A stock has a beta of 1.95 and an expected return of 12 percent. A risk-free ass

Question

A stock has a beta of 1.95 and an expected return of 12 percent. A risk-free asset currently earns 3.8 percent.

What is the expected return on a portfolio that is equally invested in the two assets? (Round your answer to 2 decimal places. (e.g., 32.16))

If a portfolio of the two assets has a beta of 0.78, what are the portfolio weights? (Round your answer to 4 decimal places. (e.g., 32.1616))

If a portfolio of the two assets has an expected return of 10 percent, what is its beta? (Do not round intermediate calculations and round your answer to 3 decimal places. (e.g., 32.161))

If a portfolio of the two assets has a beta of 3.90, what are the portfolio weights? (Negative amount should be indicated by a minus sign.)

A stock has a beta of 1.95 and an expected return of 12 percent. A risk-free asset currently earns 3.8 percent.

Explanation / Answer

Since the portfolio is equally weighted, we can sum the returns of each asset and
divide by the number of assets. The expected return of the portfolio is:

= 12%+3.8%=7.9%

b. If a portfolio of the two assets has a beta of 0.78 what are the portfolio weights?

Here we need to find the portfolio weights that result in a portfolio with a b of 0.78
We know the b of the risk-free asset is zero. We also know the weight of the risk-free
asset is one minus the weight of the stock since the portfolio weights must sum to
one, or 100 percent.
So:

       bp = 0.78 = wS(1.95) + (1 – wS)(0)
       0.78 =1.95 wS + 0 – 0wS
       wS = 0.78/1.95
       wS = 0.4   

       And, the weight of the risk-free asset is:

       wRf = 1 – 0.4
       wRf = 0.6

c. If a portfolio of the two assets has an expected return of 10 percent, what is its beta?

We need to find the portfolio weights that result in a portfolio with an expected
return of 10%. We also know the weight of the risk-free asset is one minus the
weight of the stock since the portfolio weights must sum to one, or 100 percent. So:

       E(Rp) = 0.10 = .12wS + .038(1 – wS)   
       0.10 = 0.12wS + 0.038– 0.038 wS
       0.10– 0.038= 0.082wS
       wS = 0.062/.082

       wS = 0.76

       So, the b of the portfolio will be:

       bp = 0.76(1.95) + (1 – 0.76)(0)
       bp = 1.482

d. If a portfolio of the two assets has a beta of 3.90, what are the portfolio weights.

Solving for the b of the portfolio as we did in part b, we find:

      
bp = 3.9 = wS(1.95) + (1 – wS)(0)

      
3.9 = 1.95 wS

      
wS = 3.9/1.95

      
wS = 2

      
wRf = 1 – 2
      
wRf = –1
  

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