Answer to Question #263283 in Financial Math for Kwabenya

Question

Suppose financial analysts believe that there are four equally likely states of the economy: depression, recession, normal, and boom. The returns on the Supertech Company are expected to follow the economy closely, while the returns on the Slowpoke Company are not. The return predictions are as follows:

  States of the economy

Depression

Recession Normal Boom

Allos Inc. Returns (𝑹𝑨) -20%

10% 30% 50%

Orangus Inc.Returns (𝑹𝑩)

5%

20% -12% 9%

       a. For each company calculate:

i. the expected returns

ii. the Variance

iii. the Standard deviation

b. Assuming you are an investor with GHS100 available. If you invest GHS60 and GHS40 in Allos Inc. and Orangus Inc. respectively, what will be your portfolio returns?

c. Calculate the Standard deviation of the portfolio.

Answer 1

a. (i)

Let the profit under different scenarios be accordingly:

Allos Inc. 

1:-20%

2:10%

3:30%

4:50%

$E(r)=1\times(-0.2)+2\times 0.1+3\times0.3+4\times0.5=2.9$

Orangus Inc.Returns

1:5%

4:20%

0:-12%

3:9%

$E(r)=1\times(0.05)+4\times 0.2+0\times(-0.12)+3\times0.9=1.24$

(ii)

$$\tilde{𝑹𝑨}=\frac{-0.2+0.1+0.3+0.5}{4}=0.175$$

$\tilde{𝑹B}=\frac{0.05+0.2-0.12+0.09}{4}=0.055$

$V(A)=\frac{(-0.2-0.175)^2+(0.1-0.175)^2+(0.3-0.175)^2+(0.5-0.175)^2+}{4}=0.0669$

$V(B)=\frac{(0.05-0.055)^2+(0.2-0.055)^2+(-0.12-0.055)^2+(0.09-0.055)^2+}{4}=0.0132$

(iii) $\sigma (A)=\sqrt{V}=\sqrt{0.0669}=0.2587$

$\sigma (B)=\sqrt{V}=\sqrt{0.0132}=0.1149$

b.

$\omega (A)=\frac{60}{100}=0.6$

$\omega (B)=\frac{40}{100}=0.4$

Let the expected profitability be 20% and 10% respectively

$E(Rp)=0.6\times0.2+.0.4\times0.1=0.52$

c.

$\sigma(Rp)=\sqrt{(0.6\times0.2587)^2+(0.4\times0.1149)^2}=0.1618$