Why Compare Portfolio Allocations?
An asset’s standard deviation describes the variability of its returns on its own. A portfolio combines assets, so its risk depends both on the assets’ individual standard deviations and on how their returns move together. Consequently, changing the weights can change portfolio risk even when the assets and their return behavior are held fixed.
This tutorial compares a stated set of two-asset allocations. The aim is to calculate each portfolio’s standard deviation consistently and identify the lowest value among those allocations. That conclusion is limited to the combinations evaluated; it does not, by itself, establish the lowest risk over every possible allocation.
The formula assumes that the weights apply over the return period being analyzed, and that both asset standard deviations and their correlation refer to that same period. In the examples, weights are nonnegative and sum to one, so the portfolios are fully invested without short positions or borrowing. The inputs are treated as fixed for each comparison. The formula calculates risk from those inputs; it does not guarantee that historical estimates will describe future returns.
Why the Covariance Term Appears
A portfolio return is the weighted sum of the asset returns. Its variance therefore includes each asset’s weighted variance and a term for the joint movement of the two returns. Expanding that variance gives the formula directly.
The final equality uses the definition \(\operatorname{Cov}(R_A,R_B)=\rho_{AB}\sigma_A\sigma_B\). The factor of 2 in the covariance term arises because expanding the squared deviation of a sum produces two equal cross-products. Once the variance is calculated, taking its nonnegative square root gives the standard deviation. Since the square-root function is increasing for nonnegative values, the allocation with the smallest variance also has the smallest standard deviation.
For a clear comparison, keep the units consistent: if standard deviations are entered as decimals, the variance is in squared-decimal units and the final standard deviation is converted back to a percentage. Do not compare one portfolio’s variance with another portfolio’s standard deviation.
Worked Comparisons
Worked Example: Comparing Several Allocations With a Lower-Volatility Asset
A hypothetical investor considers assets A and B. Their annual return standard deviations are \(10\%\) and \(16\%\), respectively, and their annual return correlation is \(0.50\). Compare allocations of \(100/0\), \(75/25\), \(50/50\), \(25/75\), and \(0/100\), where each pair gives the percentages in A and B. Which evaluated allocation has the lowest annual standard deviation?
Method and inputs. Use decimal standard deviations \(\sigma_A=0.10\) and \(\sigma_B=0.16\), with \(\rho_{AB}=0.50\). The asset variances are \(0.10^2=0.0100\) and \(0.16^2=0.0256\). The covariance is \(0.50(0.10)(0.16)=0.0080\).
All A. With \(w_A=1\) and \(w_B=0\), \(\sigma_p^2=1^2(0.0100)+0^2(0.0256)+2(1)(0)(0.0080)=0.0100\). Thus \(\sigma_p=\sqrt{0.0100}=0.1000\), or \(10.00\%\).
75% A, 25% B. The variance is \(0.75^2(0.0100)+0.25^2(0.0256)+2(0.75)(0.25)(0.0080)\). The three terms are \(0.005625\), \(0.001600\), and \(0.003000\), so the variance is \(0.010225\). Therefore, \(\sigma_p=\sqrt{0.010225}\), which is \(10.11\%\) to two decimal places.
50% A, 50% B. The variance is \(0.50^2(0.0100)+0.50^2(0.0256)+2(0.50)(0.50)(0.0080)\). The three terms are \(0.002500\), \(0.006400\), and \(0.004000\), giving variance \(0.012900\) and standard deviation \(\sqrt{0.012900}=11.36\%\) to two decimal places.
25% A, 75% B. The variance is \(0.25^2(0.0100)+0.75^2(0.0256)+2(0.25)(0.75)(0.0080)\). The terms are \(0.000625\), \(0.014400\), and \(0.003000\), giving variance \(0.018025\) and standard deviation \(\sqrt{0.018025}=13.43\%\) to two decimal places.
All B. With \(w_A=0\) and \(w_B=1\), the variance is \(0.0256\) and the standard deviation is \(16.00\%\).
The calculated standard deviations are \(10.00\%\), \(10.11\%\), \(11.36\%\), \(13.43\%\), and \(16.00\%\), respectively. Among these allocations, all A has the lowest standard deviation, at \(10.00\%\).
