202 Mathematics for Finance
9.2.1 Value at Risk ......................................
Let us present the basic idea using a simple example. We buy a share of stock for
S(0) = 100 dollars to sell it after one year. The selling priceS(1) is random. We
shall suffer a loss ifS(1)<100er,whereris the risk-free rate under continuous
compounding. (The purchase can either be financed by a loan, or, if the initial
sum is already at our disposal, we take into account the foregone opportunity
of a risk-free investment.) What is the probability of a loss being less than a
given amount, for example,
P(100er−S(1)<20) =?Let us reverse the question and fix the probability, 95% say. Now we seek an
amount such that the probability of a loss not exceeding this amount is 95%.
This is referred to asValue at Risk at 95% confidence level and denoted by
VaR. (Other confidence levels can also be used.) So, VaR is an amount such
that
P(100er−S(1)<VaR) = 95%.
It should be noted that the majority of textbooks neglect the time value of
money in this context, stating the definition of VaR only forr=0.
Example 9.1
Suppose that the distribution of the stock price is log normal, the logarithmic
returnk=ln(S(1)/S(0)) having normal distribution with meanm= 12%
and standard deviationσ= 30%. With probability 95% the return will satisfy
k>m+xσ∼=− 37 .50%,whereN(x)∼=5%,sox∼=− 1 .645. (HereN(x)isthe
normal distribution function (8.10) with mean 0 and variance 1.) Hence with
probability 95% the future priceS(1) will satisfy
S(1)>S(0)em+xσ∼= 68 .83 dollars,and so, given thatr=8%,
VaR =S(0)er−S(0)em+xσ∼= 39 .50 dollars.Exercise 9.8
Evaluate VaR at 95% confidence level for a one-year investment of $1, 000
into euros if the interest rate for risk-free investments in euros isrEUR=
4% and the exchange rate from euros into US dollars follows the log
normal distribution withm=1%andσ= 15%. Take into account the
foregone opportunity of investing dollars without risk, given that the
risk-free interest rate for dollars isrUSD=5%.