Why the Direction of a Weight Change Matters
A fully invested two-asset portfolio can change risk even when the total amount invested stays fixed. Increasing the allocation to one asset necessarily reduces the allocation to the other. Whether that raises or lowers portfolio variance depends not only on the assets’ individual variances, but also on how their returns move together and on the allocation before the shift.
The practical question is often comparative: if an investor transfers a specified fraction of the portfolio from asset B to asset A, does the portfolio become more or less volatile? The answer must use the joint behavior of the returns. A lower-volatility asset does not automatically reduce portfolio risk when its weight rises; its covariance with the asset being reduced also matters.
Unless otherwise stated, examples use returns over a common single period, with weights held fixed during that period and no borrowing or short selling, so \(0\leq w\leq1\). No particular return distribution is required: finite variances and covariance are enough for the variance formula. Interest rates and mortality assumptions do not apply to this portfolio calculation. The calculations describe variance, a squared-return measure; portfolio standard deviation is the square root of variance.
Reading the Variance Formula as an Allocation Changes
The first two terms in \(V(w)\) reflect the weighted variances of the individual assets. The third term reflects their joint variation. If covariance is positive, the assets tend to move in the same direction, contributing positively to portfolio variance. If covariance is negative, the covariance term reduces variance. Covariance can be found directly or from \(\sigma_{AB}=\rho_{AB}\sigma_A\sigma_B\), where \(\rho_{AB}\) is the correlation.
A weight change is a transfer, not an addition to the portfolio. If the weight in A rises from \(w\) to \(w+h\), the weight in B falls from \(1-w\) to \(1-w-h\). The number \(h\) is the size of the shift: positive \(h\) means a transfer into A, and negative \(h\) means a transfer out of A. Comparing the original and new allocations requires evaluating the same variance function at both weights.
A useful way to organize the calculation is to expand this expression as a quadratic. Define \(C=\sigma_A^2+\sigma_B^2-2\sigma_{AB}\). Then
Because \(C=\operatorname{Var}(R_A-R_B)\), it cannot be negative. This quadratic representation makes clear why the effect of a transfer can depend on the starting weight: the slope of variance changes with \(w\).
Deriving the Effect of a Finite Shift
The derivative of variance describes the direction of change for a sufficiently small shift at the current allocation. But a specified transfer may not be small. The exact finite-change formula accounts for both the initial slope and the curvature of the variance function.
To justify the result, write the expanded variance as \(V(w)=Cw^2+2(\sigma_{AB}-\sigma_B^2)w+\sigma_B^2\). Substituting \(w+h\) and subtracting \(V(w)\) gives
The term \(hV'(w)\) gives the initial marginal effect in the direction of the shift, while \(Ch^2\) accounts for curvature over the whole transfer. Since \(C\geq0\), that curvature term is nonnegative. Thus a negative initial slope does not guarantee that every larger transfer will reduce variance. For the exact change, use both terms—or calculate the variance at the new allocation directly.
Worked Applications
Worked Example: A Shift Toward the Higher-Variance Asset
A hypothetical portfolio holds assets A and B for the same one-year period. Their standard deviations are 20% and 10%, respectively, and their return correlation is 0.25. The weight in A rises from 25% to 35%, with the rest in B. Determine how variance changes and check the direction using the finite-shift formula.
Method and inputs. There is no assumed return distribution beyond finite second moments. Convert standard deviations to variances: \(\sigma_A^2=0.20^2=0.04\) and \(\sigma_B^2=0.10^2=0.01\). The covariance is \(\sigma_{AB}=0.25(0.20)(0.10)=0.005\). The shift is \(h=0.35-0.25=0.10\).
Calculate the original and new variances. Substitution gives \(V(w)=0.04w^2+0.01(1-w)^2+0.01w(1-w)\). Expanding: \(V(w)=0.04w^2-0.01w+0.01\). Thus \(V(0.25)=0.04(0.0625)-0.01(0.25)+0.01=0.0025-0.0025+0.01=0.0100\). At the new weight, \(V(0.35)=0.04(0.1225)-0.01(0.35)+0.01=0.0049-0.0035+0.01=0.0114\).
Check the change. The variance increases by \(0.0114-0.0100=0.0014\). In the finite-shift formula, \(C=0.04+0.01-2(0.005)=0.04\) and \(V'(0.25)=2(0.04)(0.25)-0.01=0.01\). Therefore \(hV'(w)+Ch^2=0.10(0.01)+0.04(0.10^2)=0.001+0.0004=0.0014\), confirming the direct comparison. Portfolio standard deviation rises from \(\sqrt{0.0100}=10.00\%\) to \(\sqrt{0.0114}\approx10.68\%\), rounded to two decimal places. The common mistake is to reason that raising the weight in a riskier asset must determine the result by itself; the covariance term is also part of the calculation.
Worked Example: A Small Shift Can Help Even When a Larger One Does Not
A hypothetical portfolio has \(\sigma_A=12\%\), \(\sigma_B=20\%\), and correlation \(\rho_{AB}=-0.25\). The weight in A is initially 40%. Compare a 2-percentage-point shift into A with a 10-percentage-point shift into A. Determine whether each transfer raises or lowers variance.
Method and inputs. Returns have finite second moments; no distribution is specified or needed. The variances are \(0.12^2=0.0144\) and \(0.20^2=0.04\). The covariance is \((-0.25)(0.12)(0.20)=-0.006\). The variance function is \(V(w)=0.0144w^2+0.04(1-w)^2-0.012w(1-w)\), which expands to \(V(w)=0.0664w^2-0.092w+0.04\).
