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

Stock ABC - Expected return computation

Scenario	Probability	Scenario return	Proability weighted return (Probability x scenario)
Boom	30%	25%	7.500%
Normal	70%	4%	2.800%
Expected return --> Sum of Probability weighted return	10.300%

Stock ABC - Variance computation

Scenario	Probability	Scenario return	Expected return	Scenario return - expected return	Square of (scenario return - expected return)	Probability x square of (scenario return - expected return)
Boom	30%	25%	10.300%	14.700%	2.16%	0.6483%
Normal	70%	4%	10.300%	-6.300%	0.40%	0.2778%
Variance ----> sum of (Probability x square of (Scenario return - expected return)	0.9261%

Stock ABC - Standard Deviation

Variance of stock ABC	0.9261%
Standard deviation of ABC ---> square root of variance	9.623%

Stock XYZ - Expected return computation

Scenario	Probability	Scenario return	Probability weighted return (Probability x scenario)
Boom	30%	10.000%	3.000%
Normal	70%	6.500%	4.550%
Expected return --> Sum of Probability weighted return	7.550%

Stock XYZ - Variance computation

Scenario	Probability	Scenario return	Expected return	Scenario return - expected return	Square of (scenario return - expected return)	Probability x square of (scenario return - expected return)
Boom	30%	10.000%	7.550%	2.450%	0.060%	0.018%
Normal	70%	6.500%	7.550%	-1.050%	0.011%	0.008%
Variance ----> sum of (Probability x square of (Scenario return - expected return)	0.0257%

Stock XYZ - Standard Deviation

Variance of stock ABC	0.0257%
Standard deviation of ABC ---> square root of variance	1.604%

Part 2 - Based on above computations we can note that stock ABC has higher return in comparison to stock XYZ. However, stock ABC carries higher risk in comparison to stock XYZ. We can note that stock ABC has higher incremental risk in comparison to incremental return. Therefore it would be optimal to invest in stock XYZ.

Part 3 - Beta computation

Stock ABC

Market return- Expected return computation

Scenario	Probability	Scenario return	Probability weighted return (Probability x scenario)
Boom	30%	12%	3.600%
Normal	70%	5%	3.500%
Expected return --> Sum of Probability weighted return	7.100%

Market - Variance computation

Scenario	Probability	Scenario return	Expected return	Scenario return - expected return	Square of (scenario return - expected return)	Probability x square of (scenario return - expected return)
Boom	30%	12%	7.100%	4.900%	0.24%	0.0720%
Normal	70%	5%	7.100%	-2.100%	0.04%	0.0309%
Variance ----> sum of (Probability x square of (Scenario return - expected return)	0.1029%

Covariance between market and stock ABC

Scenario	Market return	Expected market	Stock ABC return	Expected ABC return	Market return - expected market return	ABC return - expected ABC return	(Market return - expected market return) x (ABC return - expected ABC return)
Boom	12%	7.100%	25%	10.300%	4.900%	14.700%	0.00720
Normal	5%	7.100%	4%	10.300%	-2.100%	-6.300%	0.00132
Step 1 : Sum of (Market return - expected market return) x (ABC return - expected ABC return)	0.00853
Step 2 : No of scenarios	2.00000
Covariance ---> Step 3 : Step 1 / Step 2	0.00426

Beta computation

Covariance between market and stock ABC	0.00426
Variance of market return	0.1029%
Beta of ABC	4.14

Stock XYZ

Market return- Expected return computation

Scenario	Probability	Scenario return	Probability weighted return (Probability x scenario)
Boom	30%	12%	3.600%
Normal	70%	5%	3.500%
Expected return --> Sum of Probability weighted return	7.100%

Covariance between market and stock XYZ

Scenario	Market return	Expected market	Stock XYZ return	Expected XYZ return	Market return - expected market return	XYZ return - expected XYZ return	(Market return - expected market return) x (XYZ return - expected XYZ return)
Boom	12%	7.100%	10.000%	7.550%	4.900%	2.450%	0.00120
Normal	5%	7.100%	6.500%	7.550%	-2.100%	-1.050%	0.00022
Step 1 : Sum of (Market return - expected market return) x (XYZ return - expected XYZ return)	0.00142
Step 2 : No of scenarios	2.00000
Covariance ---> Step 3 : Step 1 / Step 2	0.00071

Beta computation

Covariance between market and stock XYZ	0.00071
Variance of market return	0.1029%
Beta of XYZ	0.69

Part 4 : CAPM return

Stock ABC

Risk free return ( T bill rate)	3%
Market return	7.100%
Beta of ABC	4.14
CAPM return of ABC ---> Risk free return + Beta (Market return - risk free return)	19.99%

Stock XYZ

Risk free return ( T bill rate)	3%
Market return	6.500%
Beta of XYZ	0.69
CAPM return of XYZ ---> Risk free return + Beta (Market return - risk free return)	5.42%

Stock	CAPM return	Expected return	Under / Over valued	Reason
ABC	19.99%	10.300%	overvalued	Since expected return is lower than CAPM return, stock is overvalued
XYZ	5.42%	7.550%	Undervalued	Since expected return is higher than CAPM return, stock is undervalued

From the above we can decide to invest in stock XYZ as it is under valued

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