0 CHAPTER 16
APPENDIX 16B
Use of the Cornish—Fisher Expansion to Estimate VaR
As shown in equation (16.8), we can approximate 6P for a portfolio containing options as
sP = dards, + 2 Bd 8x, (16B.1)i=l
i=l jalDefine o,; as the covariance between variable i and j:O75 = Pij%O7;
It can be shown that when the dx; are multivariate normal
E(6P) = So Bie;ij
ESP) 1 = So oeejo5 + Y BijBal ijn + non + Ono pH)iy i fal
EU(6P)'} =3 0; Bg (Oi ju + ORO 1 OO j—) + > Bi is Bisig Bisig 2
ijkl Fd adzdgis.te
The variable Q consists of fifteen terms of the form 0%, 04,4, %ksk¢+ Where the ky, kz, kz, ky, ks, keare permutations of i). i. 13, ig. is, ig.Define 4p and op as the mean and standard deviation of 5P, so that
Up = EP)op = El(6P)"] — [EPPThe skewness, Ep, of the probability distribution of 5P is defined as
E((6P)*] — 3EUSPY Yup + 2p
1Ep =—z E[(SP — uy] = 5Op opUsing the first three moments of 6P, the Cornish—Fisher expansion estimates the gth percentile ofthe distribution of 6P asHp + WaowhereWy = 2q + $25 — Dep
and z, is the gth percentile of the standard normal distribution (0, 1).
Example 16.2 Suppose that for a certain portfolio we calculate zp =—0.2, op = 2.2, andEp = —0.4. If we assume that the probability distribution of 5P is normal, the first percentile ofthe probability distribution of 6P is
—0.2 — 2.33 x 2.2 = —5.326Value at Risk 371
 
In other words, we are 99% certain that5P > —5.326
When we use the Cornish-Fisher expansion to adjust for skewness and set gq = 0.01, we obtainwy = 2.33 —2(2.33 — 1) x 0.4 = 2.625so that the first percentile of the distribution is—0.2 — 2.625 x 2.2 = —5.976
Taking account of skewness, therefore, changes the VaR from 5.326 to 5.976.CHAPTER 17
ESTIMATING VOLATILITIES ANDCORRELATIONS
In this chapter we explain how historical data can be used to produce estimates of the current andfuture levels of volatilities and correlations. The chapter is relevant both to the calculation of valueat risk using the model-building approach and to the valuation of derivatives. When calculatingvalue at risk, we are most interested in the current levels of volatilities and correlations because weare assessing possible changes in the value of a portfolio over a very short period of time. Whenvaluing derivatives, forecasts of volatilities and correlations over the whole life of the derivative areusually required.
The chapter considers models with imposing names such as exponentially weighted movingaverage (EWMA), autoregressive conditional hetcrosecdasticity (ARCH), and generalized auto-regressive Conditional heteroscedasticity (GARCH). The distinctive feature of the models is thatthey recognize that volatilities and correlations are not constant. During some periods a particularvolatility or correlation may be relatively low, whereas during other periods it may be relativelyhigh. The models attempt to keep track of the variations in the volatility or correlation throughtime.
17.1_ ESTIMATING VOLATILITY
 
 
Define o,, as the volatility of a market variable on day n, as estimated at the end of day n — 1. Thesquare of the volatility, 02, on day n is the variance rate.
We described the standard approach to estimating o,, from historical data in Section 12.4.Suppose that the value of the market variable at the end of day i is S;. The variable u; is defined asthe continuously compounded return during day i (between the end of day i — | and the end of
day #):(5)uj =n w=i=
An unbiased estimate of the variance rate per day, 07, using the most recent m observations on theUu; 1S
Rol idle _;- ay (47.1)
372Estimating Volatilities and Correlations 373
 
