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 of Stock ABC = Probability of Boom *Return of ABC in boom+Probability of Normal*Return of ABC in norma
=30%*25%+70%*4% =10.30%
Expected Return of Stock XYZ = Probability of Boom *Return of ABC in boom+Probability of Normal*Return of ABC in norma
=30%*10%+70%*6.5% =7.55%

Variance of Stock ABC =30%*(25%-10.30%)^2+70%*(4%-10.30%)^2 =0.9261% or 0.93%
Variance of Stock XYZ =30%*(10%-7.55%)^2+70%*(6.5%-7.55%)^2 =0.02573% or 0.03%

Standard Deviation of ABC =0.9261%^0.5 =9.62%
Standard Deviation of XYZ =0.02573%^0.5 =1.60%

2) Coefficient of Variation of ABC=Standard Deviation of ABC/Expected Return of ABC =9.62%/10.30%=0.93
Coefficient of Variation of XYZ=Standard Deviation of XYZ/Expected Return of XYZ =1.60%/7.55%=0.21
Since Coefficient of Variation of XYZ is lower it should be accepted.

3)Expected Return of Market =30%*12%+70%*5% =7.1%
Variance of Market =30%*(12%-7.1%)^2+70%*(5%-7.1%)^2 =0.1029%
Covariance of Stock ABC and Market =30%*(12%-7.1%)*(25%-10.30%)+70%*(5%-7.1%)*(4%-10.30%)=0.0030870
Beta of ABC =Covariance of Stock ABC and Market /Variance of Market =0.0030870/0.1029%=3.00

Covariance of Stock XYZ and Market =30%*(12%-7.1%)*(10%-7.55%)+70%*(5%-7.1%)*(6.50%-7.55%)=0.000515
Beta of Stock XYZ =Covariance of Stock XYZ and Market /Variance of Market =0.000515/0.1029%=0.5

4)Fair Return of ABC using CAPM =Risk free Rate+Beta*(Market Return-Risk free Rate) =3%+3*(7.1%-3%) =15.3%
Fair Return of XYZ using CAPM =Risk free Rate+Beta*(Market Return-Risk free Rate) =3%+0.5*(7.1%-3%) =5.05%

Since Expected Return of ABC is less than required rate of ABC Project should be rejected.
and Expected return of XYZ is more than the required rate of XYZ project should be accepted.