Hull J.C. Risk management and Financial institutions

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270 Chapter 11

rating. Unfortunately, credit ratings are revised relatively infrequently.

This has led some analysts to argue that equity prices can provide more

up-to-date information for estimating default probabilities.

In 1974, Merton proposed a model where a company's equity is an

option on the assets of the company.

Suppose, for simplicity, that a firm

has one zero-coupon bond outstanding and that the bond matures at

time T. Define:

Value of company's assets today

Value of company's assets at time T

Value of company's equity today

Value of company's equity at time T

Amount of debt interest and principal due to be repaid at time T

Volatility of assets (assumed constant)

Instantaneous volatility of equity

If V

< D, it is (at least in theory) rational for the company to default on

the debt at time T. The value of the equity is then zero. If V

> D, the

company should make the debt repayment at time T and the value of the

equity at this time is V

— D. Merton's model, therefore, gives the value of

the firm's equity at time T as

= max(V

- D, 0)

This shows that the equity of a company is a call option on the value of

the assets of the company with a strike price equal to the repayment

required on the debt. The Black-Scholes formula (see Appendix C at the

end of this book) gives the value of the equity today as

where

and N is the cumulative normal distribution function. The value of the

debt today is V

— E

The risk-neutral probability that the company will default on the debt

is N(-d

). To calculate this, we require V

and Neither of these are

See R. Merton "On the Pricing of Corporate Debt: The Risk Structure of Interest

Rates." Journal of Finance, 29 (1974), 449-470 .

Credit Risk: Estimating Default Probabilities 271

directly observable. However, if the company is publicly traded, we can

observe E

. This means that equation (11.4) provides one condition that

must be satisfied by V

and We can also estimate From a result in

stochastic calculus known as lemma, we have

This provides another equation that must be satisfied by V

and

Equations (11.4) and (11.5) provide a pair of simultaneous equations

that can be solved for V

and

Example 11.2

The value of a company's equity is $3 million and the volatility of the equity is

80%. The debt that will have to be paid in one year is $10 million. The risk-

free rate is 5% per annum. In this case, E

— 3, = 0.80, r = 0.05, T = 1,

and D=10. Solving equations (11.4) and (11.5) yields V

= 12.40 and

= 0.2123. The parameter d

is 1.1408, so that the probability of default

is N(-d

) = 0.127, or 12.7%. The market value of the debt is V

- E

, or 9.40.

The present value of the promised payment on the debt is 10e

-0.05xl

= 9.51.

The expected loss on the debt is therefore (9.51 — 9.40)/9.51, or about 1.2% of

its no-default value. Comparing this with the probability of default gives the

expected recovery in the event of a default as (12.7 — 1.2)/12.7, or about 91%.

Distance to Default

Moody's KMV has coined the term distance to default to describe the

output from Merton's model. This is the number of standard deviations

by which the asset price must change for default to be triggered T years in

the future. It is given by

As the distance to default decreases, the company becomes more likely to

default. In Example 11.2 the one-year distance to default is 1.14 standard

deviations.

To solve two nonlinear equations of the form F(x, y) = 0 and G(x, y) = 0, we can use

the Solver routine in Excel to find the values of x and y that minimize [F(x, y)]

272 Chapter 11

Extensions of the Basic Model

The basic Merton model we have just presented has been extended in a

number of ways. For example, one version of the model assumes that a

default occurs whenever the value of the assets falls below a barrier level.

Another allows payments on debt instruments to be required at more than

one time. Many analysts have found the implied volatility of equity issued

by a company to be a good predictor of the probability of default. (The

higher the implied volatility, the higher the probability of default.) Hull et

al. show that this is consistent with Merton's model.

They provide a way

of implementing Merton's model using two equity implied volatilities and

show that the resulting model provides results comparable to those

provided by the usual implementation of the model.

Performance of the Model

How well do the default probabilities produced by Merton's model and

its extensions correspond to actual default experience? The answer is that

Merton's model and its extensions produce a good ranking of default

probabilities (risk neutral or real world). This means that a monotonic

transformation can be estimated to convert the probability of default

output from Merton's model into a good estimate of either the real-world

or risk-neutral default probability.

It may seem strange to use a default

probability, N(—d

), that is in theory a risk-neutral default probability to

estimate a real-world default probability. Given the nature of the calibra-

tion process we have just described, the underlying assumption is that the

ranking of risk-neutral default probabilities of different companies is the

same as that of their real-world default probabilities.

