Hull J.C. Risk management and Financial institutions

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assumes that the percentage changes are the same as those on the second

day; and so on. The change in the portfolio value, is calculated for

each simulation trial, and VaR is calculated as the appropriate percentile

of the probability distribution of The procedure assumes that the

future will in some sense be like the past. The standard error for a VaR

that is estimated using historical simulation tends to be quite high. The

higher the VaR confidence level required, the higher the standard error.

There are a number of extensions of the basic historical simulation

approach. The weights given to observations can be allowed to decrease

exponentially as we look further and further into the past. Volatility

updating schemes can be used to take account of differences between the

volatilities of market variables today and their volatilities at different times

during the period covered by the historical data.

Extreme value theory is a way of smoothing the tails of the probability

distribution of portfolio daily changes calculated using historical simula-

tion. It leads to estimates of VaR that reflect the whole shape of the tail of

the distribution, not just the positions of a few losses in the tails. Extreme

value theory can also be used to estimate VaR when the VaR confidence

level is very high. For example, even if we have only 500 days of data, it

could be used to come up an estimate of VaR for a VaR confidence level

of 99.9%.

FURTHER READING

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

Risk, 11 (May 1998): 64-67.

Embrechts, P., C. Kluppelberg, and T. Mikosch, Modeling Extremal Events for

Insurance and Finance. New York: Springer, 1997.

Hendricks, D., "Evaluation of Value-at-Risk Models Using Historical Data,'

Economic Policy Review, Federal Reserve Bank of New York, Vol. 2 (April

1996): 39-69.

Hull, J. C, 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.

McNeil, A. J., "Extreme Value Theory for Risk Managers," in Internal Modeling

and CAD II, Risk Books, 1999. See also: www.math.ethz.ch/~mcneil.

Neftci, S.N., "Value at Risk Calculations, Extreme Events and Tail

Estimation," Journal of Derivatives, 1, No. 3 (Spring 2000): 23-38.

VaR: Historical Simulation Approach

231

QUESTIONS AND PROBLEMS (Answers at End of Book)

9.1. What assumptions are being made when VaR is calculated using the

historical simulation approach and 500 days of data.

9.2. Show that when approaches 1, the weighting scheme in Section 9.3

approaches the basic historical simulation approach.

9.3. Calculate the 1-day 99% VaR on February 10, 2006, for a £100 million

portfolio invested in the FTSE 100 Index. Use the previous 1,000 days of

data (available on the author's website) and historical simulation.

9.4. Repeat Problem 9.3 using the exponential weighting scheme in Section 9.3

with = 0.99.

9.5. Repeat Problem 9.3 using the volatility updating scheme discussed in

Section 9.3. Use EWMA with = 0.94 to update volatilities. Assume that

the volatility is initially equal to the standard deviation of daily returns

calculated from the whole sample.

9.6. How does extreme value theory modify your answer to Problem 9.3. Try

values of u equal to 0.005, 0.01, and 0.015.

9.7. Suppose we estimate the 1-day 95% VaR from 1,000 observations as

$5 million. By fitting a standard distribution to the observations, the

probability density function of the loss distribution at the 95% point is

estimated to be 0.01 when losses are measured in millions. What is the

standard error of the VaR estimate?

ASSIGNMENT QUESTIONS

9.8. Values for the NASDAQ composite index during the 1,500 days preceding

March 10, 2006, can be downloaded from the author's website. Calculate

the 1-day 99% VaR on March 10, 2006, for a $10 million portfolio invested

in the index using (a) the basic historical simulation approach, (b) the

exponential weighting scheme in Section 9.3 with = 0.995, (c) the volatility

updating scheme in Section 9.3 with = 0.94 (assume that the volatility is

initially equal to the standard deviation of daily returns calculated from the

whole sample), (d) extreme value theory with u = 0.03, (e) a model where

daily returns are assumed to be normally distributed (use both an approach

where observations are given equal weights and the EWMA approach with

= 0.94 to estimate the standard deviation of daily returns). Discuss the

reasons for the differences between the results you get.

