C-1) You are allocating your total wealth of $10,000 into a risky asset and a riskfree asset. The risky asset has an expected return of 10% and a standard deviation of 15%. The riskfree rate is 3%. The borrowing rate (when you need to buy the risky asset on margin) is also 3%. a) If you want to have a standard deviation of 30%, how would you allocate your money into the two assets? (3 pts) b) What is the expected return of your portfolio in part (a)? (3 pts) c) Another investor, who also has $10,000 as his wealth, decides to buy the risky asset on margin. He borrows $5,000 at the borrowing rate to form his optimal complete portfolio. What is his risk aversion A? (4 pts) d) If the portfolio in part (a) is your optimal complete portfolio, is your risk aversion higher or lower than the answer in part (c)? (No explanations/calculations necessary. Simply state whether it is higher or lower.) (3 pts)

Part (a)

If w is the proportion invested in the risk asset with standard deviation σa then,

Standard deviation of the portfolio, σp = w x σa

Hence, 30% = w x 15%

Hence, w = 30% / 15% = 2

Hence, allocation of money across two assets:

• Money allocated to risky asset = Wealth x w = 10,000 x 2 = $ 20,000
• Money allocated to risk free asset = Wealth x (1 - w) = 10,000 x (1 - 2) = - $ 10,000.

This means you need to borrow $ 10,000 and invest the borrowed amount + wealth = 10,000 + 10,000 = $ 20,000 in the siky asset.

Part (b)

The expected return of the portfolio = w x expected return of risky asset + (1 - w) x riskfree rate = 2 x 10% + (1 - 2) x 3% = 17%

Part (c)

w = proportion invested in risky asset = (Borrowed amount + wealth) / Wealth = (5,000 + 10,000) / 10,000 = 1.5

Std dev of the Complete portoflio = σp = w x σa = 1.5 x 15% = 22.50%

Expected return of the portoflio = Rp = w x expected return of risky asset + (1 - w) x riskfree rate = 1.5 x 10% + (1 - 1.5) x 3% = 13.50%

His risk aversion, A = (Rp - Risk free rate) / (w x σp²) = (13.50% - 3%) / (1.5 x 22.50%²) = 1.38

Part (d)

Your risk aversion is lower than the answer in part (c).

In part (a) we have w = 2 against w = 1.5 in part (c) i.e. a higher proportion is invested in the risky asset in part (a). Hence, the portfolio in (a) is riskier than that in part (c). Hence, the investor in part (a) has higher risk appetite and hence his risk aversion will be lower. Hence, your risk aversion is lower than the answer in part (c).