Hull J.C. Risk management and Financial institutions
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Credit Risk Losses and Credit VaR 291
revaluation losses in years 1, 2, and 3 (the CreditMetrics approach). The
losses under the second approach should in theory add up to the losses
under the first approach.
There are two key issues in the implementation of CreditMetrics. One is
handling correlations between bonds. The other is calculating credit
spreads for valuing the bonds at the end of the year.
The CreditMetrics Correlation Model
In sampling to determine credit losses, the credit rating changes for
different counterparties should not be assumed to be independent. A
Gaussian copula model can be used to construct a joint probability
distribution of rating changes (see Section 6.4 for a discussion of copula
models). The copula correlation between the rating transitions for two
companies is typically set equal to the correlation between their equity
returns using a factor model similar to that in Section 6.3.
As an illustration of the CreditMetrics approach, suppose that we are
simulating the rating change of an A-rated and a B-rated company over a
one-year period using the transition matrix in Table 12.1. Suppose that
the correlation between the equity returns of the two companies is 0.2. On
each simulation trial we would sample two variables x
A
and x
B
from
standard normal distributions, so that their correlation is 0.2. The
variable x
A
determines the new rating of the A-rated company and
variable x
B
determines the new rating of the B-rated company. Since
the A-rated company gets upgraded to Aaa if x
A
< —3.2905, it becomes
Aa if -3.2905 <x
A
< -1.9703, it stays A if -1.9703 <x
A
< 1.5779, and
so on. Similarly, since
the B-rated company becomes Aaa if x
B
< —3.7190, it becomes Aa if
-3.7190 < x
B
< -3.3528, it becomes A if -3.3528 < x
B
< -2.9290, and
so on. The A-rated company defaults if x
A
> that is, when
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Chapter 12
Figure 12.4 The CreditMetrics correlation model: transition of an A-rated and
a B-rated company to a new rating after one year. Here x
A
and x
B
are sampled
from standard normal distributions with the correlation equal to the correlation
between the equity returns of A and B.
x
A
> 3.5401. The B-rated company defaults when x
B
> N
-1
(0.9336), that
is, when x
B
> 1.5032. This example is illustrated in Figure 12.4.
Spread Changes
In order to revalue the portfolio of bonds on each simulation trial, it is
necessary to calculate spread changes. One way of proceeding is to use a
one-factor regression model to divide the spread changes for each bond
into a systematic component and an idiosyncratic component. The sys-
tematic component is the spread change that affects all bonds in the
rating category being considered. The idiosyncratic component affects
only one particular bond.
If the systematic component of spread changes is reflected in market risk
VaR calculations, it is only necessary to take account of the idiosyncratic
Credit Risk Losses and Credit VaR 293
spread changes in specific risk VaR calculations. On each trial, therefore,
we calculate not only what happens to the rating of each bond but also an
idiosyncratic spread change for the bond. For each bond on each simula-
tion trial, one of three things can happen:
1. The credit rating of the bond stays the same. In this case the bond is
revalued in a way that reflects the idiosyncratic spread change for the
bond.
2. The credit rating of the bond changes. In this case the bond is
revalued using a spread corresponding to the new rating category.
3. The bond defaults. In this case a recovery rate is sampled. Often the
recovery rate is assumed to have a beta distribution centered on the
mean recovery rate published by rating agencies (see Table 11.2).
Time Horizon
Regulators require a bank's credit VaR for specific risk to be a ten-day
99% VaR. In practice, this means a bank has two choices. One is to
calculate a one-year 99% VaR in the way we have just outlined and then
use the square-root rule to scale it to a ten-day 99% VaR. If we assume
250 days in a year, this means that the calculated VaR is divided by 5. The
other is to convert the transition matrix in Table 12.1 from a one-year
transition matrix to a ten-day transition matrix and use the procedure we
have just described. to calculate a ten-day loss distribution directly.