Find the current variance and marginal direction. At \(w=0.40\), \(V(0.40)=0.0664(0.16)-0.092(0.40)+0.04=0.010624-0.0368+0.04=0.013824\). The derivative is \(V'(w)=0.1328w-0.092\), so \(V'(0.40)=0.05312-0.092=-0.03888\). A small shift into A initially reduces variance.
Compare the two finite shifts. Here \(C=0.0664\). For \(h=0.02\), the change is \(0.02(-0.03888)+0.0664(0.02^2)=-0.0007776+0.00002656=-0.00075104\), so variance falls. For \(h=0.10\), the change is \(0.10(-0.03888)+0.0664(0.10^2)=-0.003888+0.000664=-0.003224\), so variance also falls. The new variances are \(0.013824-0.00075104=0.01307296\) and \(0.013824-0.003224=0.010600\), respectively. Both results agree with the sign of the exact change. The common mistake is to treat the positive curvature term as if it automatically made a shift riskier; it offsets, but does not necessarily outweigh, the negative marginal term.
Worked Example: A Transfer Toward the Lower-Volatility Asset
A hypothetical investment portfolio has \(\sigma_A=8\%\), \(\sigma_B=18\%\), and correlation \(0.20\). Its weight in A increases from 60% to 70%. Calculate the variance and standard deviation before and after the change, and explain why the transfer reduces risk.
Method and inputs. Assume only that the same-period returns have finite second moments. The variances are \(\sigma_A^2=0.0064\) and \(\sigma_B^2=0.0324\); covariance is \(0.20(0.08)(0.18)=0.00288\). Therefore \(V(w)=0.0064w^2+0.0324(1-w)^2+0.00576w(1-w)\), or \(V(w)=0.03304w^2-0.05904w+0.0324\).
Evaluate both allocations. At \(w=0.60\), \(V(0.60)=0.03304(0.36)-0.05904(0.60)+0.0324=0.0118944-0.035424+0.0324=0.0088704\). At \(w=0.70\), \(V(0.70)=0.03304(0.49)-0.05904(0.70)+0.0324=0.0161896-0.041328+0.0324=0.0072616\). The variance decreases by \(0.0072616-0.0088704=-0.0016088\).
Interpret the risk measure. The standard deviations are \(\sqrt{0.0088704}\approx9.42\%\) and \(\sqrt{0.0072616}\approx8.52\%\), rounded to two decimal places. The shift reduces risk here because the calculation, including the covariance term, gives a lower portfolio variance at the new weight. The fact that A has lower standalone volatility is relevant, but it is not a general substitute for comparing portfolio variances. The common mistake is to compare only 8% with 18% and overlook how the assets’ returns co-move.
A Reliable Direction Check
For a proposed reallocation, the following sequence separates the meaning of the shift from the arithmetic and makes the conclusion auditable.
Write the original allocation as \(w\) in A and \(1-w\) in B. If A’s weight rises by \(h\), B’s weight must fall by the same amount.
Square standard deviations to obtain variances. If correlation is provided, calculate covariance as \(\rho_{AB}\sigma_A\sigma_B\).
Substitute the initial and final weights into the same two-asset formula. Alternatively, use \(hV'(w)+Ch^2\) to compute the exact difference directly.
A positive difference means the shift increases variance; a negative difference means it decreases variance. If asked for standard deviation, take square roots only after finding the variances.
The derivative \(V'(w)\) is useful when assessing a small change from the current allocation: for a sufficiently small positive \(h\), a negative derivative indicates a local decrease in variance, and a positive derivative indicates a local increase. For a nonzero specified shift, use the exact finite-change formula. The derivative alone is not the answer to every finite-transfer question.
Common Traps and Practical Importance
- Changing one weight without changing the other. In a fixed-total two-asset portfolio, the weights sum to one. A shift into A is an equal shift out of B.
- Leaving covariance out. The portfolio variance includes \(2w(1-w)\sigma_{AB}\). Omitting it can reverse the direction of the calculated risk change.
- Using correlation as if it were covariance. Correlation is unitless; covariance must be calculated with both standard deviations before it is substituted into the variance formula.
- Using the marginal direction for a finite transfer. A derivative describes the local slope. A finite change also includes the curvature term \(Ch^2\).
- Confusing variance and standard deviation. Variance is expressed in squared-return units. Standard deviation is in return units and is obtained by taking the nonnegative square root.
This analysis matters whenever an investor evaluates a proposed reallocation while holding the total investment fixed. It also clarifies why risk cannot be inferred from the standalone volatility of the asset being purchased or sold. The existing allocation and the relationship between asset returns both affect the outcome. The calculation here answers whether a particular shift changes variance; it does not by itself decide whether the shift is appropriate given expected return, objectives, or constraints.
Check Your Understanding
For each problem, treat the returns as same-period variables with finite second moments and use the stated inputs. No specific return distribution is required.
- A hypothetical portfolio has \(\sigma_A=16\%\), \(\sigma_B=10\%\), and correlation \(0.25\). The weight in A rises from 30% to 40%. Find the change in variance and state its direction. (Answer: variance increases by \(0.000732\).)
- A portfolio has \(\sigma_A=12\%\), \(\sigma_B=20\%\), and correlation \(-0.25\), with \(w=0.50\). Find the exact variance change when \(w\) rises to 0.55. (Answer: variance decreases by \(0.001114\).)
- For assets with \(\sigma_A=10\%\), \(\sigma_B=15\%\), and correlation \(0.60\), compare \(V(0.20)\) and \(V(0.30)\). Does shifting 10 percentage points into A increase or decrease variance, and by how much? (Answer: variance decreases by \(0.001975\).)
- A portfolio has \(\sigma_A=25\%\), \(\sigma_B=15\%\), and covariance \(\sigma_{AB}=0.0100\). The weight in A increases from 40% to 50%. Calculate the change in variance and identify the direction. (Answer: variance increases by \(0.00335\).)