OUTLINE OF THE BOOK
3.3 ESTIMATING PARAMETRIC V A R
We can also estimate VaR using parametric approaches, the distinguishing feature of which is that they require us to specify explicitly the statistical distribution from which our data observations are drawn. We can think of parametric approaches as fitting curves through the data and then reading off the VaR from the fitted curve.
In making use of a parametric approach, we therefore need to take account of both the statistical distribution and the type of data to which it applies. We now discuss some of the more important cases that we might encounter in practice.
3.3.1 Estimating VaR with Normally Distributed Profits/Losses
If we are using P/L data to estimate VaR under the assumption that P/L is normally distributed, our VaR is:
VaR= −αclσP/L−µP/L (3.8)
whereµP/L andσP/L are the mean and standard deviation of P/L, andαcl is the standard normal variate corresponding to our chosen confidence level. Thus, if we have a confidence levelcl, we defineαclas that value of the standard normal variate such that 1−clof the probability density lies to the left, andclof the probability density lies to the right. For example, if our confidence level is 95%, our value ofαclwill be−1.645.1
In practice,µP/L andσP/L would be unknown, and we would have to estimate VaR based on estimates of these parameters. Our VaR estimate would then be:
VaRe= −αclsP/L−mP/L (3.9)
wheremP/LandsP/Lare estimates of the mean and standard deviation of P/L.
Figure 3.2 shows the VaR at the 95% confidence level for a normally distributed P/L with mean 0 and standard deviation 1. Since the data are in P/L form, the VaR is indicated by the negative of the cut-off point between the lower 5% and the upper 95% of P/L observations. The actual VaR is the negative of−1.645, and is therefore 1.645.
0.4
0.3 0.35
0.25 0.2 0.15 0.1 0.05
0−4 −3 −2 −1 0 1 2 3 4
Profit () / loss ()
Probability
Negative of VaR at 95%%cl 1.645
Figure 3.2 VaR with normally distributed profit/loss data.
Note: Obtained from Equation (3.9) withµP/L=0 andσP/L=1. Estimated with the ‘normalvarfigure’ function.
1There is also an interesting special case: the VaR is proportional to the standard deviation of P/L whenever the mean P/L is zero and the P/L is elliptically distributed. However, it would be unwise to get into a habit of identifying the standard deviation and the VaR too closely, because these conditions will often not be met.
0.4
0.3 0.35
0.25 0.2 0.15 0.1 0.05
0−4 −3 −2 −1 0 1 2 3 4
Loss () / profit ()
Probability
VaR at 95%cl 1.645
Figure 3.3 VaR with normally distributed loss/profit data.
Note: Obtained from Equation (3.10a) withµL/P=0 andσL/P=1.
If we are working with normally distributed L/P data, thenµL/P = −µP/LandσL/P =σP/L, and it immediately follows that:
VaR= −αclσL/P +µL/P (3.10a)
VaRe= −αclsL/P +mL/P (3.10b)
Figure 3.3 illustrates the corresponding VaR. In this case, the VaR is given by the cut-off point between the upper 5% and the lower 95% of L/P observations. The VaR is again 1.645, as we would (hopefully) expect.
This figure also shows why it is sometimes more convenient to work with L/P rather than P/L data when estimating VaR: with L/P data the VaR is immediately apparent, whereas with P/L data the VaR is shown as the negative of a (usually) negative quantile. The information given is the same in both cases, but is more obvious in Figure 3.3.
3.3.2 Estimating VaR with Normally Distributed Arithmetic Returns
We can also estimate VaR making assumptions about returns rather than P/L. Suppose then that we assume arithmetic returns are normally distributed with meanµrand standard deviationσr. To derive the VaR, we begin by obtaining the critical value ofrt,r*, such that the probability thatrt exceeds r* is equal to our chosen confidence level.r* is therefore:
r*=µr+αclσr (3.11)
We know that the returnrt is related to the negative of the loss/profit divided by the earlier asset value,Pt−1:
rt =(Pt−Pt−1)/Pt−1= −Losst/Pt−1 (3.12) This gives us the relationship betweenr*, the critical value ofPt,P*, corresponding to a loss equal to VaR, and the VaR itself:
r*
t =(P*
t −Pt−1)/Pt−1= −VaR/Pt−1 (3.13) Substituting Equation (3.11) into Equation (3.13) and rearranging then gives us the VaR:
VaR= −(µr+αclσr)Pt−1 (3.14) Equation (3.14) will give us equivalent answers to our earlier VaR equations. For example, if we set cl=0.95,µr =0,σr =1 and Pt−1=1, which correspond to our earlier illustrative P/L and L/P parameter assumptions, our VaR is 1.645: these three approaches give the same results, because all three sets of underlying assumptions are equivalent.
3.3.3 Estimating Lognormal VaR
Unfortunately, each of these approaches also assigns a positive probability of the asset value, Pt, becoming negative. We can avoid this drawback by working with geometric returns rather than arithmetic returns. As noted already, the geometric return is:
Rt =log[(Pt+Dt)/Pt−1]
Now assume that geometric returns are normally distributed with meanµRand standard deviation σR. If we assume that Dt is zero or reinvested continually in the asset itself (e.g., as with profits reinvested in a mutual fund), this assumption implies that the natural logarithm of Pt is normally distributed, or thatPtitself is lognormally distributed. A lognormal asset price is shown in Figure 3.4:
observe that the price is always non-negative, and its distribution is skewed with a long right-hand tail.
If we now proceed as we did earlier with the arithmetic return, we begin by deriving the critical value of R,R*, that is the direct analogue ofr*, i.e.:
R* =µR+αclσR (3.15)
We then use the definition of the geometric return to unravel the critical value P* (i.e., the value of Ptcorresponding to a loss equal to our VaR), and thence infer our VaR:
R*=logP*−logPt−1⇒logP*=R*+logPt−1
⇒P*=exp[R*+logPt−1]=exp[µR+αclσR+logPt−1]
⇒VaR=Pt−1−P* =Pt−1−exp[µR+αclσR+logPt−1] (3.16) This gives us the lognormal VaR, which is consistent with normally distributed geometric returns.
The formula for lognormal VaR is more complex than the earlier VaR equations, but the lognormal VaR has the attraction over the others of ruling out the possibility of negative asset (or portfolio) values.
The lognormal VaR is illustrated in Figure 3.5, based on the hypothetical assumptions thatµR =0, σR =1, and Pt−1=1. In this case, the VaR at the 95% confidence level is 0.807. The figure also
0.7 0.6
0.5
0.4
0.3
0.2
0.1 0
0 1 2 3 4 5 6 7 8
Asset price
Probability
Figure 3.4 A lognormally distributed asset price.
Note: Estimated with mean and standard deviation equal to 0 and 1 respectively, using the ‘lognpdf’ function in the Statistics Toolbox.
0.7
0.6 0.5 0.4 0.3
0.2 0.1
0−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Loss () / profit ()
Probability VaR at 95%%cl
0.807
Figure 3.5 Lognormal VaR.
Note: Estimated assuming the mean and standard deviation of geometric returns are 0 and 1, and for an initial investment of 1. The figure is produced using the ‘lognormalvarfigure’.
shows that the distribution of L/P is the mirror image of the distribution of P/L, which is in turn a reflection of the distribution of Ptshown earlier in Figure 3.4.
It is also worth stressing that lognormal VaR can never exceedPt−1because the L/P is bounded above by Pt−1, and (as we have seen) this is a generally desirable property because it ensures that we cannot lose more than we invest.