Hull J.C. Risk management and Financial institutions

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Credit Risk Losses and Credit VaR

281

Example 12.2

Consider an interest rate swap where a bank is receiving fixed and paying

floating. The exposure on the swap at a particular future time t

maxfV(t

), 0]

where V(t

) is the value of the swap at time t

. This is the payoff from a swap

option. The variable v

is therefore the value today of a type of European swap

option exercisable at time t

The analyses we have presented assume that the probability of default is

independent of the value of the derivative. This is likely to be a reasonable

approximation in circumstances when the derivative is a small part of the

portfolio of the counterparty or when the counterparty is using the

derivative for hedging purposes. When a counterparty wants to enter into

a large derivatives transaction for speculative purposes, a financial institu-

tion should be wary. When the transaction has a large negative value for

the counterparty (and a large positive value for the financial institution),

the probability of the counterparty declaring bankruptcy may be much

higher than when the situation is the other way round.

Interest Rate Swaps vs. Currency Swaps

The impact of default risk on interest rate swaps is considerably less than

that on currency swaps. Figure 12.2 shows the reason for this. It compares

the expected exposure for a bank on a pair of offsetting interest rate swaps

Figure 12.2 Expected exposure on a matched pair of interest

rate swaps and a matched pair of currency swaps.

282 Chapter 12

with different counterparties to the expected exposure on a pair of

offsetting currency swaps with different counterparties. The expected

exposure on the interest rate swaps starts at zero, increases, and then

decreases to zero. By contrast, expected exposure on a the currency swaps

increases steadily with the passage of time.

The reason for the difference

is largely that principals are exchanged at the end of the life of a currency

swap and there is uncertainty about the exchange rate at that time.

Two-Sided Default Risk

One of the interesting aspects of contracts in the third category (i.e.,

contracts that can become assets or liabilities) is that there is two-sided

default risk. For example, if Company X enters into a currency swap with

Company Y, Company X may lose money if Company Y defaults and

vice versa. Human nature being what it is, most financial and nonfinan-

cial companies consider that there is no chance that they themselves will

default but want to make an adjustment to contract terms for a possible

default by their counterparty. In our example, Company X wants to be

compensated for the possibility that Company Y will default and Com-

pany Y wants to be compensated for the possibility that Company X will

default. This can make it very difficult for the two companies to agree on

terms and explains why it is difficult for financial institutions that are not

highly creditworthy to be active in the derivatives market.

In theory, the value of the currency swap to Company X should be

f-y+ x

where f is the value if both sides were default-free, y is the adjustment

necessary for the possibility of Company Y defaulting, and x is the

adjustment necessary for Company X itself defaulting. Similarly, the value

to Company Y should be

-f - x + y

In practice, Company X is likely to calculate a value of f — y and

Company Y is likely to calculate a value of — f — x. Unless x and y are

small, it is unlikely that the companies will be able to agree on a price.

Continuing with our currency swap example, suppose that interest rates

in the foreign and domestic currency are the same and that Company X

The expected exposure affects the v

in equation (12.1). The u

are the same for both

types of swaps.

Currency swaps with no final exchange of principal have become more common in

recent years. These have less credit risk and market risk than traditional currency swaps.

Credit Risk Losses and Credit VaR 283

and Company Y are equally creditworthy. The swap is symmetrical in the

sense that it has just as much chance of moving in the money for

Company X as for Company Y. In this case, x = y and neither side

should in theory make any adjustment for credit risk.

12.2 CREDIT RISK MITIGATION

In many cases the analysis we have just presented overstates the credit risk

in a derivatives transaction. This is because there are a number of clauses

that derivatives dealers include in their contracts to mitigate credit risk.

Netting

We discussed netting in Section 7.5. A netting clause in a derivatives

contract states that if a company defaults on one contract it has with a

counterparty then it must default on all outstanding contracts with the

counterparty. Netting has been successfully tested in the courts in most

jurisdictions and can substantially reduce credit risk for a financial

institution. We can extend the analysis presented in the previous section

so that equation (12.1) gives the present value of the expected loss from

all contracts with a counterparty when netting agreements are in place.

This is achieved by redefining v

in the equation as the present value of a

derivative that pays off the exposure at time t

on the portfolio of all

contracts with a counterparty.

A challenging task for a financial institution when considering whether

it should enter into a new derivatives contract with a counterparty is to

calculate the incremental effect on expected credit losses. This can be

done by using equation (12.1) in the way just described to calculate

expected default costs with and without the contract. It is interesting to

note that, because of netting, the incremental effect of a new contract on

expected default losses can be negative. This tends to happen when the

value of the new contract is highly negatively correlated with the value of

eisting contracts.

