VALUE AT RISK (VAR) (PART 2)

Value-at-Risk (VaR)

by Simon Benninga and Zvi Wiener

VALUE-AT-RISK (VAR)
LOGNORMAL DISTRIBUTIONS
As explained in our previous articles, the lognormal distribution is a more reasonable distribution for many asset prices (which can not become negative) than the normal distribution. This is not a problem: Suppose that the natural logarithm of the portfolio value is normally distributed with annual mean m and annual standard deviation s. Denoting the value of the portfolio by v it follows that the logarithm of the portfolio value at time T , vt , is normally distributed:

Log[vT ] ~ Normal CLog[v] + Km - s22 OT ,sT G The term s22 T appears due to Ito’s Lemma (see [Hull1997]). In our case, this means that v = 100, m = 10%,s = 30%. Thus the end-of-year log of the portfolio value is distributed NormalDistribution[ Log[ 100 ] + (0.1-0.3ˆ2/2), 0.3] = NormalDistribution[
4.666017,0.3]. This means that the probability that the end-of-year value of the portfolio is less than 80 is
given by: CDF[NormalDistribution[Log[100]+ (0.1-0.3ˆ2/2),0.3],Log[80]] 0.176926
Similarly the VaR function has to be redefined:
ClearAll[lognormalVaR];
lognormalVaR[mu_,sigma_,x_]:=100-Exp[Quantile[NormalDistribution[
Log[100]+(mu-sigmaˆ2/2),sigma],x]];
lognormalVaR[0.10,0.30,0.01]47.4237
Thus a portfolio whose initial value is $100 million and whose annual returns are lognormally distributed with parameters mu = 10% and sigma = 30%, has an annual VaR equal to $47.42 million at 1%.
Most VaR calculations are not concerned with annual value at risk. The main regulatory and management concern is with loss of portfolio value over a much shorter time period (typically several days or perhaps weeks).
It is clear that the distribution formula Log[vT ] ~ Normal[Log[v ] + (m - s22 )T ,sT ] can be used to calculate the VaR over any horizon. Recall that T is measured in annual terms; if there are 250 business days in a year, then the daily VaR corresponds to T = 1/250 (for many fixed income instruments one should use 1/360, 1/365, or 1/365.25 depending on the market convention):
ClearAll[lognormalVaR];
lognormalVaR[mu_,sigma_,x_,T_]:=100-Exp[Quantile[NormalDistribution[Log[100]+(musigmaˆ2/2)*T,sigma*T],x]];
lognormalVaR[0.10,0.30,0.01,1/250]
lognormalVaR[0.10,0.30,0.01,5/250]
lognormalVaR[0.10,0.30,0.01,21/250]
0.256831
1.27758
5.25717
The daily VaR of the portfolio at 1% is $256,831. The probability that the firm will lose more than this amount on its portfolio over the course of a single day is less than 1%. Similarly, the weekly and the monthly VaRs at 1% are $1.28 and $5.26 million.