Hull J.C. Risk management and Financial institutions
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210 Chapter 8
Example 8.7
Suppose again that we back test a VaR model using 600 days of data when the
VaR confidence level is 99% and we observe one exception, well below the
expected number of six. Should we reject the model? The probability of one or
zero exceptions can be calculated in Excel as
BINOMDIST(l, 600, 0.01, TRUE)
It is 0.017. At a 5% confidence level, we should therefore reject the model.
However, if the number of exceptions had been two or more, we would not
have rejected the model.
The tests we have considered so far have been one-tailed tests. In
Example 8.6 we assumed that the true probability of an exception was
either 1% or greater than 1%. In Example 8.7 we assumed that it was 1%
or less than 1 %. Kupiec has proposed a relatively powerful two-tailed test.
6
If the probability of an exception under the VaR model is p and m
exceptions are observed in n trials, then
should have a chi-square distribution with one degree of freedom. Values
of the statistic are high for either very low or very high numbers of
exceptions. There is a probability of 5% that the value of a chi-square
variable with one degree of freedom will be greater than 3.84. It follows
that we should reject the model whenever the expression in equation (8.7)
is greater than 3.84.
Example 8.8
Suppose that as in the previous two examples we back test a VaR model using
600 days of data when the VaR confidence level is 99%. The value of the
statistic in equation (8.7) is greater that 3.84 when the number of exceptions, m,
is one or less and when the number of exceptions is 12 or more. We therefore
accept the VaR model when 2 m 11, and reject it otherwise.
Generally speaking the difficulty of back testing a VaR model increases as
the VaR confidence level increases. This is an argument in favor of not
using very high confidence levels for VaR.
Basel Committee Rules
The 1986 BIS Amendment (see Section 7.6) requires VaR models to be
back tested. Banks should use both actual and hypothetical changes in
6
See P. Kupiec, "Techniques for Verifying the Accuracy of Risk Management Models,
Journal of Derivatives, 3 (1995), 73-84.
The VaR Measure 211
the daily profit and loss to test a one-day VaR model that has a
confidence level of 99%. If the number of exceptions during the previous
250 days is less than 5, the regulatory multiplier for VaR is set at its
minimum value of 3. If the number of exceptions are 5, 6, 7, 8, and 9,
values of the multiplier equal to 3.4, 3.5, 3.65, 3.75, and 3.85, respectively,
are specified. The bank supervisor has some discretion as to whether the
higher multipliers are used. The penalty will normally apply when the
reason for the exceptions is identified as a deficiency in the VaR model
being used. If changes in positions during the day result in exceptions, the
higher multiplier should be considered. When the only reason that is
identified is bad luck, no guidance is provided for the supervisor. In
circumstances where the number of exceptions is 10 or more the Basel
Amendment requires the multiplier to be set at 4. The statistical tests we
have presented can be used to determine the confidence limits the Basel
Committee is implicitly using for its decision to accept or reject a model
(see Problem 8.13).
Bunching
A separate issue from the number of exceptions is bunching. If daily
portfolio changes are independent, exceptions should be spread evenly
through the period used for back testing. In practice, they are often
bunched together, suggesting that losses on successive days are not
independent. One approach for testing for bunching is to use the test
for autocorrelation in Section 5.9. Another approach is to use the
following test statistic suggested by Christofferson
7
where is the number of observations in which we go from a day where
we are in state i to a day where we are in state j. This statistic is chi-square
with one degree of freedom if there is no bunching. State 0 is a day where
there is no exception while state 1 is a day where there is an exception.
Also,
7
See P. F. Christofferson, "Evaluating Interval Forecasts," International Economic
Review, 39 (1998), 841-862.
212 Chapter 8
8.7 STRESS TESTING
In addition to requiring that a model for market risk be back tested, the
Basel Committee requires market risk VaR calculations be accompanied
by a "rigorous and comprehensive" stress-testing program. Stress testing
involves estimating how the portfolio would have performed under extreme
market moves. These extreme market moves typically have a very low
(virtually zero) probability under most VaR models—but they do happen!
Stress testing is a way of taking into account extreme events that are
virtually impossible according to the probability distributions assumed
for market variables, but do occur from time to time. A five-standard-
deviation daily move in a market variable is one such extreme event.
Under the assumption of a normal distribution, it happens about once
every 7,000 years, but, in practice, it is not uncommon to see a five-
standard-deviation daily move once or twice every ten years.