It might be thought that in well-functioning capital markets a company

wanting to enter into a derivatives transaction will get the same quote

from all dealers. Netting shows that this is not necessarily the case. The

company is likely to get the most favorable quote from a financial

institution it has done business with in the past—particularly if that

business gives rise to exposures for the financial institution that are

opposite to the exposure generated by the new transaction.

284 Chapter 12

Collateralization

Another clause frequently used to mitigate credit risks is known as

Collateralization. Suppose that a company and a financial institution have

entered into a number of derivatives contracts. A typical Collateralization

agreement specifies that the contracts be marked to market periodically

using a pre-agreed formula. If the total value of the contracts to the

financial institution is above a certain threshold level on a certain day, it

can ask the company to post collateral. The amount of collateral posted,

when added to collateral already posted, by the company is equal to the

difference between the value of the contracts to the financial institution

and the threshold level. When the contracts move in favor of the com-

pany, so that the difference between value of the contracts to the financial

institution and the threshold level is less than the total margin already

posted, the company can reclaim margin. In the event of a default by the

company, the financial institution can seize the collateral. If the company

does not post collateral as required, the financial institution can close out

the contracts. Long-Term Capital Management made extensive use of

Collateralization agreements (see Business Snapshot 12.1).

Suppose, for example, that the threshold level for the company is

$10 million and contract is marked to market daily for the purposes of

Collateralization. If on a particular day the value of the contract to the

financial institution is $10.5 million, it can ask for $0.5 million of col-

lateral. If on the next day the value of the contract rises to $11.4 million, it

can ask for a further $0.9 million of collateral. If the value of the contract

falls to $10.8 million on the following day, the company can ask for

$0.6 million of the collateral to be returned. Note that the threshold

($10 million in this case) can be regarded as a line of credit that the

financial institution is prepared to grant to the company.

Collateral must be deposited by the company with the financial institu-

tion in cash or in the form of acceptable securities such as bonds. Interest

is normally paid on cash. The securities are subject to a discount known

as a haircut applied to their market value for the purposes of collateral

calculations.

If the Collateralization agreement is a two-way agreement, a threshold

will also be specified for the financial institution. The company can then

ask the financial institution to post collateral when the marked-to-market

value of the outstanding contracts to the company exceeds the threshold.

Collateralization agreements provide a great deal of protection against

the possibility of default (just as the margin accounts discussed in

Credit Risk Losses and Credit VaR 285

Chapter 2 provide protection for people who trade on an exchange).

However, the threshold amount is not subject to protection. Furthermore,

even when the threshold is zero, the protection is not total. When a

company gets into financial difficulties it is likely to stop responding to

requests to post collateral. By the time the counterparty exercises its right

to close out contracts, their value may have moved further in its favor.

Downgrade Triggers

Another credit mitigation technique used by a financial institution is

known as a downgrade trigger. This is a clause stating that if the credit

Business Snapshot 12.1 Long-Term Capital Management's Big Loss

Long-Term Capital Management (LTCM), a hedge fund formed in the mid-

1990s, always collateralized its transactions. The hedge fund's investment

strategy was known as convergence arbitrage. A very simple example of what

it might do is the following. It would find two bonds, X and Y, issued by the

same company, promising the same payoffs, with X being less liquid (i.e., less

actively traded) than Y. The market always places a value on liquidity. As a

result, the price of X would be less than the price of Y. LTCM would buy X and

short Y, and wait, expecting the prices of the two bonds to converge at some

future time.

When interest rates increased, the company expected both bonds to move

down in price by about the same amount, so that the collateral it paid on bond

X would be about the same as that which it received on bond Y. Similarly, when

interest rates decreased, LTCM expected both bonds to move up in price by

about the same amount, so that the collateral it received on bond X would be

about the same as that which it paid on bond Y. It therefore expected no

significant outflow of funds as a result of its Collateralization agreements.

In August 1998, Russia defaulted on its debt and this led to what is termed a

"flight to quality" in capital markets. One result was that investors valued

liquid instruments more highly than usual and the spreads between the prices

of the liquid and illiquid instruments in LTCM's portfolio increased dramatic-

ally. The prices of the bonds LTCM had bought went down and the prices of

those it had shorted increased. It was required to post collateral on both. The

company was highly leveraged and unable to make the payments required

under the Collateralization agreements. The result was that positions had to be

closed out and there was a total loss of about $4 billion. If the company had

been less highly leveraged, it would probably have been able to survive the

flight to quality and could have waited for the prices of the liquid and illiquid

bonds to become closer.

