(a) Suppose f1 and f2 are the factor scores for the first two principal components. The change in the portfolio value ($ millions) in one day as a result of the first two principal components is ΔP = 0.05f1 – 3.88f2. The factor scores are uncorrelated. Given the standard deviations of the first two factor scores ($millions) are 17.55 and 4.77. Assuming the factor scores are normally distributed, what are the one-day 99% VaR and expected shortfall of the portfolio?

(b) A bank has written a European call option on one stock and a European put option on another stock. For the first option, the stock price is 50, the strike price is 51, the volatility is 28% per annum, and the time to maturity is nine months. For the second option, the stock price is 20, the strike price is 19, the volatility is 25% per annum, and the time to maturity is one year. Neither stock pays a dividend, the risk-free rate is 6% per annum, and the correlation between stock price returns is 0.2. What is a 10-day 99% VaR using only deltas? By how much does diversification reduce the VaR?