Suppose a portfolios daily log returns are normally distributed with a stan- dard deviation of 1% and a mean of 0.01% above the discount rate. Cal- culate (a) the portfolio volatility and (b) the 1% 10-day normal linear VaR of the portfolio under the assumption of iid daily log returns and under the assumption that daily log returns are autocorrelated with first order autocorrelation = 0.2.

Answer:

Under the i.i.d. assumption and assuming 250 trading days per year, the annual excess return is 0.01% 250 = 2.5% and the volatility is 1% 250 = 15.81% The 1% 10-day VaR is 2.32635 0.01 10 10 0.0001 = 0.0726. That is, the 1% 10-day VaR is 7.26% of the portfolios value. But under the assumption that daily log returns have an autocorrelation of 0.2, the volatility and the VaR will be greater. The adjustment factor, i.e. the second term on the right-hand side of (IV.2.10) is calculated to be 124.375 for h = 250, and 4.375 for h = 10. Hence, the volatility is 1% 374.375 = 19.35%. and the 1% 10-day VaR is 2.32635 0.01 14.375 10 0.0001 = 0.0872.

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