Question

Suppose an economy has three states: boom, normal, and recession. Assume that the probability of a boom state is 0.2, a normal state is 0.5, and a recession state is 0.3. And there are three stocks in this economy, called Alpha, Beta, and Gamma respectively. The return performance of these stocks has been summarized by the following table:

	Alpha	Beta	Gamma
boom	15%	28%	1%
normal	6%	12%	3%
recession	-12%	-30%	20%

(Please show your intermediate processes, instead of just a final number for your answers. Only round your final answers to two decimal places.)

(a) What is the expected return of Stock Alpha?

(b) What is the variance of Stock Beta?

(c) What is the standard deviation of Stock Gamma?

(d) Suppose you build a portfolio by including these three stocks. The weight of Stock Alpha in your portfolio is 0.2, the weight of Stock Beta is 0.3, and the weight of Stock Gamma is 0.5. What are the expected return, variance, and standard deviation of your portfolio?

(e) Based on what you observe from the calculations and what you learned from the class, could you specify what are the characteristics of portfolios?

that's all I have. can you please check. i don't know any correlation

Answer 1

(a) Calculation of Expected return of Stock Alpha:

Particulars	Probability (1)	Return (2)	Expected return(3) (1*2)
Boom	0.2	15%	3%
Normal	0.5	6%	3%
Recession	0.3	-12%	-3.6%
		Expected return of Alpha	2.4%

(b) Calculation of Expected return of Stock Beta:

Particulars	Probability (1)	Return (2)	Expected return (3) (1*2)
Boom	0.2	28%	5.6%
Normal	0.5	12%	6%
Recession	0.3	-30%	-9%
		Expected return	2.6%

Calculation of Variance of Stock Beta:

Particulars	Probability (1)	Return-Expected return (2)	Square of Return-Expected return (3)	Variance (4) (1*3)
Boom	0.2	28%-2.6%=25.4% = 0.254	(0.254)^2 = 0.0645	0.0129
Normal	0.5	12%-2.6%=9.4% = 0.094	(0.094)^2 = 0.0088	0.0044
Recession	0.3	-9%-2.6% =11.6% = 0.116	(0.116)^2 = 0.0135	0.0041
			Variance	0.0214

Standard deviation = Square root of Variance

= Square root of 0.0214

= 0.15 or 15%

(c) Calculation of Expected return of Gamma:

Particulars	Probability (1)	Return (2)	Expected return (3) (1*2)
Boom	0.2	1%	0.2%
Normal	0.5	3%	1.5%
Recession	0.3	20%	6%
		Expected return	7.7%

Calculation of Standard deviation of Stock Gamma:

Particulars	Probability (1)	Return-Expected return (2)	Square of Return-Expected return (3)	Variance (4) (1*3)
Boom	0.2	1%-7.7%=-6.7%= -0.067	(-0.067)^2 = 0.0045	0.0009
Normal	0.5	3%-7.7%=-4.7%=-0.047	(-0.047)^2= 0.0022	0.0011
Recession	0.3	20%-7.7% =12.3% = 0.123	(0.123)^2 = 0.0151	0.0045
			Variance	0.0065

Variance = 0.0065

Standard deviation = Square root of Variance

= Square root of 0.0065

= 0.08 or 8%

(d) Calculation of Expected return,Variance and standard deviation:

Particulars	Probability (1)	Return (2)	Expected return (3) (1*2)
Alpha	0.2	2.4%	0.48%
Beta	0.3	2.6%	0.78%
Gamma	0.5	7.7%	3.85%
		Expected return	5.11%

Calculation of Standard Deviation of Stock Alpha:

Particulars	Probability (1)	Return-Expected return (2)	Square of Return-Expected return (3)	Variance (4) (1*3)
Boom	0.2	15%-2.4% = 12.6% =0.126	(0.126)^2 = 0.0158	0.00316
Normal	0.5	6%-2.4%=3.6%= 0.036	(0.036)^2 = 0.0013	0.00065
Recession	0.3	-12%-2.4%=-14.4%= -0.144	(-0.144)^2 =0.0207	0.00621
			Variance	0.01002

Standard deviation of Alpha = square root of Variance

= Square root of 0.01002

= 0.10 or 10%

Calculation of Covariance of Alpha and Beta:

Particulars	Probability (1)	Return-Expeced return of Alpha (2)	Return-Expected return of Beta (3)	Covariance (4) (1*2*3)
Boom	0.2	0.126	0.254	0.0064
Normal	0.5	0.036	0.094	0.0017
Rcession	0.3	-0.144	0.116	-0.0050
			Covariance	0.0031

Correlation coefficient = covarinace/ std devi of alpha* std devia of beta

= 0.0031/0.1*0.15

= 0.0031/0.015

= 0.21

Calculation of correlation coefficient of beta and gamma:

Particulars	Probability (1)	Return-Expected return of beta (2)	Return-Expected return of Gamma (3)	Covariance (4) (1*2*3)
Boom	0.2	0.254	-0.067	-0.0034
Normal	0.5	0.094	-0.047	-0.0022
Recession	0.3	0.116	0.123	0.0043
			Covariance	-0.0013

Correlation coefficient = covarinace/std devia of beta* std devia of gamma

= -0.0013/0.15*0.08

= -0.0013/0.012

= -0.11

Calculation of correlation coefficient of gamma and alpha:

Particulars	Probability (1)	Return-Expected return of gamma (2)	Return- Expected return of Alpha (3)	Covariance (4) (1*2*3)
Boom	0.2	-0.067	0.126	-0.0017
Normal	0.5	-0.047	0.036	-0.0008
Recession	0.3	0.123	-0.144	-0.0053
			Covariance	-0.0078

Correlation coefficient = -0.0078/0.08*0.1

= -0.0078/0.008

= -0.98

Calculation of std deviation and variance

Varinace = (0.2)^2*(0.1)^2+(0.3)^2*(0.15)^2+(0.5)^2*(0.08)^2+2*0.2*0.3*0.1*0.15*0.21+2*0.3*0.5*0.15*0.08*-0.11+2*0.5*0.2*0.08*0.1*-0.98

= 0.0004+0.002025+0.0016+0.0018-0.000396-0.001568

= 0.003861

Variance = 0.003861

Standard deviation = Square root of Variance

= square root of 0.003861

=0.062 or 6.2%

(e)

Particulars	Expected return	Standard deviation
Alpha	2.4%	10%
Beta	2.6%	15%
Gamma	7.7%	8%

Portfolio expected return = 5.11%

Portfolio standard deviation = 6.2%

Based on the above calculations, we can say that investing in portfolio is better because by taking risk of 6.2% it is generating expected return of 5.11%.

So, investing in portfolio is better.