Question

Er (%) 0.2 Consider the CAPM framework. Suppose that you currently have 40% of your wealth in Treasury Bills, risk-free, and 60% in the four assets below Asset i Bi Wi i = 1 8.5 0.2 0.1 i = 2 13.1 0.8 0.1 i = 3 16.6 1.2 0.2 i = 4 18.7 1.4 Let the four assets be traded in a market M with Erm = 15% and let the risk-free rate be rf = 4%, answer the following questions: a. (1 marks) Find the expected rate of return and the beta of your current portfolio. b. (3 marks) Now, suppose that you want an expected rate of return of 13%, your strategy is to sell some of your holdings of the risk-free assets and use the proceeds to buy the portfolio market. Find beta of the new portfolio? C. (2 marks) Let us assume that you want to hold only the risk-free asset and the market portfolio. What weights would give you an expected rate of ret of 13%, what is the beta of this new portfolio?

Answer 1

a) Expected return of a portfolio is the weighted average expected return of the constituent securities

So, Expected return of portfolio = 0.4* 4% + 0.1*8.5%+0.1*13.1%+0.2*16.6%+0.2*18.7% = 10.82%

Beta of a portfolio is the weighted average beta of the constituent securities (Risk free asset has a zero beta)

So, Beta of portfolio = 0.4* 0 + 0.1*0.2+0.1*0.8+0.2*1.2+0.2*1.4= 0.62

b) Let a weight of x be sold off in Risk free asset and invested in market portfolio to get a return of 13%

= (0.4-x)* 4% + x * 15%+ 0.1*8.5%+0.1*13.1%+0.2*16.6%+0.2*18.7% = 13%

=> 10.82%+ 11%*x = 13%

=> x= 0.1982

So, the Beta of new portfolio (market has a beta of 1)

= (0.4-0.1982)* 0 + 0.1982 * 1+ 0.1*0.2+0.1*0.8+0.2*1.2+0.2*1.4

=0.8182

c) Let w be the weight invested in Risk free rate and remaining (1-w) in Market portfolio

Then,

w*4%+ (1-w)*15% = 13%

=> w= 2/11 = 0.1818

So, the beta of new portfolio

=0.1818*0 + (1-0.8182)*1

=0.8182