9.9. Suppose that a 1-day 97.5% VaR is estimated as $13 million from 2,000

observations. The observation on the 1-day changes are approximately

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Chapter 9

normal with mean 0 and standard deviation $6 million. Estimate a 99%

confidence interval for the VaR estimate.

Market Risk VaR:

Model-Building

Approach

The main alternative to the historical simulation approach for estimating

VaR for market risk is what is known as the model-building approach or

the Variance-covariance approach. In this approach, we assume a model

for the joint distribution of changes in market variables and use histor-

ical data to estimate the model parameters.

The model-building approach is based on ideas pioneered by Harry

Markowitz. We used these ideas for assessing the risk-return trade-offs

in portfolios of stocks in Section 1.1. They can also be used to calculate

VaR. Estimates of the current levels of the variances and covariances of

market variables are made using the approaches described in Chapters 5

and 6. If the probability distributions of the daily percentage changes in

market variables are assumed to be normal and the dollar change in the

value of the portfolio is assumed to be linearly dependent on percentage

changes in market variables, VaR can be obtained very quickly.

In this chapter we explain the model-building approach and show how

it can be used for portfolios consisting of stocks, bonds, forward con-

tracts, and interest rate swaps. We discuss attempts to extend it to

situations where the portfolio is not linearly dependent on the market

Variables and to situations where the distributions of daily percentage

changes in market variables are not normal. Finally, we evaluate the

strengths and weaknesses of both the model-building approach and the

historical simulation approach.

234 Chapter 10

10.1 THE BASIC METHODOLOGY

In option pricing, volatility is normally measured as "volatility per year".

When using the model-building approach to calculate VaR, we usually

measure the volatility of an asset as "volatility per day". As explained in

Section 5.1, the relationship between the volatility per day and the volatility

per year is

where is the volatility per year and is the corresponding volatility

per day. This equation shows that the daily volatility is about 6% of

annual volatility.

As also pointed out in Chapter 5, is approximately equal to the

standard deviation of the percentage change in the asset price in one day.

For the purposes of calculating VaR, we assume exact equality. We define

the daily volatility of an asset price (or any other variable) as equal to the

standard deviation of the percentage change in one day.

Single-Asset Case

We now consider how VaR is calculated using the model-building

approach in a very simple situation where the portfolio consists of a

position in a single stock. The portfolio we consider is one consisting of

$10 million in shares of Microsoft. We suppose that the time horizon is

ten days and the VaR confidence level is 99%, so that we are interested in

the loss level over ten days that we are 99% confident will not be

exceeded. Initially, we consider a one-day time horizon.

We assume that the volatility of Microsoft is 2% per day (correspond-

ing to about 32% per year). Because the size of the position is $10 million,

the standard deviation of daily changes in the value of the position is 2%

of $10 million, or $200,000.

It is customary in the model-building approach to assume that the

expected change in a market variable over the time period considered is

zero. This is not exactly true, but it is a reasonable assumption. The

expected change in the price of a market variable over a short time period

is generally small when compared to the standard deviation of the

change. Suppose, for example, that Microsoft has an expected return of

20% per annum. Over a one-day period, the expected return is 0.20/252,

or about 0.08%, whereas the standard deviation of the return is 2%. Over

a ten-day period, the expected return is 0.08 x 10, or about 0.8%'

whereas the standard deviation of the return is or about 6.3%.

VaR Model-Building Approach 235

So far, we have established that the change in the value of the

portfolio of Microsoft shares over a one-day period has a standard

deviation of $200,000 and (at least approximately) a mean of zero. We

assume that the change is normally distributed.

Since N(—2.33) = 0.01,

this means that there is a 1% probability that a normally distributed

variable will decrease in value by more than 2.33 standard deviations.

