Question

A stock costs $30. It rises to $40 with probability 0.62 and falls to $20 with probability 0.38. The risk free interest rate is 2%. The market risk premium is 8%.

What is the stock's beta? Why does the stock pay a lower risk premium than the market?

Answer 1

We will first calculate the Expected return for the stock.

Stock's Expected return calculation

Current price of the stock = $30

Probability that the price will be $40 = p1 = 0.62

Return when the price becomes $40 = R1 = (40-30)/30 = 33.33% = 1/3

Probability that he price will be $20 = p2 = 0.38

Return when the price becomes $20 = (20-30)/30 = -33.33% = -1/3

Probability	Return
0.62	33.33%
0.38	-33.33%

Expected return is calculated using the formula:

Expected return for stock = E[RS] = p1*R1 + p2*R2 = 0.62*(1/3) + 0.38*(-1/3) = 8%

Beta calculation

Risk-free rate = RF = 2%

Market risk premium = RM - RF

Beta of the stock can be calculated from CAPM equation

E[RS] = RF + βS*(RM - RF)

8% = 2% + βS*8%

8% - 2% = βS*8%

βS = 6%/8% = 0.75

Beta of the stock = βS = 0.75

Risk premium for the stock = E[RS] - RF = 8% - 2% = 6%

Market risk premium = 8%

Beta of the stock = 0.75

Beta of the market = 1 [We know that the beta of the market is 1]

The risk premium for the stock (6%) is less than the market risk premium (8%) because the beta for the stock is less than the beta of the market. This means that the stock carries lower risk relative to market and hence it pays lower risk premium than the market.

Answer -> Stock's beta = 0.75