Question

Assume the standard deviation of security A is .38 and the standard deviation of security B is .32. The correlation coefficient between A and B is .36. What is the standard deviation of a portfolio composed of 57% security A and 43% security B?

Answer 1

Answer 2

Solution: Calculating Portfolio Standard Deviation

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Formula:

$$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho}$$

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Step 1: Assign Values

• $w_A = 0.57$ → weight of security A

• $w_B = 0.43$ → weight of security B

• $\sigma_A = 0.38$ → standard deviation of security A

• $\sigma_B = 0.32$ → standard deviation of security B

• $\rho = 0.36$ → correlation coefficient between A and B

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Step 2: Square the Weights and Standard Deviations

$$w_A^2 = (0.57)^2 = 0.3249$$
$$w_B^2 = (0.43)^2 = 0.1849$$
$$\sigma_A^2 = (0.38)^2 = 0.1444$$
$$\sigma_B^2 = (0.32)^2 = 0.1024$$

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Step 3: Multiply and Add Terms

$$w_A^2 \sigma_A^2 = 0.3249 \times 0.1444 = 0.04693$$
$$w_B^2 \sigma_B^2 = 0.1849 \times 0.1024 = 0.01894$$
$$2 w_A w_B \sigma_A \sigma_B \rho = 2 \times 0.57 \times 0.43 \times 0.38 \times 0.32 \times 0.36 = 0.03169$$

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Step 4: Sum of the Terms

$$\sigma_p^2 = 0.04693 + 0.01894 + 0.03169 = 0.09756$$

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Step 5: Calculate the Standard Deviation

$$\sigma_p = \sqrt{0.09756} \approx 0.2955$$

Final Answer:

$$\text{The standard deviation of the portfolio is approximately }\mathbf{0.2955}\text{ or }\mathbf{29.55}\mathbf{\%}\mathbf{.}$$