Hull J.C. Risk management and Financial institutions

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220

Chapter 9

total of 500 scenarios in Table 9.2, we can estimate this as the fifth worst

number in the final column of the table. Alternatively, we can use extreme

value theory, which will be described later in the chapter. As mentioned in

Section 8.4, the ten-day VaR for a 99% confidence level is usually

calculated as times the one-day VaR.

Each day the VaR estimate in our example would be updated using the

most recent 500 days of data. Consider, for example, what happens on

Day 501. We find out new values for all the market variables and are able

to calculate a new value for our portfolio.

We then go through the

procedure we have outlined to calculate a new VaR. We use data on

the market variables from Day 1 to Day 501. (This gives us the required

500 observations on percentage changes in market variables; the Day 0

values of the market variables are no longer used.) Similarly, on Day 502,

we use data from Day 2 to Day 502 to determine VaR, and so on.

9.2 ACCURACY

The historical simulation approach estimates the distribution of portfolio

changes based on a finite number of observations of what happened in

the past. As a result, the estimates of quantiles of the distribution are not

perfectly accurate.

Kendall and Stuart describe how to calculate a confidence interval for

the quantile of a probability distribution when it is estimated from sample

data.

Suppose that the q-quantile of the distribution is estimated as x.

The standard error of the estimate is

where n is the number of observations and f(x) is the probability density

function of the loss evaluated at x. The latter can be estimated by fitting

the empirical data to a standard distribution.

Example 9.1

Suppose we are interested in estimating the 0.01 quantile (= 1 percentile) of a

loss distribution from 500 observations so that n = 500 and q — 0.01. We can

estimate f{x) by approximating the actual empirical distribution with a

Note that the portfolio's composition may have changed between Day 500 and Day 501.

See M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 1:

Distribution Theory, 4th edn. London: Griffin, 1972.

VaR: Historical Simulation Approach 221

If the estimate of the 0.01 quantile using historical simulation is $25 million,

a 95% confidence interval is from 25 — 1.96 x 1.67 to 25 + 1.96 x 1.67, that

is, from $21.7 million to $28.3 million.

As Example 9.1 illustrates, the standard error of a VaR estimated using

historical simulation tends to be quite high. It increases as the VaR

confidence level is increased. For example, if in Example 9.1 the VaR

confidence level had been 95% instead of 99%, the standard error would

be $0.95 million instead of $1.67 million. The standard error declines as

the sample size is increased—but only as the square root of the sample

size. If we quadrupled the sample size in Example 9.1 from 500 to 2,000

(i.e., from approximately two to approximately eight years of data), the

standard error halves from $1.67 million to about $0.83 million.

Additionally, we should bear in mind that historical simulation assumes

that the joint distribution of daily changes in market variables is stationary

through time. This is unlikely to be exactly true and creates additional

uncertainty about the value of VaR.

9.3 EXTENSIONS

In this section we cover a number of extensions of the basic historical

simulation methodology that we discussed in Section 9.1.

Weighting of Observations

The basic historical simulation approach assumes that each day in the past

is given equal weight. More formally, if we have observations for n day-to-

day changes, each of them is given a weighting of \/n. Boudoukh et al.

suggest that more recent observations should be given more weight

because they are more reflective of current volatilities and current macro-

economic variables.

The natural weighting scheme to use is one where

See J. Boudoukh, M. Richardson, and R. Whitelaw, "The Best of Both Worlds: A

Hybrid Approach to Calculating Value at Risk," Risk, 11 (May 1998), 64-67.

standard distribution. Suppose that the approximate empirical distribution is

normal with mean zero and standard deviation $10 million. Using Excel, the

0.01 quantile is NORMINV(0.01,0, 10), or 23.26. The value of f(x) is

NORMDIST(23.26, 0, 10, FALSE), or 0.0027. The standard error of the

estimate that is made is

222

Chapter 9

weights decline exponentially. We used this in Section 5.6 when developing

the exponentially weighted moving average model for monitoring vari-

ance. Suppose that we are now at the end of day n. The weight assigned to

the change in the portfolio value between day n — i and day n — i + 1 is

times that assigned to the change between day n — i + 1 and day n — i + 2.

