Question

Business Condition Probability Investment

X Fund Y Fund
Recession 0.50 120 200
Stable 0.30 100 150
Depression 0.20 250 100

1. Find out the expected value, standard deviation and coefficient variation of each investment.
2. Compute correlation between the two investments.
3. Give some managerial implications of this problem.

Answer 1

Expected value:

$$\mu=\sum x\cdot P(x)$$

$\mu_X=0.5\cdot120+0.3\cdot100+0.2\cdot250=140$

$\mu_Y=0.5\cdot200+0.3\cdot150+0.2\cdot100=165$

Standard deviation:

$$\sigma=\sqrt{\sum(x-\mu)^2\cdot P(x)}$$

$\sigma_X=\sqrt{0.5\cdot(120-140)^2+0.3\cdot(100-140)^2+0.2\cdot(250-140)^2}=55.68$

$\sigma_Y=\sqrt{0.5\cdot(200-165)^2+0.3\cdot(150-165)^2+0.2\cdot(100-165)^2}=39.05$

Coefficient of variation:

$$CV=\frac{\sigma}{\mu}$$

$CV_X=\frac{55.68}{140}=0.4$

$CV_Y=\frac{39.05}{165}=0.27$

Correlation between the two investments:

$$r=\frac{\sum (x-\mu_X)(y-\mu_Y)}{\sqrt{\sum (x-\mu_X)^2\sum (y-\mu_Y)^2}}$$

$r=\frac{(120-140)(200-165)+(100-140)(150-165)+(250-140)(100-165)}{\sqrt{((120-140)^2+(100-140)^2+(250-140)^2)((200-165)^2+(150-165)^2+(100-165)^2)}}=$

$=\frac{-20\cdot35+40\cdot15-110\cdot65}{\sqrt{((-20)^2+(-40)^2+110^2)(35^2+(-15)^2+(-65)^2)}}=\frac{-7250}{\sqrt{14100\cdot 5675}}=-0.81$

Answer:

$\mu_X=140$ $\mu_Y=165$

$\sigma_X=55.68$ $\sigma_Y=39.05$

$CV_X=0.4$ $CV_Y=0.27$

$r=-0.81$

The coefficient of variation shows the risk per unit of return. The smaller the coefficient of variation, the lower risk factor occurs. Therefore choose Y.

The two investment has fairly strong negative relationship.