risk of loss due to a

counterparty’s failure to honor an obligation in part or in full

• Credit risk can take several forms

• For banks credit risk arises fundamentally through its

lending activities

• Nonbank corporations that provide short-term credit to

their debtors face credit risk as well

• Investors who hold a portfolio of corporate bonds or

distressed equities need to get a handle on the default

probability of the assets in their portfolio

• Default risk, a key element of credit risk, introduces an

important source of nonnormality into the portfolio

• Credit risk can also arise in the form of counterparty

default in a derivatives transaction

• The chapter is structured as follows:

• Section 2 provides a few stylized facts on corporate

defaults

• Section 3 develops a model for understanding the effect on corporate debt and equity values of corporate default

• Default risk will have an important effect on how corporate debt is priced, but default risk will also impact the equity

price The model will help us understand which factors

drive corporate default risk

• Section 4 builds on single-firm model from Section 3 to

develop a portfolio model of default risk The model and its extensions provide a framework for computing credit Value-at-Risk

• Section 5 discusses further issues in credit risk including

recovery rates, measuring credit quality through ratings, and measuring default risk using credit default swaps

Figure 12.1: Exposure to Counterparty Default Risk

• Credit rating agencies such as Moody’s and Standard &

Poor’s maintain databases of corporate defaults through

Table 12.1: Largest Moody's-Rated Defaults

Company Default Volume ($ mill) Year Industry Country

Lehman Brothers $120,483 2008 Financials United States

Worldcom, Inc $33,608 2002 Telecom / Media United States

GMAC LLC $29,821 2008 Financials United States

Kaupthing Bank Hf $20,063 2008 Financials Iceland

Washington Mutual, Inc $19,346 2008 Financials United States

Glitnir Banki Hf $18,773 2008 Financials Iceland

NTL Communications $16,429 2002 Telecom / Media United Kingdom

Adelphia Communications $16,256 2002 Telecom / Media United States

Enron Corp $13,852 2001 Energy United States

Figure 12.2: Annual Average Corporate Default

Rates for Speculative Grade Firms, 1983-2010

• 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

Modeling Corporate Default

• 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, A t+T , is uncertain

of the Firm

• At time t+T when the company’s debt comes due the firm will continue to operate if A t+T > D but the firm’s debt

holders will declare the firm bankrupt if A t+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

Equity is a Call Option on the Assets

of the Firm

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

function of the asset value A t+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

13

Equity is a Call Option on the Assets of

the Firm

• 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, r f ,

are constant, and assuming that the log asset value is

normally distributed we get the current value of the equity to be

• where

of the Firm

• Note that the risk-free rate, r f , is not the rate earned on the

company’s debt; it is instead the rate earned on risk-free

debt that can be obtained from the price of a government bond

• Investors who are long options are long volatility

• The Merton model therefore provides the additional insight that equity holders are long asset volatility

• The option value is particularly large when the option is the-money; that is, when the asset value is close to the face value of debt

at-Equity is a Call Option on the Assets

of the Firm

• In this case if the manager holds equity he or she has an incentive to increase the asset value volatility (perhaps by taking on more risky projects) so as to increase the option value of equity

• This action is not in the interest of the debt holders as we shall see now

17

• The simple accounting identity states that the asset value must equal the sum of debt and equity at any point in time and so we have

• where we have used the option payoff on equity described

Figure 12.5: Market Value of Debt as a Function of

Asset Value when Face Value of Debt is $50

• Figure 12.5 shows the payoff to the debt holder of the firm as

a function of the asset value A t+T when the face value of debt

D is $50

• Comparing Figure 12.5 with the option payoffs we see that the debt holders look as if they have sold a put option

although the out-of-the-money payoff has been lifted from 0

to $50 on the vertical axis corresponding to the face value of debt in this example

Corporate Debt is a Put Option Sold

• Figure 12.5 suggests that we can rewrite the debt holder

payoff as

• which shows that the holder of company debt can be

viewed as being long a risk-free bond with face value D

and short a put option on the asset value of the company,

A t+T , with a strike value of D

• We can therefore use the model to value corporate debt;

for example, corporate bonds

• Using the put option formula from Chapter 10 the value

today of the corporate debt with face value D is

• where d is again defined by

• The debt holder is short a put option and so is short asset

volatility

• If the manager takes actions that increase the asset

volatility of the firm, then the debt holders suffer because the put option becomes more valuable

Implementing the Model

• Stock return volatility needs to be estimated for the BSM model to be implemented

• In order to implement the Merton model we need values for

σA and A t, which are not directly observable

• In practice, if the stock of the firm is publicly traded then

we do observe the number of shares outstanding and we

also observe the stock price, and we therefore do observe E t

• where N is the number of shares outstanding

• From the call option relationship earlier we know that E t is

related to σA and A t via the equation

• This gives us one equation in two unknowns

• We need another equation

• The preceding equation for E t implies a dynamic for the

stock price that can be used to derive the following

relationship between the equity and asset volatilities:

• where σ is the stock price volatility

Implementing the Model

• The stock price volatility can be estimated from historical

returns or implied from stock option prices

• We therefore now have two equations in two unknowns, A t

and σA

• The two equations are nonlinear and so must be solved

numerically using, for example, Solver in Excel

• Note that a crucially powerful feature of the Merton model is that we can use it to price corporate debt on firms even

without observing the asset value as long as the stock price is available

• The risk-neutral probability of default in the Merton model

corresponds to the probability that the put option is exercised

• It is simply

• Note that this probability of default is constructed from risk

neutral distribution of asset values and so it may well be

different from the actual physical probability

• The physical default probability could be derived in the

model but would require an estimate of the physical growth rate of firm assets

The Risk-Neutral Probability of

Default

• Default risk is also sometimes measured in terms of distance

to default, which is defined as

• The interpretation of dd is that it is the number of standard

deviations the asset value must move down for the firm to default

• As expected, distance to default is increasing in the asset

value and decreasing in the face value of debt

• The distance to default is also decreasing in the asset

volatility

• Note that the probability of default is

• The probability of default is therefore increasing in asset volatility

Portfolio Credit Risk

• The Merton model gives powerful intuition about corporate default and debt pricing

• It enables us to link the debt value to equity price and

volatility, which in the case of public companies can be

observed or estimated

• While much can be learned from the Merton model, we

have several motivations for going further

• First, we are interested in studying the portfolio implications

of credit risk

• Default is a highly nonlinear event and furthermore default

is correlated across firms and so credit risk is likely to

impose limits on the benefits to diversification

• Second, certain credit derivatives, such as collateralized debt obligations (CDOs), depend on the correlation of

defaults that we therefore need to model

• Third, for privately held companies we may not have

information necessary to implement the Merton model

• Fourth, even if we have the information needed, for a

portfolio of many loans, the implementation of Merton’s model for each loan would be cumbersome

• To keep things relatively simple, we will assume a single factor model similar to the market index model

• For simplicity, we will also assume normal distribution

Portfolio Credit Risk

• We will assume a multivariate version of Merton’s model in

which the asset value of firm i is log normally distributed

• where z i,t+T is a standard normal variable

• As before, the probability of default for firm i is

• where

• We will assume further that the unconditional probability of

default on any one loan is PD

• This implies that the distance to default is now

• for all firms

• A firm defaults when the asset value shock z i is less than

-dd i or equivalently less than Φ-1(PD)

• We will assume that the horizon of interest, T, is one year

so that T = 1 and T is therefore left out of the formulas in

this section

• For ease of notation, time subscript, t is suppressed

Factor Structure

• The relationship between asset values across firms will be crucial for measuring portfolio credit risk

• Assume that the correlation between any firm i and any

other firm j is ρ, which does not depend on i nor j

• This equi-correlation assumption implies a factor structure

on the n asset values

• We have

• where the common factor F and the idiosyncratic are

independent standard normal variables

• Note that the z is will be correlated with each other with

coefficient ρ because they are all correlated with the common

factor F with coefficient ρ

• Using the factor structure we can solve for in terms of z i and F as

• From this we know that a firm defaults when z i < Φ-1(PD)

or equivalently when is less than

Factor Structure

• The probability of firm i defaulting conditional on the

common factor F is therefore

• Note that because of the assumptions we have made this

probability is the same for all firms

• Define the gross loss (before recovery) when firm i defaults

• The credit portfolio loss rate is defined as the average of

the individual losses via

The Portfolio Loss Distribution

• Note that L takes a value between zero and one

• The factor structure assumed earlier implies that

conditional on F the L i variables are independent

• This allows the distribution of the portfolio loss rate to be derived

• It is only possible to derive the exact distribution when

assuming that the number of firms, n, is infinite

• The distribution is therefore only likely to be accurate in portfolios of many relatively small loans

• As the number of loans goes to infinity we can derive the

limiting CDF of the loss rate L to be

• where Φ-1(•) is the standard normal inverse CDF

• The portfolio loss rate distribution thus appears to have

similarities with the normal distribution but the presence of the Φ-1(x) term makes the distribution highly nonnormal

• This distribution is sometimes known as the Vasicek

distribution, from Oldrich Vasicek who derived it

The Portfolio Loss Distribution

• The corresponding PDF of the portfolio loss rate is

• The mean of the distribution is PD and the variance is

• where Φρ(•) is the CDF of the bivariate normal distribution

we used in Chapters 7–9

• Figure 12.6 clearly illustrates the nonnormality of the

credit portfolio loss rate distribution

• The loss distribution has large positive skewness that

risk-The Portfolio Loss Distribution

• Investors dislike negative skewness in the return

distribution—equivalently they dislike positive skewness

in the loss distribution

• The credit portfolio distribution in Figure 12.6 is not only important for credit risk measurement but it can also be used to value credit derivatives with multiple underlying assets such as collateralized debt obligations (CDOs)

• The VaR for the portfolio loss rate can be computed by

inverting the CDF of the portfolio loss rate defined as F L (x)

earlier

• Because we are now modeling losses and not returns, we

are looking for a loss VaR with probability (1-p), which

corresponds to a return VaR with probability p used in

previous chapters

Value-at-Risk on Portfolio Loss Rate

• which yields the following VaR formula:

Granularity Adjustment

• The model may appear to be restrictive because we have

assumed that the n loans are of equal size

• But it is possible to show that the limiting loss rate

distribution is the same even when the loans are not of the same size as long as the portfolio is not dominated by a few very large loans

• The limiting distribution, which assumes that n is infinite,

can of course only be an approximation in any real-life

applications where n is finite

• It is possible to derive finite-sample refinements to the

limiting distribution based on granularity adjustments