Question

Consider the following two scenarios for the economy and the expected returns in each scenario for the market portfolio, an aggressive stock A, and a defensive stock D.

Rate of Return

Scenario

Market

Aggressive Stock A

Defensive Stock D

Bust

–8

%

–10

%

–5

%

Boom

30

40

22

Required:
a. Find the beta of each stock.
b. If each scenario is equally likely, find the expected rate of return on the market portfolio and on each stock.
c. If the T-bill rate is 3%, what does the CAPM say about the fair expected rate of return on the two stocks?
d. Which stock seems to be a better buy on the basis of your answers to (a) through (c)?

Answer 1

Answer a)

Given data in question,

Rate of Return in %

Scenario

Market

Aggressive Stock A

Defensive Stock D

Bust

-8

-10

-5

Boom

30

40

22

Beta of Agg. Stock A and Defensive Stock D can be found out by the formula,

Beta Of a Stock = (Absolute Change in the Returns of a stock) / (Absolute Change in the Returns of the Market)

Absolute change in Market's Return = 30 - (-8) = 38%

Absolute change in A's Return = 40 - (10) = 50%

Absolute change in D's Return = 22 - (-5) = 27%

Therefore,

Beta of A= (50/38) = 1.315789

Beta of D = 17/38      = 0.71052

Answer B)

We can calculate the expected returns using the following formula:

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Therefore, if both scenarios are equally likely, i.e. a probability of 0.5 for each event,

E(R) of market = (0.5* -8%) + (0.5 * 30%) = 11%

E(R) of Stock A = (0.5* -10%) + (0.5 * 40%) = 15%

E(R) of Stock D = (0.5* -5%) + (0.5 * 22%) = 8.5%

Answer C)

According to the CAPM, if we calculate the expected returns on the individual stocks taking into consideration the risk free treasury rate and the Beta's of

the individual stocks, the formula to calculate the same is,

Fair Expected (Return of a Stock) =   Risk Free Rate + Beta ( Expected return of the Market - Risk Free Rate )

(Exp. Return of Market taken as solved in the previosu question)

Therefore, using the CAPM, the Exp. Returns for Stock A and D are,

Stock A Fair Return = 3 + 1.315789 (11 - 3)   = 13.52 %

Stock D FairReturn = 3 + 0.71052 (11 - 3)   = 8.68 %

Hence, according to the returns calculated by the CAPM, the fair returns for A is 13.52 while we are expecting to generate 15%,

which means the stock is currently undervalued and we should buy and go long on the same,

whereas for Stock B the fair return is 8.68 while we expect to generate only 8.5%, which means that the stock is overvalued and we should sell or go short on the same.

Answer D)

Stock A seems to be a better buy on the basis of my answers from (a) to (c), as it is expected to generate a rate of return greater than the fair value for the same risk exposure,

whilst Stock D seems to be overvalued as it has a lower expected rate of return for the risk undertaken and is priced higher than its fair value.