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Weighted Euler volatility contribution

For positive σP, weighted contribution of asset i is wi Cov(Ri,RP)/σP.

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Differentiate the homogeneous risk function

For a fixed covariance matrix, portfolio standard deviation is the square root of wᵀΣw. Its derivative in weight i is Cov(Ri,RP)/σP when σP>0. The derivative is the marginal contribution.

Form the Euler component

Multiplying by wi produces the weighted component. The components sum to σP for this homogeneous calculation.

RCi=wiCov(Ri,RP)σP,∑iRCi=σP\mathrm{RC}_i=w_i\frac{\mathrm{Cov}(R_i,R_P)}{\sigma_P},\qquad \sum_i\mathrm{RC}_i=\sigma_P

Weighted contributions can be signed; σP must be positive.

Reconcile the original Asset A component

In the 40/60 pair, Cov(A,P)=0.4×0.04+0.6×0.006=0.0196. Multiply by weight 0.4 and divide by σP≈0.113490 to obtain approximately 0.069081, or 6.908092 percentage points.

Distinguish the share and the zero case

Dividing the weighted component by σP gives its risk share. At σP=0 the volatility derivatives/shares used here are undefined. Taking absolute values of signed components would change the decomposition.

Further references