**Solved by: AllAcademicHelp.com**

Chapter Ten

CookMyProject

Market Risk

Chapter Outline

Introduction

Market Risk Measurement

Calculating Market Risk Exposure

The RiskMetrics Model

The

Market Risk of Fixed-Income Securities

Foreign

Exchange

Equities

Portfolio

Aggregation

Historic or Back Simulation

The

Historic (Back Simulation) Model versus RiskMetrics

The

Monte Carlo Simulation Approach

Regulatory Models: The BIS

Standardized Framework

Fixed

Income

Foreign

Exchange

Equities

The BIS Regulations and Large Bank

Internal Models

Summary

Solutions for End-of-Chapter Questions and Problems: Chapter Ten

1. What is meant by market risk?

Market

risk is the uncertainty of the effects of changes in economy-wide

systematic factors that affect earnings and stock prices of different

firms in a similar manner. Some of these market-wide risk factors

include volatility, liquidity, interest-rate and inflationary

expectation changes.

2. Why is the measurement of market risk important to the manager of

a financial institution?

Measurement of market risk can help an FI manager in the following

ways:

a. Provide information on the risk positions taken by individual

traders.

b. Establish limit positions on each trader based on the market

risk of their portfolios.

c. Help allocate resources to departments with lower market risks

and appropriate returns.

d. Evaluate performance based on risks undertaken by traders in

determining optimal bonuses.

e. Help develop more efficient internal models so as to avoid using

standardized regulatory models.

3. What is meant by daily earnings at risk (DEAR)?

What are the three measurable components? What is the price

volatility component?

DEAR or Daily Earnings at Risk is defined as the estimated potential

loss of a portfolio’s value over a one-day unwind period as a result

of adverse moves in market conditions, such as changes in interest

rates, foreign exchange rates, and market volatility. DEAR is

comprised of (a) the dollar value of the position, (b) the price

sensitivity of the assets to changes in the risk factor, and (c) the

adverse move in the yield. The product of the price sensitivity of

the asset and the adverse move in the yield provides the price

volatility component.

4. Follow Bank has a $1 million position in a five-year, zero-coupon

bond with a face value of $1,402,552. The bond is trading at a yield

to maturity of 7.00 percent. The historical mean change in daily

yields is 0.0 percent, and the standard deviation is 12 basis points.

a. What is the modified duration of the bond?

MD = 5 ÷ (1.07) = 4.6729 years

b. What is the maximum adverse daily yield move given that we desire

no more than a 5 percent chance that yield changes will be greater

than this maximum?

Potential adverse move in yield at 5 percent = 1.65

= 1.65 x 0.0012 = .001980

c. What is the price volatility of this bond?

Price volatility = -MD x potential adverse move in yield

= -4.6729 x .00198 = -0.009252 or -0.9252 percent

d. What is the daily earnings at risk for this bond?

DEAR = ($ value of position) x (price volatility)

= $1,000,000 x 0.009252 = $9,252

5. What is meant by value at risk (VAR)? How is VAR

related to DEAR in J.P. Morgan’s RiskMetrics model? What would be

the VAR for the bond in problem (4) for a 10-day period? With what

statistical assumption is our analysis taking liberties? Could this

treatment be critical?

Value at Risk or VAR is the cumulative DEARs over a specified period

of time and is given by the formula VAR = DEAR x [N]½.

VAR is a more realistic measure if it requires a longer period to

unwind a position, that is, if markets are less liquid. The value

for VAR in problem four above is $9,252 x 3.1623 = $29,257.39.

The relationship according to the above formula assumes that the

yield changes are independent. This means that losses incurred on one

day are not related to the losses incurred the next day. However,

recent studies have indicated that this is not the case, but that

shocks are autocorrelated in many markets over long periods of time.

6. The DEAR for a bank is $8,500. What is the VAR for a 10-day

period? A 20-day period? Why is the VAR for a 20-day period not

twice as much as that for a 10-day period?

For the 10-day period: VAR = 8,500 x [10]½ = 8,500 x

3.1623 = $26,879.36

For the 20-day period: VAR = 8,500 x [20]½ = 8,500 x

4.4721 = $38,013.16

The reason that VAR20

(2 x VAR10) is because [20]½

(2 x [10]½). The interpretation is that the daily

effects of an adverse event become less as time moves farther away

from the event.