where # is the mean of the u,’s:
For the purposes of calculating VaR, the formula in equation (17.1) is usually changed in anumber of ways:
1. u; is defined as the percentage change in the market variable between the end of day i- 1and the end of day i, so that!ue S; — Sint
j Tar (17.2)
2
$
2. # is assumed to be zero.23.
— 1 is grenlaced by a37 — 1 is repiaccd
by i.
These three changes make very little difference to the variance estimates that are calculated, butthey allow us to simplify the formula for the variance rate to
2 I 2y . 17.3On Un; ( )
where u; 18 given by equation (17.2).4
Weighting Schemes
Equation (17.3) gives equal weight to all u?’s. Our objective is to estimate the current level ofvolatility, o,,. It therefore makes sense to give more weight to recent data. A model that does this is
on = aur; (17.4)i=]
The variable @; is the amount of weight given to the observation i days ago. The a’s are positive. Ifwe choose them so that a; < a; when i > j, less weight is given to older observations, The weightsmust sum to unity, so that
mYa =It=]
' This is consistent with the point made in Section 16.3 about the way that volatility is defined for the purposes ofVaR calculations.
? As explained in Section 16.3, this assumption usually has very little effect on estimates of the variance because theexpected change in a variable in one day is very small when compared with the standard deviation of changes.
3 Replacing m — 1 by m moves us from an unbiased estimate of the variance to a maximum likelihood estimate.Maximum likelihood estimates are discussed later in the chapter.
* Note that the w’s in this chapter play the same role as the 5x’s in Chapter 16. Both are daily percentage changes inmarket variables. In the case of the w’s, the subscripts count observations made on different days on the same marketvariable. In the case of the 5x’s, they count observations made on the same day on different market variables. The useof subscripts for o is similarly different between the two chapters. In this chapter, the subscripts refer to days; inChapter 16, they referred to market variables.374 CHAPTER 17
An extension of the idea in equation (17.4) is to assume that there is a long-run average variancerate and that this should be given some weight. This leads to the model that takes the form
mon = Ve + Yau; j (17.5)i=l
where V, is the long-run variance rate and y is the weight assigned to V,. Because the weights mustsum to unity,myt >> a= 1i=1
This is known as an ARCHG@n) model. It was first suggested by Engle.> The estimate of thevariance is based on a long-run average variance and m observations. The older an observation,the less weight it is given. Defining w = yV_,, the model in equation (17.5) can be written
o=ot Seti (17.6)i=l
This is the version of the model that is used when parameters are estimated.In the next two sections we discuss two important approaches to monitoring volatility using theideas in equations (17.4) and (17.5).
17.2. THE EXPONENTIALLY WEIGHTED MOVINGAVERAGE MODEL
 
