Question

Hyacinth Macaw invests 55% of her funds in stock I and the rest in stock J. The standard deviation of returns on I is 16%, and on J it is 23%.

    a) Calculate the variance of portfolio returns, assuming the correlation between the returns is 1.

    b) Calculate the variance of portfolio returns, assuming the correlation is 0.6.

    c) Calculate the variance of portfolio returns, assuming the correlation is zero.

Answer 1

The standard deviation of a two-asset portfolio is calculated as:

σ_P = (w_I² * σ_I² + w_J² * σ_J² + 2 * w_I * w_J * σ_I * σ_J *ρ_IJ)^1/2

w_I = weight of stock I in the portfolio = 55% = 0.55
w_J = weight of stock J in the portfolio = (1 - 0.55) = 0.45

σ_I = standard deviation of asset I = 16%

σ_J = standard deviation of asset J = 23%

Steps used to formulate the above answers are:

1) Put in the values of weights and standard deviation returns of the respective stock

2) Put in the value of the correlation between the returns

3) Use the variance of portfolio returns formula in the above screenshot to calculate the variance

You can also solve the above question manually using the steps given below:

a) correlation between the returns is 1, ρ_IJ = 1

σ_P = (0.55² * 0.16² + 0.45² * 0.23² + 2 * 0.55* 0.45* 0.16* 0.23* 1)^1/2

σ_P = (0.3025 * 0.0256 + 0.2025 * 0.0529 + 0.018216)^1/2

σ_P = (0.01845625 + 0.018216)^1/2

σ_P = (0.03667225)^1/2 =0.1915 = 19.15%

Variance = σ_P² =  0.03667225 = 0.0367

b) correlation between the returns is 0.6, ρ_IJ = 0.6

σ_P = (0.55² * 0.16² + 0.45² * 0.23² + 2 * 0.55* 0.45* 0.16* 0.23* 0.6)^1/2

σ_P = (0.3025 * 0.0256 + 0.2025 * 0.0529 + 0.0109296)^1/2

σ_P = (0.01845625 + 0.0109296)^1/2

σ_P = (0.02938585)^1/2 = 0.171423 = 17.14%

Variance = σ_P² = 0.02938585 = 0.0294

c) correlation between the returns is 0, ρ_IJ = 0

σ_P = (0.55² * 0.16² + 0.45² * 0.23² + 2 * 0.55* 0.45* 0.16* 0.23* 0)^1/2

σ_P = (0.3025 * 0.0256 + 0.2025 * 0.0529 + 0)^1/2

σ_P = (0.01845625)^1/2 = 0.135853 = 13.59%

Variance = σ_P² = 0.01845625 = 0.0185