A3.3 RECOMMENDED READING
5.4 EXTREME VALUE DISTRIBUTIONS
−5 −4 −3 −2 −1 0 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Probability
Loss () / profit () on long position
Long VaR at 95 cl= 0.807
Short VaR at 95 cl= 4.180
Figure 5.8 Lognormal VaRs for long and short positions.
Note: Estimated with the ‘lognormalvarfigure’ function, assuming the mean and standard deviation of geometric returns are 0 and 1, and with initial positions worth plus or minus $1.
process for the underlying asset price — a process that has a number of nice features and is widely used in the derivatives industry. However, it also has potential disadvantages, of which the most important are that it does not accommodate fat tails in (geometric) returns,7and extreme value (EV) theory tells us that it is not suitable for VaR and ETL at extreme confidence levels.
Fisher–Tippett theorem, which tells us that ifXhas a suitably ‘well-behaved’ distribution function F(x), then the distribution function of extreme values ofXconverges asymptotically to the GEV distribution function:
Hξ,a,b=
exp[−(1+ξ(x−a)/b)−1/ξ]
exp[−exp(−(x−a)/b)] if ξ =0
ξ =0 (5.8)
where 1+ξ(x−a)/b>0,aandbare location and scale parameters, andξ is the tail index. The relationship of the location and scale parameters to the mean and variance is explained in Tool No. 5.
We are usually interested in two special cases of this distribution: the Gumbel distribution (if ξ =0) and the Fr´echet distribution (ifξ >0). The Gumbel and Fr´echet VaRs are:
VaR=
a−(b/ξ)[1−(−logcl)−ξ]
a−blog(log(1/cl)) if ξ >0
ξ =0 (5.9)
We can estimate these VaRs by inserting estimates of the parameters into the relevant formulas, and obtain parameter estimates by ML or semi-parametric methods.
In estimating EV VaR, we must also take care to use the correct formula(s). For example, the formulas given in Equation (5.9) could apply to P/L, L/P or just high losses. However, as they involve no variable representing the size of our portfolio or the amount invested — we should multiply them by the value of such a variable when our data are in rate-of-return form: if we have a $1 investment, we would multiply Equation (5.9) by 1, if we had a $2 investment, we would multiply Equation (5.9) by 2, etc.
We should also take care to apply the (correct) formula(s) in the correct way: we need to decide whether we are fitting losses, profits/losses, returns, etc., to the EV distribution, and in doing so, we need to take account of any constraints on the values that our data can take; we need to be clear which tail of the EV distribution — the lower tail or the upper tail — we are interested in; and we need to take account of whether our position is a long or a short one, which is also one factor in determining which tail we should be interested in.
These decisions will have a critical effect on the VaR and on the shape of the VaR surface. And, as we might expect from the earlier cases, the shape of the VaR surface will also depend in an important way on the mean parameter. For example, if we apply the Gumbel to P/L, and take the mean to be positive, we get the Gumbel VaR surface shown in Figure 5.9. This surface is reminiscent of the normal positive-mean VaR surface shown in Figure 5.3, and has much the same explanation (i.e., the mean term becomes more important as the holding period rises, and eventually pulls the VaR down, etc.). However, if we assume a zero mean, we get a Gumbel VaR surface reminiscent of Figure 2.9, in which the VaR continues to rises as the holding period increases. We would also get a different VaR surface if we apply the Gumbel to, say, geometric returns, because geometric returns imply that the VaR of any positive-value position is limited by the value of that position: we would then get VaR surfaces reminiscent of the lognormal VaR surfaces in Figures 5.6 and 5.7. In short, the shape of the VaR surface depends on how we apply EV theory, and on the parameter values involved.
5.4.2 The Peaks Over Threshold (Generalised Pareto) Approach
We can also apply EV theory to the distribution of excess losses over a (high) threshold. This leads us to the peaks over threshold (POT) or generalised Pareto approach. IfXis a suitable random loss
80 60 40 20
1000
0 0.99
0.995
Confidence level Holding period
VaR
500 1
−20 0
Figure 5.9 A Gumbel VaR surface.
Note: Estimated with the ‘gumbelvarplot3D’ function, assuming the mean and standard deviation of P/L are 0.1 and 1.
with distribution function F(x), andu is a threshold value ofX, then we can define a distribution of excesses overuas:
Fu(y)=Pr{X−u≤ y|X>u} (5.10)
This gives the probability that a loss exceeds the thresholduby at mosty, given that it does exceed the threshold. The distribution ofXitself can be any of the commonly used distributions, but asu gets large, the distributionFu(y) converges to a generalised Pareto distribution:
Gξ,β(x)=
1−(1+ξx/β)−1/ξ
1−exp(−x/β) if ξ =0
ξ =0 (5.11)
whereβ >0 is a scale parameter,ξis a shape or tail parameter (see, e.g., McNeil (1999a, p. 4)). We are usually interested in the case whereξ >0, corresponding to our returns being fat-tailed.
As discussed in Tool No. 5: Extreme Value VaR and ETL, the POT approach gives rises to the following formulas for VaR and ETL:
VaR=u+(β/ξ){[(n/Nu)(1−cl)]−ξ −1} (5.12a)
ETL=[VaR+(β−ξu)]/(1−ξ) (5.12b)
provided thatξ <1, wherenis the sample size andNuis the number of excess values.
In applying the POT approach, we should also take account of the same ancillary factors as we would if we were using a GEV approach: which variable — excess loss, excess return, etc. — we are fitting to the GP distribution; the need to multiply the formulas by a position size variable if we are dealing with rates of return; the need to decide which tail we should be interested in; and the need to take account of whether our position is long or short.
Box 5.3 Estimating Confidence Intervals for Parametric VaR and ETL
There are various ways we can estimate confidence intervals for parametric VaR. Leaving aside simple cases where we can easily derive confidence intervals analytically — for an example see Chappell and Dowd (1999) — we can obtain confidence intervals using the following approaches:
r
We can estimate confidence intervals for VaR using the quantile standard error approach outlines in Section 4.3.1r
We can obtain confidence intervals using order statistics theory, explained in Tool No. 1:Estimating VaR and ETL Using Order Statistics, and Section 4.3.2. The OS approach is also useful for estimating confidence intervals for ETL.