 
The exponentially weighted moving average (EWMA) model is a particular case of the model inequation (17.4) where the weights, a;, decrease exponentially as we move back through time.Specifically, a,.; = A@;, where A is a constant between zero and one.
It turns out that this weighting scheme leads to a particularly simple formula for updatingvolatility estimates. The formula is
of =hot_, t+ (1-22, (7.72)
The estimate o, of the volatility for day » (made at the end of day # — 1) is calculated from o,_,(the estimate that was made at the end of day n ~ 2 of the volatility for day n — 1) and u,_, (themost recent daily percentage change in the market variable).
To understand why equation (17.7) corresponds to weights that decrease exponentially, wesubstitute for o2_, to get
Oy — Alon 2 +L = Adup-a] + (1 = 2)orOn = (1 = AYU + Atta) + Pon 2
> See R. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation,”Econometrica, 50 (1982), 987-1008.Estimating Volatilities and Correlations 375
Substituting in a similar way for 07_» givesoF = (1 AN + Aten +P u3_4) + ores
Continuing in this way, we see thatuu .Oe = (LA) a hs + Oei=]
For large m, the term 2”"o7_,, is sufficiently small to be ignored, so that equation (17.7) is the same
as equation (17.4) with a, = (1 — yar, The weights for the u,’s decline at rate A as we move backthrough time. Each weight is A times the previous weight.
Example17.1 Suppose that 2 is 0.90, the volatility estimated for a market variable for day n — 1 is1% per day, and during day »—1 the market variable increased by 2%. This means thata,-; = 9.01 — 0.0001 and u7_,; — 0.02? — 0.0004. Equation (17.7) gives
On-i
o? = 0.9 x 0.0001 +. 0.1 x 0.0004 = 0.00013
The estimate of the volatility, o,, for day n is therefore /0.00013, or 1.14% per day. Note that theexpected value of u2_, is o?_, or 0.0001. In this example, the realized value of we, is greater than theexpected value, and as a result our volatility estimate increases. If the realized value of u2_, had beenless than its expected value, our estimate of the volatility would have decreased.
The EWMA approach has the attractive feature that relatively little data need to be stored. At anygiven time, we need to remember only the current estimate of the variance rate and the most recentobservation on the value of the market variable. When we get a new observation on the value ofthe market variable, we calculate a new daily percentage change and use equation (17.7) to updateour estimate of the variance rate. The old estimate of the variance rate and the old value of themarket variable can then be discarded.
The EWMA approach is designed to track changes in the volatility. Suppose that there is a bigmove in the market variable on day n — 1, so that u2_, is large. From equation (17.7) this causes ourestimate of the current volatility to move upward. The value of A governs how responsive the estimateof the daily volatility is to the most recent daily percentage change. A low value of A leads to a greatdeal of weight being given to the u7_; when o, is calculated. In this case, the estimates produced forthe volatility on successive days are themselves highly volatile. A high value of A (ie., a value closeto 1.0) produces estimates of the daily volatility that respond relatively slowly to new informationprovided by the daily percentage changes.
The RiskMetrics database, which was created by J.P. Morgan and made publicly available in1994, uses the EWMA model with A = 0.94 for updating daily volatility estimates. The companyfound that, across a range of different market variables, this value of 4 gives forecasts of thevariance rate that come closest to the realized variance rate.® The realized variance rate on aparticular day was calculated as an equally weighted average of the ur on the subsequent 25 days.(See Problem 17.17.)
® See J.P. Morgan, RiskMertrics Monitor, Fourth Quarter, 1995. We will explain an alternative (maximumlikelihood) approach to estimating parameters later in the chapter.376 CHAPTER 17
17.3. THE GARCH(1,1) MODEL
 
