of our portfolio.The fifth-worst daily change is the first percentile of the distribution. The estimate of VaR is theloss when we are at this first percentile point. Assuming that the last 500 days are a good guide towhat could happen during the next day, we are 99% certain that we will not take a loss greater thanour VaR estimate.
The historical simulation methodology is illustrated in Tables 16.1 and 16.2. Table 16.1 showsobservations on market variables over the last 500 days. The observations are taken at someparticular point in time during the day (usually the close of trading). We denote the first day forwhich we have data as Day 0; the second as Day 1; and so on. Today is Day 500; tomorrow isDay 501.
Table 16.2 shows the values of the, market variables tomorrow if their percentage changesbetween today and tomorrow are the same as they were between Day i—1 and Day i for1 <i < 500. The first row in Table 16.2 shows the values of market variables tomorrow assumingtheir percentage changes between today and tomorrow are the same as they were between Day 0and Day 1; the second row shows the values of market variables tomorrow assuming theirpercentage changes between Day | and Day 2 occur; and so on. The 500 rows in Table 16.2 arethe 500 scenarios considered.
Define v; as the value of a market variable on Day i and suppose that today is Day m. The ithscenario assumes that the value of the market variable tomorrow will be
v;
 
UnU-]
In our example, m = 500. For the first variable, the value today, vsgq, is 25.85. Also vp = 20.33 andv, = 20.78. It follows that the value of the first market variable in the first scenario is20.7825.85 x 0337 26.42The final column of Table 16.2 shows the value of the portfolio tomorrow for each of the 500scenarios. The value of the portfolio today is known. Suppose this is $23.50 million. We cancalculate the change in the value of the portfolio between today and tomorrow for all the different350 CHAPTER 16
scenarios. For Scenario 1, it is +$210,000; for Scenario 2 it is -$380,000; and so on. These changesin value are then ranked. The fifth-worst loss is the one-day 99% VaR. As mentioned in the previoussection, the N-day VaR for a 99% confidence level is calculated as ./N times the one-day VaR.
Each day the VaR estimate in our example would be updated using the most recent 500 days ofdata. Consider, for example, what happens on Day 501. We find out new values for all the marketvariables and are able to calculate a new value for our portfolio. We then go through theprocedure we have outlined to calculate a new VaR. We use data on the market variables fromDay 1 to Day 501. (This gives us the required 500 observations on the percentage changes inmarket variables; the Day 0 values of the market variables are no longer used.) Similarly, onDay 502 we use data from Day 2 to Day 502 to determine VaR, and so on.
 
