Practice Problems There are two assets A and B with returns ra and r, respectively. Given the covariance matrix as follows. ( I. Find out the composition (weights) of the lowest risk portfolio. Use w denote the weight in asset A II. Consider a risk averse investor with exponential utility function as follow U(x) = 1 - e-Ax in which x is the net wealth and 1 is the degree of risk aversion. Now assume x is normally distributed due to risky investment, i.e. x~N(u,0%). The investor is rational and she will make the investment decision by the following rule: max E(U(x)) Please show that the above optimization program is equivalent to the below one 1 max p +3 102 in which u = E(x). Hint: the unity in the exponential utility function could be ignored in derivation III. Assume the asset A in the first question is a risk free asset. The expected return for asset B is up. Please use the result in the second question to solve the optimal portion in asset Aw, for the investor. 1 Practice Problems There are two assets A and B with returns ra and r, respectively. Given the covariance matrix as follows. ( I. Find out the composition (weights) of the lowest risk portfolio. Use w denote the weight in asset A II. Consider a risk averse investor with exponential utility function as follow U(x) = 1 - e-Ax in which x is the net wealth and 1 is the degree of risk aversion. Now assume x is normally distributed due to risky investment, i.e. x~N(u,0%). The investor is rational and she will make the investment decision by the following rule: max E(U(x)) Please show that the above optimization program is equivalent to the below one 1 max p +3 102 in which u = E(x). Hint: the unity in the exponential utility function could be ignored in derivation III. Assume the asset A in the first question is a risk free asset. The expected return for asset B is up. Please use the result in the second question to solve the optimal portion in asset Aw, for the investor. 1