Value at Risk (VaR) is a statistic that measures and quantifies the level of financial risk within a firm, portfolio or position over a specific time frame.

QUESTION #4: PORTFOLIO OPTIMIZATION AND CAPM (25%) Setup: Suppose the investment universe consists of only two risky assets with returns as follows: - Asset A: FA = 10% and A = 15% - Asset B: Pb = 15% and ob = 30% - cov(ra,TB) = 0 A,B = 0.018 (a): Write out the Markowitz (mean-variance) portfolio optimization problem and the system of equations associated with the Lagrangian solution method. (b): Find the weights, expected return, and standard deviation for the minimum variance portfolio (Hint: set = 0 and u = 1 using the Lagrangian method) (c): Find a second portfolio in the minimum variance set (Hint: set 1=1 and =0) (d): Plot the individual assets and the two efficient portfolios found in (a) & (b). Roughly sketch the efficient frontier (e): Suppose a risk-free asset is also available with rf = 5%. Find the optimal fund of risky assets and the new efficient frontier (f): Now assume the portfolio found in (e) corresponds to the CAPM market portfolio and this portfolio is efficient. Find BA & BB (g): Suppose you discover a new asset, C, with Bc = BA and oc = 25%. Ignoring any impact on the composition of the market portfolio, calculate ic, and the systematic and idiosyncratic risk (standard deviation) for rc (h): Calculate VaR at the a = 95% & a = 99% confidence levels for market portfolio found in (e) assuming the portfolio has a normal distribution. (Hint: The 5% and 1% critical values for a standard normal random variable are -1.645 & -2.326, respectively.) (*): Reconsider the risky portfolio found in (e). If FA, CA, FB, and ob are as in the problem setup, rf = 5%, but o A, B is unknown, what is the range of possible values for the expected return of the optimal risky portfolio? (Hint: Find an expression for the weights as a function of o A, B and recall o A,B = PA,BOAOB) QUESTION #4: PORTFOLIO OPTIMIZATION AND CAPM (25%) Setup: Suppose the investment universe consists of only two risky assets with returns as follows: - Asset A: FA = 10% and A = 15% - Asset B: Pb = 15% and ob = 30% - cov(ra,TB) = 0 A,B = 0.018 (a): Write out the Markowitz (mean-variance) portfolio optimization problem and the system of equations associated with the Lagrangian solution method. (b): Find the weights, expected return, and standard deviation for the minimum variance portfolio (Hint: set = 0 and u = 1 using the Lagrangian method) (c): Find a second portfolio in the minimum variance set (Hint: set 1=1 and =0) (d): Plot the individual assets and the two efficient portfolios found in (a) & (b). Roughly sketch the efficient frontier (e): Suppose a risk-free asset is also available with rf = 5%. Find the optimal fund of risky assets and the new efficient frontier (f): Now assume the portfolio found in (e) corresponds to the CAPM market portfolio and this portfolio is efficient. Find BA & BB (g): Suppose you discover a new asset, C, with Bc = BA and oc = 25%. Ignoring any impact on the composition of the market portfolio, calculate ic, and the systematic and idiosyncratic risk (standard deviation) for rc (h): Calculate VaR at the a = 95% & a = 99% confidence levels for market portfolio found in (e) assuming the portfolio has a normal distribution. (Hint: The 5% and 1% critical values for a standard normal random variable are -1.645 & -2.326, respectively.) (*): Reconsider the risky portfolio found in (e). If FA, CA, FB, and ob are as in the problem setup, rf = 5%, but o A, B is unknown, what is the range of possible values for the expected return of the optimal risky portfolio? (Hint: Find an expression for the weights as a function of o A, B and recall o A,B = PA,BOAOB)