Equivalently, it means that we are 99% certain that a normally dis-

tributed variable will not decrease in value by more than 2.33 standard

deviations. The one-day 99% VaR for our portfolio consisting of a

$10 million position in Microsoft is therefore

2.33 x 200,000 = $466,000

Assuming that the changes in Microsoft's stock price on successive days

are independent, the N-day VaR is calculated as times the one-day

VaR. The ten-day 99% VaR for Microsoft is therefore

Consider next a portfolio consisting of a $5 million position in AT&T,

and suppose the daily volatility of AT&T is 1% (approximately 16% per

year). A similar calculation to that for Microsoft shows that the standard

deviation of the change in the value of the portfolio in one day is

5,000,000 x 0.01 = 50,000

Assuming that the change is normally distributed, the one-day 99% VaR is

50,000x2.33 = $116,500

and the ten-day 99% VaR is

Two-Asset Case

Now consider a portfolio consisting of both $10 million of Microsoft

shares and $5 million of AT&T shares. We suppose that the returns on

the two shares have a bivariate normal distribution with a correlation

of 0.3. A standard result in statistics tells us that, if two variables X and Y

have standard deviations equal to and with the coefficient of

We could assume that the price of Microsoft is lognormal tomorrow. Since one day is

such a short period of time, this is almost indistinguishable from the assumption we do

make—that the change in the stock price between today and tomorrow is normal.

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Chapter 10

correlation between them being equal to the standard deviation of

X + Y is given by

To apply this result, we set X equal to the change in the value of the

position in Microsoft over a one-day period and Y equal to the change in

the value of the position in AT&T over a one-day period, so that

The standard deviation of the change in the value of the portfolio

consisting of both stocks over a one-day period is therefore

The mean change is assumed to be zero. The change is normally dis-

tributed. So the one-day 99% VaR is therefore

220,227 x 2.33 = $513,129

The ten-day 99% VaR is times this or $1,622,657.

The Benefits of Diversification

In the example we have just considered:

1. The ten-day 99% VaR for the portfolio of Microsoft shares is

$1,473,621.

2. The ten-day 99% VaR for the portfolio of AT&T shares is $368,405.

3. The ten-day 99% VaR for the portfolio of both Microsoft and

AT&T shares is $1,622,657.

The amount

(1,473,621 + 368,405) - 1,622,657 = $219,369

represents the benefits of diversification. If Microsoft and AT&T were

perfectly correlated, the VaR for the portfolio of both Microsoft and

AT&T would equal the VaR for the Microsoft portfolio plus the VaR for

the AT&T portfolio. Less than perfect correlation leads to some of the

risk being "diversified away".

VaR reflects the benefits of diversification when the distribution of the portfolio-value

changes is normal. As we saw in Section 8.3, VaR does not always reflect the benefits of

diversification. The VaR of two portfolios can be greater than the sum of their separate

VaRs.

VaR: Model-Building Approach 237

10.2 THE LINEAR MODEL

The examples we have just considered are simple illustrations of the use

of the linear model for calculating VaR. Suppose that we have a port-

folio worth P consisting of n assets with an amount being invested in

asset i (1 i n). Define as the return on asset i in one day. The

dollar change in the value of our investment in asset i in one day is

and

where is the dollar change in the value of the whole portfolio in

one day.

In the example considered in the previous section, $10 million was

invested in the first asset (Microsoft) and $5 million was invested in the

second asset (AT&T) so that (in millions of dollars) =10, =5,

and

If we assume that the in equation (10.1) are multivariate normal,

is normally distributed. To calculate VaR, we therefore need to

calculate only the mean and standard deviation of We assume, as

discussed in the previous section, that the expected value of each is

zero. This implies that the mean of is zero.

To calculate the standard deviation of we define as the daily

volatility of the ith asset and as the coefficient of correlation between

returns on asset i and asset j.

