Portfolio Standard Deviation with Varying Correlation

Finance · Easy · Free problem

Two assets have daily return standard deviations $\sigma_A = 2\%$ and $\sigma_B = 3\%$. You hold an equally-weighted portfolio ($w_A = w_B = 0.5$).

  1. What is the portfolio's standard deviation when the correlation is $\rho = 0.6$?
  1. What is it when $\rho = -0.6$?

Comment on the effect of correlation on portfolio risk.

Hints

  1. Start with the portfolio variance formula: $\sigma_P^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho \sigma_A \sigma_B$. Which term changes between the two cases?
  2. Separate the computation into the fixed part ($w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2$) and the correlation-dependent cross-term. The cross-term flips sign when $\rho$ flips sign.
  3. With $\rho = 0.6$, the variance is $0.000505$; with $\rho = -0.6$, it is $0.000145$. Take square roots to get the standard deviations.

Worked Solution

How to Think About It: This is the most fundamental calculation in portfolio theory. The portfolio variance formula has three pieces: the weighted variances of each asset plus a cross-term that depends on correlation. When correlation is positive, the cross-term adds to risk. When negative, it subtracts. The key intuition: diversification works because the cross-term can be negative, reducing total portfolio risk below the weighted average of individual risks.

Quick Estimate: The weighted average of the two standard deviations is $0.5 \times 2\% + 0.5 \times 3\% = 2.5\%$. With positive correlation, the portfolio std should be somewhat below

.5\%$ (diversification still helps but is limited). With negative correlation, it should be much lower -- maybe around
\%$ to
.5\%$. Let us compute exactly.

Approach: Apply the two-asset portfolio variance formula directly.

Formal Solution:

The portfolio variance is:

$\sigma_P^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho \sigma_A \sigma_B$

With $w_A = w_B = 0.5$, $\sigma_A = 0.02$, $\sigma_B = 0.03$:

The fixed terms are: - $w_A^2 \sigma_A^2 = 0.25 \times 0.0004 = 0.0001$ - $w_B^2 \sigma_B^2 = 0.25 \times 0.0009 = 0.000225$ -

w_A w_B \sigma_A \sigma_B = 2 \times 0.25 \times 0.02 \times 0.03 = 0.0003$

Case 1: $\rho = 0.6$

$\sigma_P^2 = 0.0001 + 0.000225 + 0.6 \times 0.0003 = 0.000325 + 0.00018 = 0.000505$

$\sigma_P = \sqrt{0.000505} \approx 2.25\%$

Case 2: $\rho = -0.6$

$\sigma_P^2 = 0.0001 + 0.000225 + (-0.6) \times 0.0003 = 0.000325 - 0.00018 = 0.000145$

$\sigma_P = \sqrt{0.000145} \approx 1.20\%$

Comparison:

| Correlation | Portfolio Std Dev | vs. Weighted Avg (2.5%) | |---|---|---| | $\rho = 0.6$ | 2.25% | 10% reduction | | $\rho = -0.6$ | 1.20% | 52% reduction |

With positive correlation, diversification provides modest risk reduction. With negative correlation, the portfolio std drops dramatically -- nearly halved relative to the individual assets.

Answer: $\sigma_P \approx 2.25\%$ when $\rho = 0.6$, and $\sigma_P \approx 1.20\%$ when $\rho = -0.6$. Negative correlation dramatically amplifies the diversification benefit.

Intuition

This problem is the quantitative foundation for why diversification is called the only free lunch in finance. When two assets are negatively correlated, their movements partially cancel out in a portfolio, reducing volatility far below what you would get by simply averaging the individual volatilities. The extreme case is $\rho = -1$, where you can construct a zero-volatility portfolio (a perfect hedge).

In practice, finding assets with genuinely negative correlation is rare and valuable. Most equities are positively correlated (they tend to move together, especially in crises). This is why portfolio managers prize assets like bonds, gold, or volatility products that historically have low or negative correlation with equities. The portfolio variance formula quantifies exactly how much each unit of correlation costs you in terms of risk -- and it is the starting point for mean-variance optimization, risk budgeting, and hedging.

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