Question

Suppose the yield on short-term government securities (perceived to be risk-free) is about 4%. Suppose also that the expected return required by the market for a portfolio with a beta of 1 is 12%. According to the capital asset pricing model:

a. What is the expected return on the market portfolio?

b. What would be the expected return on a zero-beta stock?

c. Suppose you consider buying a share of stock at a price of \(40. The stock is expected to pay a dividend of \)3 next year and to sell then for $41. The stock risk has been evaluated atβ= - .5. Is the stock overpriced or underpriced?

Short answer

The correct answer would be:

a. 12%

b. 4%

c. Underpriced

Step-by-step solution

Given Information

E ( r ) = 12%

Beta = 1.0

Solution for ‘a’ Calculation of expected return of the portfolio

E(r) = r_f+β[E(r_M) - r_f ]

On substitutingr_f= 4%,(r_M)= 12% and β = 1.0

E(r) = 4+1[12 - 4 ] = 12%

Since the market portfolio has a Beta of 1.0 (given), its expected rate of return would be 12%

Solution for ‘b’ Calculation of expected return of zero beta stock

Sinceβ = 0 (given), this would mean that the stock has no systematic risk.

In this case, the expected rate of return would be risk free rate = 4% i.e.E(r) = 4+ 0[12 - 4 ] = 4%

Solution for ‘c’ Calculation to check if stock is overpriced or underpriced

β = - 0.5 (given),

(r_M) = 12%

r_f= 4%

So, using the SML, the fair rate of return E ( r ) =r_f + β_i[E(r_M) - r_f]

= 4% + (-0.5) (12%- 4%) = 0.0 %

The E ( r ) using expected price and dividend for next year = (P₁ + D₁ – P_{0 )}/ P₀

=(41 + 3 - 40) / 40

= 4/40 = 10%

In the above scenario, since expected return exceeds the fair return, the stock must be underpriced.