Question

The analysis of outcomes for sales and the associated rate of return on common stocks for companies X and Y are shown below. You intend to form a portfolio by allocating $3750 of your total wealth of $5000 in company X, and the remainder in company Y. The covariance between the two companies is -0.000927. Show all work to receive credit.

DECLINING	FLAT	RISING
probability	40%	25%	35%
% of return X	-0.4%	10.8%	21.7%
% of return Y	5.2%	18.3%	23.5%

1. What is the expected return and standard deviation for common stock in each company?

2. What is the portfolio return and standard deviation for this two stock portfolio? Explain in words how and why portfolio variance is different than the individual standard deviation of the stocks?

Answer 1

1. Expected Return of X  =Probability of Flat*Return of X+Probability of Rising*Return of X in Rising+Probability of Other Scenario*Return of X =40%*-0.4%+25%*10.80%+35%*21.7% =10.135% or 10.14%
Standard Deviation of X =(40%*(-0.4%-10.135%)^2+25%*(10.80%-10.135%)^2+35%*(21.7%-10.135%)^2)^0.5=9.5560%
or 9.56%
Expected Return of Y  =Probability of Flat*Return of Y+Probability of Rising*Return of Y in Rising+Probability of Other Scenario*Return of Y=40%*5.2%+25%*18.30%+35%*23.50% =14.88%
Standard Deviation of Y=(40%*(5.2%-14.88%)^2+25%*(18.30%-14.88%)^2+35%*(23.50%-14.88%)^2)^0.5=8.1493%
or 8.15%

2. Weight of X =3750/5000 =75%
Weight of Y =25%
Expected Return =Weight of X*Expected Return of X+Weight of Y*Expected Return of Y
=75%*10.135%+25%*14.88% =11.32%
Standard Deviation with Covariance =((Weight of X*Standard Deviation of X)^2+(Weight of Y*Standard Deviation of Y)^2+2*Weight of X*Weight of Y*Covariance)^0.5 =((75%*9.5560%)^2+(25%*8.1493%)^2+2*75%*25%*-0.000927)^0.5 =
7.21%
Portfolio standard Deviation is different from individual stocks because of following reasons
1. The covariance is negative which reduces the overall standard deviation as compared to individual stocks. Higher the covariance higher is the standard deviation of portfolio and lower the covariance lower is the standard deviation of portfolio.
2. Weights of the stocks are different,which creates the variation the standard deviation.