This means that is the standard

deviation of and is the coefficient of correlation between

and The variance of which we will denote by is given by

This equation can also be written as

The are sometimes calculated using a factor model (see Section 6.3).

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Chapter 10

The standard deviation of the change over N days is and the 99%

VaR for an N-day time horizon is

In the example considered in the previous section, = 0.02, =0.01,

and = 0.3. As already noted, = 10 and = 5, so that

= 10

x 0.02

+ 5

x 0.01

+ 2 x 10 x 5 x 0.3 x 0.02 x 0.01 = 0.0485

and = 0.220. This is the standard deviation of the change in the

portfolio value per day (in millions of dollars). The ten-day 99% VaR

is 2.33 x 0.220 x = $1,623 million. This agrees with the calculation

in the previous section.

10.3 HANDLING INTEREST RATES

It is not possible to define a separate market variable for every single

bond price or interest rate to which a company is exposed. Some

simplifications are necessary when the model-building approach is used.

One possibility is to assume that only parallel shifts in the yield curve

occur. It is then necessary to define only one market variable: the size of

the parallel shift. The changes in the value of bond portfolio can then be

calculated using the approximate duration relationship in equation (4.15):

where P is the value of the portfolio, is the change in P in one day, D is

the modified duration of the portfolio, and is the parallel shift in one

day. This approach gives a linear relationship between and but it

does not usually give enough accuracy because the relationship is not

exact and does not take account of nonparallel shifts in the yield curve.

The procedure usually followed is to choose as market variables the

prices of zero-coupon bonds with standard maturities: 1 month, 3 months,

6 months, 1 year, 2 years, 5 years, 7 years, 10 years, and 30 years. For the

purposes of calculating VaR, the cash flows from instruments in the

portfolio are mapped into cash flows occurring on the standard maturity

dates.

Consider a $1 million position in a Treasury bond lasting 0.8 years

that pays a coupon of 10% semiannually. A coupon is paid in 0.3 years

and 0.8 years and the principal is paid in 0.8 years. This bond is

therefore in the first instance regarded as a $50,000 position in 0.3-year

zero-coupon bond plus a $1,050,000 position in a 0.8-year zero-coupon

VaRs Model-Building Approach

239

bond. The position in the 0.3-year bond is then replaced by an equiva-

lent position in 3-month and 6-month zero-coupon bonds and the

position in the 0.8-year bond is replaced by an equivalent position in

6-month and 1-year zero-coupon bonds. The result is that the position

in the 0.8-year coupon-bearing bond is for VaR purposes regarded as a

position in zero-coupon bonds having maturities of 3 months, 6 months,

and 1 year. This procedure is known as cash-flow mapping.

Illustration of Cash-Flow Mapping

We now illustrate how cash-flow mapping works by continuing with the

example we have just introduced. It should be emphasized that the pro-

cedure we use is just one of several that have been proposed.

Consider first the $1,050,000 that will be received in 0.8 years. We

suppose that zero rates, daily bond price volatilities, and correlations

between bond returns are as shown in Table 10.1. The first stage is to

interpolate between the 6-month rate of 6.0% and the 1-year rate of 7.0%

to obtain a 0.8-year rate of 6.6% (annual compounding is assumed for all

rates). The present value of the $1,050,000 cash flow to be received in

0.8 years is

We also interpolate between the 0.1 % volatility for the 6-month bond

and the 0.2% volatility for the 1-year bond to get a 0.16% volatility for

the 0.8-year bond.

Maturity

Zero rate

(% with ann. comp.):

Bond price volatility

(% per day):

Correlation between

daily returns

3-month bond

6-month bond

1-year bond

3-month

bond

5.50

0.06

3-month

bond

1.0

0.9

0.6

6-month

bond

6.00

0.10

6-month

bond

0.9

1.0

0.7

1-year

bond

7.00

0.20

1-year

bond

0.6

0.7

1.0

Table 10.1 Data to illustrate cash-flow mapping procedure.