7. The mean change in the daily yields of a 15-year, zero-coupon bond

has been five basis points (bp) over the past year with a standard

deviation of 15 bp. Use these data and assume the yield changes are

normally distributed.

a. What is the highest yield change expected if a 90 percent

confidence limit is required; that is, adverse moves will not occur

more than one day in 20?

If yield changes are normally distributed, 90 percent of the area of

a normal distribution will be 1.65 standard deviations (1.65)

from the mean for a one-tailed distribution. In this example, it

means 1.65 x 15 = 24.75 bp. Thus, the maximum adverse yield change

expected for this zero-coupon bond is an increase of 24.75 basis

points in interest rates.

b. What is the highest yield change expected if a 95 percent

confidence limit is required?

If a 95 percent confidence limit is required, then 95 percent of the

area will be 1.96 standard deviations (1.96)

from the mean. Thus, the maximum adverse yield change expected for

this zero-coupon bond is an increase of 29.40 basis points (1.96 x

15) in interest rates.

8. In what sense is duration a measure of market risk?

The market risk calculations typically are based on the trading

portion of an FIs fixed-rate asset portfolio because these assets

must reflect changes in value as market interest rates change. As

such, duration or modified duration provides an easily measured and

usable link between changes in the market interest rates and the

market value of fixed-income assets.

9. Bank Alpha has an inventory of AAA-rated, 15-year zero-coupon

bonds with a face value of $400 million. The bonds currently are

yielding 9.5% in the over-the-counter market.

a. What is the modified duration of these bonds?

Modified duration = (MD) = D/(1 + r) = 15/(1.095) = -13.6986.

b. What is the price volatility if the potential adverse move in

yields is 25 basis points?

Price volatility = (-MD) x (potential adverse move in yield)

= (-13.6986) x (.0025) = -0.03425 or -3.425 percent.

c. What is the DEAR?

Daily earnings at risk (DEAR) = ($ Value of position) x (Price

volatility)

Dollar value of position = 400/(1 + 0.095)15 =

$102.5293million. Therefore,

DEAR = $102.5293499 million x -0.03425 = -$3.5116 million, or

-$3,511,630.

d. If the price volatility is based on a 90 percent confidence limit

and a mean historical change in daily yields of 0.0 percent, what is

the implied standard deviation of daily yield changes?

The potential adverse move in yields (PAMY) = confidence limit value

x standard deviation value. Therefore, 25 basis points = 1.65 x ,

and = .0025/1.65 = .001515 or

15.15 basis points.

10. Bank Two has a portfolio of bonds with a market value of $200

million. The bonds have an estimated price volatility of 0.95

percent. What are the DEAR and the 10-day VAR for these bonds?

Daily earnings at risk (DEAR) = ($ Value of position) x (Price

volatility)

= $200 million x .0095

= $1.9million, or $1,900,000

Value at risk (VAR) = DEAR x N

= $1,900,000 x 10

= $1,900,000 x 3.1623 = $6,008,327.55

11. Bank of Southern Vermont has determined that its inventory of 20

million euros (€) and 25 million British pounds (£) is subject to

market risk. The spot exchange rates are $0.40/€ and $1.28/£,

respectively. The ’s of the

spot exchange rates of the € and £, based on the daily changes of

spot rates over the past six months, are 65 bp and 45 bp,

respectively. Determine the bank’s 10-day VAR for both currencies.

Use adverse rate changes in the 95th percentile.

FX position of € = 20m x 0.40 = $8 million

FX position of £ = 25m x 1.28 = $32 million

FX volatility € = 1.65 x 65bp = 107.25, or 1.0725%

FX volatility £ = 1.65 x 45bp = 74.25, or 0.7425%

DEAR = ($ Value of position) x (Price volatility)

DEAR of € = $8m x .010725 = $0.0860m, or $85,800

DEAR of £ = $32m x .007425 = $0.2376m, or $237,600

VAR of € = $138,000 x 10 =

$85,800 x 3.1623 = $271,323.42

VAR of £ = $237,600 x 10 =

$237,600 x 3.1623 = $751,357.17

12. Bank of Alaska’s stock portfolio has a market value of

$10,000,000. The beta of the portfolio approximates the market

portfolio, whose standard deviation (m)

has been estimated at 1.5 percent. What is the 5-day VAR of this

portfolio, using adverse rate changes in the 99th

percentile?