 
16.3 MODEL-BUILDING APPROACH
The main alternative to historical simulation is the model-building approach (sometimes alsocalled the variance-covariance approach). Before getting into the details of the approach, it isappropriate to mention one issue concerned with the units for measuring volatility.
Daily Volatilities
In option pricing we usually measure time in years, and the volatility of an asset is usually quotedas a “volatility per year”. When using the model-building approach to calculate VaR, we usuallymeasure time in days and the volatility of an asset is usually quoted as a “volatility per day”.
What is the relationship between the volatility per year used in option pricing and the volatilityper day used in VaR calculations? Let us define oy, as the volatility per year of a certain asset andOday 28 the equivalent volatility per day of the asset. Assuming 252 trading days in a year, we canuse equation (12.2) to write the standard deviation of the continuously compounded return on theasset in one year as either o,, or Oday 252. It follows that
Oyr = FgayV 252
orOyr
J252so that daily volatility is about 6% of annual volatility.
As pointed out in Section 12.4, og, is approximately equal to the standard deviation of thepercentage change in the asset price in one day. For the purposes of calculating VaR, we assumeexact equality. We define the daily volatility of an asset price (or any other variable) as equal to thestandard deviation of the percentage change in one day.
Our discussion in the next few sections assumes that we have estimates of daily volatilities andcorrelations. In Chapter 17 we explain how the estimates are produced.
Oday
Single-Asset Case
We now consider how VaR is calculated using the model-building approach in a very simplesituation where the portfolio consists of a position in a single stock. The portfolio we consider is
4 Note that the portfolio’s composition may have changed between Day 500 and Day 501.Value at Risk 351
one consisting of $10 million in shares of Microsoft. We suppose that N = 10 and X¥ = 99, so thatwe are interested in the loss level over 10 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 (corresponding to about 32% peryear). Because the size of the position is $10 million, the standard deviation of daily changes in thevalue 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 marketvariable over the time period considered is zero. This is not exactly true, but it is a reasonableassumption. The expected change in the price of a market variable over a short time period isgenerally small when compared with 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 expectedreturn is 0.20/252, or about 0.08%, whereas the standard deviation of the return is 2%. Overa 10-day period, the expected return is 0.20/25.2, or about 0.8%, whereas the standard deviationof the return is 2/10, or about 6.3%.
So far, we have established that the change in the value of the portfolio of Microsoft shares overa 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.> From the tables at the end of this book, wesee that N(—2.33) = 0.01. This means that there is a 1% probability that a normally distributedvariable will decrease in value by more than 2.33 standard deviations. Equivalently, it means thatwe are 99% certain that a normally distributed variable will not decrease in value by more than2.33 standard deviations. The 1-day 99% VaR for our portfolio consisting of a $10 millionposition in Microsoft is therefore
2.33 x 200, 000 = $466, 000
As discussed earlier, the N-day VaR is calculated as \/N times the one-day VaR. The 10-day 99%VaR for Microsoft is therefore466, 000 x V10 = $1, 473, 621
Consider next a portfolio consisting of a $5 million position in AT&T, and suppose the dailyvolatility of AT&T is 1% (approximately 16% per year), A similar calculation to that forMicrosoft shows that the standard deviation of the change in the value of the portfolio in oneday is
5,000,000 x 0.01 = 50,000
Assuming the change is normally distributed, the 1-day 99% VaR is
50,000 x 2.33 = $116,500and the 10-day 99% VaR is116,500 x /10 = $368,405
Two-Asset Case
Now consider a portfolio consisting of both $10 million of Microsoft shares and $5 million ofAT&T shares. We suppose that the returns on the two shares have a bivariate normal distributionwith a correlation of 0.3. A standard result in statistics tells us that, if two variables X and Y have
5 To be consistent with the option pricing assumption in Chapter 11, we could assume that the price of Microsoft islognormal tomorrow. Because one day is such a short period of time, this is almost indistinguishable from theassumption we do make—that the change in the stock price between today and tomorrow is normal.352 CHAPTER 16
standard deviations equal to oy and oy with the coefficient of correlation between them beingequal to p, then the standard deviation of X + Y is given by
Oxuy =y oy + OF + 2poyoy
To apply this result, we set X equal to the change in the value of the position in Microsoft over aone-day period and Y equal to the change in the value of the position in AT&T over a one-dayperiod, so that
ox = 200,000, oy = 50,000
The standard deviation of the change in the value of the portfolio consisting of both stocks over aone-day period is therefore
 
200,000 + 50,000? + 2 x 0.3 x 200,000 x 50,000 = 220,227
The mean change is assumed to be zero. The 1-day 99% VaR is therefore
220,227 X 2.33 = Sd135,129
The 10-day 99% VaR is ./10 times this or $1,622,657.
The Benefits of DiversificationIn the example we have just considered:1. The 10-day 99% VaR tor the portfolio of Microsoft shares is $1,473,621.2. The 10-day 99% VaR for the portfolio of AT&T shares is $368,405.3. The 10-day 99% VaR for the portfolio of both Microsoft and AT&T shares is $1,622,657.
The amount(1473,621 + 368,405) — 1,622,657 = $219,369
represents the benefits of diversification. If Microsoft and AT&T were perfectly correlated, the VaRfor the portfolio of both Microsoft and AT&T would equal the VaR for the Microsoft portfolioplus the VaR for the AT&T portfolio. Less than perfect correlation leads to some of the risk being“diversified away.” ©
16.4 LINEAR MODEL
 
