Question

Two assets P and Q, E(rp) = 10%, σp = 10%, E(rq) = 15%, σq = 25%. The risk-free rate of interest (on bills) is 5%.

Now, let’s allow borrowing on margin. Investors can buy P and Q on margin (but not both). The borrowing rate is the risk-free rate. State whether the following statements are true or false and give reasons:

(a) “Investor who prefers P to Q must be risk-averse.”

(b) “Investor who prefers Q to P cannot be risk-averse.”

(c) “No rational, risk-averse investor will borrow to buy P on margin.”

(d) “No rational, risk-averse investor will combine bills with Q.”

Answer 1

To compare the two assets, we calculate the sharpe ratio for both. Sharpe Ratio tells us excess returns per unit of additional risk

Sharpe Ratio = (Exp Return-Risk Free Return) / Std Dev

Sharpe Ratio for P = (10-5)/10 = 0.5

Sharpe Ratio for Q = (15-5)/25 = 0.4

a) Investor who prefers P to Q must be risk-averse. The statement is true as risk averse investors need higher returns for taking additional risk. From sharpe ratio we conclude that Asset P yield higher return per additional unit of risk take i.e 0.5% as compared to 0.4%

b) Investor who prefers Q to P cannot be risk-averse. The statement is false as if Investor who prefers Q over P can still be Risk Averse. This is because the returns are marginally less and Q still yield a 0.4% extra return by taking additional risk.

c) No rational, risk-averse investor will borrow to buy P on margin. This is false, as buying Asset on margin implies taking additional risk that is not a characteristic of a Risk Averse Investor.

d)No rational, risk-averse investor will combine bills with Q. This statement is true because combining Asset q with Bills means reducing portfolio risk by investing a portion in risk free asset.