 
We now move on to discuss what is known as the GARCH(1, 1) model proposed by Bollerslev in1986.’ The difference between the GARCH(I, 1) model and the EWMA model is analogous tothe difference between equation (17.4) and equation (17.5). In GARCH(I, 1), o is calculatedfrom a long-run average variance rate, V,, as well as from a,_, and u,_,. The equation for
GARCH(L, 1) is ~~ :a; =yV, tau; + Baz (17.8)
where y is the weight assigned to V,, a is the weight assigned to u2_,, and £ is the weight assignedto of. Because the weights must sum to one,
vla@!4@=YPi@ti pr
1The EWMA model is a particular case of the GARCH(1,1) model where y=0, a=1-4,and B=.
The “(1,1)” in GARCH(I, 1) indicates that of is based on the most recent observation of wand the most recent estimate of the variance rate. The more general GARCH(p,q)} modelcalculates o from the most recent p observations on u? and the most recent q estimates of thevariance rate.® GARCH(I, 1) is by far the most popular of the GARCH models.
Setting w = yV,, the GARCH(I, |) model can also be written
Oo; =O + aU 5 + POn-1 (17.9)
This is the form of the model that is usually used for the purposes of estimating the parameters.Once w, a, and B have been estimated, we can calculate y as 1 — w — £. The long-term variance V;,can then be calculated as w/y. For a stable GARCH(I, 1) process, we require a + 8 < 1. Otherwisethe weight applied to the long-term variance is negative.
Example 17.2 Suppose that a GARCH(, 1) model is estimated from daily data asoF = 0.000002 + 0.1342_, + 0.8607_,
This corresponds to a = 0.13, 6 = 0.86, and w = 0.000002. Because y = 1 — a — f, it follows thaty =0.01. Because w= yV,, it follows that V, = 0.0002. In other words, the long-run averagevariance per day implied by the model is 0.0002. This corresponds to a volatility of.f0.0002 — 0.014, or 1.4% per day.
7 See T. Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,” Journal of Econometrics, 31 (1986),307-27.
5 Other GARCH models have been proposed that incorporate asymmetric news. These models are designed so thato, depends on the sign of u,_,. Arguably, the models are more appropriate for equities than GARCH(I, 1). Asmentioned in Chapter 15, the volatility of an equity’s price tends to be inversely related to the price so that a negative4 ehawld bawe «2 bigges effect an 5 than the came nective up Tae 5 dissiccinn of models far Gardunes#,-| should have a bigger effect on o, than the same positive w,_;. Tor a discussion of models for handlingasymmetric news, see D. Nelson, “Conditional Heteroscedasticity and Asset Returns: A New Approach,”Econometrica, 59 (1990), 347-70; R. F. Engle and V. Ng, ‘““Measuring and Testing the Impact of News on Volatility,”Journal of Finance, 48 (1993), 1749-78.Estimating Volatilities and Correlations 377
Suppose that the estimate of the volatility on day n- 1 is 1.6% per day, so that o7_) =0.0167 = 0.000256, and that on day n—1 the market variable decreased by 1%, so thatw_, = 0.017 = 0.0001. Then
a7 = 0.000002 + 0.13 x 0.0001 + 0.86 x 0.000256 = 0.00023516The new estimate of the volatility is therefore V0.00023516 = 0.0153, or 1.53% per day.
The Weights
Substituting for o2_, in equation (17.9), we obtain
op =wtauz_, + (w+ aur_y + Boz_y)orOy = 0+ Peat oun,» + @Bug > + Bray n
Substituting for On, we getOn = 0 + Bo + Bw + au,» + afin > + Of Uy a + Bons
Continuing in this way, we see that the weight applied to u2_; is a6’ !. The weights declineexponentially at rate 6. The parameter 6 can be interpreted as a “decay rate”. It is similar to d inthe EWMA model. It defines the relative importance of the observations on u’s in determining thecurrent variance rate. For example, if 8 = 0.9, u2_, is only 90% as important as w2_,: u2_3 is 81%as important as u,_;; and so on. The GARCH(I, 1) model is similar to the EWMA model exceptthat, in addition to assigning weights that decline exponentially to past u’, it also assigns some
weight to the long-run average volatility.
as mp 2Niean nmeversion
The GARCH (1, 1) model recognizes that over time the variance tends to get pulled back to a long-run average level of V,. The amount of weight assigned to V, is y= | —@ — 8. The GARCH(1, 1)is equivalent to a model where the variance V follows the stochastic process
dV =a(v, — V)dt+éV dz
where time is measured in days, a = 1 — a — f, and & = a2 (see Problem 17.14). This is a mean-reverting model. The variance has a drift that pulls it back to V; at rate a. When V > V,, thevariance has a negative drift: when V, < V, it has a positive drift. Superimposed on the drift is avolatility €. We will discuss this type of model further in Section 20.3.
17.4 CHOOSING BETWEEN THE MODELS
 
In practice, variance rates do tend to be mean reverting. The GARCH(1, 1) model incorporatesmean reversion, whereas the EWMA model does not. GARCH (1, !) is therefore theoreticallymore appealing than the EWMA model.
In the next section, we will discuss how best-fit parameters w, a, and 8 in GARCH(1, 1) can beestimated. When the parameter w is zero, the GARCH(I, 1) reduces to EWMA. In circumstances378 CHAPTER 17
 
where the best-fit value of @ turns out to be negative the GARCH(1, 1) model is not stable and itmakes sense to switch to the EFWMA model.
17.5 MAXIMUM LIKELIHOOD METHODS
 
