Question

Arbor Systems and Gencore stocks both have a volatility of 33%. Compute the volatility of a portfolio with 50% invested in each stock if the correlation between the stocks is ​(a​) +1.00​, ​(b​) 0.50​, ​(c​) 0.00​, ​(d​) −0.50​, and ​(e​) −1.00.
In which of the cases is the volatility lower than that of the original​ stocks?

Answer 1

Answer:

In case of b, c, d ,e volatility is less than that of original stock

Explanation:

The formula to compute the volatility of a portfolio

$=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}$

Here,

The standard deviation of the first stock is σ₁

The standard deviation of the second stock is σ₂

The weight of the first stock W₁

The weight of the second stock W₂

The correlation between the stock c

a) If the correlation between the stock is +1

$=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}$

$=\sqrt{(0.5\times0.33)^2+(0.5\times0.33)^2+(2\times(0.5\times 0.33)\times(0.5\times0.33)\times1} \\\\=0.33$

Hence, the volatility of the portfolio is 0.33 0r 33%

b) If the correlation between the stock is 0.50

$=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}$

$=\sqrt{(0.5\times0.33)^2+(0.5\times0.33)^2+(2\times(0.5\times 0.33)\times(0.5\times0.33)\times0.5} \\\\=0.29$

Hence, the volatility of the portfolio is 0.29 0r 29%

c) If the correlation between the stock is 0.00

$=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}$

$=\sqrt{(0.5\times0.33)^2+(0.5\times0.33)^2+(2\times(0.5\times 0.33)\times(0.5\times0.33)\times0.0} \\\\=0.23$

Hence, the volatility of the portfolio is 0.23 0r 23%

d) If the correlation between the stock is -0.50

$=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}$

$=\sqrt{(0.5\times0.33)^2+(0.5\times0.33)^2+(2\times(0.5\times 0.33)\times(0.5\times0.33)\times-0.5} \\\\=0.17$

Hence, the volatility of the portfolio is 0.17 or 17%

e) If the correlation between the stock is -1

$=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}$

$=\sqrt{(0.5\times0.33)^2+(0.5\times0.33)^2+(2\times(0.5\times 0.33)\times(0.5\times0.33)\times-1} \\\\=0$

Hence, the volatility of the portfolio is 0

In case of b, c, d ,e volatility is less than that of original stock