286 Chapter 12

rating of the counterparty falls below a certain level, say Baa, then the

financial institution has the option to close out a derivatives contract at its

market value. (A procedure for determining the value of the contract

must be agreed to in advance.)

Downgrade triggers do not provide protection from a big jump in a

company's credit rating (e.g., from A to default). Also, downgrade

triggers work well only when relatively little use is made of them. If a

company has entered into downgrade triggers with many counterparties'

they are liable to provide relatively little protection to the counterparties

(see Business Snapshot 12.2).

Business Snapshot 12.2 Downgrade Triggers and Enron's Bankruptcy

In December 2001, Enron, one of the largest companies in the United States,

went bankrupt. Right up to the last few days, it had an investment-grade credit

rating. The Moody's rating immediately prior to default was Baa3 and the

S&P rating was BBB—. The default was, however, anticipated to some extent

by the stock market because Enron's stock price fell sharply in the period

leading up to the bankruptcy. The probability of default estimated by models

such as the one described in Section 11.6 increased sharply during this period.

Enron had entered into a huge number of derivatives contracts with down-

grade triggers. The downgrade triggers stated that, if its credit rating fell below

investment grade (i.e., below Baa3/BBB—), its counterparties would have the

option of closing out contracts. Suppose that Enron had been downgraded to

below investment grade in, say, October 2001. The contracts that counter-

parties would choose to close out would be those with a negative values to

Enron (and positive values to the counterparties). As a result Enron would

have been required to make huge cash payments to its counterparties. It would

not have been able to do this and immediate bankruptcy would result.

This example illustrates that downgrade triggers provide protection only

when relatively little use is made of them. When a company enters into a huge

number of contracts with downgrade triggers, they may actual cause a com-

pany to go bankrupt prematurely. In Enron's case we could argue that it was

going to go bankrupt anyway and accelerating the event by two months would

not have done any harm. In fact, Enron did have a chance of survival in

October 2001. Attempts were being made to work out a deal with another

energy company, Dynergy, and so forcing bankruptcy in October 2001 was

not in the interests of either creditors or shareholders.

The credit rating companies found themselves in a difficult position. If they

downgraded Enron to recognize its deteriorating financial position, they were

signing its death warrant. If they did not do so, there was a chance of Enron

surviving.

Credit Risk Losses and Credit VaR 287

12.3 CREDIT VaR

Credit value at risk can be defined in the same way as we defined market

value at risk in Chapter 8. For example, a credit VaR with a confidence

level of 99.9% and a one-year time horizon is the credit loss that we are

99.9% confident will not be exceeded over one year. Whereas the time

horizon for market risk is usually between one day and one month that

for credit risk is usually much longer—often one year.

For regulatory purposes banks using the internal ratings based

approach must calculate credit VaR for items in the banking book using

the methodology prescribed by the Basel Committee. Banks are given

freedom in making their own estimates of probability of default, PD.

However, they must use the correlation model and correlation parameters

specified by the Basel Committee.

When calculating credit VaR for specific risk, banks have more free-

dom. Specific risk is the risk in the trading book that is related to the

credit quality of individual companies. Although standard rules for

determining specific risk have been specified, banks can, with regulatory

approval, use their own models. For a model to be approved, the bank

supervisor must be satisfied that concentration risk, spread risk, down-

grade risk, and default risk are appropriately captured. The capital charge

for specific risk is the product of a multiplier and the ten-day 99% VaR,

with the minimum level for the multiplier being 4.

12.4 VASICEK'S MODEL

Vasicek's model, which we discussed in Sections 6.5 and 7.8, provides an

easy way to estimate credit VaR for a loan portfolio. From

equation (6.12), there is a probability X that the percentage of defaults

on a large portfolio by time T is, in a one-factor Gaussian copula model,

less than

where Q(T) is the cumulative probability of each loan defaulting by time T

and is the copula correlation. When multiplied by the average exposure

per loan and by the average loss given default, this gives the T-year VaR for

As mentioned earlier, the probability of default in credit VaR calculations should be a

real-world probability of default, not a risk-neutral probability of default.

288 Chapter 12

an X% confidence level. As explained in Chapter 7, the Basel Committee

has based the capital it requires for credit risk in the banking book on this

model. The time horizon T is one year, the confidence level X is 99.9%

and the value of is determined by the Basel Committee. As we saw in

Chapter 7, in some cases depends on the one-year probability of default.

12.5 CREDIT RISK PLUS

In 1998 Credit Suisse Financial Products proposed a methodology for

calculating VaR that it termed Credit Risk Plus.

