Rate of Return if State Occurs |
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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
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%
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