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%

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