SUMMARY

The probability that a company will default during a particular period 0f

time in the future can be estimated from historical data, bond prices, or

equity prices. The default probabilities calculated from bond prices are

risk-neutral probabilities, whereas those calculated from historical data

See J. Hull, I. Nelken, and A. White, "Merton's Model, Credit Risk, and Volatility

Skews," Journal of Credit Risk, 1, No. 1 (2004), 1-27.

Moody's KMV provides a service that transforms a default probability produced by

Merton's model into a real-world default probability (which it refers to as an EDF, short

for expected default frequency). CreditGrades uses Merton's model to estimate credit

spreads, which are closely linked to risk-neutral default probabilities.

Credit Risk: Estimating Default Probabilities 273

are real-world probabilities. The default probabilities calculated from

equity prices using Merton's model are in theory risk-neutral default

probabilities. However, the output from the model can be calibrated so

that either risk-neutral or real-world default probabilities are produced.

Real-world probabilities should be used for scenario analysis and the

calculation of credit VaR. Risk-neutral probabilities should be used for

valuing credit-sensitive instruments. Risk-neutral default probabilities are

usually significantly higher than real-world probabilities.

FURTHER READING

Altman, E. I., "Measuring Corporate Bond Mortality and Performance,"

Journal of Finance, 44 (1989): 902-922.

Duffie, D., and K. Singleton, "Modeling Term Structures of Defaultable

Bonds," Review of Financial Studies, 12 (1999): 687-720.

Hull, J., M. Predescu, and A. White, "Relationship between Credit Default

Swap Spreads, Bond Yields, and Credit Rating Announcements," Journal of

Banking and Finance, 28 (November 2004): 2789-2811.

Hull, J., M. Predescu, and A. White, "Bond Prices, Default Probabilities, and

Risk Premiums," Journal of Credit Risk, 1, No. 2 (2004): 53-60.

Kealhofer, S., "Quantifying Default Risk I: Default Prediction," Financial

Analysts Journal, 59, No. 1 (2003): 30-44.

Kealhofer, S., "Quantifying Default Risk II: Debt Valuation," Financial

Analysts Journal, 59, No. 3 (2003), 78-92.

Litterman, R., and T. Iben, "Corporate Bond Valuation and the Term Structure

of Credit Spreads," Journal of Portfolio Management, Spring 1991: 52-64.

Merton, R. C, "On the Pricing of Corporate Debt: The Risk Structure of

Interest Rates," Journal of Finance, 29 (1974): 449-470.

Rodriguez, R. J., "Default Risk, Yield Spreads, and Time to Maturity," Journal

of Financial and Quantitative Analysis, 23 (1988): 111-117.

QUESTIONS AND PROBLEMS (Answers at End of Book)

11.1. How many ratings does Moody's use for companies that have not

defaulted? What are they?

11.2. How many ratings does S&P use for companies that have not defaulted?

What are they?

11.3. Calculate the average default intensity for B-rated companies during the

first year from the data in Table 11.1.

274 Chapter \\

11.4. Calculate the average default intensity for Ba-rated companies during the

third year from the data in Table 11.1.

11.5. The spread between the yield on a 3-year corporate bond and the yield on

a similar risk-free bond is 50 basis points. The recovery rate is 30%,

Estimate the average default intensity per year over the 3-year period.

11.6. The spread between the yield on a 5-year bond issued by a company and

the yield on a similar risk-free bond is 80 basis points. Assume a recovery

rate of 40%. Estimate the average default intensity per year over the

5-year period. If the spread is 70 basis points for a 3-year bond, what do

your results indicate about the average default intensity in years 4 and 5?

11.7. Should researchers use real-world or risk-neutral default probabilities for

(a) calculating credit value at risk and (b) adjusting the price of a

derivative for defaults?

11.8. How are recovery rates usually defined?

11.9. Verify (a) that the numbers in the second column of Table 11.4 are

consistent with the numbers in Table 11.1 and (b) that the numbers in

the fourth column of Table 11.5 are consistent with the numbers in

Table 11.4 and a recovery rate of 40%.

11.10. A 4-year corporate bond provides a coupon of 4% per year payable

semiannually and has a yield of 5% expressed with continuous compound-

ing. The risk-free yield curve is flat at 3% with continuous compounding.

Assume that defaults can take place at the end of each year (immediately

before a coupon or principal payment) and the recovery rate is 30%.

