Euler volatility contributions

Calculation contract

RC_i = w_i*(S*w)_i / sigma_p; risk_share_i = RC_i / sigma_p

Method and interpretation

The marginal derivative of volatility is scaled by each capital weight to obtain its Euler contribution. In the default example, the contribution numerators are 0.01584 and 0.00304. Dividing each by total variance gives risk shares of approximately 83.8983% and 16.1017%, despite capital weights of 60% and 40%.

A negative contribution is possible when an exposure offsets the rest of the portfolio strongly enough. The signed contributions still add to total volatility; forcing every value to be nonnegative would destroy that reconciliation.

Where the reference stops

At zero total volatility the division is undefined, so contribution arrays are null rather than invented zeros. These are local variance-based allocations, not loss attribution in a stress event or a recommendation to remove a hedging exposure.

Published synthetic inputs

{
  "weights": [
    0.6,
    0.4
  ],
  "covariance": [
    [
      0.04,
      0.006
    ],
    [
      0.006,
      0.01
    ]
  ]
}

Frozen calculation output

{
  "variance": 0.01888,
  "volatility": 0.13740451229854134,
  "component_volatility": [
    0.11528005692843722,
    0.02212445537010412
  ],
  "risk_shares": [
    0.8389830508474576,
    0.1610169491525424
  ]
}

Signed Euler contributions sum to portfolio volatility; individual contributions can be negative.

Two aligned long-only weights and a same-frequency covariance matrix. No additional annualization is applied.

Verification coverage

• Euler sum and risk-share sum reconciled; negative contribution fixture tested.
• Zero-volatility outputs explicitly withheld.

Unit and independent-reference checks test arithmetic and rejection rules, not future investment performance.

Method references

• MOSEK: risk budgeting and contributions

Read the assumptions in context