In order for the weights to add up to 1, the weight given to the change

between day n — i and n — i + 1 is

where n is the number of days. As this weighting scheme

approaches the basic historical simulation approach, where all observa-

tions are given a weight of \/n (see Problem 9.2).

VaR is calculated by ranking the observations from the worst outcome

to the best. Starting at the worst outcome, weights are summed until the

required quantile of the distribution is reached. For example, if we are

calculating VaR with a 99% confidence level, we continue summing

weights until the sum just exceeds 0.01. We have then reached the 99%

VaR level. The best value of can be obtained by experimenting to see

which value back tests best. One disadvantage of the exponential weight-

ing approach relative to the basic historical simulation approach is that the

effective sample size is reduced. However, we can compensate for this by

using a larger value of n. Indeed it is not really necessary to discard old

days as we move forward in time because they are given very little weight.

Incorporating Volatility Updating

Hull and White suggest a way of incorporating volatility updating into the

historical simulation approach.

A volatility updating scheme, such as

EWMA or GARCH(1,1) (both of which were described in Chapter 5) is

used in parallel with the historical simulation approach for all market

variables. Suppose that the daily volatility for a particular market vari-

able estimated at the end of day i — 1 is This is an estimate of the daily

volatility between the end of day i — 1 and the end of day i. Suppose it is

now day n. The current estimate of the volatility of the market variable is

This applies to the time period between today and tomorrow, which

is the time period over which we are calculating VaR.

Suppose that is twice This means that we estimate the daily

See J. Hull and A. White, "Incorporating Volatility Updating into the Historical

Simulation Method for Value-at-Risk," Journal of Risk, 1, No. 1 (Fall 1998), 5-19.

VaR: Historical Simulation Approach 223

volatility of this particular market variable to be twice as great today as on

day i — 1. This means that we expect to see changes between today and

tomorrow that are twice as big as changes between day i — 1 and day i.

When carrying out the historical simulation and creating a sample of what

could happen between today and tomorrow based on what happened

between day i — 1 and day i, it therefore makes sense to multiply the latter

by 2. In general, when this approach is used, the expression in equa-

tion (9.1) for the value of a market variable under the ith scenario becomes

Each market variable is handled in the same way. This approach takes

account of volatility changes in a natural and intuitive way and produces

VaR estimates that incorporate more current information. The VaR

estimates can be greater than any of the historical losses that would have

occurred for our current portfolio on the days we consider. Hull and White

produce evidence using exchange rates and stock indices to show that this

approach is superior to traditional historical simulation and to the expo-

nential weighting scheme described earlier. More complicated models can

be developed where observations are adjusted for the latest information on

correlations as well as for the latest information on volatilities.

Bootstrap Method

The bootstrap method is another variation on the basic historical simula-

tion approach. It involves creating a set of changes in the portfolio value

based on historical movements in market variables in the usual way. We

then sample with replacement from these changes to create many new

similar data sets. We calculate the VaR for each of the new data sets. Our

95% confidence interval for VaR is the range between the 2.5 and the

97.5 percentile point of the distribution of the VaRs calculated from the

data sets.

Suppose, for example, that we have 500 days of data. We could sample

with replacement 500,000 times from the data to obtain 1,000 different

sets of 500 days of data. We calculate the VaR for each set. We then rank

the VaRs. Suppose that the 25th largest VaR is $5.3 million and the 475th

largest VaR is $8.9 million. The 95% confidence limit for VaR is

$5-3 million to $8.9 million. Usually the 95% confidence range calculated

for VaR using the bootstrap method is less than that calculated using the

Procedure in Section 9.2.