Some stress tests focus on particular market variables. Examples of
stress tests that have been recommended include:
1. Shifting a yield curve by 100 basis points
2. Changing implied volatilities for an asset by 20% of current values
3. Changing an equity index by 10%
4. Changing an exchange rate for a major currency by 6% or changing
the exchange rate for a minor currency by 20%
Stress tests more often involve making changes to several market vari-
ables. A common practice is to use historical scenarios. For example, to
test the impact of an extreme movement in US equity prices, a company
might set the percentage changes in all market variables equal to those on
October 19, 1987 (when the S&P 500 moved by 22.3 standard deviations).
If this is considered too extreme, the company might choose January 8,
1988 (when the S&P 500 moved by 6.8 standard deviations). To test the
effect of extreme movements in UK interest rates, the company might set
the percentage changes in all market variables equal to those on April 10,
1992 (when ten-year bond yields moved by 8.7 standard deviations).
The scenarios used in stress testing are also sometimes generated by
senior management. One technique sometimes used is to ask senior
management to meet periodically and "brainstorm" to develop extreme
scenarios that might occur given the current economic environment and
global uncertainties. Whatever the procedure used to generate the stress
tests, there should be a "buy in" to the idea of stress testing by senior
The VaR Measure 213
management and it should be senior management that reviews the results
of stress tests.
If movements in only a few variables are specified in a stress test, one
approach is to set changes in all other variables to zero. Another
approach is to regress the nonstressed variables on the variables that
are being stressed to obtain forecasts for them, conditional on the
changes being made to the stressed variables. These forecasts can be
incorporated into the stress test. This is known as conditional stress testing
and is discussed by Kupiec.
8
SUMMARY
A value-at-risk (VaR) calculation is aimed at making a statement of the
form: "We are X percent certain that we will not lose more than V dollars
in the next N days." The variable V is the VaR, X% is the confidence
level, and N days is the time horizon. It has become a very popular risk
measure. An alternative measure that has rather better theoretical proper-
ties is expected shortfall. This is the expected loss conditional on the loss
being greater than the VaR level.
When changes in a portfolio value are normally distributed, a VaR
estimate with one confidence level can be used to calculate a VaR level
with another confidence level. Also, if one-day changes have independent
normal distributions, an N-day VaR equals the one-day VaR multiplied
by When the independence assumption is relaxed other somewhat
more complicated formulas can be used to go from the one-day VaR to
the N-day VaR.
The marginal VaR with respect to the ith position is the partial
derivative of VaR with respect to the size of the position. The incremental
VaR with respect to a particular position is the incremental effect of that
position on VaR. There is a formula that can be used for dividing VaR
into components that correspond to the positions taken. The component
VaRs sum to VaR and each component is, for a large portfolio of
relatively small positions, approximately equal to the corresponding
incremental VaR.
Back testing is an important part of a VaR system. It examines how
well the VaR model would have performed in the past. There are two
ways in which back testing may indicate weaknesses in a VaR model. One
8
P. Kupiec, "Stress Testing in a Value at Risk Framework," Journal of Derivatives, 6
(1999), 7-24.
214
Chapter 8
is in the percentage of exceptions, that is, the percentage of times the
actual loss exceeds VaR. The other is in the extent to which exceptions are
bunched. There are statistical tests to determine whether a VaR model
should be rejected because of the percentage of exceptions or the amount
of bunching. Regulators have rules for increasing the VaR multiplier
when market risk capital is calculated if they consider the results from
back testing over 250 days to be unsatisfactory.
Stress testing is an important complement to VaR calculations. It
considers scenarios that either have occurred in the past or are considered
possibilities for the future. Typically, the scenarios have a very low
probability of occurring under the models used for calculating VaR.
FURTHER READING
Artzner P., F. Delbaen, J.-M. Eber, and D. Heath, "Coherent Measures of
Risk," Mathematical Finance, 9 (1999): 203-228.
Basak, S., and A. Shapiro, "Value-at-Risk-Based Risk Management: Optimal
Policies and Asset Prices," Review of Financial Studies, 14, No. 2 (2001):
371-405.
Beder, T., "VaR: Seductive But Dangerous," Financial Analysts Journal, 51,
No. 5 (1995): 12-24.
Boudoukh, J., M. Richardson, and R. Whitelaw, "The Best of Both Worlds,"
Risk, May 1998: 64-67.
Dowd, K., Measuring Market Risk. 2nd edn. New York: Wiley, 2005.
Duffie, D., and J. Pan, "An Overview of Value at Risk," Journal of Derivatives,
4, No. 3 (Spring 1997): 7-49.
Hopper, G., "Value at Risk: A New Methodology for Measuring Portfolio
Risk," Business Review, Federal Reserve Bank of Philadelphia, July/August
1996: 19-29.
Hua P., and P. Wilmott, "Crash Courses," Risk, June 1997: 64-67.
