Why can equal portfolio weights produce unequal variance allocations?

Define the quantity and allocation convention

Write portfolio variance as V = wᵀΣw, where w contains portfolio weights and Σ is the covariance matrix for returns measured over the same period. Define the allocation for asset i as Cᵢ = wᵢ(Σw)ᵢ. Summing these allocations gives V by construction. Their units are squared return units, not dollars or percentage-point losses.

William Sharpe’s original discussion of risk budgeting explains the covariance relationships and cautions against treating such decompositions as independently caused pieces of total risk when components are correlated. Here the word allocation means this declared accounting convention. It does not identify what a particular holding would cause in an uncontrolled future market.

Reconcile an original equal-weight example

Stipulate two fictional assets with weights 0.5 each, standard deviations 20% and 10% per observation period, and correlation 0.25. In decimal-return units, their variances are 0.04 and 0.01. The covariance is 0.25 × 0.20 × 0.10 = 0.005. These are invented model inputs, not estimates from market data.

Multiplying Σ by w gives 0.04 × 0.5 + 0.005 × 0.5 = 0.0225 for asset A and 0.005 × 0.5 + 0.01 × 0.5 = 0.0075 for asset B. Multiplying again by each weight gives Cₐ = 0.01125 and Cᵦ = 0.00375. Their sum is V = 0.015, with modeled standard deviation √0.015 ≈ 12.247449% per period.

Read the shares without turning them into probabilities

The declared variance shares are 0.01125 / 0.015 = 75% and 0.00375 / 0.015 = 25%, despite dollar weights of 50% each. Those percentages describe this covariance worksheet. They do not say that A has a 75% chance of losing money, causes 75% of every drawdown, or should receive a particular portfolio weight.

Nor can the two assets’ standalone standard deviations simply be added or averaged to obtain portfolio standard deviation. The covariance cross-term matters. Changes in weights or covariance assumptions change the allocations; an allocation is not an immutable characteristic of the security.

Test a negative-allocation boundary case

Keep both weights and standalone standard deviations, but stipulate correlation −0.8, making covariance −0.016. Now Σw = (0.012, −0.003), so the allocations are 0.006 and −0.0015. They sum to positive variance 0.0045, whose square root is approximately 6.708204% per period.

The corresponding shares are about 133.333333% and −33.333333%. A negative value is possible under the stated convention because one component has negative covariance with the modeled portfolio. It is not a negative probability or guaranteed insurance. If modeled variance were zero, dividing by it to form percentage shares would be undefined. Estimated covariance and future relationships can change.

Method, sources, and limits

This original two-case worksheet was recomputed on 27 October 2026. It excludes tail behavior, liquidity, leverage constraints and estimation uncertainty, and makes no allocation recommendation. AI assisted the draft; separate AI verification is required before publication and is not qualified human financial review.