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:## 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

[tex]=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}[/tex]

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

[tex]=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}[/tex]

[tex]=\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[/tex]

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

b) If the correlation between the stock is 0.50

[tex]=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}[/tex]

[tex]=\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[/tex]

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

c) If the correlation between the stock is 0.00

[tex]=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}[/tex]

[tex]=\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[/tex]

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

d) If the correlation between the stock is -0.50

[tex]=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}[/tex]

[tex]=\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[/tex]

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

e) If the correlation between the stock is -1

[tex]=\sqrt{W_1^2\sigma_1^2+W_2^2\sigma_2^2+2W_1W_2\sigma_1\sigma_2*c}[/tex]

[tex]=\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[/tex]

Hence, the volatility of the portfolio is 0

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