Jackson, P., D. J. Maude, and W. Perraudin, "Bank Capital and Value at Risk,'
Journal of Derivatives, 4, No. 3 (Spring 1997): 73-90.
Jorion, P., Value at Risk. 2nd edn. New York: McGraw-Hill, 2001.
Longin, F.M., "Beyond the VaR," Journal of Derivatives, 8, No. 4 (Summer
2001): 36-48.
Marshall, C, and M. Siegel, "Value at Risk: Implementing a Risk Measurement
Standard," Journal of Derivatives, 4, No. 3 (Spring 1997): 91—111.
The VaR Measure
215
QUESTIONS AND PROBLEMS (Answers at End of Book)
8.1. What is the difference between expected shortfall and VaR? What is the
theoretical advantage of expected shortfall over VaR?
8.2. What is a spectral risk measure? What conditions must be satisfied by a
spectral risk measure for the Subadditivity condition in Section 8.3 to be
satisfied?
8.3. A fund manager announces that the fund's 1-month 95% VaR is 6% of
the size of the portfolio being managed. You have an investment of
$100,000 in the fund. How do you interpret the portfolio manager's
announcement?
8.4. A fund manager announces that the fund's one-month 95% expected
shortfall is 6% of the size of the portfolio being managed. You have an
investment of $100,000 in the fund. How do you interpret the portfolio
manager's announcement?
8.5. Suppose that each of two investments has a 0.9% chance of a loss of
$10 million, a 99.1% of a loss of $1 million, and zero probability of a gain.
The investments are independent of each other. (a) What is the VaR for
one of the investments when the confidence level is 99%? (b) What is the
expected shortfall for one of the investments when the confidence level is
99%? (c) What is the VaR for a portfolio consisting of the two investments
when the confidence level is 99%? (d) What is the expected shortfall for a
portfolio consisting of the two investments when the confidence level is
99%? (e) Show that in this example VaR does not satisfy the Subadditivity
condition whereas expected shortfall does.
8.6. Suppose that the change in the value of a portfolio over a 1-day time
period is normal with a mean of zero and a standard deviation of
$2 million, What is (a) the 1-day 97.5% VaR, (b) the 5-day 97.5% VaR,
and (c) the 5-day 99% VaR?
8.7. What difference does it make to your answers to (b) and (c) of Problem 8.6
if there is first-order daily autocorrelation with correlation parameter
equal to 0.16?
8.8. Explain carefully the differences between marginal VaR, incremental VaR,
and component VaR for a portfolio consisting of a number of assets.
8.9. Suppose that we back test a VaR model using 1,000 days of data. The VaR
confidence level is 99% and we observe 17 exceptions. Should we reject the
model at the 5% confidence level? Use a one-tailed test.
8.10. Explain what is meant by bunching.
8.11. Describe two ways extreme scenarios can be developed for stress testing.
8.12. Prove equation (8.3).
216 Chapter 8
8.13. The back-testing rules of the Basel Committee can lead to questions about
a VaR model when there are 5 or more exceptions in 250 trials. What is the
chance of this if the VaR methodology is perfectly accurate?
ASSIGNMENT QUESTIONS
8.14. Suppose that each of two investments has a 4% chance of a loss of
$10 million, a 2% chance of a loss of $1 million, and a 94% chance of
a profit of $1 million. They are independent of each other. (a) What is the
VaR for one of the investments when the confidence level is 95%? (b) What
is the expected shortfall when the confidence level is 95%? (c) What is the
VaR for a portfolio consisting of the two investments when the confidence
level is 95%? (d) What is the expected shortfall for a portfolio consisting of
the two investments when the confidence level is 95%? (e) Show that, in
this example, VaR does not satisfy the Subadditivity condition whereas
expected shortfall does.
8.15. Suppose that daily changes for a portfolio have first-order correlation with
correlation parameter 0.12. The 10-day VaR, calculated by multiplying the
1-day VaR by is $2 million. What is a better estimate of the VaR that
takes account of autocorrelation?
8.16. The probability that the loss from a portfolio will be greater than
$10 million in 1 month is estimated to be 5%. (a) What is the 1-month
99% VaR assuming the change in value of the portfolio is normally
distributed. (b) What is the 1-month 99% VaR assuming that the power
law described in Section 5.4 applies with = 3.
8.17. Suppose that we back test a VaR model using 1,000 days of data. The VaR
confidence level is 99% and we observe 15 exceptions. Should we reject the
model at the 5% confidence level. Use Kupiec's two-tailed test.
In this chapter and the next we cover the two main approaches for
calculating VaR for market risk. The approach we consider in this
chapter is known as historical simulation. This involves using historical
day-to-day changes in the values of market variables in a direct way to
estimate the probability distribution of the change in the value of the
current portfolio between today and tomorrow.
