Question

You are the new investment manager of Bob and Jim. You received from the previous investment manager of Bob and Jim’s partial information regarding their portfolios:

- Both have an optimal portfolio.

- The expected return in Bob's portfolio is 6%.

- The SD in Jim’s portfolio is 12%.

You know that the current risk-free interest rate is 5% and the market portfolio has an expected return of 8% and an SD of 10%.

a. What is the proportion of each brother’s investment in a risk-free asset out of their portfolio?

Answer: the proportion of Bob's investment in a risk-free asset out of his portfolio is __________%

the proportion of Jim’s investment in a risk-free asset out of his portfolio is __________%

Answer 1

a. Bob's investment in risk free asset

weight of risk free asset = w

weight of risky asset = 1 - w

Expected return of Bob = w * return of risk free rate + (1 - w) * return of market portfolio

6% = w * 5% + (1 - w) * 8%

6% = - w * 3% + 8%

3% * w = 2%

w = 66.67%

the proportion of Bob's investment in a risk-free asset out of his portfolio is 66.67%

2. Jim's investment in risk free asset

expected return of Jim portfolio = Risk free rate + SD of Jim portfolio * (market rate - risk free rate) / SD of market

expected return of Jim portfolio = 5% + 12% * (8% - 5%) / 10%

expected return of Jim portfolio = 5% + 3.60%

expected return of Jim portfolio = 8.60%

weight of risk free asset = w

weight of risky asset = 1 - w

Expected return of Bob = w * return of risk free rate + (1 - w) * return of market portfolio

8.60% = w * 5% + (1 - w) * 8%

8.60% = - w * 3% + 8%

3% * w = -0.60%

w = -20%

the proportion of Jim’s investment in a risk-free asset out of his portfolio is -20%

it means jim borrowed money at risk free rate of 5% and invested into market at 8%