Question

? Suppose the investment budget is 300,000$ and investor borrows an additional 120,000, investing the whole money in the risky asset. In fact, we borrow at risk?free rate to buy more of risky asset. The risk?free rate and return on the risky asset are 7% and 15% respectively. The standard deviation of the risky asset is 22%. Using the information provided answer the following questions: 12.1? What are the weights you are investing in risky and riskless asset? 12.2? What is the standard deviation of your complete portfolio? 12.3? What is the sharp ratio of your portfolio? 12.4? What is your Risk Aversion based on the choice you have made? 12.5? How would you change the allocation if your Risk Aversion was 5? (A=5) Answer the next two questions based on the result you got for 12.5: 12.6? How would you compare the expected return and standard deviation of your complete portfolio to expected return and standard deviation of the risky asset? 12.7? Draw a Capital Allocation Line for your portfolio on expected return/ standard deviation diagram. What is the slope of the CAL?

Answer 1

Risk free rate (r_f) = 7% ; Return on risk asset (r_a) = 15% ; Standard deviation risk free asset (SD_r) = 0 (by definition that is why it is called risk free; SD_a = 22%

12.1 : The investment in risk asset is (300000+120000) = $420000 and the borrowing in risk free are $. 120000. That is if we denote risk asset as 'a' and risk free as 'f' then we have:

420000 a - 120000 f = 300000; where the weights of a will be (420000/300000) = 140% and f will be (-120000/300000) = - 40%

12.2: Variance of portfolio is given by :

Covariance can be defined as product of correlation and respective SD. Since the SD for f is going to be zero the above formula just reduced to and SD of portfolio will be square root of this variance. Hence SD = [(140%)² * (22%)²]^(1/2) = 30.80%

12.3 : Sharpe ratio = [Expected Return - risk free rate]/SD

Expected return of portfolio = 140% * 15% - 40% * 7% = 18.20%

Sharpe ratio = (18.20%-7%)/30.80% = 36.36

12.4 : U = Expected Return - 1/2 * A * (SD)² ; now for risk free asset, the U will simply equal to its return which is 7% or 0.07

For risk asset portfolio U = 18.20% - 1/2 * A * (30.80%)² = 18.20% - 0.09486/2 * A

For the investor to prefer risky portfolio, the U value should be higher than U of risk free asset at 0.07, hence we have : 18.20% - 0.09486/2 A > 0.07 or A < 2.36

12.5 : At 5, the value of U for the above portfolio with 140% a and 40% f will be negative. We need to find the value of w_a which will atleast make the value of U equal to 7%.

ER = w_a * 15% + w_f * 7% and SD = (w²_a * SD²_a)²; pluggin these value in the U equation we get:

U = (w_a * 15% + w_f * 7%) - 1/2 * A * w²_a * SD²_a ; given A = 5 and U has to be greater than 7% then w_a should be atmost 66% and w_f = 34%

12.6 : New portfolio return : 66% * 15% + 34% * 7% = 12.280%

New portfolio SD = [(66%)² * (22%)²]^(1/2) = 14.52%