A3.3 RECOMMENDED READING
5.3 THE LOGNORMAL DISTRIBUTION
Another popular alternative is to assume that geometric returns are normally distributed. As explained in Chapter 3, this is tantamount to assuming that the value of our portfolio at the end of our holding period is lognormally distributed. Hence, this case is often referred to as lognormal. The pdf of the end-period value of our portfolio is illustrated in Figure 5.5. The value of the portfolio is always positive, and the pdf has a distinctive long tail on its right-hand side.
0 1 2 3 4 5 6 7 8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Probability
Portfolio value
Figure 5.5 A lognormally distributed portfolio.
Note: Estimated with mean and standard deviation equal to 0 and 1, and a current portfolio value of $1, using the ‘lognpdf’
function in the Statistics Toolbox.
5We might also use thet-distribution for another reason: if we have a normal P/L, but don’t know the parametersµand σ, then we would have to work with estimates of these parameters instead of their true values. In these circumstances we might sometimes use at-distribution instead of a normal one, where the number of degrees of freedom is equal to the number of observations in our sample minus 2. For more on this use of the Studentt-distribution, see Wilson (1993). However, this exception aside, the main reason for using at-distribution is to accommodate excess kurtosis, as explained in the text.
1 0.9 0.8 0.7
150
0 0.9 0.92 0.94
Confidence level Holding period
VaR
0.96 0.98 1 50
100 0.5
0.6
Figure 5.6 A lognormal VaR surface.
Note: Estimated with the ‘lognormalvarplot3D’ function, assuming the mean and standard deviation of daily geometric returns are 0.1 and 1, and with an initial investment of $1.
The lognormal VaR is given by the following formula:
VaR=Pt−1−exp[hpµR+αcl
hpσR+logPt−1] (5.7)
where Pt−1 is the current value of our portfolio. Equation (5.7) generalises the earlier lognormal VaR equation (i.e., Equation (3.16)) by allowing for an arbitrary holding periodhp. The lognormal assumption has the attraction of ruling out the possibility of a positive-value portfolio becoming a negative-value one: in this case, the VaR can never exceed the value of our portfolio.
The lognormal VaR was illustrated earlier in Figure 3.5, and the typical lognormal VaR surface — that is, the VaR surface with positiveµR— is shown in Figure 5.6. The VaR initially rises with the con- fidence level and holding period until it reaches an upper bound. This bound is given by the initial value of our portfolio, assumed here to be $1. The surface then flattens out along this ceiling for a while. As hpcontinues to rise, the surface eventually turns down again, enters negative territory, and then be- comes ever more strongly negative as the holding period gets bigger. As we would expect, the surface falls off at lower confidence levels first, and it can take a long time for it to fall off at higher confidence levels. However, the VaR surface always turns down eventually, regardless of the confidence level, so long as the mean return is positive. The reason for this is the same as with normal VaR: as the holding period continues to rise, the mean term becomes more important than the standard deviation term be- cause it grows at a faster rate; eventually, therefore, it kicks in to pull the VaR down, then make it neg- ative, and subsequently make it ever more strongly negative as the holding period continues to rise.6
6Of course, the parameter values on which Figure 5.6 is based are merely illustrative and are not meant to be empirically realistic. However, the general surface will always take the form shown in Figure 5.6 provided the mean is positive, and the only real issue is how long it takes for the surface to start falling ashpcontinues to rise. Note also that the identification of our basic time period as a day is merely a convention, and our time period could be any length we choose.
1 0.95
0.85 0.8
150
0 0.9
0.92 0.94
Confidence level Holding period
VaR
0.96 0.98 1 50
100 0.7
0.75 0.9
Figure 5.7 A lognormal VaR surface with zero mean return.
Note: Estimated with the ‘lognormalvarplot3D’ function, assuming the mean and standard deviation of geometric returns are 0 and 1, and with an initial investment of $1.
Once again, the mean term is important in determining the shape of the VaR surface. A lognor- mal VaR surface with a zero mean term would, by contrast, have the VaR rise — and rise more quickly — to hit its ceiling and then stay there for ever. This is shown in Figure 5.7, which is the direct analogue to the normal zero-mean case shown in Figure 2.9. Again, the mean term can make a big difference to estimated risks, particularly over long holding periods.
One other point to note about a lognormal distribution is its asymmetry, which is obvious from Figure 5.5. One important implication of any asymmetric P/L or return distribution is that long and short positions have asymmetric risk exposures. A long position loses if the market goes down, and a short position loses if the market goes up, but with any symmetric distribution the VaR on a long position and the VaR on a short position are mirror images of each other, reflecting the symmetry of the lower and upper tails of the distribution.
The situation can be very different with asymmetric distributions. With the lognormal, for example, the worst the long position can do is lose the value of its investment — the VaR and ETL are bounded above in a natural way — but a short position can make much larger losses. To illustrate, Figure 5.8 shows the VaRs for long and short positions in a lognormally distributed asset. The long position loses when the market falls, but the fall in the market is limited by the value of the initial investment. A $1 investment — a long position — has a VaR of 0.807. However, the corresponding short position — a short $1 position has a VaR of 4.180. The short side has a potentially unlimited VaR, and its VaRs will be particularly high because it gets hit very hard by the long right-hand tail in the asset price, shown in Figure 5.5 — a tail that translates into very high profits for the long position, and very high losses for the short position.
In short, the lognormal approach has the attraction of taking account of maximum loss constraints on long positions. It also has the advantage of being consistent with a geometric Brownian motion
−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.