After describing the mechanics of the historical simulation approach,
we explain how to calculate the standard error of the VaR estimate and
how to modify the procedure to take into account the latest information
about volatility. We also describe how extreme value theory can be used
in conjuction with a historical simulation to improve VaR estimates and
to deal with situations where the VaR confidence level is very high.
9.1 THE METHODOLOGY
Historical simulation involves using past data as a guide to what will
happen in the future. Suppose that we want to calculate VaR for a
Portfolio using a one-day time horizon, a 99% confidence level, and
500 days of data. (The time horizon and confidence level are those
typically used for a market risk VaR calculation; 500 is a popular choice
for the number of days of data used.) The first step is to identify the
market variables affecting the portfolio. These will typically be exchange
Market Risk VaR:
Historical Simulation
Approach
218
Chapter 9
rates, equity prices, interest rates, and so on. We then collect data on the
movements in these market variables over the most recent 500 days. This
provides us with 500 alternative scenarios for what can happen between
today and tomorrow. Denote the first day for which we have data as
Day 0, the second day as Day 1, and so on. Scenario 1 is where the
percentage changes in the values of all variables are the same as they were
between Day 0 and Day 1; Scenario 2 is where they are the same as
between Day 1 and Day 2; and so on. For each scenario we calculate the
dollar change in the value of the portfolio between today and tomorrow.
This defines a probability distribution for daily changes in the value of
our portfolio. The first percentile of the distribution can be estimated as
the fifth worst outcome.
1
The estimate of VaR is the loss when we are at
this first percentile point. We are 99% certain that we would not have
taken a loss greater than our VaR estimate if the changes in market
variables are a random sample from the last 500 days.
The historical simulation methodology is illustrated in Tables 9.1
and 9.2. Table 9.1 shows observations on market variables over the last
500 days. The observations are taken at some particular point in time
during the day (usually the close of trading). There are assumed to be a
total of 1,000 market variables.
Table 9.2 shows the values of the market variables tomorrow if their
percentage changes between today and tomorrow are the same as they
Table 9.1 Data for VaR historical simulation calculation.
Day
0
1
2
3
498
499
500
Market
variable 1
20.33
20.78
21.44
20.97
25.72
25.75
25.85
Market
variable 2
0.1132
0.1159
0.1162
0.1184
0.1312
0.1323
0.1343
Market
variable 1,000
65.37
64.91
65.02
64.90
62.22
61.99
62.10
1
There are alternatives here. A case could be made for using the fifth worst outcome or
the sixth worst outcome, or an average of the two, as the first percentile of the
distribution when there are 500 outcomes. In Excel's PERCENTILE function, when there
are n observations and k is an integer, the k/(n — 1) percentile is the observation ranked
k + 1. Other percentiles are calculated using linear interpolation.
VaR: Historical Simulation Approach 219
Table 9.2 Scenarios generated for tomorrow (Day 501) using data in Table 9.1.
Value of portfolio on Day 500 is $23.50 million
Scenario
number
1
2
3
499
500
Market
variable 1
26.42
26.67
25.28
25.88
25.95
Market
variable 2 ..
0.1375 ••
0.1346 ••
0.1368
0.1354
0.1363
Market
,. variable 1,000
61.66
62.21
61.99
61.87
62.21
Portfolio
value
($ millions)
23.71
23.12
22.94
23.63
22.87
Change
in value
($ millions)
0.21
-0.38
-0.56
0.13
-0.63
were between Day i — 1 and Day i for 1 i 500. The first row in
Table 9.2 shows the values of market variables tomorrow assuming their
percentage changes between today and tomorrow are the same as they
were between Day 0 and Day 1; the second row shows the values of
market variables tomorrow assuming their percentage changes are the
same as those between Day 1 and Day 2; and so on. The 500 rows in
Table 9.2 are the 500 scenarios considered.
Define v
i
as the value of a market variable on Day i and suppose that
today is Day n. The ith scenario assumes that the value of the market
variable tomorrow will be
In our example, n = 500. For the first variable, the value today, is
25.85. In addition, = 20.33 and = 20.78. It follows that the value of
the first market variable in the first scenario is
The penultimate column of Table 9.2 shows the value of the portfolio
tomorrow for each of the 500 scenarios. We suppose the value of the
Portfolio today is $23.50 million. This leads to the numbers in the final
column for the change in the value between today and tomorrow for all
the different scenarios. For Scenario 1 the change in value is +$210,000;
for Scenario 2 it is -$380,000; and so on.
We are interested in the one-percentile point of the distribution of
changes in the portfolio value. As indicated earlier, because there are a