Common mistake avoided. The \(75/25\) portfolio’s \(10.11\%\) is not lower than the all-A portfolio’s \(10.00\%\): it is higher. Also, for the \(50/50\) portfolio, the cross-term is \(2(0.50)(0.50)(0.0080)=0.004000\), not \(0.0080\). Omitting the multiplication by the weights and the factor of 2 gives an incorrect variance.
Worked Example: A Lower-Volatility Allocation Is Not Necessarily All in One Asset
A hypothetical portfolio uses assets A and B with annual standard deviations of \(20\%\) and \(12\%\), respectively, and correlation \(0.25\). Compare allocations \(100/0\), \(75/25\), \(50/50\), \(25/75\), and \(0/100\). Identify the lowest-risk allocation among them.
Method and inputs. Use \(\sigma_A=0.20\), \(\sigma_B=0.12\), and \(\rho_{AB}=0.25\). The variances are \(0.0400\) and \(0.0144\). The covariance is \(0.25(0.20)(0.12)=0.0060\).
For \(100/0\), the standard deviation is \(20.00\%\). For \(0/100\), it is \(12.00\%\). The mixed allocations are calculated as follows.
For \(75/25\), the variance terms are \(0.75^2(0.0400)=0.022500\), \(0.25^2(0.0144)=0.000900\), and \(2(0.75)(0.25)(0.0060)=0.002250\). Their sum is \(0.025650\), so the standard deviation is \(\sqrt{0.025650}=16.02\%\) to two decimal places.
For \(50/50\), the terms are \(0.50^2(0.0400)=0.010000\), \(0.50^2(0.0144)=0.003600\), and \(2(0.50)(0.50)(0.0060)=0.003000\). The variance is \(0.016600\), and the standard deviation is \(\sqrt{0.016600}=12.88\%\) to two decimal places.
For \(25/75\), the terms are \(0.25^2(0.0400)=0.002500\), \(0.75^2(0.0144)=0.008100\), and \(2(0.25)(0.75)(0.0060)=0.002250\). The variance is \(0.012850\), and the standard deviation is \(\sqrt{0.012850}=11.34\%\) to two decimal places.
The five standard deviations are \(20.00\%\), \(16.02\%\), \(12.88\%\), \(11.34\%\), and \(12.00\%\). The \(25/75\) allocation has the lowest calculated standard deviation, at \(11.34\%\). It is lower than the all-B portfolio’s \(12.00\%\), although B alone has the lower standalone standard deviation.
Common mistake avoided. Choosing the asset with the lowest standalone standard deviation without calculating the mixed portfolios can miss the lowest-risk allocation in the evaluated set. Here, the \(25/75\) portfolio’s covariance and weighted variance terms together produce less risk than either the all-B portfolio or the other listed combinations.
Worked Example: Ranking Specified Allocations Without Claiming a Global Minimum
A hypothetical investor compares three allocations in assets A and B: \(100/0\), \(80/20\), and \(40/60\). Annual standard deviations are \(8\%\) for A and \(14\%\) for B, and their annual return correlation is \(0.30\). Calculate the standard deviation of each allocation and select the lowest among these three.
Method and inputs. In decimals, \(\sigma_A=0.08\), \(\sigma_B=0.14\), and \(\rho_{AB}=0.30\). Their variances are \(0.0064\) and \(0.0196\); covariance is \(0.30(0.08)(0.14)=0.00336\).
For \(100/0\), variance is \(0.0064\), so standard deviation is \(8.00\%\).
For \(80/20\), the variance terms are \(0.80^2(0.0064)=0.004096\), \(0.20^2(0.0196)=0.000784\), and \(2(0.80)(0.20)(0.00336)=0.0010752\). The sum is \(0.0059552\). Thus the standard deviation is \(\sqrt{0.0059552}=7.72\%\) to two decimal places.
For \(40/60\), the terms are \(0.40^2(0.0064)=0.001024\), \(0.60^2(0.0196)=0.007056\), and \(2(0.40)(0.60)(0.00336)=0.0016128\). The sum is \(0.0096928\), giving standard deviation \(\sqrt{0.0096928}=9.85\%\) to two decimal places.
