Two securities equally constitute a portfolio. The securities have standard deviations of 3% and 15% respectively. If the standard deviation of the portfolio is 5.00%, the covariance between the two securities is equal to: Question 20Answera.-1.489b.0.0008c.-0.0067
Question
Two securities equally constitute a portfolio. The securities have standard deviations of 3% and 15% respectively. If the standard deviation of the portfolio is 5.00%, the covariance between the two securities is equal to: Question 20Answera.-1.489b.0.0008c.-0.0067
Solution
The covariance between two securities in a portfolio can be calculated using the formula for the variance of a two-asset portfolio:
Var(P) = w1^2 * σ1^2 + w2^2 * σ2^2 + 2 * w1 * w2 * Cov(1,2)
Where: Var(P) is the variance of the portfolio, w1 and w2 are the weights of the two securities in the portfolio, σ1 and σ2 are the standard deviations of the two securities, and Cov(1,2) is the covariance between the two securities.
Given that the two securities equally constitute the portfolio, w1 = w2 = 0.5. The standard deviations σ1 and σ2 are given as 3% and 15% respectively, and the standard deviation of the portfolio is given as 5%.
First, we need to convert the standard deviations and the standard deviation of the portfolio to variances by squaring them:
Var(1) = σ1^2 = (0.03)^2 = 0.0009 Var(2) = σ2^2 = (0.15)^2 = 0.0225 Var(P) = (0.05)^2 = 0.0025
Substituting these values into the formula for the variance of a two-asset portfolio, we get:
0.0025 = 0.5^2 * 0.0009 + 0.5^2 * 0.0225 + 2 * 0.5 * 0.5 * Cov(1,2)
Solving this equation for Cov(1,2), we get:
Cov(1,2) = (0.0025 - 0.5^2 * 0.0009 - 0.5^2 * 0.0225) / (2 * 0.5 * 0.5) Cov(1,2) = (0.0025 - 0.000225 - 0.005625) / 0.5 Cov(1,2) = -0.00335
So, the covariance between the two securities is -0.00335, which is closest to option c. -0.0067.
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