A)Suppose you have two stocks (A and B) in your portfolio, worth $120,000 and $250,000 respectively. The annual volatility is 0.28 and 0.22 respectively. The correlation between the two stocks is 0.45. Please calculate the 5-day, 99% VaR and ES (expected shortfall) for the two positions separately and for the portfolio. How much is the VaR reduction in the portfolio? How much is the ES reduction in the portfolio? Re-calculate the portfolio VaR and ES and the corresponding reductions assuming a correlation of -0.45 and 1.0. What can you conclude? (Hint: for accuracy and convenience, it is a good idea to set up the calculation in a spreadsheet. Also, assume 252 days in a year.)

B)Suppose you add Stock C to your portfolio. Stock C has an annual volatility of 0.16. The additional correlations are _AC=0.55 and _BC=0.45. Keeping the total investment at $370,000, and assuming that achieving the lowest VaR (or ES) is the only objective, what is optimal allocation among the three stocks? Do you need to short any stocks? What would be the optimal allocation if shorting is not allowed?