The examples we have just considered are simple illustrations of the use of the linear model forcalculating VaR. Suppose that we have a portfolio worth P consisting of » assets with an amount q;being invested in asset i (1 < i <n). We define 6x; as the return on asset i in one day. It follows that
© Harry Markowitz was one of the first researchers to study the benefits of diversification to a portfolio manager, Hewas awarded a Nobel prize for this research in 1990, See H. Markowitz, “Portfolio Selection,” Journal of Finance, 7,no, 1 (March 1952), 77-91.Value at Risk 353
 
the dollar change in the value of our investment in asset i in one day is a; dx; andn8P =) a,x; (16.1)i=1
where 5P 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)a, = 10, a, = 5, and
bP = 10x; + 58x5
If we assume that the dx; in equation (16.1) are multivariate normal, then SP is normallydistributed. To calculate VaR, we therefore need to calculate only the mean and standard deviationof 5P. We assume, as discussed in the previous section, that the expected value of each 8x, is zero.This implies that the mean of 8P is zero.
To calculate the standard deviation of 5P, we define o; as the daily volatility of the ith assetand ,; as the coefficient of correlation between returns on asset i and asset j. This means that a; isthe standard deviation of 5x;, and p,,; is the coefficient of correlation between éx; and 6x,. Thevariance of 5P, which we will denote by o7, is given by
non2op= > PjjgjO jO;;i=l j=lThis equation can also be written
aps y aro; +2 ” > Pj jOljO jFjO; (16.2)i=1
i=l jai
The standard deviation of the change over N days is op./N, and the 99% VaR for an N-day timehorizon is 2.330p/N.
In the example considered in the previous section, a; = 0.02, a, =0.01, and pj). =0.3. Asalready noted, a; = 10 and a = 5, so that
op = 10? x 0.027 + 5? x 0.017 +2 x 10x 5x 0.3 x 0.02 x 0.01 = 0.0485
and op = 0.220. This is the standard deviation of the change in the portfolio value per day(in millions of dollars). The 10-day 99% VaR is 2.33 x 0.220 x /10 = $1.623 million. This agreeswith the calculation in the previous section.
Handling tnterest Rates
It is out of the question to define a separate market variable for every single bond price or interesttate to which a company is exposed. Some simplifications are necessary. One possibility is toassume that only parallel shifts in the yield curve occur. It is then necessary to define only onemarket variable: the size of the parallel shift. The changes in the value of a bond portfolio can thenbe caiculated using the duration relationship
8P =—DP by354 CHAPTER 16
where P is the value of the portfolio, 6P is the change in P in one day, D is the modified durationof the portfolio, and dy is the parallel shift in one day.
This approach does not usually give enough accuracy. The procedure usually followed is tochoose as market variables the prices of zero-coupon bonds with standard maturities: | month,3 months, 6 months, | year, 2 years, 5 years, 7 years, 10 years, and 30 years. For the purposes ofcalculating VaR, the cash flows trom instruments in the portfolio are mapped into cash flowsoccurring on the standard maturity dates.
Consider a $1 million position in a Treasury bond lasting 1.2 years that pays a coupon of 6%semiannually. Coupons are paid in 0.2, 0.7, and |.2 years, and the principal is paid in 1.2 years. Thisbond is therefore, in the first instance, regarded as a $30,000 position in 0.2-year zero-coupon bondplus a $30,000 position in a 0.7-year zero-coupon bond plus a $1.03 million position in a 1.2-yearzero-coupon bond. The position in the 0.2-year bond is then replaced by an equivalent position inl-month and 3-month zero-coupon bonds; the position in the 0.7-year bond is replaced by anequivalent position in 6-month and 1-year zero-coupon bonds; and the position in the 1.2-year bondis replaced by an equivalent position in |-year and 2-year zero-coupon bonds. The result is that theposition in the |.2-year coupon-bearing bond is for VaR purposes regarded as a position in zero-coupon bonds having maturities of 1 month, 3 months, 6 months, | year, and 2 years.