Assuming that ratings transitions in successive periods are independent,
the second approach involves finding a matrix B such that
B
25
= A
where A is the matrix in Table 12.1. A procedure for doing this is outlined
in Appendix E at the end of the book.
12.7 INTERPRETING CREDIT CORRELATIONS
Care should be taken in interpreting credit correlations. Different ways of
calculating a credit correlation can give quite different answers. We
illustrate this by considering the binomial correlation measure that is
sometimes used by rating agencies and comparing it with the Gaussian
copula correlation measure that underlies Vasicek's model.
294
Chapter 12
For two companies, A and B, the binomial correlation measure is the
coefficient of correlation between:
1. A variable that equals 1 if Company A defaults between times 0 and
T and zero otherwise; and
2. A variable that equals 1 if Company B defaults between times 0 and
T and zero otherwise.
The measure is
where P
AB
(T) is the joint probability of A and B defaulting between time
0 and time T, Q
A
{T) is the cumulative probability that Company A will
default by time T, and Q
B
(T) is the cumulative probability that Company
B will default by time T. Typically depends on T, the length of the
time period considered. Usually it increases as T increases.
In the Gaussian copula model,
P
AB
(T)
= M[x
A
(T),
x
B
(T);
], where
x
A
(T) = and x
B
(T) = are the transformed times
to default for companies A and B, and is the Gaussian copula
correlation for the times to default for A and B. The quantity
is the probability that, in a bivariate normal distribution where
the correlation between the variables is the first variable is less than a
and the second variable is less than b.
9
It follows that
This shows that, if Q
A
(T) and Q
B
(T) are known, can be calculated
from and vice versa. Usually is markedly greater than as
is illustrated by the following example.
Example 12.3
Suppose that the probability of Company A defaulting in one-year period
is 1% and the probability of Company B defaulting in a one-year period
is also 1%. In this case, =-2.326. If is
0.20, M(x
A
(l),x
B
(l), ) = 0.000337, and equation (12.2) shows that
= 0.024 when T=1.
9
An Excel function for calculating is on the author's website.
Credit Risk Losses and Credit VaR
295
SUMMARY
The credit risk on a loan depends on the probability of default and the
recovery rate (or equivalently the loss given default). The credit risk in a
derivatives transaction is more complicated than that in a loan because
the exposure at the time of default is uncertain. Some derivatives trans-
actions (e.g., written options) are always liabilities and give rise to no
credit risk. Some (e.g., long positions in options) are always assets and
entail significant credit risks. The most difficult types of derivatives
transactions to deal with from a credit risk perspective are those that
may become either assets or liabilities during their life. Examples are
forward contracts and swaps.
The over-the-counter market has developed a number of ways of
mitigating credit risk. The most important of these is netting. This is a
clause in most contracts written by a financial institution stating that, if a
counterparty defaults on one contract it has with the financial institution,
then it must default on all contracts it has with that financial institution.
Another credit mitigation technique is Collateralization. This requires a
counterparty to post collateral. If the value of the contract moves against
the counterparty, more collateral is required. In the event that collateral is
not posted in a timely fashion the contract is closed out using a pre-agreed
procedure for valuation. A third credit mitigation technique is downgrade
trigger. This gives a company the option to close out a contract if the credit
rating of the counterparty falls below a certain level.
Credit VaR can be defined similarly to the way VaR is defined for market
risk. It is the credit loss that will not be exceeded over some time horizon
with a specified confidence level. Basel II calculates credit VaR for the
banking book using a one-factor Gaussian copula model of time to default
that was originally developed by Vasicek. An approach for calculating
credit VaR that is similar to procedures used in the insurance industry is
Credit Risk Plus which was proposed by Credit Suisse Financial Products
in 1997. For specific risk in the trading book most large banks use
CreditMetrics which was proposed by J.P. Morgan in 1997. This involves
simulating rating changes for companies. The correlation between different
companies is handled using a Gaussian copula model for rating changes.
FURTHER READING
Credit Suisse Financial Products, "Credit Risk Management Framework,"
October, 1997.
