Question

Consider 2 scenarios: Boom Economy and Normal Economy. The Boom economy has 30% chance of happening, while Normal economy has 70% chance of happening.

For each scenario (Boom and Normal), stock ABC has a return of 25%, and 4%, respectively; stock XYZ has a return of 10% and 6.5%, respectively; the market portfolio has a return of 12% and 5% respectively.

1) Calculate Expected return, Variance and Standard deviation for stock ABC and XYZ

2) Based on your results in part (1), can you decide which stock to invest?

3) Calculate Beta for stock ABC and XYZ

4) If the T-bill rate is 3%, what does the CAPM say about the fair expected rate of return on the two stocks? How does this result influence your investment decision?

Answer 1

1)

Expected return = Summation of (Probability of the scenario*Expected return in that scenario)

Expected return (mean) of Stock ABC = 0.3*25% + 0.7*4% = 10.3%

Expected return (mean) of Stock XYZ = 0.3*10% + 0.7*6.5% = 7.55%

Variance of returns = Summation of (Probability of the scenario*(Expected return in that scenario-mean)^2)

Variance of Stock ABC returns = 0.3*(0.25-0.103)^2 + 0.7*(0.04-0.103)^2 = 0.009261

Variance of Stock XYZ returns = 0.3*(0.1-0.0755)^2 + 0.7*(0.065-0.0755)^2 = 0.00025725

Standard deviation of return = Square root of variance

Standard deviation of Stock ABC returns = 0.009261^0.5 = 0.096

Standard deviation of Stock XYZ returns = 0.00025725^0.5 = 0.016

2)

We take the ratio of Expected return to standard deviation to find which stocks provide superior returns for the risk

ratio of Expected return to standard deviation Stock ABC = 10.3%/0.096 = 1.073

ratio of Expected return to standard deviation Stock XYZ = 7.55%/0.016 = 4.72

Hence, Stock XYZ would be preferred

3) Beta of slope of stock return to the market returns

Beta of ABC = (0.7*25%-0.3*4%)/(0.7*12%-0.3*5%) = 2.36

Beta of XYZ = (0.7*10%-0.3*6.5%)/(0.7*12%-0.3*5%) = 0.73

4) Expected return on market = 0.3*12% + 0.7*5% = 7.1%

The required rate of return R(e) is calculated by CAPM model

R(e) = r(f) + Beta*(R(m) - r(f))

fair expected rate of return on ABC = 0.03 + 2.36*(0.071-0.03) = 0.1268

fair expected rate of return on XYZ = 0.03 + 0.73*(0.071-0.03) = 0.0600

Here, the fair expected rate using is less than the actual expected rate in case of Stock XYZ. Hence, stock XYZ is a better investment.