Why Portfolio Standard Deviation Requires More Than an Average
A portfolio combines asset returns, but its risk is not generally the weighted average of the assets’ standard deviations. Portfolio risk also depends on how the returns move together. To calculate it, use each asset’s weight and standard deviation along with the correlation between the two assets’ returns.
This tutorial considers a one-period portfolio of two assets. The weights are the proportions of portfolio value invested in each asset at the start of the period. The portfolio return is therefore the weighted sum of the asset returns. The calculations assume the returns, standard deviations, and correlation all refer to the same period, and that the weights remain fixed for that period. If the inputs refer to different periods, they must be made consistent before applying the formula.
The formula permits negative weights, which represent short positions, as long as the portfolio weights and return model are interpreted consistently. A correlation must lie between \(-1\) and \(1\). Standard deviations are nonnegative; the correlation term can be positive, zero, or negative depending on the inputs.
Why the Formula Has These Terms
The squared-weight terms represent the variance contribution from each asset on its own. The final term accounts for joint variation: it is the covariance contribution, written using correlation. The following identity gives the formula directly.
With fixed weights over the period, \(R_p=w_1R_1+w_2R_2\).
For any two random variables with finite variances, \(\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y)\). Applying this identity gives \(\operatorname{Var}(R_p)=w_1^2\operatorname{Var}(R_1)+w_2^2\operatorname{Var}(R_2)+2w_1w_2\operatorname{Cov}(R_1,R_2)\).
By definition, \(\operatorname{Var}(R_j)=\sigma_j^2\), and \(\operatorname{Cov}(R_1,R_2)=\rho_{12}\sigma_1\sigma_2\). Substitution yields the stated portfolio variance formula. Standard deviation is the nonnegative square root of variance.
This also explains why the cross term contains both standard deviations, not their variances: covariance is correlation multiplied by \(\sigma_1\sigma_2\). The weights in that term are not squared. Omitting the cross term would amount to treating the assets’ returns as uncorrelated, which is justified only when their correlation is zero.
A Reliable Calculation Method
When the data are given as percentages, convert them to decimals before calculating. For example, a standard deviation of 12% is \(0.12\), so its variance is \(0.12^2=0.0144\). Keep the units consistent through the calculation. The resulting variance is in squared-return units; its square root is in return units and can be reported as a percentage.
Worked Example: A Fully Invested Portfolio With Positive Correlation
A hypothetical investment account allocates 30% to Asset 1 and 70% to Asset 2. Their one-period standard deviations are 12% and 8%, respectively, and their return correlation is \(0.50\). Calculate the portfolio variance and standard deviation. Use decimal returns in the calculation and round the final standard deviation to two decimal places as a percentage.
Method. Set \(w_1=0.30\), \(w_2=0.70\), \(\sigma_1=0.12\), \(\sigma_2=0.08\), and \(\rho_{12}=0.50\). The weights sum to \(0.30+0.70=1.00\), as required for this fully invested account.
The variance contribution from Asset 1 is \(w_1^2\sigma_1^2=(0.30)^2(0.12)^2=0.09(0.0144)=0.001296\). The contribution from Asset 2 is \(w_2^2\sigma_2^2=(0.70)^2(0.08)^2=0.49(0.0064)=0.003136\).
The covariance contribution is \(2w_1w_2\rho_{12}\sigma_1\sigma_2 =2(0.30)(0.70)(0.50)(0.12)(0.08) =0.42(0.50)(0.0096)=0.002016\). Therefore, \(\operatorname{Var}(R_p)=0.001296+0.003136+0.002016=0.006448\). Taking the nonnegative square root gives \(\sigma_p=\sqrt{0.006448}\approx0.08030\), or **8.03%** to two decimal places.
Check. The variance is positive, and the standard deviation is reported as a decimal return converted to a percentage. The common mistake of omitting the covariance term would give variance \(0.001296+0.003136=0.004432\) and standard deviation about 6.66%. That is not the answer here because the correlation is \(0.50\), not zero.
Handling Negative Correlation and Short Positions
A negative correlation makes the covariance term negative when both weights are positive. If one weight is negative, the sign of the covariance contribution depends on the product \(w_1w_2\rho_{12}\). A negative term reduces the sum of the variance components, but it does not mean that portfolio variance itself is negative. For valid inputs, the full variance remains nonnegative.
Worked Example: Negative Correlation With Positive Weights
A hypothetical reserve account places 60% in a short-term bond fund and 40% in an infrastructure fund. Their one-period standard deviations are 10% and 15%, and their return correlation is \(-0.20\). Calculate the variance and standard deviation. Round the final standard deviation to two decimal places as a percentage.
Method. Use \(w_1=0.60\), \(w_2=0.40\), \(\sigma_1=0.10\), \(\sigma_2=0.15\), and \(\rho_{12}=-0.20\). The weights sum to one.
