Signed risk contributions and marginal transfers

What you will learn

Derive signed Euler volatility contributions from statewise covariance, reconcile negative and above-total shares, and distinguish independent exposure derivatives from feasible funded-transfer slopes.

Prerequisites: Portfolio covariance and minimum variance · Portfolio weights, drift and rebalancing

Separate capital allocation from risk attribution

Two sleeves can receive equal capital without contributing equal risk. A volatile sleeve may offset another sleeve so effectively that its signed contribution is negative. Attribution asks how the risk of the current combined position is distributed under a declared decomposition. It does not ask which portfolio maximizes a return objective, nor does it automatically prescribe a trade. We use portfolio standard deviation as the risk measure. This is a model of dispersion, not a complete description of downside, liquidity, suitability or future performance. A decomposition must name its risk measure before its numbers can be interpreted.

Assume one period, divisible exposures, four equally probable population states, fixed initial capital weights, no fees, flows, shorting, borrowing or intermediate trading. Sleeve returns are already centered at zero and measured in percentage points. A has states (−2,−2,2,2). An orthogonal disturbance Z has states (−1,1,−1,1). In the opposed model B=−A+Z=(1,3,−3,−1). Each coordinate is a simultaneous state, not a successive time observation. A and Z have zero covariance because their products (2,−2,−2,2) average zero. Thus Var(A)=4, Var(B)=5 and Cov(A,B)=−4. These are stipulated population moments; no sample correction or empirical forecast is involved.

Derive a signed decomposition from covariance

Let a and b be nonnegative capital fractions summing to one, and R=aA+bB. Write V=Var(R) and sigma=sqrt(V). Linearity in the second argument of covariance gives Cov(A,R)=4a−4b and Cov(B,R)=−4a+5b. Multiplying by each exposure and adding yields a Cov(A,R)+b Cov(B,R)=Cov(R,R)=V. Define the two variance contributions as vA=a Cov(A,R) and vB=b Cov(B,R). The shared covariance enters once in each component, making twice the cross term in the total. Assigning only a²Var(A) and b²Var(B) leaves that interaction unaccounted for.

For positive sigma, differentiate the square root of the quadratic exposure function while holding the other exposure fixed: partial sigma/partial a=Cov(A,R)/sigma and partial sigma/partial b=Cov(B,R)/sigma. Multiply each marginal derivative by its own exposure to obtain volatility contributions cA=vA/sigma and cB=vB/sigma. Their sum is V/sigma=sigma. This proves the Euler identity directly for this finite model. The derivative temporarily treats exposures as independent coordinates; it does not silently renormalize the portfolio. Variance contributions have squared-percentage-point units. Volatility contributions have percentage-point units. Adding the former and calling the result volatility changes the risk measure.

Work a hedge example all the way through

Choose a=3/4 and b=1/4. The four portfolio returns are (−5/4,−3/4,3/4,5/4). They average zero, and their squared values sum to (25+9+9+25)/16=68/16. Dividing by four gives V=17/16 and sigma=sqrt(17)/4, approximately 1.030776 percentage points. Independently expand the quadratic: 4(9/16)+5(1/16)−8(3/16)=36/16+5/16−24/16=17/16. Agreement between the state calculation and the moment expansion checks both the population denominator and the covariance sign.

The statewise A×R products are (5/2,3/2,3/2,5/2), averaging 2. The B×R products are (−5/4,−9/4,−9/4,−5/4), averaging −7/4. Consequently vA=(3/4)2=3/2 and vB=(1/4)(−7/4)=−7/16. Their sum is 24/16−7/16=17/16. Divide by sigma: cA=6/sqrt(17), approximately 1.455214, and cB=−7/(4sqrt(17)), approximately −0.424437. Their signed sum is 1.030776. The normalized shares are 24/17 and −7/17, totaling one despite exceeding 100% and falling below zero individually. Negative contribution means offsetting covariance in this position, not negative standalone volatility or a guaranteed favorable payoff.

Distinguish independent additions from funded transfers

At the worked position the B marginal derivative is negative, so adding a small independent B exposure reduces volatility locally when A exposure stays fixed. A fully invested portfolio instead finances an increase in B by reducing A. Parameterize this feasible direction by (a−h,b+h). Its volatility derivative at h=0 is [Cov(B,R)−Cov(A,R)]/sigma. Here the numerator is −7/4−2=−15/4, so a sufficiently small positive transfer lowers volatility. This conclusion uses a directional derivative, not the signed contribution alone. At zero A capital, a positive A-to-B transfer is unavailable even if an algebraic slope can be computed.

For a counterexample, set a=1/4 and b=3/4 in the opposed model. The covariance averages are −2 and 11/4. Adding B alone increases volatility. More subtly, at a=1/2 and b=1/2 the A marginal is zero but the B marginal is 1/2. A funded transfer from B to A lowers volatility although A contributes zero at the starting point. Always specify the direction and what finances it. Local slopes also do not justify arbitrarily large moves. From the worked position an A-to-B transfer h changes variance exactly to 17/16−(15/2)h+17h². The positive quadratic term eventually reverses the initial improvement. A derivative is a local comparison, not an endpoint guarantee.

Build state witnesses and report the limits

In the workshop, move eight conserved capital packets and switch B between opposed and aligned common shocks. In the aligned model B=A+Z=(−3,−1,1,3); standalone variances remain 4 and 5 but covariance becomes positive four. This isolates dependence from individual sleeve dispersion. Mark every state where B×R is strictly negative before certifying the ledger. Selecting states merely because B itself is negative fails when the portfolio also loses there: a product of two negatives is positive. Zero products must stay unselected. The state witness makes the sign of the average visible instead of trusting a covariance label.

Construct the opposed six-packet A position, certify it, then switch alignment without moving capital. Save both verified ledgers to document how attribution changes while the standalone risks stay fixed. Try equal capital and both endpoints as well. A zero holding has zero weighted contribution even when its independent marginal is nonzero. In any different model with sigma=0 the displayed division formula needs separate treatment: variance accounting still exists, but the volatility gradient need not be defined. Our supported states always have positive volatility. Risk attribution is conditional on the supplied law, horizon and exposures. Estimated covariances, nonlinear positions and alternative risk measures require their own analysis. Equalizing contributions is another objective and is not established as optimal or desirable by this accounting identity.

Hands-on: Build a signed risk ledger

Change common-shock alignment and capital weights, construct negative state-product witnesses and certify signed covariance contributions and funded-transfer slopes.

Common mistakes

• Dropping the shared covariance from component totals.
• Summing variance contributions and labeling the result volatility.
• Clipping negative contributions to zero or treating them as guaranteed profits.
• Using an independent marginal for a capital-conserving transfer.
• Extending a local slope to a large trade or dividing by zero volatility.

Keep these ideas

• Weighted covariances sum to variance.
• For positive volatility, divide weighted covariances by volatility to obtain Euler contributions.
• Negative attribution describes a hedge relative to the combined position.
• A funded A-to-B slope subtracts the removed A marginal; check feasibility and finite curvature.

Further reading

• MOSEK Portfolio Optimization Cookbook: Risk contribution, section 9.1 ↗

  Read section 9.1 Euler identity and marginal sensitivity definition, and section 9.2 distinction between capital and risk budgets. Supports the decomposition definition only. Original four-state construction, signed hedge ledger, exact arithmetic, funded-transfer counterexamples and activity were independently derived. No optimization software, market data, source example, quotation or video reviewed. Checked 2026-10-25.