Covariance Units and Frequency Must Travel With the Matrix

Covariance has squared return units

A covariance matrix is not unitless. If returns are represented as decimals, a 1% move is 0.01; if represented in percentage points, the same move is 1. Multiplying every return by 100 multiplies each covariance entry by 10,000 and each standard deviation by 100. Dimensionless portfolio weights do not undo that conversion. Consequently, w' Sigma w inherits the matrix's squared-return units, and its square root inherits the return unit. A displayed variance without this convention is ambiguous. Reading a percentage-point covariance matrix as a decimal-return matrix overstates the corresponding volatility by a factor of 100, not by 10,000.

Construct covariance from compatible ingredients

Correlation is dimensionless, while covariance between A and B equals correlation times sigmaA times sigmaB. In the synthetic daily example, volatilities are 1% and 2%, and correlation is 0.15. Decimal-return covariance is therefore 0.15 times 0.01 times 0.02, or 0.00003. Combining an annual volatility with a daily volatility produces a number that is neither an ordinary daily nor annual covariance. Combining percent and decimal volatility likewise creates mixed units. The portfolio-variance reference can multiply the resulting matrix perfectly and still produce a meaningless interpretation. Unit reconciliation belongs before the calculation, not in a cosmetic output label afterward.

Equivalent representations must reconcile

At weights of 60% in A and 40% in B, the daily decimal-return variance is 0.0001144. Its components are 0.000036 from A, 0.000064 from B and 0.0000144 from the two covariance terms. The square root is approximately 0.0106957936, equivalent to 1.069579% daily volatility. The same computation in percentage-point units gives variance 1.144 and standard deviation 1.069579 percentage points. These are different numerical representations of the same assumed risk. The table places both beside their annualized representations so that unit conversion and time scaling can be inspected separately instead of being folded into one unexplained multiplier.

Annualization is a conditional operation

For an additive return process with stable one-period covariance and no cross-period covariance, the covariance of a sum over K periods is K times the one-period matrix. Volatility then scales by the square root of K. The table uses K = 252 as a hypothetical annualization convention, not a universal number of observations in every market or calendar year. This gives annualized decimal-return variance 0.0288288 and volatility 16.979046%. Multiplying covariance by the square root of 252 would confuse variance scaling with volatility scaling. Multiplying an already annualized matrix by 252 again would count the time conversion twice.

Serial dependence adds terms

If returns are serially correlated, the variance of their sum includes covariances between different dates. For a stationary scalar return process over K periods, the expression is K times gamma_0 plus twice the sum of (K-h) times gamma_h over lags h from one through K-1. Here gamma_h is the lag-h autocovariance. With two days of variance 0.0001144 each and lag-one covariance 0.00002, the sum's variance is 0.0002688 rather than 0.0002288. Positive serial covariance raises this aggregation; negative serial covariance lowers it. Cross-asset horizon aggregation also requires cross-date covariance terms, which a same-day covariance matrix alone cannot supply.

Return definition changes which sums are exact

Log returns add across time for one asset, while simple returns compound. For a one-period portfolio with starting capital weights, simple portfolio return is the weighted sum of asset simple returns, before costs and external cash flows. The weighted sum of asset log returns is not generally the exact portfolio log return. Therefore an exact time-additivity argument for log returns cannot automatically justify an exact portfolio calculation using the same weights. The annualized table is explicitly a covariance-scaling illustration under additive, serially uncorrelated increments. It is not a claim that the variance of a compounded annual portfolio return equals the displayed quantity.

Frequency also defines the observations

Two daily series can cover different clock intervals if markets close at different times. Matching date labels does not guarantee synchronized return windows, and aggregating overlapping multiday returns introduces dependence by construction. Sample normalization is a separate issue: NumPy's covariance documentation distinguishes division by the observation count from division by one less than that count, along with row and column orientation. Those choices do not change the physical units but do change estimated values. A usable matrix therefore travels with its ordered assets, return representation, observation interval and estimation convention. Its numeric entries alone are insufficient to reproduce or compare a portfolio-risk result.

Primary references

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

• NumPy: numpy.cov, covariance definition, normalization and observation orientation
• J.P. Morgan and Reuters: RiskMetrics Technical Document, fourth edition, return definitions and horizon scaling

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