Hull J.C. Risk management and Financial institutions

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Answers to Problems and Questions 437

For example, the Student t-copula gives more tail dependence than the

Gaussian copula.

6.11. Sample from a bivariate Student t-distribution as in Figure 6.5.

Convert each sample to a normal distribution on a "percentile-to-

percentile" basis.

6.12. The probability that V

< 0.1 is 0.05. The conditional probability

that V

<0.1 is 0.006/0.05 = 0.12. The conditional probability that

< 0.2 is 0.017/0.05 = 0.34, etc.

6.13. When V

= 0.2, we have U

= —0.84. From the properties of the

bivariate normal distribution, the median of U

is —0.5 x 0.84 = —0.42.

This translates into a median value for V

of 0.458.

6.14. In this case,

The "99.5% worst case" is that there is a loss of 500 x 0.7 x 0.127 =

44.62, or $44.62 million.

CHAPTER 7

7.1. The removal of a competitor may be beneficial. However, banks

enter into many contracts with each other. When one bank goes bank-

rupt, other banks are liable to lose money on the contracts they have with

the bank. Also, other banks will be adversely affected if the bankruptcy

reduces the public's overall level of confidence in the banking system.

7.2. Deposit insurance means that depositors are safe regardless of the

risks taken by their financial institution. It is liable to lead to financial

institutions taking more risks than they otherwise would because they can

do so without the risk of losing deposits. This in turn leads to more bank

failures and more claims under the deposit insurance system. Regulation

requiring the capital held by a bank to be related to the risks taken is

necessary to avoid this happening.

7.3. The credit risk on the swap is the risk that the counterparty defaults

at some future time when the swap has a positive value to the bank.

7.4. The value of a currency swap is liable to deviate further from zero

than the value of an interest rate swap because of the final exchange of

principal. As a result the potential loss from a counterparty default is

higher.

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7.5. There is some exposure. If the counterparty defaulted now there

would be no loss. However, interest rates could change so that at a future

time the swap has a positive value to the financial institution. If the

counterparty defaulted at that time there would be a loss to the financial

institution.

7.6. The risk-weighted assets for the three transactions are (a) $1.875

million, (b) $2 million, and (c) $3 million, for a total of $6.875 million.

Capital is 0.08 x 6.875, or $0.55 million.

7.7. The NRR is 2.5/4.5 = 0.556. The credit equivalent amount is

2.5+ (0.4 +0.6 x 0.556) x 9.25, or $9.28 million. The risk-weighted

assets is $4.64 million and the capital required is $0.371 million.

7.8. In this case there is no value to the netting provisions.

7.9. This converts the estimated capital requirement to an estimated risk-

weighted assets. Capital required equals 8% of risk-weighted assets.

7.10. The trading book consists of instruments that are actively traded

and marked to market daily. The banking book consists primarily of

loans and is not market to market daily. Prior to the change the bank

keeps credit risk capital calculated according to Basel I or Basel II. The

effect of the change is to move the clients borrowings from the banking

book to the trading book. The bank will be required to hold specific risk

capital for the securities reflecting the credit exposure, as well as market

risk capital reflecting the market risk exposure. The previous credit risk

capital is no longer required.

7.11. Under Basel I the capital charged for lending to a corporation is the

same regardless of the credit rating of the corporation. This leads to a

bank's return on capital being relatively low for lending to highly credit-

worthy corporations. Under Basel II the capital requirements of a loan are

tied much more carefully to the creditworthiness of the borrower. As a

result lending to highly creditworthy companies may become attractive

again.

7.12. Regulatory arbitrage involves entering into a transaction or series

of transactions solely to reduce regulatory capital requirements.

7.13. EAD is the estimated exposure at default. LGD is the loss given

default, that is, the proportion of the exposure that will be lost if a default

occurs. WCDR is the one-year probability of default in a bad year that

occurs only one time in 1,000. PD is the probability of default in an

average year. MA is the maturity adjustment. The latter allows for the fact

that in the case of instruments lasting longer than a year there may be

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439

losses arising from a decline in the creditworthiness of the counterparty

during the year as well as from a default during the year.

7.14. Under the simple approach, the risk weight of the counterparty is

replaced by the risk weight of the collateral for the part of the exposure

covered by the collateral. Under the comprehensive approach, the exposure

is adjusted for possible increases and the collateral is adjusted for possible

decreases in value. The counterparty's risk weight is applied to the excess

of the adjusted exposure over the adjusted collateral.

7.15. The standardized approach uses external ratings to determine

capital requirements (but in a more sophisticated way than in Basel I).