 
It is now appropriate to discuss how the parameters in the models we have been considering areestimated from historical data. The approach used is the maximum likelihood method. It involveschoosing values for the parameters that maximize the chance (uf likelihood) of the data vccurriny.
To illustrate the method, we start with a very simple example. Suppose that we sample ten stocksat random on a certain day and find that the price of one of them declined on that day and theprices of the other nine either remained the same or increased. What is our best estimate of theproportion of all stocks with price declines? The natural answer is 10%. Let us see if this is whatthe maximum likelihood method gives.
Suppose that the proportion of stocks with price declines is p. The probability that onepatticular stock declines in price and the other nine do not is p(1 — p)’. Using the maximumlikelihood approach, the best estimate of p is the one that maximizes p(1 — p)’. Differentiating thisexpression with respect to p and setting the result equal to zero, we find that p = 0.1 maximizes theexpression. This shows that the maximum likelihood estimate of p is 10% as expected.
Estimating a Constant Variance
As our next example of maximum likelihood methods, we consider the problem of estimating thevariance of a variable X from m observations on X when the underlying distribution is normal withzero mean. We assume that the observations are 4), 42,..., 4, and that the mean of the underlyingdistribution is zero. Denote the variance by v. The likelihood of u; being observed is the value of theprobability density function for X when X = u;. This is
 
Lx (=)200 P\ 3p
The likelihood of the m observations occurring in the order in which they are observed is
Rt 1 yeal = oo(5")| (17.10)i=1
Using the maximum likelihood method, the best estimate of v is the value that maximizes thisexpression.
Maximizing an expression is equivalent to maximizing the logarithm of the expression. Takinglogarithms of the expression in equation (17.10) and ignoring constant multiplicative factors, it can
be seen that we wish to maximizem 2»(- ) -“) (7.1)v
t=l
m “we—m In(v) — >i=l
 
orEstimating Volatilities and Correlations 379
Differentiating this expression with respect to » and setting the result equation to zero, we see thatthe maximum likelihood estimator of v is”
Estimating GARCH(1, 1) Parameters
We now consider how the maximum likelihood method can be used to estimate the parameterswhen GARCH(1,1) or some other volatility updating scheme is used. Define v, = o, as thevariance estimated for day i. We assume that the probability distribution of u; conditional onthe variance is normal. A similar analysis to the one just given shows that the best parameters are
the ones that maximizeI aae(=s)]exp{ ——jap LV 270; 20;
Taking logarithms, we see that this is equivalent to maximizing
m ueYo(-ineey -=) (17.12)
i=l i
This is the same as the expression in equation (17.11). except that v is replaced by v;. We searchiteratively to find the parameters in the modei that maximize the expression in equation (17.12).
The spreadsheet in Table 17.1 indicates how the calculations could be organized for theGARCH(I, 1} model. The table analyzes data on the Japanese yen exchange rate betweenJanuary 6, 1988, and August 15, 1997. The numbers in the table are based on trial estimatesof the three GARCH(I, 1) parameters w, a, and f. The first column in the table records the date.The second column counts the days. The third column shows the exchange rate, S;, at the end ofday i. The fourth column shows the proportional change in the exchange rate between the end ofday i— 1 and the end of day i. This is u; = (S; — 5;-\)/S;-,. The fifth column shows the estimateof the variance rate, v; = 07, for day i made at the end of day i — 1. On day 3, we start things offby setting the variance equal to uz. On subsequent days equation (17.9) is used. The sixth columntabulates the likelihood measure, — In(v;) — u?/v;. The values in the fifth and sixth columns arebased on the current trial estimates of w, w, and §. We are interested in choosing w, a, and B tomaximize the sum of the numbers in the sixth column. This involves an iterative searchprocedure.!°
In our example, the optimal values of the parameters turn out to be
w = 0.00000176. a= 90.0626, 6B =0.8976
and the maximum value of the function in equation (17.12) is 22,063.5763. The numbers shown inTable 17.1 were calculated on the final iteration of the search for the optimal w, w, and £.
° This confirms the point made in footnote 3.
'° A general purpose algorithm such as Solver in Microsoft’s Excel is liable to provide a local rather than globalmaximum of the likelihood function. A special purpose algorithm, such as Levenberg—Marquardt, should ideally beused, See W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art ofScientific Computing, Cambridge University Press, Cambridge, 1988.380 CHAPTER 17
Table 17.1 Estimation of parameters in GARCH(1, 1) model
 
