When Inverse Volatility Is, and Is Not, Risk Parity

Two equalities with different meanings

For strictly positive standalone volatilities, inverse-volatility allocation gives asset i a weight proportional to 1/sigma_i, normalized so all weights sum to one. This makes every product w_i times sigma_i identical. That product describes weighted standalone volatility, not the asset's contribution to joint portfolio volatility. Equal risk contribution instead asks each asset to account for the same share of a specified portfolio risk measure. Here that measure is volatility. The distinction matters because a holding can offset or reinforce other holdings. A rule using only individual volatilities cannot generally observe those interactions, however accurately those individual inputs were estimated.

The missing information is in correlation

Write the covariance matrix as Sigma and portfolio variance as V = w' Sigma w. For positive V, asset i's relative volatility contribution is w_i times (Sigma w)_i divided by V. Under inverse-volatility weights, set every weighted standalone volatility to the common value k. The numerator then becomes k squared times the sum of row i of the correlation matrix, including its diagonal entry. Thus equal row sums produce equal contributions, while unequal row sums generally do not. This algebra identifies exactly what the heuristic leaves unspecified: not merely how risky an asset is alone, but how its standardized return relates to all the others.

Important cases where the rules coincide

Uncorrelated assets have a correlation matrix with identical row sums, so inverse volatility gives equal risk contributions. A valid constant-correlation matrix also has equal row sums. With just two assets, the row sums are necessarily equal, which makes positive inverse-volatility weights an equal-contribution solution whenever portfolio volatility is nonzero. This includes correlated two-asset cases, not only independent ones. Perfect offsetting that creates zero portfolio volatility makes relative contributions undefined and needs separate treatment. These special cases explain why a two-asset demonstration cannot establish that inverse volatility remains risk parity after additional, differently correlated assets enter the universe.

A three-asset counterexample

The synthetic table uses three assets with identical 20% annual volatilities. A and B have correlation -0.75; C has zero correlation with both. The correlation matrix is positive definite, so this is not an invalid-input trick. Inverse volatility assigns one third to every asset. The offsetting A and B positions each contribute one sixth of variance, whereas C contributes two thirds. For equal contributions, weights of 40%, 40% and 20% produce variance allocations of 0.0016 apiece. Total variance is 0.0048 and volatility is approximately 6.928203%. These numbers are consequences of the stipulated covariance matrix, not observations about any traded assets.

Inverse variance answers another question

Replacing 1/sigma_i with 1/sigma_i squared does not improve the same heuristic; it changes the objective implied by the formula. With a diagonal positive covariance matrix and only a full-investment constraint, minimum-variance weights are proportional to inverse variance. Equal risk contributions in that setting instead use inverse volatility. Confusing the two can substantially overweight the lowest-volatility input. Neither formula should be described as a general solution to the other objective. Once correlations are nonzero, minimum variance uses covariance interactions too, and constraints can move its solution away from the familiar closed form.

Constraints and problematic inputs

A maximum allocation or a minimum holding can prevent exact equal risk contributions. Applying a cap after computing inverse-volatility weights and then renormalizing is a new allocation procedure, not evidence that a risk-budgeting problem was solved. Likewise, a zero volatility estimate makes a reciprocal undefined. Replacing zero with a chosen floor changes relative weights and should be described as regularization, not an arithmetic identity. Stale prices can generate implausibly small estimates, while mixed annual and daily inputs can distort the entire normalization. These are reasons to inspect the volatility inputs before interpreting either method's apparent preference for a holding.

Read weights and contributions together

The inverse-volatility card can demonstrate the reciprocal-and-normalize calculation without claiming to solve a correlation-aware allocation problem. Passing the resulting weights and an explicitly specified covariance matrix to a component-risk calculation reveals whether contributions actually match. For a numerical risk-parity solution, compare achieved contribution shares with the intended budgets and report any residual mismatch caused by constraints or solver tolerance. Equal budgets are themselves a design choice, and asset-level equality does not establish equality across underlying economic factors. An arithmetic match verifies the definition for those inputs; it supplies no evidence of future diversification, superior returns or reliable stress protection.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• MOSEK Portfolio Optimization Cookbook: Risk budgeting, sections 9.1 and 9.2
• Daniel P. Palomar: Portfolio Optimization, section 11.3, Risk Contributions

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