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Market Risk Measurement
Calculating Market Risk Exposure
The RiskMetrics Model
Market Risk of Fixed-Income Securities
Historic or Back Simulation
Historic (Back Simulation) Model versus RiskMetrics
Monte Carlo Simulation Approach
Regulatory Models: The BIS
The BIS Regulations and Large Bank
Solutions for End-of-Chapter Questions and Problems: Chapter Ten
1. What is meant by market risk?
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
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
a. Provide information on the risk positions taken by individual
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
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
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 ½ = 8,500 x
3.1623 = $26,879.36
For the 20-day period: VAR = 8,500 x ½ = 8,500 x
4.4721 = $38,013.16
The reason that VAR20
(2 x VAR10) is because ½
(2 x 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
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
Dollar value of position = 400/(1 + 0.095)15 =
DEAR = $102.5293499 million x -0.03425 = -$3.5116 million, or
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
= $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
$10,000,000. The beta of the portfolio approximates the market
portfolio, whose standard deviation (m)
has been estimated at 1.5 percent. What is the 5-day VAR of this
portfolio, using adverse rate changes in the 99th
DEAR = ($ 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.
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
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
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 =
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
Swiss Francs: 1.01 x current exchange rate = 1.01 x Swf1.414 =
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
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
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
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
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%
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
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
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
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
Charges against unsystematic risk or firm-specific risk:
Gross position in all stocks = $45 + $55 + $20 + $25 + $12 + $15 =
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
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
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
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
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
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