Why Imperfect Correlation Matters
Combining two assets does not generally produce a portfolio whose risk is just the weighted average of the assets’ risks. The returns may rise and fall together, or they may move differently. That joint movement affects portfolio risk. When correlation is less than one, a portfolio with positive holdings in both assets can have lower standard deviation than the weighted average of their standalone standard deviations.
The size of this reduction depends on the assets’ weights, their individual standard deviations, and their correlation. Imperfect correlation does not mean that either asset is risk-free, nor does it guarantee that the portfolio has lower risk than each asset on its own. It means that the assets’ returns do not move in exact lockstep, so their fluctuations can partly offset in the combined portfolio.
The formula applies to returns measured over the same period, with weights set for that period. The weights may be positive or negative in general, but the comparisons with weighted-average standalone risk below assume positive holdings and weights summing to one. No assumption that returns are normally distributed is needed to calculate variance from the stated standard deviations and correlation.
Why the Formula Produces a Diversification Benefit
The portfolio return is a weighted sum. To find its variance, subtract its expected value and square the result:
The cross-product in the squared expression contributes twice the covariance. Since \(\operatorname{Cov}(R_A,R_B)=\rho_{AB}\sigma_A\sigma_B\), substituting this identity gives the two-asset formula. This also explains the role of correlation: it determines the sign and size of the cross-term.
For positive weights, compare the portfolio standard deviation with the weighted average \(w_A\sigma_A+w_B\sigma_B\). Squaring that weighted average and subtracting the portfolio variance gives:
If both weights and both standard deviations are positive and \(\rho_{AB}<1\), the right-hand side is positive. Both quantities being compared are nonnegative, so the portfolio standard deviation is strictly less than the weighted-average standalone standard deviation. The equality case is \(\rho_{AB}=1\). This comparison does not say that portfolio risk is below each asset’s risk; that depends on the weights and the asset risks.
Worked Applications
Worked Example: Positive but Imperfect Correlation
A hypothetical portfolio holds 60% in Asset A and 40% in Asset B. Their annual expected returns are 8% and 5%, their annual standard deviations are 20% and 10%, and their return correlation is \(0.25\). Calculate the portfolio’s expected return and standard deviation, then compare its risk with weighted-average standalone risk. Use decimal rates in the variance calculation and round final percentages to two decimal places. No distributional assumption is required.
Method. First calculate the weighted expected return. Then use the two-asset variance formula, including its covariance term.
Expected return:
Variance: The three terms are
Thus \(\operatorname{Var}(R_P)=0.0144+0.0016+0.0024=0.0184\), and \(\sigma_P=\sqrt{0.0184}=0.135646\ldots\), or 13.56% to two decimal places. Weighted-average standalone risk is \(0.60(20\%)+0.40(10\%)=16.00\%\). The portfolio standard deviation is therefore 2.44 percentage points lower than that benchmark.
Check. The variance is positive, and the portfolio standard deviation is below 16%, as the imperfect-correlation result predicts. The covariance term is positive because the correlation is positive, but smaller than it would be at correlation \(1\).
Common mistake avoided: Treating 16% as the portfolio standard deviation. It is only the weighted average of the standalone standard deviations; the covariance term is needed to calculate actual portfolio variance.
Worked Example: Comparing Positive and Negative Correlation
A hypothetical investor places half of a portfolio in each of two assets. Their annual standard deviations are 12% and 8%. Compare portfolio standard deviation when their correlation is \(0.75\) with the result when it is \(-0.25\). The weights and standalone risks are unchanged. Use decimal rates and round final percentages to two decimal places; no distributional assumption is needed.
Method. Calculate the two squared-risk terms once, then change only the correlation-dependent cross-term.
The first two variance terms are \((0.50)^2(0.12)^2=0.25(0.0144)=0.0036\) and \((0.50)^2(0.08)^2=0.25(0.0064)=0.0016\). The cross-term is \(2(0.50)(0.50)\rho_{AB}(0.12)(0.08)=0.0048\rho_{AB}\).
When \(\rho_{AB}=0.75\), the cross-term is \(0.0048(0.75)=0.0036\). Therefore, the portfolio variance is \(0.0036+0.0016+0.0036=0.0088\), and the standard deviation is \(\sqrt{0.0088}=0.093808\ldots\), or 9.38%.
When \(\rho_{AB}=-0.25\), the cross-term is \(0.0048(-0.25)=-0.0012\). The variance is \(0.0036+0.0016-0.0012=0.0040\), and the standard deviation is \(\sqrt{0.0040}=0.063245\ldots\), or 6.32%.
The weighted-average standalone risk is \(0.50(12\%)+0.50(8\%)=10.00\%\) in both cases. Both portfolios have lower standard deviation than that benchmark, but the lower-correlation portfolio has lower risk. The negative cross-term in the second calculation reflects returns tending to move in opposite directions.
Check. The lower-correlation case has the same two standalone-risk terms and a smaller cross-term, so its variance must be lower. Both calculated variances are positive, and both standard deviations are below 10%.
Common mistake avoided: Using the absolute value of a negative correlation. Correlation’s sign matters: \(-0.25\) produces a negative cross-term, not a positive one.
