NORMAL V A R AND ETL

A3.3 RECOMMENDED READING

5.1 NORMAL V A R AND ETL

5.1.1 General Features

The normal (or Gaussian) distribution was briefly introduced in Chapter 3. It is very widely used, and has plausibility in many contexts because of the central limit theorem. Loosely speaking, this theorem says that if we have a random variable from an unknown but well-behaved distribution, then the means of samples drawn from that distribution are asymptotically (i.e., in the limit) normally distributed.

Consequently, the normal distribution is often used when we are concerned about the distribution of sample means and, more generally, when we are dealing with quantiles and probabilities near the centre of the distribution.

The normal distribution is also attractive because it has only two independent parameters — a mean, µ, and a standard deviation, σ (or its square, the variance, σ²). The third moment of the normal distribution, the skewness, is zero (i.e., so the normal distribution is symmetric) and the fourth moment, the kurtosis (which measures tail fatness), is 3. To apply the normal distribution, we therefore need estimates of onlyµandσ.

The normal distribution is also convenient because it produces straightforward formulas for both VaR and ETL. If we apply a normal distribution to P/L,¹then the VaR and ETL²are respectively:

VaR= −αclσP/L−µP/L (5.1a)

ETL=σP/Lφ(−αcl)/F(αcl)−µP/L (5.1b) whereµP/LandσP/Lhave their obvious meanings,αclis the standard normal variate corresponding to our chosen confidence level (e.g., αcl= −1.645 if we have a 95% confidence level), andφ(·) andF(·) are the values of the normal density and distribution functions. However, in most cases the mean and standard deviation are not known, so we have to work with estimates of them,mands.

Our estimates of VaR and ETL are therefore:³

VaR^e= −αclsP/L−mP/L (5.2a)

ETL^e=sP/Lφ(−αcl)/F(αcl)−mP/L (5.2b) Figure 5.1 shows the standard normal L/P pdf curve and normal VaR and ETL at the 95% confidence level. The normal pdf has a distinctive bell-shaped curve, and the VaR cuts off the top 5% tail whilst the ETL is the probability-weighted average of the tail VaRs.

One of the nice features of parametric approaches is that the formulas they provide for VaR (and, where they exist, ETL) also allow us to estimate these risk measures at any confidence level or holding period we like. In the normal case, we should first note that Equations (5.1a and b) give the VaR and ETL formulas for a confidence level reflected in the value of αcl, and for a holding period equal to the period over which P/L is measured (e.g., a day). If we change the

1As discussed already in Chapter 3, we can apply a normality assumption to portfolio P/L, L/P or arithmetic returns. The formulas then vary accordingly: if we assume that L/P is normal, our VaR and ETL formulas are the same as in Equations (5.1a and b), except for the mean terms having a reversed sign; if we assume that arithmetic returns are normal, then theµ andσterms refer to returns, rather than P/L, and we need to multiply our VaR and ETL formulas by the current value of our portfolio.

2Following on from the last footnote, if L/P is normal, with meanµL/Pand standard deviationσL/P, then our ETL is σL/Pφ(−αcl)/F(αcl)+ µL/P; and if our portfolio returnris normal, with meanµrand standard deviationσr, our ETL is [σrφ(−αcl)/F(αcl)−µr]P, wherePis the current value of our portfolio.

3The normality assumption has the additional attraction of making it easy for us to get good estimators of the parameters. As any econometrics text will explain, under normality, least squares (LS) regression will give us best linear unbiased estimators of our parameters, and these are also the same as those we would get using a maximum likelihood approach.

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0−5 −4 −3 −2 −1 0 1 2 3 4 5

Loss () / profit ()

Probability ETL=2.061

VaR at 95 cl

= 1.645

Figure 5.1 VaR and ETL for normal loss/profit.

Note: Based on a 1-day holding period with L/P having a mean 0 and standard deviation 1. The figure is obtained using the

‘normaletlfigure’ function.

confidence level, we therefore change the value ofαcl to correspond to the new confidence level:

for example, if we change the confidence level from 95% to 99%, αcl changes from−1.645 to

−2.326.