This procedure is known as cash-flow mapping. One way of implementing the procedure isexplained in Appendix 16A.
Applications of the Linear Model
The simplest application of the linear model is to a portfolio with no derivatives consisting ofpositions in stocks, bonds, foreign exchange, and commodities. In this case the change in thevalue of the portfolio is linearly dependent on the percentage changes in the prices of the assetscomprising the portfolio. Note that, for the purposes of VaR calculations, all asset prices aremeasured in the domestic currency. The market variables considered by a large bank in theUnited States are therefore likely to include the Nikkei 225 index measured in dollars, the price ofa 10-year sterling zero-coupon bond measured in dollars, and so on.
An example of a derivative that can be handled by the linear model is a forward contract to buya foreign currency. Suppose the contract matures at time T. It can be regarded as the exchange of aforeign zero-coupon bond maturing at time T for a domestic zero-coupon bond maturing at time T.For the purposes of calculating VaR, the forward contract is therefore treated as a long position inthe foreign bond combined with a short position in the domestic bond. Each bond can be handledusing a cash-flow mapping procedure.
Consider next an interest rate swap. As explained in Chapter 6, this can be regarded as theexchange of a floating-rate bond for a fixed-rate bond. The fixed-rate bond is a regular coupon-bearing bond. The floating-rate bond is worth par just after the next payment date. It can beregarded as a zero-coupon bond with a maturity date equal to the next payment date. The interestrate swap therefore reduces to a portfolio of long and short positions in bonds and can be handledusing a cash-flow mapping procedure.
The Linear Model and Options
We now consider how the linear model can be used when there are options. Consider first aportfolio consisting of options on a single stock whose current price is S. Suppose that the delta ofthe position (calculated in the way described in Chapter 14) is A. Because A is the rate of change ofValue at Risk 355
the value of the portfolio with S, it is approximately true that
A _ 6P~ 8SorSP=A8S (16.3)
where 8S is the dollar change in the stock price in one day and §P is, as usual, the dollar change inthe portfolio in one day. We define 5x as the percentage change in the stock price in one day, so that
adTt follows that an approximate relationship between §P and dx is5P= SAd&x
When we have a position in several underlying market variables that includes options, we canderive an approximate linear relationship between 5P and the 8x,’s similarly. This relationship is
x (16.4)i (16.4)
where S; is the value of the ith market variable and 4; is the delta of the portfolio with respect tothe ith market variable. This corresponds to equation (16.1):
dP =) a; 8x; (16.5)f=l
with a; = S; A;. Equation (16.2) can therefore be used to calculate the standard deviation of 8P.
Example16.1_ A portfolio consists of options on Microsoft and AT&T. The options on Microsofthave a delta of 1,000, and the options on AT&T have a delta of 20,000. The Microsoft share price is$120, and the AT&T share price is $30. From equation (16.4), it is approximately true that
8P = 120 x 1,000 x dx, + 30 x 20,000 x dxor8P = 120,0008x, + 600,0008x,
where 6x, and dx; are the returns from Microsoft and AT&T in one day and $P is the resultantchange in the value of the portfolio. (The portfolio is assumed to be equivalent to an investment of$120,000 in Microsoft and $600,000 in AT&T.) Assuming that the daily volatility of Microsoft is 2%and the daily volatility of AT&T is 1%, and the correlation between the daily changes is 0.3, thestandard deviation of 5P (in thousands of dollars) is
 