DEAR = ($ Value of portfolio) x (2.33 x m) = $10m x (2.33 x .015)

= $10m x .03495 = $0.3495m or $349,500

VAR = $349,500 x 5 = $349,500 x

2.2361 = $781,505.76

Jeff Resnick, vice president of operations of Choice Bank, is

estimating the aggregate DEAR of the bank’s portfolio of assets

consisting of loans (L), foreign currencies (FX), and common stock

(EQ). The individual DEARs are $300,700, $274,000, and $126,700

respectively. If the correlation coefficients ij

between L and FX, L and EQ, and FX and EQ are 0.3, 0.7, and 0.0,

respectively, what is the DEAR of the aggregate portfolio?

14. Calculate the DEAR for the following portfolio with and without

the correlation coefficients.

Estimated

Assets DEAR S,FXS,BFX,B

Stocks (S) $300,000 -0.10 0.75 0.20

Foreign Exchange (FX) $200,000

Bonds (B) $250,000

What is the amount of risk reduction resulting from the lack of

perfect positive correlation between the various assets groups?

The DEAR for a portfolio with perfect correlation would be $750,000.

Therefore the risk reduction is $750,000 – $559,464 = $190,536.

15. What are the advantages of using the back simulation approach to

estimate market risk? Explain how this approach would be

implemented.

The advantages of the back simulation approach to estimating market

risk are that (a) it is a simple process, (b) it does not require

that asset returns be normally distributed, and (c) it does not

require the calculation of correlations or standard deviations of

returns. Implementation requires the calculation of the value of the

current portfolio of assets based on the prices or yields that were

in place on each of the preceding 500 days (or some large sample of

days). These data are rank-ordered from worst case to best and

percentile limits are determined. For example, the five percent

worst case provides an estimate with 95 percent confidence that the

value of the portfolio will not fall more than this amount.

16. Export Bank has a trading position in Japanese Yen and Swiss

Francs. At the close of business on February 4, the bank had

¥300,000,000 and Sf10,000,000. The exchange rates for the most

recent six days are given below:

Exchange Rates per U.S. Dollar at the Close of Business

2/4 2/3 2/2 2/1 1/29

1/28

Japanese Yen 112.13 112.84 112.14 115.05 116.35 116.32

Swiss Francs 1.4140 1.4175 1.4133 1.4217 1.4157 1.4123

a. What is the foreign exchange (FX) position in dollar equivalents

using the FX rates on February 4?

Japanese Yen: ¥300,000,000/¥112.13 = $2,675,465.98

Swiss Francs: Swf10,000,000/Swf1.414 = $7,072,135.78

b. What is the definition of delta as it relates to the FX position?

Delta measures the change in the dollar value of each FX position if

the foreign currency depreciates by 1 percent against the dollar.

c. What is the sensitivity of each FX position; that is, what is the

value of delta for each currency on February 4?

Japanese Yen: 1.01 x current exchange rate = 1.01 x ¥112.13 =

¥113.2513/$

Revalued position in $s = ¥300,000,000/113.2513 = $2,648,976.21

Delta of $ position to Yen = $2,648,976.21 – $2,675,465.98

= -$26,489.77

Swiss Francs: 1.01 x current exchange rate = 1.01 x Swf1.414 =

Swf1.42814

Revalued position in $s = Swf10,000,000/1.42814 = $7,002,114.64

Delta of $ position to Swf = $7,002,114.64 – $7,072,135.78

= -$70,021.14

d. What is the daily percentage change in exchange rates for each

currency over the five-day period?

Day Japanese Yen: Swiss Franc

2/4 -0.62921% -0.24691% % Change = (Ratet/Ratet-1)

– 1 * 100

2/3 0.62422% 0.29718%

2/2 -2.52934% -0.59084%

2/1 -1.11732% 0.42382%

1/29 0.02579% 0.24074%

e. What is the total risk faced by the bank on each day? What is

the worst-case day? What is the best-case day?

Japanese Yen Swiss

Francs Total

Day Delta % Rate Risk Delta %

Rate Risk Risk

2/4 -$26,489.77 -0.6292% $166.68 -$70,021.14 -0.2469% $172.88 $339.56

2/3 -$26,489.77 0.6242% -$165.35 -$70,021.14 0.2972% -$208.10 -$373.45

2/2 -$26,489.77 -2.5293% $670.01 -$70,021.14 -0.5908% $413.68 $1,083.69

2/1 -$26,489.77 -1.1173% $295.97 -$70,021.14 0.4238% -$296.75 -$0.78

1/29 -$26,489.77 0.0258% -$6.83 -$70,021.14 0.2407% -$168.54 -$175.37

The worst-case day is February 3, and the best-case day is February

2.

f. Assume that you have data for the 500 trading days preceding

February 4. Explain how you would identify the worst-case scenario

with a 95 percent degree of confidence?