To take account of a change in the holding period, we need formulas for the mean and standard deviation of P/L over arbitrary periods. If we now defineµP/L andσP/L as the mean and standard deviation of P/L over a given period (e.g., a day), then the mean and standard deviation of P/L over hpsuch periods are:

µP/L(hp)=hpµP/L (5.3a)

σP²/L(hp)=hpσP²/L ⇒σP/L(hp)=

hpσP/L (5.3b)

We now substitute these into Equations (5.1a and b) to get the formulas for VaR and ETL over an arbitrary holding periodhpand confidence levelcl:

VaR(hp,cl)= −αcl

hpσP/L−hpµP/L (5.4a)

ETL(hp,cl)=

hpσP/Lφ(−αcl)/F(αcl)−hpµP/L (5.4b) These formulas make it very easy to measure VaR and ETL once we have values (or estimates) of µP/LandσP/Lto work with. These formulas tell us that VaR and ETL will rise with the confidence level, as shown earlier in Figure 2.7. However, the effects of a rising holding period are ambiguous, as the first terms in each formula rise withhp, but the second terms fall ashprises. Since the first terms relate toσ, and the second toµ, the effects of a risinghpon VaR or ETL depend on the relative sizes ofσ andµ. Furthermore, since the first terms rise with the square root ofhp, whilst the second terms rise proportionately withhp, we also know that the second terms will become more

6 8

4

0 2

−2

−4

−6

−8

0 50 100 150 200 250 300 350 400

Holding period (days)

VaR

Figure 5.2 Normal VaR and holding period.

Note: This figure is obtained using the ‘normalvarplot2D hp’ function and shows the VaR at the 95% confidence level for a normal P/L distribution with mean 0.1 and standard deviation 1.

prominent ashpgets larger. If we assume plausible parameter values (e.g., a confidence level of 95%

andµpositive but ‘small’ relative toσ), then we get the following:

r

_Whenhpis very low, the first term dominates the second, so the VaR and ETL are positive.

r

_Ashp gets bigger, the second terms grow at a faster rate, so VaR and ETL will rise but at a diminishing rate.

r

_Ashpcontinues to rise, VaR and ETL will turn down, and eventually become negative.

r

Thereafter, they will move further away from zero ashpcontinues to rise.

Figure 5.2 gives an example of how VaR behaves under these conditions. (We get a similar looking chart for the ETL.) In this particular case (with daily parameters ofµ=0.1 andσ =1), the VaR peaks at a holding period of around 70 days, and becomes negative at a holding period of around 250 days. VaRs beyond that holding period move further and further away from zero as the holding period continues to rise.

As an aside, this VaR–holding period chart is very different from the one we would get under the well-known ‘square root rule’, which is now enshrined in the Basle regulations on bank capital adequacy. According to this rule, we can obtain VaRs over longer holding periods by taking a VaR measured over a short holding period and scaling it up by the square root of the longer holding period. If our VaR over a 1-day holding period isVaR(1,cl), say, then the VaR over a holding period ofhpdays,VaR(hp,cl), is given by:

VaR(hp,cl)=

hp VaR(1,cl) (5.5)

20 10 0

−10

500

0 0.9

0.92 0.94

Confidence level Holding period

VaR

0.96 0.98 1 100

200 300 400

−30

−20

Figure 5.3 A normal VaR surface.

Note: This figure shows the VaR surface for a normal P/L distribution with daily mean 0.1 and daily standard deviation 1. It is produced using the ‘normalvarplot3D’ function.

This formula produces a VaR that always rises as the holding period increases, although at a decreasing rate, as illustrated earlier in Figure 2.8. If we compare the two figures, we can see that the ‘true’ normal VaR becomes increasingly strongly negative, whilst the ‘square root VaR’ becomes increasingly strongly positive ashpgets large. It follows, then, that we should never use the ‘square root’ ex- trapolation rule, except in the special case whereµ=0 — and even then, the ‘square root VaR’ will only be correct for certain P/L distributions. My advice is to forget about it altogether, and use the correct parametric VaR formula instead.