¥(120 x 0,02)? + (600 x 0.01)? +2 x 120 x 0.02 x 600 x 0.01 x 0.3 = 7.099
Because N(—1.65) = 0.05, the 5-day 95% value at risk is
1.65 x V5 x 7,099 = $26, 193356 CHAPTER 16
 
16.5 QUADRATIC MODEL
 
When a portfolio includes options, the linear model is an approximation. It does not take accountof the gamma of the portfolio. As discussed in Chapter 14, delta is defined as the rate of change ofthe portfolio value with respect to an underlying market variable and gamma is defined as the rateof change of the delta with respect to the market variable. Gamma measures the curvature of therelationship between the portfolio value and an underlying market variable.
Figure 16.3 shows the impact of a nonzero gamma on the probability distribution of the value ofthe portfolio. When gamma is positive, the probability distribution tends to be positively skewed;when gamma is negative, it tends to be negatively skewed. Figures 16.4 and 16.5 illustrate thereason for this result. Figure 16.4 shows the relationship between the value of a Jong call optionand the price of the underlying asset. A long call is an example of an option position with positivegamma. The figure shows that, when the probability distribution for the price of the underlyingasset at the end of one day is normal, the probability distribution for the option price is positivelyskewed.’ Figure 16.5 shows the relationship between the value of a short call position and the priceof the underlying asset. A short call position has negative gamma. In this case we see that a normaldistribution for the price of the underlying asset at the end of one day gets mapped into anegatively skewed distribution for the vaiue of the option position.
The VaR for a portfolio is critically dependent on the left tail of the probability distribution ofthe portfolio value. For example, when the confidence level used is 99%, the VaR is calculatedfrom the value in the left tail below which there is only 1% of the distribution. As indicated inFigures 16.3a and 16.4, a positive-gamma portfolio tends to have a less heavy left tail than thenormal distribution. If we assume the distribution is normal, we will tend to calculate a VaR that istoo high. Similarly, as indicated in Figures 16.3b and 16.5, a negative-gamma portfolio tends tohave a heavier ieft tail than the normal disiribution. If we assume the distribution is normal, wewill tend to calculate a VaR that is too low.
For a more accurate estimate of VaR than that given by the linear model, we can use both deltaand gamma measures to relate 5P to the 5x;’s. Consider a portfolio dependent on a single assetwhose price is S. Suppose that the delta of a portfolio is A and its gamma is I. FromAppendix 14A, the equation
8P = ASS + iT (8SY
(a) . (b)Figure 16.3 Probability distribution for value of portfolio: (a) positive gamma, (b) negative gamma
7 As mentioned in footnote 5, we can use the normal distribution as an approximation to the lognormal distributionin VaR calculations.Value at Risk 357
 
           
Value oflong call
Underlying asset
Figure 16.4 Transiation of normai probability distribution for asset into probabilitydistribution for value of a long call on asset
Value ofshort call
Underlying asset_
 
wee enn dene
a ee
\
Figure 16.5 Translation of normal probability distribution for asset into probabilitydistribution for value of a short call on asset358 CHAPTER 16
is an improvement over the approximation in equation (16.3).® Setting
bS6x = —FESreduces this to8P = SA8x + 48°F (8x)? (16.6)
The variable 5P is not normal. Assuming that 5x is normal with mean zero and standard deviation o,the first three moments of 5P areE(8P) = 4S°To?
E[(8P)] = S770? +4384P7o4E((8P)*] = 284A7To4 + 5°T308
The first two moments can be fitted to a normal distribution. This is better than ignoring gammaaltogether, but, as already pointed out, the assumption that dP is normal is less than ideal. Analternative approach is to use the three moments in conjunction with the Cornish—Fisher expansionas described in Appendix 16B.?
For a portfolio with n underlying market variables, with each instrument in the portfolio beingdependent on only one of the market variables, equation (16.6) becomes
8P = }~ SA; 8x; + ¥ 4.571; (8x)
i=1 i=l
where S; is the value of the ith market variable, and A; and I; are the delta and gamma of theportfolio with respect to the ith market variable. When individual instruments in the portfolio maybe dependent on more than one market variable, this equation takes the more general form
 
a n na8P = >> S,A; 8x; + 5S STi; 8x; 8x; (16.7)i=] i=l j=lwhere T’,; is a “cross gamma” defined asro vP7 a8; aS;
Equation (16.7) can be written as
6P= ya 5x; +20 Bu bx; (16.8)i=1
fai j=lwhere a; = S;A; and 6; = 45,51 j;.
® The Taylor series expansion in Appendix 14A suggests the approximation6P = O51 + ASS + {Psywhen terms of order higher than ér are ignored. In practice the © dt term is so small that it is usually ignored.
° The Cornish—Fisher expansion provides a relationship between the moments of a distribution and its percentiles.For a description, see N. L. Johnson and 8. Kotz, Distributions in Statistics: Continuous Univariate Distributions 1,Wiley, New York, 1972.Value at Risk 359
Equation (16.8) is not as easy to work with as equation (16.5), but it can be used to calculatemoments for 6P. Appendix 16B gives the first three moments and describes how they can be usedin conjunction with the Cornish—-Fisher expansion to estimate the percentiles of the probabilitydistribution of 5P that correspond to the required VaR.
16.6 MONTE CARLO SIMULATION
 