296
Chapter 12
Finger, C. C, "The One-Factor CreditMetrics Model in the New Basel Capital
Accord," RiskMetrics Journal, Summer 2001.
J.P. Morgan, "CreditMetrics-Technical Document," April, 1997.
QUESTIONS AND PROBLEMS (Answers at End of Book)
12.1. A bank already has one transaction with a counterparty on its books.
Explain why a new transaction by a bank with a counterparty can have
the effect of increasing or reducing the bank's credit exposure to the
counterparty.
12.2. Suppose that the measure in equation (12.2) is the same in the real
world and the risk-neutral world. Is the same true of the Gaussian copula
measure
12.3. What is meant by a "haircut" in a Collateralization agreement. A com-
pany offers to post its own equity as collateral. How would you respond?
12.4. Explain the difference between Vasicek's model, the Credit Risk Plus
model, and CreditMetrics as far as the following are concerned: (a) when
a credit loss is recognized, and (b) the way in which default correlation is
modeled.
12.5. Suppose that the probability of Company A defaulting during a 2-year
period is 0.2 and the probability of Company B defaulting during this
period is 0.15. If the Gaussian copula measure of default correlation is
0.3, what is the binomial correlation measure?
12.6. Suppose that a financial institution has entered into a swap dependent on
the sterling interest rate with counterparty X and an exactly offsetting
swap with counterparty Y. Which of the following statements are true
and which are false? (a) The total present value of the cost of defaults is
the sum of the present value of the cost of defaults on the contract with X
plus the present value of the cost of defaults on the contract with Y.
(b) The expected exposure in 1 year on both contracts is the sum of the
expected exposure on the contract with X and the expected exposure on
the contract with Y. (c) The 95% upper confidence limit for the exposure
in 1 year on both contracts is the sum of the 95% upper confidence limit
for the exposure in 1 year on the contract with X and the 95% upper
confidence limit for the exposure in 1 year on the contract with Y
Explain your answers.
12.7. A company enters into a 1-year forward contract to sell $100 for AUD 150
The contract is initially at the money. In other words, the forward
exchange rate is 1.50. The 1-year dollar risk-free rate of interest is 5%
per annum. The 1-year dollar rate of interest at which the counterparty can
Credit Risk Losses and Credit VaR 297
borrow is 6% per annum. The exchange rate volatility is 12% per annum.
Estimate the present value of the cost of defaults on the contract. Assume
that defaults are recognized only at the end of the life of the contract.
12.8. Suppose that in Problem 12.7 the 6-month forward rate is also 1.50 and
the 6-month dollar risk-free interest rate is 5% per annum. Suppose
further that the 6-month dollar rate of interest at which the counterparty
can borrow is 5.5% per annum. Estimate the present value of the cost of
defaults assuming that defaults can occur either at the 6-month point or
at the 1-year point? (If a default occurs at the 1-month point, the
company's potential loss is the market value of the contract.)
12.9. "A long forward contract subject to credit risk is a combination of a
short position in a no-default put and a long position in a call subject to
credit risk." Explain this statement.
12.10. Explain why the credit exposure on a pair of offsetting forward contracts
with different counterparties resembles a straddle.
12.11. "When a bank is negotiating a pair of offsetting currency swaps, it should
try to ensure that it is receiving the lower interest rate currency from a
company with a low credit risk." Explain.
ASSIGNMENT QUESTIONS
12.12. Explain carefully the distinction between real-world and risk-neutral
default probabilities. Which is higher? A bank enters into a credit
derivative where it agrees to pay $100 at the end of 1 year if a certain
company's credit rating falls from A to Baa or lower during the year. The
1-year risk-free rate is 5%. Using Table 12.1, estimate a value for the
derivative. What assumptions are you making? Do they tend to overstate
or understate the value of the derivative.
12.13. Suppose that a bank has a total of $10 million of exposures of a certain
type. The one-year probability of default averages 1 % and the recovery
rate averages 40%. The copula correlation parameter is 0.2. Estimate the
1-year 99.5% credit VaR.