The appropriate procedure would be to repeat the process illustrated

in part (e) above for all 500 days. The 500 days would be ranked on

the basis of total risk from the worst-case to the best-case. The

fifth percentile from the absolute worst-case situation would be day

25 in the ranking.

g. Explain how the five percent value at risk (VAR) position would

be interpreted for business on February 5.

Management would expect with a confidence level of 95 percent that

the total risk on February 5 would be no worse than the total risk

value for the 25th worst day in the previous 500 days.

This value represents the VAR for the portfolio.

h. How would the simulation change at the end of the day on February

5? What variables and/or processes in the analysis may change? What

variables and/or processes will not change?

The analysis can be upgraded at the end of the each day. The values

for delta may change for each of the assets in the analysis. As

such, the value for VAR may also change.

17. What is the primary disadvantage to the back simulation approach

in measuring market risk? What affect does the inclusion of more

observation days have as a remedy for this disadvantage? What other

remedies are possible to deal with the disadvantage?

The primary disadvantage of the back simulation approach is the

confidence level contained in the number of days over which the

analysis is performed. Further, all observation days typically are

given equal weight, a treatment that may not reflect accurately

changes in markets. As a result, the VAR number may be biased upward

or downward depending on how markets are trending. Possible

adjustments to the analysis would be to give more weight to more

recent observations, or to use Monte Carlo simulation techniques.

18. How is Monte Carlo simulation useful in addressing the

disadvantages of back simulation? What is the primary statistical

assumption underlying its use?

Monte Carlo simulation can be used to generate additional

observations that more closely capture the statistical

characteristics of recent experience. The generating process is

based on the historical variance-covariance matrix of FX changes.

The values in this matrix are multiplied by random numbers that

produce results that pattern closely the actual observations of

recent historic experience.

19. In the BIS Standardized Framework for regulating risk exposure

for the fixed-income portfolios of banks, what do the terms specific

risk and general market risk mean? Why does the capital

charge for general market risk tend to underestimate the true

interest rate or price risk exposure? What additional offsets, or

disallowance factors, are included in the analysis?

Specific risk is the risk unique to the issuing party for long-term

bonds in the trading portfolio of a financial institution. Specific

risk measures the decline in the liquidity or credit risk quality of

the portfolio. General market risk measures reflect the product of

duration and possible interest rate shocks to determine the

sensitivity of the portfolio to market rate movements.

The capital charge for market risk tends to underestimate interest

rate risk because of (a) maturity timing differences in offsetting

securities in the same time band, and (b) basis point risk for

different risk assets that may not be affected in a similar manner by

interest rate changes. Thus the capital charges may be adjusted for

basis risk. These adjustments also reflect the use of excess

positions in one time zone to partially offset positions in another

time band.

20. An FI has the following bonds in its portfolio: long 1-year U.S.

Treasury bills, short 3-year Treasury bonds, long 3-year AAA-rated

corporate bonds, and long 12-year B-rated (nonqualifying) bonds worth

$40, $10, $25, and $10 million, respectively (market values). Using

Table 10-5, determine the following:

a. Charges for specific risk = $1.20 million (See below.)

AAA = Qualifying bonds; B = Nonqualifying bonds

Time Specific Risk General Market Risk

band Issue Position Weight% Charge Weight%

Charge

1 year Treasury bill $40m 0.00 0.00 1.25 0.5000

3-year Treasury bond ($10m) 0.00 0.00 2.25 (0.2250)

3-year AAA – rated $25m 1.60 0.40 2.25 0.5625

12-year BB – rated $10m 8.0 0.80 4.50 0.4500

1.20 1.2875

b. Charges for general market risk.

General market risk charges = $1.2875 million (From table above.)

c. Charges for basis risk: vertical offsets within same time-bands

only (i.e., ignoring horizon effects).

Time-band Longs Shorts Residuals Offset Disallowance Charge

3-year $0.5625m ($0.225m) ($0.3375m) $0.2250m 10% $0.0225m

d. What is the total capital charge, using the information from

parts (a) through (c)?