As an alternative to the approaches described so far, we can implement the model-buildingapproach using Monte Carlo simulation to generate the probability distribution for 6P. Supposewe wish to calculate a one-day VaK for a portfolio. Ihe procedure is as follows:
I. Value the portfolio today in the usual way using the current values of market variables.
2. Sample once from the multivariate normal probability distribution of the dx,’s.'°
3. Use the values of the 6x,’s that are sampled to determine the value of each market variable atthe end of one day.
4. Revalue the portfolio at the end of the day in the usual! way.5. Subtract the value calculated in step 1 from the value in step 4 to determine a sample 5P.6. Repeat steps 2 to 5 many times to build up a probability distribution for 8P.
The VaR is calculated as the appropriate percentile of the probability distribution of 6P. Suppose,for example, that we calculate 5,000 different sample values of 8P in the way just described. TheI-day 99% VaR is the value of 5P for the 50th worst outcome; the 1-day VaR 95% is the value of 6Pfor the 250th worst outcome: and so on.!! The N-day VaR is usually assumed to be the 1-day VaRmultiplied by /N.!?
The drawback of Monte Carlo simulation is that it tends to be slow because a company’scomplete portfolio (which might consist of hundreds of thousands of different instruments) has tobe revalued many times.'? One way of speeding things up is to assume that equation (16.8)describes the relationship between 5P and the 5x;’s. We can then jump straight from step 2 tostep 5 in the Monte Carlo simulation and avoid the need for a complete revaluation of theportfolio. This is sometimes referred to as the partial simulation approach.
16.7 COMPARISON OF APPROACHES
 
We have discussed two methods for estimating VaR: the historical simulation approach and themodel-building approach. The advantages of the model-building approach are that results can be
‘© One way of doing so is given in Chapter 18.
"| Extreme value theory provides a way of “smoothing the tails” so that better estimates of extreme percentiles areobtained. See, for example, P. Embrechts, C. Kluppelberg, and T. Mikosch, Modeling Extremal Events for Insuranceand Finance, Springer, New York, 1997.
"2 This is only approximately true when the portfolio includes options, but it is the assumption that is made inpractice for all VaR calculation methods.
8 Ay approach for limiting the number of portfolio revaluations is proposed in F. Jamshidian and Y. Zhu,“Scenario Simulation Model: Theory and Methodology,” Finance and Stochastics, 1 (1997), 43-67.360 CHAPTER 16
produced very quickly and it can be used in conjunction with volatility updating schemes such asthose we will describe in the next chapter. The main disadvantage of the model-building approachis that it assumes that the market variables have a multivariate normal distribution. In practice,daily changes in market variables often have distributions that are quite different from normal (see,for example, Table 15.1).
The historical simulation approach has the advantage that historical data determine the jointprobability distribution of the market variables. It also avoids the need for cash-flow mapping (seeProblem 16.2). The main disadvantages of historical simulation are that it is computationally slowand does not easily allow volatility updating schemes to be used.
One disadvantage of the model-building approach is that it tends to give poor results for low-delta portfolios (see Problem 16.22).
16.8 STRESS TESTING AND BACK TESTING
 
 
In addition to calculating a VaR, many companies carry out what is known as a stress test of theirportfolio. Stress testing involves estimating how the portfolio would have performed under some ofthe most extreme market moves seen in the last 10 to 20 years.
For example, to test the impact of an extreme movement in U.S. equity prices, a company mightset the percentage changes in all market variables cqual to those on October 19, 1987 (when theS&P 500 moved by 22.3 standard deviations). If this is considered to be too extreme, the companymight choose January 8, 1988 (when the S&P 500 moved by 6.8 standard deviations). To test theeffect of extreme movements in U.K. interest rates, the company might set the percentage changesin all market variables equal to those on