In the IRB approach the Basel II correlation model is used with PD being

determined by the bank. In the advanced IRB approach, the Basel II

correlation model is used with PD, LGD, EAD, and MA being deter-

mined by the bank.

7.16. In the basic indicator approach total capital is 15% of the average

total annual gross income. In the standardized approach, gross income is

calculated for different business lines and capital as a percentage of gross

income is different for different business lines. In the advanced measure-

ment approach, the bank uses internal models to determine the 1-year

99.9% VaR.

7.17. =0.1216, WCDR = 0.0914, and the capital requirement is

200 x 0.7 x 0.0814, or $11.39 million. At least half of this must be Tier I.

CHAPTER 8

8.1. VaR is the loss that is not expected to be exceeded with a certain

confidence level. Expected shortfall is the expected loss conditional that

the loss is worse than the VaR level. Expected shortfall has the

advantage that it always satisfies the Subadditivity (diversification is

good) condition.

8.2. A spectral risk measure is a risk measure that assigns weights to the

quantiles of the loss distribution. For the Subadditivity condition to be

satisfied the weight assigned to the qth quantile must be a nondecreasing

function of q.

8.3. There is a 5% chance that you will lose $6,000 or more during a one-

month period.

8.4. Your expected loss during a "bad month" is $6,000. Bad months are

defined as the worst 5% of months.

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8.5. (a) $1 million, (b) $9.1 million, (c) $11 million, (d) $11.07 million

(e) 1 + 1 < 11 but 9.2 + 9.2 > 11.07.

8.6. (a) $3.92 million, (b) $8.77 million, (c) $10.40 million.

8.7. (b) becomes $9.96 million and (c) becomes $11.82 million.

8.8. Marginal VaR is the rate of change of VaR with the amount invested

in the ith asset. Incremental VaR is the incremental effect of the ith asset

on VaR (i.e., the difference between VaR with and without the asset).

Component VaR is the part of VaR that can be attributed the ith asset

(the sum of component VaRs equals the total VaR).

8.9. The probability of 17 or more exceptions is

1 - BINOMDIST(16,1000,0.01,TRUE)

or 2.64%. The model should be rejected at the 5% confidence level.

8.10. Bunching is the tendency for exceptions to be bunched rather than

occurring randomly throughout the time period considered.

8.11. Either historical data or brainstorming by senior management can

be used to develop extreme scenarios.

8.12. We are interested in the standard deviation of R

+ R

+ ... + R

where R

is the return on day i. This is where

is the standard deviation of R

and is the correlation between R

and

Rj. In this case, for all i, and when i > j. After further

algebraic manipulations this leads directly to equation (8.3).

8.13. The probability of 5 or more exceptions is

1 - BINOMDIST(4,250,0.01,TRUE)

or 10.8%.

CHAPTER 9

9.1. The assumption is that the statistical process driving changes in

market variables over the next day is the same as that over the last 500 days.

9.2.

This shows that, as approaches 1, the weights approach \/n.

9.3. The tenth-worst outcome is a return of —3.78%. The estimate of the

1-day 99% VaR is therefore $3.78 million.

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9.4. The 1-day 99% VaR is estimated as $1.36 million. This is much less

than that given in Problem 9.3 because most of the really bad returns were

more than 500 days ago and carry relatively little weight.

9.5. The volatility is initially 1.14% per day. It varies from 0.428% per

day to 2.97% per day. After adjustment, the tenth-worst outcome is

1.77%. The VaR estimate is therefore $1.77 million.

9.6. The VaR estimates given by u equal to 0.005, 0.01, and 0.015 are

3.34, 3.34, and 3.30, respectively.

9.7. The standard error of the VaR estimate is

or $0.69 million.

CHAPTER 10

10.1. The standard deviation of the daily change in the investment in

each asset is $1,000. The variance of the portfolio's daily change is

1,000

+ 1,000

+ 2 x 0.3 x 1,000 x 1,000 = 2,600,000

The standard deviation of the portfolio's daily change is the square root

of this, or $1,612.45. The 5-day 99% VaR is therefore

2.33 x x 1,612.45 = $8,401

10.2. The three alternative procedures mentioned in the chapter for

handling interest rates when the model-building approach is used to

calculate VaR involve (a) the use of the duration model, (b) the use of

cash-flow mapping, and (c) the use of principal components analysis.

10.3. When a final exchange of principal is added in, the floating side is

equivalent to a zero-coupon bond with a maturity date equal to the date

of the next payment. The fixed side is a coupon-bearing bond, which is

equivalent to a portfolio of zero-coupon bonds. The swap can therefore

be mapped into a portfolio of zero-coupon bonds with maturity dates

corresponding to the payment dates. Each of the zero-coupon bonds can

then be mapped into positions in the adjacent standard-maturity zero-

coupon bonds.

