Question

150% of his money in the risky portfolio P is allocated by investor A.  Portfolio P has 3% excess return and 7% standard deviation, both in annualized terms. Borrowing costs are rising, and investor A is worried about that.

Investor B invests also in Portfolio P. But Investor B is not worried about rising borrowing rates.  Why would that be the case?

Assume that the utility is, U = E(r) – 0.5As²   to the portfolio with expected return E(r) and variance of return  s², where A measures the investor’s risk aversion.

1. What is the risk aversion of Investor A?

2. If one chooses to invest 100% of Portfolio P, what is the risk aversion of him?

3. Why is Investor B not worried about rising borrowing rates?

Answer 1

The Utility Function as

Risk free rate= R

Return from Portfolio P = R+3%

For Investor A,

In case , investment of 150% in portfolio P i.e borrowing of 50% at Risk free rate R , return = 1.5*(R+3)- 0.5*R = 0.5 R+4.5

For Investor B,

In case , investment of 100% in portfolio P , return = (R+3)

Answer a) The Investor B wouldn't be worried about rising borrowing rates , as his investment strategy is free from any type of borrowing. Only 100%, i.e owned, wealth has been invested in portfolio P.

Answer 1) investor indifferent between the risky portfolio and the risk-free asset at level of A ( risk Aversion ) in above utility function ,

for Investor A,

R = 0.5R+4.5 - A*0.5*(0.07^2)

=> 0.5 R=4.5-A*4.497

=> A =(0.5R-4.5) / -4.497 = 1- 0.11 R

Risk Aversion of Investor A, = 1-0.11* risk free rate

Answer 2)

for Investor B,

R = R+3 - A*0.5*(0.07^2)

=> A*4.497= 3

=> A =3 /4.497 = 0.667

Risk Aversion of Investor A, = 0.667.

Answer 3) As value of risk aversion of Investor A depend on risk free rate R(borrowing rate) , while Investor B is free from R.That's why B is not worried about the growing interest rate.