224 Chapter 9

9.4 EXTREME VALUE THEORY

In Section 5.4 we introduced the power law and explained that it can be

used to estimate the tails of a wide range of distributions. We now provide

the theoretical underpinnings for the power law and present more sophis-

ticated estimation procedures than those used in Section 5.4. The term

used to describe the science of estimating the tails of a distribution is

extreme value theory. In this section we show how extreme value theory

can be used to improve VaR estimates and to deal with situations where

the VaR confidence level is very high. It provides a way of smoothing and

extrapolating the tails of an empirical distribution.

The Key Result

A key result in extreme value theory was proved by Gnedenko in 1943.

This concerns the properties of the tails of a wide range of different

probability distributions.

Suppose that F(x) is the cumulative distribution function for a variable

x and that u is a value of x in the right-hand tail of the distribution. The

probability that x lies between u and u + y (y > 0) is F(u + y) — F(u).

The probability that x is greater than u is 1 — F(u). Define F

(y) as the

probability that x lies between u and u + y conditional on x > u. This is

The variable F

(y) defines the right tail of the probability distribution. It

is the cumulative probability distribution for the amount by which x

exceeds u given that it does exceed u.

Gnedenko's result states that, for a wide class of distributions F(x), the

distribution of F

(y) converges to a generalized Pareto distribution as the

threshold u is increased. The generalized Pareto distribution is

The distribution has two parameters that have to be estimated from the

data. These are and The parameter is the shape parameter and

See D.V. Gnedenko, "Sur la distribution limite du terme d'une serie aleatoire," Ann-

Math., 44 (1943), 423-453.

VaR: Historical Simulation Approach

225

determines the heaviness of the tail of the distribution. The parameter is

a scale parameter.

When the underlying variable x has a normal distribution, = 0.

the tails of the distribution become heavier, the value of increases. For

most financial data, is positive and in the range 0.1 to 0.4.

Estimating and

The parameters and can be estimated using maximum-likelihood

methods (see Section 5.9 for a discussion of these methods). The

probability density function of the cumulative distribution in

equation (9.3) is calculated by differentiating with respect to y. It is

We first choose a value for u. This could be a value close to the 95 percentile

point of the empirical distribution. We then rank the observations on x

from the highest to the lowest and focus our attention on those observa-

tions for which x > u. Suppose there are n

such observations and they are

i n

). The likelihood function (assuming that 0) is

Maximizing this function is the same as maximizing its logarithm:

Standard numerical procedures can be used to find the values of and

that maximize this expression.

When = 0, the generalized Pareto distribution becomes

One of the properties of the distribution in equation (9.3) is that the kth moment E(x

)

x is infinite for For a normal distribution, all moments are finite. When

= 0.25, only the first three moments are finite; when = 0.5, only the first moment is

finite; and so on.

226

Chapter 9

Estimating the Tail of the Distribution

The probability that x > u + y conditional that x > u is The

probability that x > u is 1 — F(u). The unconditional probability that

x > u + y is therefore

If n is the total number of observations, an estimate of 1 — F{u) calcu-

lated from the empirical data is n

/n. The unconditional probability that

x > u + y is therefore

This means that our estimator of the tail of the cumulative probability

distribution of x when x is large is

Equivalence to the Power Law

If we set equation (9.5) reduces to

so that the probability of the variable being greater than x is

where

and This shows that equation (9.5) is consistent with the power

law introduced in Section 5.4.

The Left Tail

The analysis so far has assumed that we are interested in the right tail of

the probability distribution. If we are interested in the left tail, we can use

the methodology just presented on the variable —x.