10.4. = 56 x The standard deviation of is 56 x 1.5 x

0.007 = 0.588. It follows that the 10-day 99% VaR for the portfolio is

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Answers to Problems and Questions

10.6. The 6. 5-year cash flow is equivalent to a position of $48.56 in a

5-year zero-coupon bond and a position of $605.49 in a 7-year zero-

coupon bond. The equivalent 5-year and 7-year cash flows are

48.56 x 1.06

= 64.98 and 605.49 x 1.07

= 972.28.

10.7. A similar calculation to that in the text shows that $37,397 of the

value is allocated to the 3-month bond worth and $11,793 of the value is

allocated to the 6-month bond.

10.8. The daily variance of the portfolio is

x 20

+ 4

x 8

= 15,424

and the daily standard deviation is = $124.19. Since

N(-1.282) = 0.9, the 5-day 90% VaR is

124.19 x x 1.282 = $356.01

10.9. (a) 3.26, (b) 63.87.

10.10. The delta of the options is the rate of change of the value of the

options with respect to the price of the asset. When the asset price

increases by a small amount, the value of the options decreases by 30

times this amount. The gamma of the options is the rate of change of

their delta with respect to the price of the asset. When the asset price

increases by a small amount, the delta of the portfolio decreases by five

times this amount.

In this case, = -0.10, = 36.03, and = -32.415.

The mean change in the portfolio value in 1 day is —0.1 and the standard

deviation of the change in 1 day is = 6.002. The Skewness is

Using only the first two moments, we find that the 1-day 99% VaR is

$14.08. When three moments are considered in conjunction with a

Cornish-Fisher expansion, it is $14.53.

10.11. Define as the volatility per year, as the change in in 1 day,

and as the proportional change in in 1 day. We measure in as a

multiple of 1% so that the current value of is 1 x = 15.87. The

delta-gamma-vega model is

Answers to Problems and Questions 443

where which simplifies to

The change in the portfolio value now depends on two market variables.

Once the daily volatility of and the correlation between a nd S have

been estimated, we can estimate moments of and use a Cornish-

Fisher expansion.

10.12. The change in the value of an option is not linearly related to the

change in the value of the underlying variables. When the change in the

values of underlying variables is normal, the change in the value of the

option is not normal. The linear model assumes that it is normal and is,

therefore, only an approximation.

10.13. The contract is a long position in a sterling bond combined with a

short position in a dollar bond. The value of the sterling bond is

or $1.492 million. The value of the dollar bond is

or $1.463 million. The variance of the change in the value

of the contract in 1 day is

1.492

x 0.0006

+ 1.463

x 0.0005

- 2 x 0.8 x 1.492 x 0.0006 x 1.463 x 0.0005 = 0.000000288

The standard deviation is therefore $0.000537 million. The 10-day 99%

VaR is 0.000537 x x 2.33 = $0.00396 million.

CHAPTER 11

11.1 Moody's has 19 ratings (excluding the "in default" rating): Aaa,

Aal, Aa2, Aa3, Al, A2, A3, Baal, Baa2, Baa3, Bal, Ba2, Ba3, Bl, B2,

B3, Caal, Caa2, Caa3.

11.2. S&P has 19 ratings (excluding the "in default" rating): AAA, AA+,

AA, AA-, A+, A, A-, BBB+, BBB, BBB-, BB+, BB, B-, B+, B, B-,

CCC+, CCC, CCC-.

11.3. Average default intensity is where = 0.9379. It is 6.41% per

year.

11.4. Conditional on no default by year 2, the probability of no default

by year 3 is 0.8926/0.9526 = 0.9370. Average default intensity for the third

year is where = 0.9370. It is 6.51% per year.

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Answers to Problems and Questions

11.5. From equation (11.3) the average default intensity over the 3 years

is 0.0050/(1 - 0.3) = 0.0071, or 0.71% per year.

11.6. From equation (11.3) the average default intensity over 5 years is

0.0080/(1-0.4), or 1.333% per year. Similarly the average default

intensity over 3 years is 1.1667% per year. This means that the average

default intensity for years 4 and 5 is 1.58%.

11.7. Real-world probabilities of default should be used for calculating

credit value at risk. Risk-neutral probabilities of default should be used

for adjusting the price of a derivative for default.

11.8. The recovery rate for a bond is the value of the bond immediately

after the issuer defaults as a percentage of its face value.

11.9. The first number in the second column of Table 11.4 is calculated as

or 0.04% per year. Other numbers in the column are calculated similarly.

