CHAPTER 20
Value at Risk
Practice Questions
Problem 20.8.
A company uses an EWMA model for forecasting volatility. It decides to change the
parameter  from 0.95 to 0.85. Explain the likely impact on the forecasts.
Reducing  from 0.95 to 0.85 means that more weight is put on recent observations of u i2
and less weight is given to older observations. Volatilities calculated with  = 085 will react
more quickly to new information and will “bounce around” much more than volatilities
calculated with  = 095 .
Problem 20.9.
Explain the difference between value at risk and expected shortfall.
Value at risk is the loss that is expected to be exceeded (100 – X)% of the time in N days
for specified parameter values, X and N . Expected shortfall is the expected loss conditional
that the loss is greater than the Value at Risk.
Problem 20.10.
Consider a position consisting of a $100,000 investment in asset A and a $100,000
investment in asset B. Assume that the daily volatilities of both assets are 1% and that the
coefficient of correlation between their returns is 0.3. What is the 5-day 99% value at risk for
the portfolio?
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 0002 + 1 0002 + 2  03 1 000 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 standard deviation of the 5-day change is
1,612.45 5 = $3,605.55
From the tables of N ( x) we see that N (−233) = 001 . This means that 1% of a normal
distribution lies more than 2.33 standard deviations below the mean. The 5-day 99 percent
value at risk is therefore 2.33×3,605.55 = $8,400.93.
Problem 20.11.
The volatility of a certain market variable is 30% per annum. Calculate a 99% confidence
interval for the size of the percentage daily change in the variable.
The volatility per day is 30  252 = 189% . There is a 99% chance that a normally
distributed variable will be within 2.57 standard deviations. We are therefore 99% confident
that the daily change will be less than 257 189 = 486% .
Problem 20.12.
Explain how an interest rate swap is mapped into a portfolio of zero-coupon bonds with
standard maturities for the purposes of a VaR calculation.
When a final exchange of principal is added in, the floating side is equivalent a zero-coupon
bond with a maturity date equal to the date of the next payment. The fixed side is a couponbearing bond, which is equivalent to a portfolio of zero-coupon bonds. The swap can
therefore be mapped into long and short positions in zero-coupon bonds with maturity dates
corresponding to the payment dates. A cash flow mapping procedure can then be used to map
each of the zero-coupon bonds to positions in the adjacent standard-maturity zero-coupon
bonds.
Problem 20.13.
Explain why the linear model can provide only approximate estimates of VaR for a portfolio
containing options.
The change in the value of an option is not linearly related to the percentage change in the
value of the underlying variable. The linear model assumes that the change in the value of a
portfolio is linearly related to percentage changes in the underlying variables. It is therefore
only an approximation for a portfolio containing options.
Problem 20.14.
Some time ago a company entered into a forward contract to buy £1 million for $1.5 million.
The contract now has six months to maturity. The daily volatility of a six-month zero-coupon
sterling bond (when its price is translated to dollars) is 0.06% and the daily volatility of a
six-month zero-coupon dollar bond is 0.05%. The correlation between returns from the two
bonds is 0.8. The current exchange rate is 1.53. Calculate the standard deviation of the
change in the dollar value of the forward contract in one day. What is the 10-day 99% VaR?
Assume that the six-month interest rate in both sterling and dollars is 5% per annum with
continuous compounding.
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 153e −00505 or $1.492 million. The value of the dollar
bond is 15e−00505 or $1.463 million. The variance of the change in the value of the contract
in one day is
14922  000062 + 14632  000052 − 2  08 1492  00006 1463  00005
= 0000000288
The standard deviation is therefore $0.000537 million. The 10-day 99% VaR is
0.000537 10  2.33 = $0.00396 million.
Problem 20.15.
The most recent estimate of the daily volatility of the U.S. dollar–sterling exchange rate is
0.6%, and the exchange rate at 4 p.m. yesterday was 1.5000. The parameter  in the EWMA
model is 0.9. Suppose that the exchange rate at 4 p.m. today proves to be 1.4950. How would
the estimate of the daily volatility be updated?
The daily return is −0005  15000 = −0003333 . The current daily variance estimate is
00062 = 0000036 . The new daily variance estimate is
09  0000036 + 01 00033332 = 0000033511
The new volatility is the square root of this. It is 0.00579 or 0.579%.
Problem 20.16.
Suppose that the daily volatilities of asset A and asset B calculated at close of trading
yesterday are 1.6% and 2.5%, respectively. The prices of the assets at close of trading
yesterday were $20 and $40, and the estimate of the coefficient of correlation between the
returns on the two assets made at close of trading yesterday was 0.25. The parameter  used
in the EWMA model is 0.95.
