Question

If the value of a portfolio follows a geometric Brownian motion with drift rate 6% and volatility
20%, then the log return of the portfolio from time ? to time ? is normally distributed with
mean 6% − 0.5∗(20%)^2 (? – ?) and variance 0.04∗(? – ?).

What is the 10-day, 1% VaR of the portfolio? You should give your answer in terms of log-returns.
You are also given the following number: For a standard normal random variable with zero mean and
unit variance, the probability that z is less than or equal to -2.33 is approximately 1%.

Answer 1

he log return of the portfolio from time ? to time ? is normally distributed with
mean 6% − 0.5(20%)²(? – ?) and variance 0.04(? – ?).

T - t = 10 days = 10/365 year

Hence, Mean = 6% − 0.5(20%)²(? – ?) = 6% − 0.5 x (20%)² x 10/365 = 5.95%

Variance = 0.04(T - t) = 0.04 x 10/365 = 0.001096

Hence, standard deviation = Variance^1/2 = 0.001096^1/2 = 3.31%

For a standard normal random variable with zero mean and unit variance, the probability that z is less than or equal to -2.33 is approximately 1%.

Hence,1 day Var = X such that

(X - Mean) / Std dev ≤ -2.33

Hence, X ≤ Mean - 2.33 x Std dev = 5.95% - 2.33 x 3.31% = -1.768%

Hence, value at risk = 1.768%