Estimate the risk-neutral default probability on the assumption that it is

the same each year.

11.11. A company has issued 3- and 5-year bonds with a coupon of 4% per

annum payable annually. The yields on the bonds (expressed with

continuous compounding) are 4.5% and 4.75%, respectively. Risk-free

rates are 3.5% with continuous compounding for all maturities. The

recovery rate is 40%. Defaults can take place halfway through each year.

The risk-neutral default rates per year are Q

for years 1 to 3 and Q

for

years 4 and 5. Estimate Q

and Q

11.12. Suppose that in an asset swap B is the market price of the bond per dollar

of principal, B* is the default-free value of the bond per dollar 01

principal, and V is the present value of the asset swap spread per dollar

of principal. Show that V = B* — B.

11.13. Show that under Merton's model in Section 11.6 the credit spread on a

T-year zero-coupon bond is

-ln[N{d2) + N(-d

)/L]/T

where L = De

-rT

Credit Risk: Estimating Default Probabilities

275

11.14. The value of a company's equity is $2 million and the volatility of its

equity is 50%. The debt that will have to be repaid in one year is

$5 million. The risk-free interest rate is 4% per annum. Use Merton's

model to estimate the probability of default. {Hint: The Solver function

in Excel can be used for this question.)

11.15. Suppose that the LIBOR/swap curve is flat at 6% with continuous

compounding and a 5-year bond with a coupon of 5% (paid semiannu-

ally) sells for 90.00. How would an asset swap on the bond be

structured? What is the asset swap spread that would be calculated in

this situation?

ASSIGNMENT QUESTIONS

11.16. Suppose a 3-year corporate bond provides a coupon of 7% per year

payable semiannually and has a yield of 5% (expressed with semiannual

compounding). The yields for all maturities on risk-free bonds is 4% per

annum (expressed with semiannual compounding). Assume that defaults

can take place every 6 months (immediately before a coupon payment)

and the recovery rate is 45%. Estimate the default probabilities, assuming

(a) that the unconditional default probabilities are the same on each

possible default date and (b) that the default probabilities conditional on

no earlier default are the same on each possible default date.

11.17. A company has 1- and 2-year bonds outstanding, each providing a

coupon of 8% per year payable annually. The yields on the bonds

(expressed with continuous compounding) are 6.0% and 6.6%, respect-

ively. Risk-free rates are 4.5% for all maturities. The recovery rate is

35%. Defaults can take place halfway through each year. Estimate the

risk-neutral default rate each year.

11.18. The value of a company's equity is $4 million and the volatility of its

equity is 60%. The debt that will have to be repaid in 2 years is

$15 million. The risk-free interest rate is 6% per annum. Use Merton's

model to estimate the expected loss from default, the probability of

default, and the recovery rate in the event of default. Explain why

Merton's model gives a high recovery rate. {Hint: The Solver function

in Excel can be used for this question.)

Credit Risk Losses

and Credit VaR

This chapter starts by discussing the nature of the credit risk in deriva-

tives transactions. It shows how expected credit losses can be calculated

and looks at the various ways in which a financial institution can

structure its contracts so as to reduce these expected credit losses.

The last part of the chapter covers the calculation of credit VaR. The

1986 amendment to Basel I allows banks to develop their own models for

calculating VaR for market risk in the trading book. However, Basel II

does not give banks quite the same freedom to calculate credit VaR for

the banking book. The development of internal models for calculating

credit VaR is nevertheless an important activity for banks. These models

can be used to determine regulatory capital for credit risk in the trading

book (the specific risk capital charge). They can also be used when

economic capital is calculated, as we will explain in Chapter 16.

Chapter 11 covered the important difference between real-world and

risk-neutral default probabilities (see Section 11.5). Real-world default

Probabilities can be estimated from historical data. Risk-neutral default

Probabilities can be estimated from bond prices. In the first part of this

chapter, when calculating expected credit losses, we will use risk-neutral

default probabilities. This is because we are calculating the present value

of future cash flows. Later in the chapter, when calculating credit VaR,

we will use real-world default probabilities. This is because we are looking

at future scenarios and not calculating present values.

278

Chapter 12

12.1 ESTIMATING CREDIT LOSSES

Credit losses on a loan depend primarily on the probability of default and

the recovery rate. We covered estimates of the probability of default in

Chapter 11. The estimate of recovery rate (or equivalently loss given

default) depends on the nature of the collateral, if any. When making

estimates, a financial institution is likely to use a mixture of its own

experience and statistics published by rating agencies such as those in

Table 11.2.