The comparison is \(8.00\%\) for \(100/0\), \(7.72\%\) for \(80/20\), and \(9.85\%\) for \(40/60\). Of the three specified allocations, \(80/20\) has the lowest calculated standard deviation.
Common mistake avoided. The conclusion is only about the three allocations stated in the problem. It does not prove that \(80/20\) is the minimum-risk allocation among every possible weight combination.
Reading and Reporting a Comparison
A useful comparison shows enough intermediate work to make the result auditable. For each allocation, record the weights, calculate the two weighted variance terms and the covariance term, add them to obtain portfolio variance, and take the square root. A compact table can then make the ranking clear.
| Allocation | Portfolio variance | Portfolio standard deviation |
|---|---|---|
| All A | \(w_A^2\sigma_A^2\), with \(w_A=1\) | \(\sigma_A\) |
| Mixed A and B | Both weighted variances plus the covariance term | Square root of the calculated variance |
| All B | \(w_B^2\sigma_B^2\), with \(w_B=1\) | \(\sigma_B\) |
In a mixed portfolio, the covariance contribution is not an optional adjustment: it is part of the variance formula. A positive covariance term increases the variance relative to the sum of the two weighted variance terms alone. This observation describes the calculation for the stated inputs; a full study of how changing correlation affects risk belongs to a separate topic.
Report the final standard deviations at a consistent precision, such as two decimal places in percentage form, and compare the rounded results only after retaining sufficient precision in the calculations. If two displayed values are close, compare the unrounded variances or square roots before deciding whether they differ. State both the winning allocation and its risk, and qualify the result as the lowest among the allocations evaluated.
Common Comparison Errors
- Reversing the ranking. A standard deviation of \(10.11\%\) is greater than \(10.00\%\), not lower. Read the computed values before writing the conclusion.
- Dropping the factor of 2. The covariance term is \(2w_Aw_B\rho_{AB}\sigma_A\sigma_B\). It is not just \(w_Aw_B\) times covariance.
- Using correlation as covariance. Correlation is dimensionless; covariance equals correlation times both standard deviations.
- Mixing units or time periods. Use decimal standard deviations consistently, and make sure both assets’ volatilities and correlation describe the same return period.
- Comparing unlike measures. Compare standard deviation with standard deviation, or variance with variance, not one with the other.
- Overstating the result. A lowest value among the listed allocations is not automatically a minimum over all possible weights.
Check Your Understanding
For each problem, use decimal standard deviations in the two-asset variance formula. Round final standard deviations to two decimal places.
- Assets A and B have annual standard deviations \(9\%\) and \(15\%\), with correlation \(0.40\). Compare the \(60/40\) allocation with all A. Give both standard deviations and identify the lower-risk allocation. (Answer: \(60/40\) is \(9.54\%\); all A is \(9.00\%\); all A is lower.)
- Assets A and B have annual standard deviations \(18\%\) and \(10\%\), with correlation \(0.20\). Compare a \(50/50\) allocation with all B. (Answer: \(50/50\) is \(11.14\%\); all B is \(10.00\%\); all B is lower.)
- Assets A and B have annual standard deviations \(11\%\) and \(13\%\), with correlation \(0.60\). Compare allocations \(70/30\) and \(30/70\), showing which has lower risk. (Answer: \(70/30\) is \(10.51\%\); \(30/70\) is \(11.39\%\); \(70/30\) is lower.)
- Assets A and B have annual standard deviations \(7\%\) and \(17\%\), with correlation \(0.10\). Compare allocations \(80/20\) and \(20/80\). (Answer: \(80/20\) is \(6.84\%\); \(20/80\) is \(13.81\%\); \(80/20\) is lower.)
- Assets A and B have annual standard deviations \(12\%\) and \(20\%\), with correlation \(0.50\). Compare \(25/75\) with \(50/50\), including the covariance contribution in each calculation. (Answer: \(25/75\) is \(16.70\%\); \(50/50\) is \(14.00\%\); \(50/50\) is lower.)