It is often useful to look at VaR and ETL surfaces, as these convey much more information than single point estimates or even curves such as Figure 5.2. The usual (i.e.,µ >0 case) normal VaR surface is shown in Figure 5.3. (Again, the ETL equivalent is similar.) The magnitudes will vary with the parameters, but we always get the same basic shape: the VaR rises with the confidence level, and initially rises with the holding period, but as the holding period continues to rise, the VaR eventually peaks, turns down and becomes negative; the VaR is therefore highest when the confidence level is highest and the holding period is high but not too high. Away from its peak, the VaR surface also has nicely curved convex isoquants: these are shown in the figure by the different shades on the VaR surface, each representing a different VaR value.

Again, it is instructive to compare this surface with the one we would obtain ifµ=0 (i.e., where the square root approach is valid). The zero-µnormal VaR surface was shown earlier in Figure 2.9, and takes a very different shape. In this case, VaR rises with both confidence level and holding period. It therefore never turns down, and the VaR surface spikes upwards as the confidence level and holding period approach their maximum values. It is important to emphasise that the difference between the surfaces in Figures 2.9 and 5.3 is due entirely to the fact thatµis zero in the first case and positive in the second. The lesson? The meanµmakes a big difference to the risk profile.

5.1.2 Disadvantages of Normality

The normality assumption — whether applied to P/L or returns — also has a number of potential disadvantages. The first is that it allows P/L (or returns) to take any value, and this means that it might produce losses so large that they more than wipe out our capital: we could lose more than the value of our total investment. However, it is usually the case (e.g., due to limited liability and similar constraints) that our losses are bounded, and the failure of the normality assumption to respect constraints on the maximum loss can lead to gross overestimates of what we really stand to lose.

A second potential problem is one of statistical plausibility. As mentioned already, the normality assumption is often justified by reference to the central limit theorem, but the central limit theorem applies only to the central mass of the density function, and not to its extremes. It follows that we can justify normality by reference to the central limit theorem only when dealing with more central quantiles and probabilities. When dealing with extremes — that is, when the confidence level is either very low or very high — we should refer to the extreme value theorem, and that tells us very clearly that we shouldnotuse normality to model extremes.

A third problem is that most financial returns have excess kurtosis, or fatter than normal tails, and a failure to allow for excess kurtosis can lead to major problems. The implications of excess kurtosis are illustrated in Figure 5.4. This figure shows both the standard normal pdf and a particular type of fat-tailed pdf, a Studenttpdf with five degrees of freedom. The impact of excess kurtosis is seen very clearly in the tails: excess kurtosis implies that tails are heavier than normal, and this means that VaRs (at the relatively high confidence levels we are usually interested in) will be bigger.

For example, if we take the VaR at the 95% confidence level, the standard normal VaR is 1.645,

0.35 0.4

0.3

0.2 0.25

0.15 0.1 0.05

0−4 −3 −2 −1 0 1 2 3 4

Loss () / profit ()

Probability

VaRs at 95% cl:

Normal VaR 1.645

t VaR 2.015 Normal pdf

tpdf

Figure 5.4 Normal VaR vs. fat-tailed VaR.

Note: This figure shows VaRs at the 95% confidence level for standard normal P/L and StudenttP/L, where the latter has five degrees of freedom.

but thetVaR is 2.015 — which is 22% bigger.⁴ Furthermore, it is obvious from the figure that the proportional difference between the two VaRs gets bigger with the confidence level (e.g., at the 99%

confidence level, the normal VaR is 2.326, but thetVaR is 3.365, which is almost 44% bigger). What this means is that if we assume that P/L is normal when it is actually fat-tailed, then we are likely to underestimate our VaRs (and, indeed, ETLs), and these underestimates are likely to be particularly large when dealing with VaRs at high confidence levels.

Box 5.2 The Cornish–Fisher Approximation

If our distribution is not normal, but the deviations from normality are ‘small’, we can approximate our non-normal distribution using the Cornish–Fisher expansion, which tells us how to adjust the standard normal variateαcl to accommodate non-normal skewness and kurtosis. To use it, we therefore replace the standard variateαclin our normal VaR or ETL formula by:

αcl+(1/6) αcl² −1

ρ3+(1/24) αcl³ −3αcl

ρ4−(1/36) 2α³cl−5αcl

ρ₃²

When using the Cornish–Fisher approximation, we should keep in mind that it will only provide a ‘good’ approximation if our distribution is ‘close’ to being normal, and we cannot expect it to be much use if we have a distribution that is too non-normal.