Question

You are the new investment manager of Michael and Ariel. You received from the previous investment manager of Michael and Ariel’s partial information regarding their portfolios:
- Both have an optimal portfolio.
- The expected return in Michael’s portfolio is 6%.
- The SD in Ariel’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 a SD of 10%.
a. What is the proportion of each brother’s investment in a risk-free asset out of their portfolio?

b. Is it possible to rank the brothers’ risk preference? Explain.

c. What is the beta in each brother’s portfolio?

d. Suppose that today Michael and Ariel received an offer to buy a share with an expected return of 6% and beta of 0.3. Will they accept the offer?

Answer 1

a) We can safely assume that CAPM follows and each brother's portfolio is composed of Risk free asset and market portfolio.

IF Michael has w invested in risk free asset and (1-w) in Market portfolio

Expected return on Michaels' portfolio = 6%

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

=> w =2/3 or 0.67

and 1-w = 1/3 or 0.3333

So, Michael has 2/3 or 66.67% of his portfolio in Risk free asset

IF Ariel has w invested in risk free asset and (1-w) in Market portfolio

Standard deviation of Ariel's portfolio = 12%

=> (1-w)*10% = 12% (as risk free asset has a 0 standard deviation)

=> 1-w = 1.2

or w =-0.2

So, Ariel has -20% investment in Risk free Asset i.e. Ariel has borrowed 20% at risk free rate

b) Yes, Michael prefers low risk and is satisfied with lesser returns

Ariel is a risk taker and is ready to take higher risk for higher returns

c) As Beta of a portfolio is weighted average beta of the constituent assets. and the Beta of Risk free asset is 0 and beta of market portfolio is 1

Michael's portfolio's beta = 2/3*0+1/3*1 = 0.33

Ariel's portfolio's beta = -0.2*0+1.2*1 = 1.2

d) Required return of brothers as per CAPM = 5%+0.3*(8%-5%) = 5.9%

As the expected return of 6% is higher than the required return, Ariel and Michael will accept the offer