Question

Consider 2 scenarios: Boom Economy and Normal Economy. The Boom economy has 20% chance of happening, while Normal economy has 80% 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

Given,

Particulars	Probability	stock ABC	Stock XYZ	Market portfolio
Boom	20%	25%	10%	12%
Normal	80%	4%	6.5%	5%

1) Calculation of Expected return of Stock ABC:

Particulars	Probability (1)	stock ABC (2)	Expected return (3) (1*2)
Boom	20% or 0.2	25% or 0.25	0.05
Normal	80% or 0.8	4% or 0.04	0.032
		Expected return	0.082

Expected return of stock ABC = 0.082 or 8.2%

Calculation of Expected return of stock XYZ:

Particulars	Probability (1)	Stock XYZ (2)	Expected return (3) (1*2)
Boom	0.2	10% or 0.1	0.02
Normal	0.8	6.5% or 0.065	0.052
		Expected return	0.072

Expected return of stock XYZ = 0.072 or 7.2%

Calculation of Variance and Standard deviation of Stock ABC:

Particulars	Probability (1)	Return-Expeced return (2)	Square of Return-Expeced return (3)	Variance (4) (1*3)
Boom	0.2	0.25-0.082=0.168	(0.168)^2 = 0.028	0.0056
Normal	0.8	0.04-0.082 = -0.042	(-0.042)^2 =0.002	0.0016
			Varinace	0.0072

Variance = 0.0072

Std deviation = Square root of Variance

= Square root of 0.0072

= 0.085 or 8.5%

Calculation of Varinace and standard deviation of Stock XYZ:

Particulars	Probability (1)	Return-Expected return (2)	Square of Return-Expeced return (3)	Variance (4) (1*3)
Boom	0.2	0.1-0.072 =0.028	(0.028)^2 = 0.0008	0.00016
Normal	0.8	0.065-0.072 =-0.007	(-0.007)^2 = 0.00005	0.00004
			Variance	0.0002

Variance = 0.002

Std deviation = Square root of Variance

= Square root of 0.002

= 0.014 or 1.4%

2) Analysis of which stock to invest

Particulars	ABC	XYZ
Expected return	8.2%	7.2%
Standard deviation	8.5%	1.4%

Since by taking less risk of 1.4% stock of XYZ has generated return of 7.2% Hence investing is better than ABC, if the investor is taking into mind only risk but not returns.

If the investor is such that who doesn't concentrate on risk but only focus on return generated then the investing in stock ABC is better since it is generating more return than XYZ.

3) Calculation of beta:

Beta = Covariance of stock and market/Variance of market

Calculation of Expected return of market portfolio:

Particulars	Probability (1)	Market portfolio (2)	Expected return (3) (1*2)
Boom	0.2	12% or 0.12	0.024
Normal	0.8	5% or 0.05	0.04
		Expected return	0.064

Expected return of Market = 0.064 or 6.4%

Calculation of Variance and standard deviation of market portfolio:

Particulars	Probability (1)	Return-Expeced return (2)	Square of Return-Expected return (3)	Variance
Boom	0.2	0.12-0.064=0.056	(0.056)^2 = 0.00313	0.000626
Normal	0.8	0.05-0.064= -0.014	(-0.014)^2 = 0.00020	0.00016
			Variance	0.000786

Varinace = 0.000786

Standard deviation = Square root of Variance

= Square root of 0.000786

= 0.028 or 2.8%

Calculation of covariance of stock ABC and market:

Particulars	Probability (1)	Return-Expected return of ABC (2)	Return-Expected return of Market (3)	Covariance (4) (1*2*3)
Boom	0.2	0.168	0.056	0.0019
Normal	0.8	-0.042	-0.014	0.0005
			Covariance of ABC and Market	0.0024

Calculation of Covarinace of Stock XYZ and Market:

Particulars	Probability (1)	Return-Expected return of XYZ (2)	Return-Expected return of market (3)	Covariance (4) (1*2*3)
Boom	0.2	0.028	0.056	0.00031
Normal	0.8	-0.007	-0.014	0.00008
			Covarinace of stock XYZ and Market	0.00039

Covariance of ABC and market = 0.0024

Covariance of XYZ and market = 0.00039

Variance of market = 0.000786

Beta of ABC = Covariance of ABC and market/ Variance of market = 0.0024/0.000786 = 3.05

Beta of XYZ = Covarinace of XYZ and market/ variance of market = 0.00039/0.000786 = 0.50

4) Calculation of CAPM expected rate of return:

T- bill (risk free rate Rf) = 3%

Expected return of market (Rm) = 6.4%

Beta of ABC = 3.05

Beta of XYZ = 0.50

Required rate of return = Rf+beta*(Rm-Rf)

Required rate of return of ABC = 3%+3.05*(6.4%-3%)

=3%+10.37% = 13.37%

Required rate of return of XYZ = 3%+0.50*(6.4%-3%)

=3%+1.7% = 4.7%

What does CAPM fair Expected rate of return say about on stocks:

In CAPM, beta indicates of potential investment is a measure of how much risk the investment will add to a portfolio that looks like the market. If a stock is riskier than the market, it will have beta greater than one. If a stick has beta of less than one, the formula assumes that it will reduce the risk of a portfolio.

In the given case, beta of XYZ is less than one hence it reduces the risk of portfolio whereas ABC has beta greater than one, and hence ABC stock is riskier than market.

How does CAPM affects investment decisions:

Investment decisions are based on the risk return patterns.Appropriate measures of risk and returns are of great concern to investors. CAPM, based on market beta addresses this concern quite well. CAPM states that if the earnings of a security are fixed and steady, its risk is zero and if earnings fluctuate it is considered risk. The CAPM also states that the expected risk premium on each investment is proportional to its beta I.e., systematic risk. In a nutshell, the CAPM model states the differences in the systematic risk. Therefore the securities with higher systematic risk will offer higher return than the securities with lower systematic return.