The standard deviation of a two-asset portfolio is given by the formula:

$\sigma_p = \sqrt{w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y \rho_{XY} \sigma_X \sigma_Y}$

where:

• $w_X = 0.5$, $w_Y = 0.5$ (portfolio weights)
• $\sigma_X = 15\% = 0.15$ (standard deviation of Asset X)
• $\sigma_Y = 20\% = 0.20$ (standard deviation of Asset Y)
• $\rho_{XY} = 0.3$ (correlation coefficient)

Step-by-step Calculation:

1. Compute the squared terms:

$(0.5)^2 (0.15)^2 = 0.5^2 \times 0.0225 = 0.25 \times 0.0225 = 0.005625$

$(0.5)^2 (0.20)^2 = 0.5^2 \times 0.04 = 0.25 \times 0.04 = 0.01$

2. Compute the interaction term:

$2 (0.5)(0.5)(0.3)(0.15)(0.20) = 2 \times 0.25 \times 0.3 \times 0.15 \times 0.20$

$= 0.5 \times 0.3 \times 0.15 \times 0.20 = 0.5 \times 0.009 = 0.0045$

3. Sum up all terms:

$0.005625 + 0.01 + 0.0045 = 0.020125$

4. Take the square root:

$\sigma_p = \sqrt{0.020125} \approx 0.142$

$\sigma_p = 14.2\%$

Since none of the options match, let's recheck the calculations carefully in Python to ensure accuracy. math

Given data

w_X = 0.5 w_Y = 0.5 sigma_X = 0.15 sigma_Y = 0.20 rho_XY = 0.3

Portfolio standard deviation formula

sigma_p_squared = (w_X2 * sigma_X2) + (w_Y2 * sigma_Y2) + (2 * w_X * w_Y * rho_XY * sigma_X * sigma_Y) sigma_p = math.sqrt(sigma_p_squared)

Convert to percentage

sigma_p * 100The calculated portfolio standard deviation is approximately 14.19%, which does not match any of the provided answer choices. There may be an issue with the given options or a mistake in interpretation. Let me know if you need further verification or explanation.

Related Questions:

1. How does correlation affect portfolio risk?
2. What happens to portfolio standard deviation when correlation is -1?
3. How do you compute the standard deviation for a three-asset portfolio?
4. Why is diversification important in portfolio management?
5. How does increasing the weight of a lower-risk asset affect portfolio risk?

Tip: A lower correlation between assets generally reduces portfolio risk due to diversification benefits.