VALUE AT RISK (VAR) (PART 1)

Value-at-Risk (VaR)

by Simon Benninga and Zvi Wiener


VALUE-AT-RISK
Value-at-Risk (VaR) measures the worst expected loss under normal market conditions over a specific time interval at a given confidence level. As one of our references states: “VaR answers the question: how much can I lose with x% probability over a pre-set horizon” (J.P. Morgan, RiskMetrics–Technical Document). Another way of expressing this is that VaR is the lowest quantile of the potential losses that can occur within a given portfolio during a specified time period. The basic time period T and the confidence level (the quantile) q are the two major parameters that should be chosen in a way appropriate to the overall goal of risk measurement. The time horizon can differ from a few hours for an active trading desk to a year for a pension fund. When the primary goal is to satisfy external regulatory requirements, such as bank capital requirements, the quantile is typically very small (for example, 1% of worst outcomes). However for an
internal risk management model used by a company to control the risk exposure the typical number is around 5% (visit the internet sites in references for more details). A general introduction to VaR can be found in Linsmeier, [Pearson 1996] and [Jorion 1997]. In the jargon of VaR, suppose that a portfolio manager has a daily VaR equal to $1 million at 1%. This means that there is only one chance in 100 that a daily loss bigger than $1 million occurs under normal market conditions.

A REALLY SIMPLE EXAMPLE
Suppose portfolio manager manages a portfolio which consists of a single asset. The return of the asset is normally distributed with annual mean return 10% and annual standard deviation 30%. The value of the portfolio today is $100 million. We want to answer various simple questions about the end-of-year distribution of portfolio value:
1. What is the distribution of the end-of-year portfolio value?
2. What is the probability of a loss of more than $20 million dollars by year end (i.e., what is the probability
that the end-of-year value is less than $80 million)?
3. With 1% probability what is the maximum loss at the end of the year? This is the VaR at 1%.
We start by loading Mathematica ’s statistical package:
Needs["Statistics‘Master‘"]
Needs["Statistics‘MultiDescriptiveStatistics‘"]
We first want to know the distribution of the end-ofyear portfolio value:
Plot[PDF[NormalDistribution[110,30],x],{x,0,200}];
50 100 150 200
0.002
0.004
0.006
0.008
0.01
0.012
The probability that the end-of-year portfolio value is less than $80 is about 15.9%.
CDF[NormalDistribution[110.,30],80]0.158655
With a probability of 1% the end-of-year portfolio value will be less than 40.2096; this means that the VaR of the distribution is 100 - 40.2096 = 59.7904.
Quantile[NormalDistribution[110.,30],0.01]40.2096
We can formalize this by defining a VaR function which takes as its parameters the mean mu and standard deviation sigma of the distribution as well as the VaR level x.
ClearAll[VaR];
VaR[mu_,sigma_,x_]:=
100-Quantile[NormalDistribution[mu,sigma],x]
VaR[110,30,0.01]59.7904
Vol. 7 No. 4 1998 MathematicainEducationandResearch 1