VaR: Historical Simulation Approach

227

Calculation of VaR

To calculate VaR with a confidence level of q it is necessary to solve the

equation

From equation (9.5), we have

so that

9.5 APPLICATION

We now illustrate the results in the previous section using data for daily

returns on the S&P 500 between July 11, 1988, and July 10, 1998. During

this period the total number of observations, n, on the daily return were

2,256 and the observations ranged from —6.87% to +5.12%. We consider

the left tail of the distribution of returns. This means that in the equations

given above the variable x is the negative of the daily return on the

S&P 500. We choose a value for u equal to 0.02. There were a total of

28 returns less than —2%. This means than n

= 28. The returns and the

x-values are shown in the first two columns of Table 9.3. The third

column shows the value of

for particular values of and (The test values of and in Table 9.3

are = 0.2 and = 0.01.) The sum of the numbers in the third column is

the log-likelihood function in equation (9.4). Once we have set up the

spreadsheet, we search for the best-fit values of and that maximize the

tog-likelihood function.

It turns out that these are

= 0.3232, = 0.0055

and maximum log-likelihood is 108.48.

The Solver routine in Excel works well provided that the spreadsheet is set up so that

the values being searched for are similar in magnitude. In this example, we could set up

the spreadsheet to search for and

Daily return

-0.068667

-0.061172

-0.036586

-0.034445

-0.031596

-0.030827

-0.029979

-0.029654

-0.029084

-0.027283

-0.025859

-0.025364

-0.024675

-0.024000

-0.023485

-0.023397

-0.023234

-0.022675

-0.022542

-0.022343

-0.022249

-0.022020

-0.021813

-0.021025

-0.020843

-0.020625

-0.020546

-0.020243

0.068667

0.061172

0.036586

0.034445

0.031596

0.030827

0.029979

0.029654

0.029084

0.027283

0.025859

0.025364

0.024675

0.024000

0.023485

0.023397

0.023234

0.022675

0.022542

0.022343

0.022249

0.022020

0.021813

0.021025

0.020843

0.020625

0.020546

0.020243

0.5268

1.0008

2.8864

3.0825

3.3537

3.4291

3.5133

3.5460

3.6035

3.7893

3.9403

3.9937

4.0689

4.1434

4.2009

4.2108

4.2291

4.2925

4.3076

4.3304

4.3412

4.3676

4.3915

4.4835

4.5049

4.5306

4.5400

4.5761

106.1842

Trial estimatesof EVT parameters

0.2

0.01

228 Chapter 9

Table 9.3 Estimation of extreme value theory parameters.

VaR: Historical Simulation Approach 229

Suppose that we wish to estimate the probability that x will be less than

0.04. From equation (9.5), this is

This means that we estimate the probability that the daily return will be

less than -4% to be 1 - 0.9989 = 0.0011. (This is more accurate than an

estimate obtained by counting observations.) The probability that x will

be less than 0.06 is similarly 0.9997. This means that we estimate the

probability that the daily return will be less than -6% to be

1 - 0.9997 = 0.0003.

From equation (9.6) the value of the one-day 99% VaR for a portfolio

where $1 million is invested in the S&P 500 is $1 million times

or $21,200. More generally, our estimate of the one-day 99% VaR for a

portfolio invested in the S&P 500 is 2.12% of the portfolio value.

Choice of u

A natural question is how the results depend on the choice of u. In our

example the values of and do depend on u, but the estimates of F(x)

remain roughly the same. For example, if we choose u = 0.015, the best-

fit values of and are 0.264 and 0.0046, respectively. The estimate for

F(x) when x = 0.04 and x = 0.06 are 0.9989 and 0.9997 (much the same

as before). The estimate of VaR also does not change too much provided

that the confidence level is not too high. The one-day 99% VaR for an

investment in the S&P 500 when u = 0.015 is 2.13% (compared with

2.12% when u = 0.02) of the value of the portfolio.

SUMMARY

Historical simulation is a very popular approach for estimating VaR. It

involves creating a database consisting of the daily movements in all

market variables over a period of time. The first simulation trial assumes

that the percentage change in each market variable is the same as that

on the first day covered by the database; the second simulation trial