The first squared-weight term is \((0.60)^2(0.10)^2=0.36(0.01)=0.0036\). The second is \((0.40)^2(0.15)^2=0.16(0.0225)=0.0036\). The cross term is \(2(0.60)(0.40)(-0.20)(0.10)(0.15) =0.48(-0.20)(0.015)=-0.00144\).
Thus, \(\operatorname{Var}(R_p)=0.0036+0.0036-0.00144=0.00576\). The portfolio standard deviation is \(\sigma_p=\sqrt{0.00576}=0.075894\ldots\), or **7.59%** to two decimal places.
Check. The negative cross term lowers the variance relative to the sum of the two squared-weight terms, which is \(0.0072\). It does not eliminate the standalone terms: the resulting variance is still \(0.00576>0\). A common error is to replace the negative covariance contribution with its absolute value, which would incorrectly increase the variance.
A negative weight is handled by the same formula; it does not call for a different variance rule. In particular, do not square the product of the two weights in the cross term. The next example uses a short position to make the signs explicit.
Worked Example: A Short Position and a Negative Cross Term
A hypothetical portfolio takes a long position of 125% in Asset A and a short position of 25% in Asset B, so its net investment is 100%. The one-period standard deviations are 8% for A and 20% for B, and the return correlation is \(0.30\). Find the portfolio variance and standard deviation. Round the final standard deviation to two decimal places as a percentage.
Method. The weights are \(w_A=1.25\) and \(w_B=-0.25\), with \(1.25-0.25=1.00\). Use \(\sigma_A=0.08\), \(\sigma_B=0.20\), and \(\rho_{AB}=0.30\).
The two squared-weight terms are \((1.25)^2(0.08)^2=1.5625(0.0064)=0.0100\) and \((-0.25)^2(0.20)^2=0.0625(0.04)=0.0025\). The cross term is \(2(1.25)(-0.25)(0.30)(0.08)(0.20) =(-0.625)(0.30)(0.016)=-0.0030\).
Therefore, \(\operatorname{Var}(R_p)=0.0100+0.0025-0.0030=0.0095\), and \(\sigma_p=\sqrt{0.0095}=0.097467\ldots\), or **9.75%** to two decimal places.
Check. The short weight makes \(w_Aw_B\) negative. Since correlation is positive, the cross term is negative. The total variance remains positive. Squaring the weight product in the cross term would erase this sign and produce an incorrect result.
Units, Checks, and Common Exam Traps
A useful way to organize the arithmetic is to calculate three terms separately: \(w_1^2\sigma_1^2\), \(w_2^2\sigma_2^2\), and \(2w_1w_2\rho_{12}\sigma_1\sigma_2\). Add those terms for variance, then take one square root for standard deviation. This separation makes sign and unit errors easier to identify.
- Confusing standard deviation with variance. If a standard deviation is 12%, the standard deviation input is \(0.12\), while the variance term uses \((0.12)^2\). Do not square an already calculated variance again.
- Omitting or mis-signing covariance. The cross term is part of the formula unless correlation is zero. Its sign follows the product \(w_1w_2\rho_{12}\).
- Taking the square root too early. Add all three terms first. The formula is not the sum of the square roots of the individual contributions.
- Mixing percentage points and decimals. Either representation can work if used consistently, but mixing, for example, \(12\) with \(0.08\) produces meaningless units and a wrong result.
- Reporting variance as a percentage. Variance is in squared-return units. Report the standard deviation as a percentage if that is the requested measure of risk.
Before accepting a result, check that the weights sum to the intended total, the correlation is between \(-1\) and \(1\), and all returns and risk inputs refer to the same period. Confirm that the variance is nonnegative before taking its square root. Round only after calculating the final standard deviation.
Check Your Understanding
For each hypothetical portfolio, calculate the requested variance and standard deviation. Use decimal returns in intermediate calculations; report final standard deviations to two decimal places as percentages.
- A portfolio invests 25% in Asset A and 75% in Asset B. Their standard deviations are 10% and 6%, and their correlation is \(0.25\). Find the portfolio variance and standard deviation. (Answer: variance \(0.0032125\); standard deviation \(5.67\%\))
- A portfolio invests equally in two assets with standard deviations of 9% and 13%, and their correlation is \(-0.40\). Find the variance and standard deviation. (Answer: variance \(0.00391\); standard deviation \(6.25\%\))
- A portfolio has weights \(0.80\) and \(0.20\), asset standard deviations of 7% and 16%, and correlation \(0.50\). Find the variance and standard deviation. (Answer: variance \(0.005952\); standard deviation \(7.71\%\))
- A fully invested portfolio is long 110% in Asset A and short 10% in Asset B. Their standard deviations are 5% and 20%, and their correlation is \(0.25\). Find the variance and standard deviation. (Answer: variance \(0.002875\); standard deviation \(5.36\%\))
- A portfolio has weights \(0.40\) and \(0.60\), asset standard deviations of 3% and 5%, and covariance \(0.00024\). First find the implied correlation, then calculate portfolio variance and standard deviation. (Answer: correlation \(0.1600\); variance \(0.0011592\); standard deviation \(3.40\%\))