(a) Calculate the current estimate of the covariance between the assets.
(b) On the assumption that the prices of the assets at close of trading today are $20.5 and
$40.5, update the correlation estimate.
a) The volatilities and correlation imply that the current estimate of the covariance is
025 0016  0025 = 00001.
b) If the prices of the assets at close of trading today are $20.5 and $40.5, the returns are
05  20 = 0025 and 05  40 = 00125 . The new covariance estimate is
095 00001+ 005 0025 00125 = 00001106
The new variance estimate for asset A is
095  00162 + 005  00252 = 000027445
so that the new volatility is 0.0166. The new variance estimate for asset B is
095  00252 + 005  001252 = 0000601562
so that the new volatility is 0.0245. The new correlation estimate is
00001106
= 0272
00166  00245
Problem 20.17.
Suppose that the daily volatility of the FT-SE 100 stock index (measured in pounds sterling)
is 1.8% and the daily volatility of the dollar/sterling exchange rate is 0.9%. Suppose further
that the correlation between the FT-SE 100 and the dollar/sterling exchange rate is 0.4. What
is the volatility of the FT-SE 100 when it is translated to U.S. dollars? Assume that the
dollar/sterling exchange rate is expressed as the number of U.S. dollars per pound sterling.
(Hint: When Z = XY , the percentage daily change in Z is approximately equal to the
percentage daily change in X plus the percentage daily change in Y .)
The FT-SE expressed in dollars is XY where X is the FT-SE expressed in sterling and Y is
the exchange rate (value of one pound in dollars). Define xi as the proportional change in X
on day i and yi as the proportional change in Y on day i . The proportional change in XY is
approximately xi + yi . The standard deviation of xi is 0.018 and the standard deviation of yi
is 0.009. The correlation between the two is 0.4. The variance of xi + yi is therefore
00182 + 00092 + 2  0018  0009  04 = 00005346
so that the volatility of xi + yi is 0.0231 or 2.31%. This is the volatility of the FT-SE
expressed in dollars. Note that it is greater than the volatility of the FT-SE expressed in
sterling. This is the impact of the positive correlation. When the FT-SE increases, the value of
sterling measured in dollars also tends to increase. This creates an even bigger increase in the
value of FT-SE measured in dollars. A similar result holds for a decrease in the FT-SE.
Problem 20.18.
Suppose that in Problem 20.17 the correlation between the S&P 500 Index (measured in
dollars) and the FT-SE 100 Index (measured in sterling) is 0.7, the correlation between the
S&P 500 index (measured in dollars) and the dollar-sterling exchange rate is 0.3, and the
daily volatility of the S&P 500 Index is 1.6%. What is the correlation between the S&P 500
Index (measured in dollars) and the FT-SE 100 Index when it is translated to dollars? (Hint:
For three variables X , Y , and Z , the covariance between X + Y and Z equals the
covariance between X and Z plus the covariance between Y and Z .)
Continuing with the notation in Problem 20.17, define zi as the proportional change in the
value of the S&P 500 on day i . The covariance between xi and zi is
07  0018  0016 = 00002016 . The covariance between yi and zi is
03 0009  0016 = 00000432 . The covariance between xi + yi and zi equals the
covariance between xi and zi plus the covariance between yi and zi . It is
00002016 + 00000432 = 00002448
The correlation between xi + yi and zi is
00002448
= 0662
0016  00231
Problem 20.19.
The one-day 99% VaR is calculated for the four-index example in Section 20.2 as $253,385.
Look at the underlying spreadsheets on the author’s web site and calculate a) the 95% oneday VaR and b) the 97% one-day VaR.
The 95% one-day VaR is the 25th worst loss. This is $168,612. The 97% one-day VaR is the
15th worst loss. This is $188,758.
Problem 20.20.
Use the spreadsheets on the author’s web site to calculate the one-day 99% VaR, using the
basic methodology in Section 20.2 if the four-index portfolio considered in Section 20.2 is
equally divided between the four indices.
In the “Scenarios” worksheet the portfolio investments are changed to 2500 in cells L2:O2.
The losses are then sorted from the largest to the smallest. The fifth worst loss is $258,355.
This is the one-day 99% VaR.
Problem 20.21.
At the end of Section 20.6, the VaR for the four-index example was calculated using the
model-building approach. How does the VaR calculated change if the investment is $2.5
million in each index? Carry out calculations when a) volatilities and correlations are
estimated using the equally weighted model and b) when they are estimated using the EWMA
model with  = 094 . Use the spreadsheets on the author’s web site.