Worked Example: Unequal Weights and the Size of the Benefit
A hypothetical fund allocates 40% to Asset A and 60% to Asset B. Their annual expected returns are 11% and 6%, their annual standard deviations are 18% and 12%, and their correlation is \(0.60\). Find the expected return and standard deviation and calculate the reduction from weighted-average standalone risk. Use decimal rates for variance and round final percentages to two decimal places. No distributional assumption is needed.
Expected return: \(E[R_P]=0.40(0.11)+0.60(0.06)=0.044+0.036=0.080\), or 8.00%.
Variance: The Asset A term is \((0.40)^2(0.18)^2=0.16(0.0324)=0.005184\). The Asset B term is \((0.60)^2(0.12)^2=0.36(0.0144)=0.005184\). The cross-term is \(2(0.40)(0.60)(0.60)(0.18)(0.12)=0.48(0.60)(0.0216)=0.0062208\).
Adding the terms gives \(\operatorname{Var}(R_P)=0.005184+0.005184+0.0062208=0.0165888\). Thus \(\sigma_P=\sqrt{0.0165888}=0.128797\ldots\), or 12.88%. Weighted-average standalone risk is \(0.40(18\%)+0.60(12\%)=7.20\%+7.20\%=14.40\%\). The reduction is \(14.40\%-12.88\%=1.52\) percentage points, using the rounded portfolio standard deviation.
Check. The cross-term is positive because correlation is positive, yet it is less than the cross-term at perfect positive correlation. The portfolio standard deviation is below the 14.40% weighted average, consistent with the result for correlation below one.
Common mistake avoided: Assuming that a positive correlation rules out a diversification benefit. The relevant comparison is with perfect correlation: \(0.60\) is below \(1\), so the portfolio risk is below the weighted average despite the positive covariance.
Interpreting the Result Carefully
Correlation affects portfolio variance through the cross-term, but it does not enter the weighted-average expected return. For fixed weights and expected returns, changing correlation changes calculated portfolio risk without changing the portfolio’s expected return. This is why comparing correlation cases is useful: it isolates the effect of joint return movement.
The weighted-average standalone standard deviation is a benchmark, not a formula for portfolio standard deviation. A different quantity, the weighted average of standalone variances, is also not generally the portfolio variance. The squared weights and the covariance term matter. Keeping these quantities separate is essential:
| Quantity | Calculation | What it represents |
|---|---|---|
| Weighted-average standalone standard deviation | \(w_A\sigma_A+w_B\sigma_B\) | A comparison benchmark when weights are positive |
| Portfolio variance | \(w_A^2\sigma_A^2+w_B^2\sigma_B^2+2w_Aw_B\rho_{AB}\sigma_A\sigma_B\) | The variance of the combined return |
| Portfolio standard deviation | \(\sqrt{\operatorname{Var}(R_P)}\) | The portfolio’s standard deviation on the same return scale |
A reliable calculation keeps all rates in consistent units, uses the correlation with its sign, and takes the square root only after adding the variance terms. If weights are positive and sum to one, a correlation below one ensures a strict reduction relative to weighted-average standalone standard deviation, provided both assets have positive standard deviations. The size of the reduction is not fixed: it depends on weights, standalone risks, and correlation.
This comparison is useful in portfolio analysis because it makes the value of imperfect correlation measurable. It does not imply that diversification eliminates risk, or that any particular allocation is optimal. It establishes a more limited and precise point: combining assets whose returns are not perfectly correlated can lower portfolio risk relative to the weighted-average standalone-risk benchmark.
Check Your Understanding
For each problem, calculate portfolio standard deviation using the stated two-asset model. Unless stated otherwise, returns cover the same annual period, weights are positive and sum to one, and no distributional assumption is needed. Round final standard deviations to two decimal places.
- A portfolio invests 70% in an asset with annual standard deviation 10% and 30% in an asset with annual standard deviation 20%. Their return correlation is \(0.50\). Find portfolio standard deviation and compare it with weighted-average standalone risk. (Answer: portfolio standard deviation \(11.27\%\); weighted-average standalone risk \(13.00\%\).)
- Two assets each receive 50% of a portfolio and have annual standard deviations of 10% and 14%, respectively. Their correlation is zero. Find the portfolio standard deviation and weighted-average standalone risk. (Answer: portfolio standard deviation \(8.60\%\); weighted-average standalone risk \(12.00\%\).)
- A portfolio invests 60% in an asset with annual standard deviation 16% and 40% in an asset with annual standard deviation 10%. The return correlation is \(-0.50\). Find portfolio standard deviation and weighted-average standalone risk. (Answer: portfolio standard deviation \(8.35\%\); weighted-average standalone risk \(13.60\%\).)
- A portfolio invests 25% in an asset with annual standard deviation 8% and 75% in an asset with annual standard deviation 12%. Their return correlation is \(0.40\). Find the portfolio standard deviation and the reduction from weighted-average standalone risk. (Answer: portfolio standard deviation \(9.97\%\); weighted-average standalone risk \(11.00\%\), a reduction of \(1.03\) percentage points.)
- Two assets have positive standard deviations and are held with positive weights summing to one. Without calculating a numerical example, state when portfolio standard deviation equals the weighted average of the standalone standard deviations, and when it is strictly lower. (Answer: They are equal when correlation is \(1\); portfolio standard deviation is strictly lower when correlation is less than \(1\).)