 
Date Day i Si Ki vj = oF —In(v;) — u?/v;
06-Jan-88 1 9.007728
07-Jan-88 2 0.007779 0.006599
08-Jan-88 3 0.007746 —0.004242 0.00004355 9.6283
11-Jan-88 4 0.007816 0.009037 0.000041 98 8.1329
12-Jan-88 5 0.007837 0.002687 0.00004455 9.8568
13-Jan-88 6 9.007924 0.011101 0.00004220 7.1529
13-Aug-97 2421 0.008643 0.003374 0.00007626 9.3321
14-Aug-97 2422 0.008493 0.017309 0.00007092 5.3294
15-Aug-97 2423 0.008495 0.000144 0.000084 17 9.382422,063.5763
Trial estimates of GARCH parameterso a B0.00000176 0.0626 0.8976
The long-term variance rate, V,, in our example is
w 0.00000176Toons a> —)0308 = 0.00004422The long-term volatility is /0.00004422, or 0.665% per day.
Figure {7.1 shows the way the GARCH (1,1) volatility for the Japanese yen changed over the10-year period covered by the data. Most of the time, the volatility was between 0.4% and 0.8%per day, but volatilities over 1% were experienced during some periods.
An alternative more robust approach to estimating parameters in GARCH(I, 1) is known asvariance targeting. This involves setting the long-run average variance rate, V;,, equal to thesample variance calculated from the data (or to some other value that is believed to bereasonable). The value of w then equals V,(1 —-a@—) and only two parameters have to beestimated. For the data in Table 17.1, the sample variance is 0.00004341, which gives a dailyvolatility of 0.659%. Setting V, equal to the sample variance, the values of a and £ thatmaximize the objective function in equation (17.12) are 0.0607 and 0.8990, respectively. Thevalue of the objective function is 22,063.5274, only marginally below the value of 22,063.5763obtained using the earlier procedure.’?
When the EWMA model is used, the estimation procedure is relatively simple. We set w = 0,a= 1-4, and 6 =A, and only one parameter has to be estimated. In the data in Table !7.1, thevalue of 2 that maximizes the objective function in equation (17.12) is 0.9686 and the value of theobjective function is 21,995.8377.
"’ See R. Engle and J. Mezrich, “GARCH for Groups,” RISK, August 1996, pp. 36-40.
2 The Solver routine in Microsoft's Excel can be used to give good results when variance targeting is used.Estimating Volatilities and Correlations 381
14
1.2
1 | | | { ié> 083£9a .6Z
0.4
0.2
0 + + ~ ~ — pa
 
Jan-88 Jan-89 Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97  Jan-98Figure 17.1 Daily volatility of the yen-USD exchange rate, 1987-97
How Good Is the Model?
The assumption underlying a GARCH model is that volatility changes with the passage of time.During some periods volatility is relatively high; during other periods it is relatively low. To putthis another way, when uw is high, there is a tendency for u7,1, Urn, ... to be high; when uw is low,there is a tendency for wits ur, - .. to be low. We can test how true this is by examining theautocorrelation structure of the u?.
Let us assume that the u? do exhibit autocorrelation. If a GARCH model is working well, itshould remove the autocorrelation. We can test whether it has done this by considering theautocorrelation structure for the variables u?/o?. If these show very little autocorrelation, ourmodel for o; has succeeded in explaining autocorrelations in the u?.
Table 17.2 shows results for the yen—dollar exchange rate data referred to earlier. The firstcolumn shows the lags considered when the autocorrelation is calculated. The second columnshows autocorrelations for uw: the third column shows autocorrelations for