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

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?

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  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

    [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

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