Derivatives Transactions

The credit exposure on a derivatives transaction is more complicated than

that on a loan. This is because the claim that will be made in the event of

a default is more uncertain than in the case of a loan. Consider a financial

institution that has one derivatives contract outstanding with a counter-

party. We can distinguish three possible situations:

1. The contract is always a liability to the financial institution.

2. The contract is always an asset to the financial institution.

3. The contract can become either an asset or a liability to the financial

institution.

An example of a derivatives contract in the first category is a short option

position; an example in the second category is a long option position; an

example in the third category is a forward contract or a swap.

Derivatives in the first category have no credit risk to the financial

institution. If the counterparty goes bankrupt, there will be no loss. The

derivative is one of the counterparty's assets. It is likely to be retained,

closed out, or sold to a third party. The result is no loss (or gain) to the

financial institution.

Derivatives in the second category always have credit risk to the

financial institution. If the counterparty goes bankrupt, a loss is likely

to be experienced. The derivative is one of the counterparty's liabilities.

The financial institution has to make a claim against the assets of the

counterparty and may eventually realize some percentage of the value of

the derivative.

Derivatives in the third category may or may not have credit risk. If

the counterparty defaults when the value of the derivative is positive to

Counterparties to derivatives transactions usually rank equally with unsecure

creditors in the event of a liquidation.

Credit Risk Losses and Credit VaR

279

the financial institution, then a claim will be made against the assets of

the counterparty and a loss is likely to be experienced. If the counter-

party defaults when the value is negative to the financial institution, no

loss is made because the derivative will be retained, closed out, or sold to

a third party.

Adjusting Derivatives Valuations for Counterparty Default Risk

How should a financial institution (or end-user of derivatives) adjust the

value of a derivative to allow for counterparty credit risk? Consider a

derivative that has a value of f

today assuming no defaults. Let us

suppose that defaults can take place at times

, t

,...,

and that the

value of the derivative to the financial institution (assuming no defaults)

at time t

is f

. Define the risk-neutral probability of default at time t

and the expected recovery rate as R.

The exposure at time t

is the financial institution's potential loss. This is

max(f

, 0) as illustrated in Figure 12.1. Assume that the expected recovery

in the event of a default is R times the exposure. Assume also that the

recovery rate and the probability of default are independent of the value of

the derivative. The risk-neutral expected loss from default at time t

where denotes expected value in a risk-neutral world. Taking present

values leads to the cost of defaults being

where u

equals q

{1 — R) and v

is the value today of an instrument that

Pays off the exposure on the derivative under consideration at time t

Consider again the three categories of derivatives mentioned earlier.

The first category (where the derivative is always a liability to the financial

institution) is easy to deal with. The value of f

is always negative, and so

the total expected loss from defaults given by equation (12.1) is always

zero. The financial institution needs to make no adjustments for the cost

A company usually defaults because of the total amount of its liabilities, not because of

the value of any one transaction. At the time a company defaults it is likely that some of

its contracts will have positive values.

This probability of default can be calculated from bond prices in the way described in

Section 11.4.

280

Chapter 12

Figure 12.1 Exposure on a derivative as a function of

its no-default value.

of defaults. (Of course, the counterparty may want to take account of the

possibility of the financial institution defaulting in its own analysis.)

For the second category (where the derivative is always an asset to the

financial institution), the value of f

is always positive. The expression

max(f

, 0) is always equal to f

. Since v

is the present value of f

, it

always equals f

The expected loss from default is therefore f

times the

total probability of default during the life of the derivative times 1 — R.

Example 12.1

Consider a two-year over-the-counter option with a value (assuming no

defaults) of $3. Suppose that the company selling the option has a risk-

neutral probability of defaulting during the two-year period of 4% and the

recovery in the event of a default is 25%. The expected cost of defaults is

3 x 0.04 x (1 - 0.25), or $0.09. The buyer of the option should therefore be

prepared to pay only $2.91.

For the third category of derivatives, the sign of f

is uncertain. The

variable v

is a call option on f

with a strike price of zero. One way of

calculating all the v

with a single analysis is to simulate the underlying

market variables over the life of the derivative. Sometimes approximate

analytic calculations are possible (see Problems 12.7, 12.8, and 12.14).

This assumes no payoffs from the derivative prior to time t

. The analysis can be

adjusted to cope with situations where there are intermediate payoffs.