The alphas (row 21 for equal weights and row 7 for EWMA) should be changed to 2,500.
This changes the one-day 99% VaR to $238,022 when volatilities and correlations are
estimated using the equally weighted model and to $510,459 when EWMA with  = 094 is
used.
Problem 20.22.
What is the effect of changing  from 0.94 to 0.97 in the EWMA calculations in the fourindex example at the end of Section 20.6? Use the spreadsheets on the author’s web site.
The parameter  is in cell N3 of the EWMA worksheet. Changing it to 0.97 changes the oneday 99% VaR from $488,217 to $404,661. This is because less weight is given to recent
observations.
Further Problems
Problem 20.23.
Consider a position consisting of a $300,000 investment in gold and a $500,000 investment
in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%,
respectively, and that the coefficient of correlation between their returns is 0.6. What is the
10-day 97.5% value at risk for the portfolio? By how much does diversification reduce the
VaR?
The variance of the portfolio (in thousands of dollars) is
00182  3002 + 00122  5002 + 2  300  500  06  0018  0012 = 10404
The standard deviation is 10404 = 102 . Since N (−196) = 0025 , the 1-day 97.5% VaR is
102 196 = 1999 and the 10-day 97.5% VaR is 10 1999 = 6322 . The 10-day 97.5%
VaR is therefore $63,220. The 10-day 97.5% value at risk for the gold investment is
5 400  10 196 = 33 470 . The 10-day 97.5% value at risk for the silver investment is
6 000  10 196 = 37188 . The diversification benefit is
33 470 + 37188 − 63 220 = $7 438
Problem 20.24.
Consider a portfolio of options on a single asset. Suppose that the delta of the portfolio is 12,
the value of the asset is $10, and the daily volatility of the asset is 2%. Estimate the 1-day
95% VaR for the portfolio. Suppose that the gamma of the portfolio is −26 . Derive a
quadratic relationship between the change in the portfolio value and the percentage change
in the underlying asset price in one day.
An approximate relationship between the daily change in the value of the portfolio, P and
the return on the asset x is
P = 10 12x = 120x
The standard deviation of x is 0.02. It follows that the standard deviation of P is 2.4. The
1-day 95% VaR is 2.4×1.65 = $3.96.
From equation (20.5) the quadratic relationship between P and x is
1
P = 10 12x − 102  26(x) 2
2
or
P = 120x − 130(x) 2
Problem 20.25.
A bank has written a call option on one stock and a put option on another stock. For the first
option the stock price is 50, the strike price is 51, the volatility is 28% per annum, and the
time to maturity is nine months. For the second option the stock price is 20, the strike price is
19, the volatility is 25% per annum, and the time to maturity is one year. Neither stock pays a
dividend, the risk-free rate is 6% per annum, and the correlation between stock price returns
is 0.4. Calculate a 10-day 99% VaR using DerivaGem and the linear model.
My answer follows the usual practice of assuming that the 10-day 99% value at risk is 10
times the 1-day 99% value at risk. Some students may try to calculate a 10-day VaR directly,
which is fine. From DerivaGem, the values of the two option positions are –5.413 and –
1.014. The deltas are –0.589 and 0.284, respectively. An approximate linear model relating
the change in the portfolio value to proportional change, x1 , in the first stock price and the
proportional change, x2 , in the second stock price is
P = −0589  50x1 + 0284  20x2
or
P = −2945x1 + 568x2
The daily volatility of the two stocks are 028  252 = 00176 and 025  252 = 00157 ,
respectively. The one-day variance of P is
29452  001762 + 5682  00157 2 − 2  2945  00176  568  00157  04 = 02396
The one day standard deviation is, therefore, 0.4895 and the 10-day 99% VaR is
233  10  04895 = 361 .
Problem 20.26.
Suppose that the price of gold at close of trading yesterday was $600, and its volatility was
estimated as 1.3% per day. The price at the close of trading today is $596. Update the
volatility estimate using the EWMA model with  = 094 .
The return on gold is −4  3600 = −000667 . Using the EWMA model the variance is updated
to
094  00132 + 006  000667 2 = 000016154
so that the new daily volatility is 000016154 = 001271 or 1.271% per day.
Problem 20.27.
Suppose that in Problem 20.28 the price of silver at the close of trading yesterday was $16,
its volatility was estimated as 1.5% per day, and its correlation with gold was estimated as
0.8. The price of silver at the close of trading today is unchanged at $16. Update the volatility
of silver and the correlation between silver and gold using the EWMA model with  = 094 .