Total capital charges = $1.20m + $1.2875 + $0.0225m = $2.51 million

21. Explain how the capital charge for foreign exchange risk is

calculated in the BIS Standardized model. If an FI has an $80

million long position in Euros, a $40 million short position in

British pounds, and a $20 million long position in Swiss francs, what

would be the capital charge required against FX market risk?

Total long position = $80 m of Euros + $20 m of Swiss franks = $100

million

Total short position = $40 million British pounds

Higher of long or short position = $100 million

Capital charge = 0.08 x $100 = $8 million

22. Explain the BIS capital charge calculation for unsystematic and

systematic risk for an FI that holds various amounts of equities in

its portfolio. What would be the total capital charge required for

an FI that holds the following portfolio of stocks? What criticisms

can be levied against this treatment of measuring the risk in the

equity portfolio?

Company Long Short

Texaco $45 million $25 million

Microsoft $55 million $12 million

Robeco $20 million

Cifra $15 million

The capital charge against common shares consists of two parts: those

for unsystematic risk (x-factor) and those for systematic risk

(y-factor). Unsystematic risk is unique to the firm in the capital

asset pricing model sense. The risk charge is found by multiplying a

four percent risk charge times the total (not net) of the long and

short positions.

Charges against unsystematic risk or firm-specific risk:

Gross position in all stocks = $45 + $55 + $20 + $25 + $12 + $15 =

$172 million

Capital charges = 4 percent x $172m = $6.88m, or $6,880,000

Systematic risk refers to market risk. The capital charge is found by

multiplying the net long or short position by eight percent.

Charges against systematic risk or market risk:

Net Positions Texaco $20m

Microsoft $43m

Robeco $20m

Cifra $15m

Total $98m

Capital charges = 8 percent x $98m = $7.84m, or $7,840,000

Total capital charges = $6.88m + $7.84m = $14.72m, or $14,720,000

This approach assumes that each stock has the same amount of

systematic risk, and that no benefits of diversification exist.

23. What conditions were introduced by BIS in 1998 to allow large

banks to use internally generated models for the measurement of

market risk? What types of capital can be held to meet the capital

charge requirements?

Large banks are allowed to utilize internally generated models to

measure market risk after receiving approval from the regulators.

The models must consider a 99 percent confidence level, the minimum

holding period for VAR estimates is 10 days, and the average

estimated VAR will be multiplied by a minimum factor of 3. Further,

the minimum capital charge must be the higher of the previous day’s

VAR or the average VAR over the previous 60 days. Thus calculation

of the capital charge is more conservative.

However, FIs are allowed to hold three types of capital to meet this

more conservative requirement. First, Tier I capital includes common

equity and retained earnings, Tier II capital includes long-term

subordinated debt with a maturity of over five years, and Tier III

capital includes short-term subordinated debt with a maturity of at

least two years.

24. Dark Star Bank has estimated its average VAR for the previous 60

days to be $35.5 million. DEAR for the previous day was $30.2

million.

a. Under the latest BIS standards, what is the amount of capital

required to be held for market risk?

Under the latest BIS standards, the proposed capital charge is the

higher of:

Previous day’s VAR = DEAR x 10

= 30.2 x 10 = $95.5008m

Average VAR x 3 = 35.5 x 3 = $106.5million

Capital charge = Higher of 1 and 2 = $106.5million

b. Dark Star has $15 million of Tier 1 capital, $37.5 million of

Tier 2 capital, and $55 million of Tier 3 capital. Is this amount of

capital sufficient? In not, what minimum amount of new capital

should be raised? Of what type?

Total capital needed = 106.5

Tier 1 + Tier 2 + Tier 3 = $15m + $37.5 + $54m = $106.5m

However, the capital is not sufficient because Tier 3 capital cannot

exceed 250% of Tier 1 capital. Thus, Tier 1 capital (X) needs to be:

X + 2.5X = $106.5m – $37.5 = $69m

X = 69/3.5 = $19.7143m

If Tier 1 capital is increased by 19.7143 – 15 = $4.7143m, then the

capital charge will be met, for example, $19.7143 + $37.5 + $49.2857

= $106.5m.

Let’s block ads! (Why?)

**READY TO PLACE AN ORDER**

CLICK HERE TO ORDER 100% ORIGINAL PAPERS FROM AllAcademicHelp.com <<