The numbers in the fourth column of Table 11.5 are the numbers in the

second column of Table 11.4 multiplied by one minus the expected

recovery rate. In this case, the expected recovery rate is 0.4.

11.10. The bond's market value is 96.19. Its risk-free value is 103.66. If Q

is the default probability per year, the loss from defaults is 272.69Q. The

implied probability of default is therefore 2.74% per year.

11.11. The market price of the first bond is 98.35 and its risk-free value is

101.23. If Q

is the default probability at times 0.5, 1.5, and 2.5 years the

loss from defaults for the first bond is 178.31Q

. It follows that

=0.0161. If Q

is the probability of default at times 3.5 and 4.5,

the loss from default from the second bond is 180.56Q

+ 108.53Q

. The

market price for the second bond is 96.24 and its risk-free value is 101.97.

It follows that 180.56Q

+ 108.5302 = 5.73 and Q

= 0.0260.

11.12. We can assume that the principal is paid and received at the end of

the life of the swap without changing the swap's value. If the spread were

zero, then the present value of the floating payments per dollar of principal

would be 1. The payment of LIBOR plus the spread therefore has a present

value of 1 + V. The payment of the bond cash flows has a present value per

dollar of principal of B*. The initial payment required from the payer of

the bond cash flows per dollar of principal is 1 — B. (This may be negative;

an initial amount of B — 1 is then paid by the payer of the floating rate.)

Because the asset swap is initially worth zero, we have

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so that

V = B* -B

11.13. The value of the debt in Merton's model is V

— E

, or

If the credit spread is s, this should equal so that

Substituting we get

so that

11.14. In this case, E

= 2, = 0.50, D = 5, r = 0.04, and T=1.

Solving the simultaneous equations gives V

= 6.80 and = 14.82.

The probability of default is N(-d

), or 1.15%.

11.15. Suppose that the principal is $100. The asset swap is structured so

that the $10 is paid initially. After that, $2.50 is paid every 6 months. In

return, LIBOR plus a spread is received on the principal of $100. The

present value of the fixed payments is

The spread over LIBOR must therefore have a present value of 5.3579.

The present value of $1 received every 6 months for 5 years is 8.5105. The

spread received every 6 months must therefore be

5.3579/8.5105 = $0.6296

The asset swap spread is therefore 2 x 0.6296 = 1.2592% per annum.

CHAPTER 12

12.1. The new transaction will increase the bank's exposure to the

counterparty if it tends to have a positive value whenever the existing

contract has a positive value and a negative value whenever the existing

contract has a negative value. However, if the new transaction tends to

offset the existing transaction, it is likely to have the incremental effect of

reducing credit risk.

12.2. Equation (12.3) gives the relationship between and This

involves Q

(T) and Q

(T). These change as we move from the real world

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to the risk-neutral world. It follows that the relationship between

and in the real world is not the same as in the risk-neutral world. If

is the same in the two worlds, then is not, and vice versa.

12.3. When securities are pledged as collateral, the haircut is the discount

applied to their market value for margin calculations. A company's own

equity would not be good collateral. When the company defaults on its

contracts, its equity is likely to be worth very little.

12.4. In Vasicek's model and Credit Risk Plus, a credit loss recognized

when a default occurs. In CreditMetrics, both downgrades and defaults

lead to credit losses. In Vasicek's model, a Gaussian copula model of time

to default is used. In Credit Risk Plus, a probability distribution is

assumed for the default rate per year. In CreditMetrics, a Gaussian

copula model is used to define rating transitions.

12.5. The binomial correlation measure is 0.156.

12.6. The statements in (a) and (b) are true. The statement in (c) is not.

Suppose that and are the exposures to X and Y. The expected value

of is the expected value of plus the expected value of The

same is not true of 95% confidence limits.

12.7. The cost of defaults is where u is percentage loss from defaults

during the life of the contract and v is the value of an option that pays off

max(150S

- 100, 0), where S

is the AUD/USD exchange rate in 1 year

(USD per AUD). The value of u is

The variable v is 150 times the value of a call option to buy 1 AUD for

0.6667. This is 4.545. It follows that the cost of defaults is

4.545 x 0.009950, or 0.04522.

12.8. In this case, the cost of defaults is where

is the value of a 6-month call option on 150 AUD with a strike price of

$100, and is the value of a similar 1-year option. = 3.300 and

= 4.545. The cost of defaults is 0.04211.

12.9. Assume that defaults happen only at the end of the life of the

forward contract. In a default-free world, the forward contract is the

combination of a long European call and a short European put where the

strike price of the options equals the delivery price and the maturity of the