The return on silver is zero. Using the EWMA model the variance is updated to
094  00152 + 006  0 = 00002115
so that the new daily volatility is 00002115 = 001454 or 1.454% per day. The initial
covariance is 08  0013 0015 = 0000156 Using EWMA the covariance is updated to
094  0000156 + 006  0 = 000014664
so that the new correlation is 000014664  (001454  001271) = 07933
Problem 20.28. (Excel file)
An Excel spreadsheet containing daily data on a number of different exchange rates and
stock indices can be downloaded from the author’s Web site:
http://www.rotman.utoronto.ca/ hull/data
Choose one exchange rate and one stock index. Estimate the value of  in the EWMA model
that minimizes the value of
 (vi − i )2
i
where vi is the variance forecast made at the end of day i −1 and  i is the variance
calculated from data between day i and i + 25 . Use Excel’s Solver tool. Set the variance
forecast at the end of the first day equal to the square of the return on that day to start the
EWMA calculations.
In the spreadsheet the first 25 observations on (vi-)2 are ignored so that the results are not
unduly influenced by the choice of starting values. The best values of  for EUR, CAD, GBP
and JPY were found to be 0.947, 0.898, 0.950, and 0.984, respectively. The best values of
 for S&P500, NASDAQ, FTSE100, and Nikkei225 were found to be 0.874, 0.901, 0.904,
and 0.953, respectively.
Problem 20.29.
A common complaint of risk managers is that the model building approach (either linear or
quadratic) does not work well when delta is close to zero. Test what happens when delta is
close to zero in using Sample Application E in the DerivaGem Application Builder software.
(You can do this by experimenting with different option positions and adjusting the position
in the underlying to give a delta of zero.) Explain the results you get.
We can create a portfolio with zero delta in Sample Application E by changing the position in
the stock from 1,000 to 513.58. (This reduces delta by 1 000 − 51358 = 48642 .) In this case
the true VaR is 48.86; the VaR given by the linear model is 0.00; and the VaR given by the
quadratic model is -35.71.
Other zero-delta examples can be created by changing the option portfolio and then zeroing
out delta by adjusting the position in the underlying asset. The results are similar. The
software shows that neither the linear model nor the quadratic model gives good answers
when delta is zero. The linear model always gives a VaR of zero because the model assumes
that the portfolio has no risk. (For example, in the case of one underlying asset P = S .)
When there are no cross gammas the quadratic model assumes that P is always positive.
(For example, in the case of one underlying asset P = 05(S ) 2 .) This gives a negative
VaR.
In practice many portfolios do have deltas close to zero because of the hedging activities
described in Chapter 17. This has led many financial institutions to prefer historical
simulation to the model building approach.
Problem 20.30 (Excel file)
Suppose that the portfolio considered in Section 20.2 has (in $000s) 3,000 in DJIA, 3,000 in
FTSE, 1,000 in CAC40, and 3,000 in Nikkei 225. Use the spreadsheet on the author’s web
site to calculate what difference this makes to the one-day 99% VaR that is calculated in
Section 20.2.
First the investments worksheet is changed to reflect the new portfolio allocation. (see row 2
of the Scenarios worksheet for historical simulation). The losses are then sorted from the
greatest to the least. The one-day 99% VaR is the fifth worst loss or $230,897.
Problem 20.31 (Excel file)
The calculations for the four-index example at the end of Section 20.6 assume that the
investments in the DJIA, FTSE 100, CAC40, and Nikkei 225 are $4 million, $3 million, \$1
million, and $2 million, respectively. How does the VaR calculated change if the investment
are $3 million, $3 million, $1 million, and $3 million, respectively? Carry out calculations
when a) volatilities and correlations are estimated using the equally weighted model and b)
when they are estimated using the EWMA model. What is the effect of changing  from 0.94
to 0.90 in the EWMA calculations? Use the spreadsheets on the author's web site.
(a) The portfolio investment amounts have to be changed in row 21 of the Equal Weights
worksheet for the model building approach. The worksheet shows that the new oneday 99% VaR is $229,683. This is slightly higher than the one-day VaR for the
original portfolio.
(b) The portfolio investment amounts have to be changed in row 7 of the EWMA
worksheet. The worksheet shows that one-day 99% VaR is $478,895. This is slightly
lower than the one-day VaR for the original portfolio. Changing  to 0.90 (see cell
N3) changes VaR to $537,828. This is higher because recent returns are given more
weight.