It utilizes ideas that are

well established in the insurance industry. We will present a simplified

version of the approach.

Suppose that a financial institution has N counterparties of a certain

type and the probability of default by each counterparty in time T is p.

The expected number of defaults, for the whole portfolio is given by

Assuming that default events are independent and that p is

small, the probability of n defaults is given by the Poisson distribution as

This can be combined with a probability distribution for the losses

experienced on a single counterparty default (taking account of the

impact of netting) to obtain a probability distribution for the total default

losses from the counterparties. To estimate the probability distribution

for losses from a single counterparty default, we can look at the current

probability distribution of our exposures to counterparties and adjust this

according to historical recovery rates.

In practice, it is likely to be necessary for the financial institution to

consider several categories of counterparties. This means that the analysis

just described must be done for each category and the results combined.

Another complication is that default rates vary significantly from year to

year. Data provided by Moody's show that the default rate per year for all

bonds during the 1970 to 1999 period ranged from 0.09% in 1979 to 3.52%

in 2001. To account for this, we can assume a probability distribution for

the overall default rate based on historical data such as that provided by

See Credit Suisse Financial Products, "Credit Risk Management Framework,

October, 1997.

Credit Risk Losses and Credit VaR 289

Figure 12.3 Probability distribution of default losses.

Moody's. The probability of default for each category of counterparty can

then be assumed to be linearly dependent on the overall default rate.

Credit Suisse Financial Products show that, if certain assumptions are

made, the total loss probability distribution can be calculated analytically.

To accommodate more general assumptions, Monte Carlo simulation can

be used. The procedure is as follows:

1. Sample an overall default rate.

2. Calculate a probability of default for each category.

3. Sample a number of defaults for each category.

4. Sample a loss for each default.

5. Calculate total loss.

6. Repeat Steps 1 to 5 many times.

The effect of assuming a probability distribution for default rates in the

way just described is to build in default correlations. This makes the

Probability distribution of total default losses positively skewed, as

indicated in Figure 12.3.

12-6 CREDITMETRICS

Credit Risk Plus estimates the probability distribution of losses arising

from defaults. As indicated earlier, regulators like the internal models

used for specific risk to reflect losses from downgrades and credit spread

290

Chapter 12

changes as well as defaults. This has led many banks to implement

CreditMetrics for specific risk calculations.

CreditMetrics was proposed by J.P. Morgan in 1997. It is based on an

analysis of credit migration. This is the probability of a company moving

from one rating category to another during a certain period of time. A

typical one-year ratings transition matrix is shown in Table 12.1. \\

indicates that the percentage probability of a bond moving from one

rating category to another during a one-year period. For example, a bond

that starts with an A credit rating has a 91.83% chance of still having an

A rating at the end of one year. It has a 0.02% chance of defaulting

during the year, a 0.13% chance of dropping to B, and so on.

Consider a bank with a portfolio of corporate bonds. Calculating a one-

year VaR for the portfolio using CreditMetrics involves carrying out

Monte Carlo simulation of ratings transitions for bonds in the portfolio

over a one-year period. On each simulation trial the final credit rating of all

bonds is calculated and the bonds are revalued to determine total credit

losses for the year. The 99% worst result is the one-year 99% VaR for the

portfolio.

It is interesting to note that if both the CreditMetrics and the Credit

Risk Plus models are accurate, they should predict the same probability

distribution for losses over the long term. It is the timing of losses that is

different. Suppose, for example, that you hold a certain bond in your

portfolio. In year 1 it gets downgraded from A to BBB; in year 2 it gets

downgraded from BBB to B; in year 3 it defaults. You could assume that

there are no losses in years 1 and 2 and calculate the loss in year 3 (the

Credit Risk Plus approach). Alternatively, you can calculate separate

Table 12.1 One-year ratings transition matrix (probabilities expressed as

percentages). From results reported by Moody's in 2004.

Initial

rating

Aaa

Baa

Caa

Default

Rating at year-end

Aaa

92.18

1.17

0.05

0.01

0.00

7.06

90.85

2.39

0.24

0.05

0.03

0.00

0.73

7.63

91.83

5.20

0.50

0.13

0.00

Baa

0.00

0.26

5.07

88.48

5.45

0.43

0.58

0.00

0.02

0.07

0.50

4.88

85.13

6.52

1.74

0.00

0.01

0.13

0.80

7.05

83.20

4.18

0.00

Caa

0.00

0.01

0.16

0.55

3.04

67.99

0.00

Default

0.00

0.02

0.18

1.2?

6.64

25.50

100.00