Explanation

When the correlation between two assets is less than 1 (i.e., less than perfect positive correlation), the portfolio standard deviation will be less than the weighted average of the individual asset standard deviations. This is due to the diversification benefit.

Mathematical Explanation

The formula for portfolio variance for a two-asset portfolio is:

$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}$

Where:

• $\sigma_p$ = portfolio standard deviation
• $w_1$, $w_2$ = weights of assets 1 and 2
• $\sigma_1$, $\sigma_2$ = standard deviations of assets 1 and 2
• $\rho_{12}$ = correlation coefficient between assets 1 and 2

Key Points:

1. When $\rho = 1$ (perfect positive correlation): $\sigma_p = w_1\sigma_1 + w_2\sigma_2$ Portfolio standard deviation equals the weighted average.

2. When $\rho < 1$: The last term $2w_1w_2\sigma_1\sigma_2\rho_{12}$is smaller than when$\rho = 1$, making$\sigma_p^2 smaller. $$\sigma_p < w_1\sigma_1 + w_2\sigma_2$3. When $\rho = -1$ (perfect negative correlation): Maximum diversification benefit, portfolio standard deviation can be minimized to potentially zero with appropriate weights.

Diversification Benefit:

The reduction in portfolio risk (standard deviation) when combining assets with less than perfect positive correlation is the fundamental principle of diversification. This is why investors combine assets with different return patterns - to achieve a better risk-return tradeoff.

Correct Answer: A - The portfolio standard deviation is less than the weighted average standard deviation of the individual assets when correlation is less than one.