12.14. Consider an option on a non-dividend-paying stock where the stock price
is $52, the strike price $50, the risk-free rate is 5%, the volatility is 30%,
and the time to maturity is 1 year. (a) What is the value of the option
assuming no possibility of a default? (b) What is the value of the option
to the buyer if there is a 2% chance that the option seller will default at
maturity? (c) Suppose that, instead of paying the option price up front,
the option buyer agrees to pay the forward value of the option price at
the end of the life of the contract. By how much does this reduce the cost
of defaults to the option buyer in the case where there is a 2% chance of
298 Chapter 12
the option seller defaulting? (d) If in case (c) the option buyer has a 1 %
chance of defaulting at the end of the life of the contract, what is the
default risk to the option seller? Discuss the two-sided nature of default
risk in this case and the value of the option to each side.
12.15. Can the existence of downgrade triggers increase default risk? Explain
your answer.
Credit Derivatives
The credit derivatives market has seen huge growth in recent years.
In 2000 the total notional principal for outstanding credit derivative
contracts was about $800 billion. By 2005 this had risen to $12 trillion.
Credit derivatives are contracts where the payoff depends on the
creditworthiness of one or more companies or countries. They allow
companies to trade credit risks in much the same way that they trade
market risks. Banks and other financial institutions, once they had
assumed a credit risk, used to be in the position where they could do
little except wait (and hope for the best). Now they can actively manage
their portfolios of credit risks, keeping some and entering into credit
derivative contracts to protect themselves from others.
13.1 CREDIT DEFAULT SWAPS
The most popular credit derivative is a credit default swap (CDS). This is a
contract that provides insurance against the risk of a default by particular
company. The company is known as the reference entity and a default by
the company is known as a credit event. The buyer of the insurance
obtains the right to sell bonds issued by the company for their face value
when a credit event occurs and the seller of the insurance agrees to buy
the bonds for their face value when a credit event occurs.
1
The total face
1
The face value (or par value) of a bond is the principal amount that the issuer will repay
at maturity if it does not default.
300 Chapter 13
value of the bonds that can be sold is known as the credit default swap's
notional principal.
The buyer of the CDS makes periodic payments to the seller until the
end of the life of the CDS or until a credit event occurs. These payments
are typically made in arrears every quarter, every half year, or every year.
The settlement in the event of a default involves either physical delivery of
the bonds or a cash payment.
An example will help to illustrate how a typical deal is structured.
Suppose that two parties enter into a five-year credit default swap on
March 1, 2006. Assume that the notional principal is $100 million and the
buyer agrees to pay 90 basis points annually for protection against default
by the reference entity.
The CDS is shown in Figure 13.1. If the reference entity does not
default (i.e., there is no credit event), the buyer receives no payoff and
pays $900,000 on March 1 of each of the years 2007, 2008, 2009, 2010,
and 2011. If there is a credit event a substantial payoff is likely. Suppose
that the buyer notifies the seller of a credit event on June 1, 2009
(a quarter of the way into the fourth year). If the contract specifies
physical settlement, the buyer has the right to sell bonds issued by the
reference entity with a face value of $100 million for $100 million. If the
contract requires cash settlement, an independent calculation agent will
conduct a poll of dealers at a predesignated number of days after the
credit event to determine the mid-market value of the cheapest deliver-
able bond. Suppose this bond is worth $35 per $100 of face value. The
cash payoff would be $65 million.
The regular quarterly, semiannual, or annual payments from the buyer
of protection to the seller of protection cease when there is a credit event.
However, because these payments are made in arrears, a final accrual
payment by the buyer is usually required. In our example, the buyer would
be required to pay to the seller the amount of the annual payment accrued
between March 1,2009, and June 1,2009 (approximately $225,000), but no
further payments would be required.
The total amount paid per year, as a percent of the notional principal.
to buy protection is known as the CDS spread. Several large banks are
Figure 13.1 Credit default swap.