Question

		Rate of Return if State Occurs
State of Economy	Probability of State of Economy	Stock A	Stock B	Stock C
Boom	0.30	50.0%	12.0%	20.0%
Average	0.45	15.0%	-5.0%	6.0%
Recession	0.25	-8.0%	2.0%	-3.2%

Your portfolio manager has invested 30% of your money in Stock A, 50% in Stock B, and the rest in Stock C.

1. What is the correlation coefficient between Stocks B and C?

2. What is the standard deviation of your portfolio? Hint: Instead of using the portfolio variance formula for three stocks, you can save time by calculating the return on the portfolio for all three states of the economy

Answer 1

1.

E(R_b) = 0.3*12% + 0.45*(-5%) + 0.25*2%= 1.85%

E(Rc) = 0.3*20% + 0.45*6% + 0.25*(-3.2%)= 7.9%

SD(R_b) = [0.3*(12%-1.85%)² + 0.45*(-5%-1.85%)² + 0.25*(2%-1.85%)²]^0.5 = 7.213%

SD(Rc) = [0.3*(20%-7.9%)² + 0.45*(6%-7.9%)² + 0.25*(-3.2%-7.9%)²]^0.5 = 8.7378%

COV(R_b,R_c) =  [0.3*(12%-1.85%)*(20%-7.9%) + 0.45*(-5%-1.85%)*(6%-7.9%) + 0.25*(2%-1.85%)*(-3.2%-7.9%)]

=0.42285%

Correlation(R_b,R_c) = COV(R_b,R_c) / [SD(R_b) *SD(R_c)] = 0.0042285/(0.07213*0.087378) = 0.6709

2. Weights of A, B, C given to be 30%, 50%, 20%.

R_p(Boom) = 30%*50% + 50%* 12%+ 20%*20% = 25%

R_p(Average) = 30%*15% + 50%* (-5%)+ 20%*6% = 3.2%

R_p(Recession) = 30%*(-8%) + 50%* 2%+ 20%*(-3.2%) = -2.04%

E(R_p) = 0.3*25% + 0.45*3.2% + 0.25*(-2.04%) = 8.43%

SD(R_p) = [0.3*(25%-8.43%)² + 0.45*(3.2%-8.43%)² + 0.25*(-2.04%-8.43%)²]^0.5 = 11.0491%