Dimensions methodology: how Portfolio Lab counts independent risks

Advanced8 min read · updated 28 October 2026 · 7 sources

The Dimensions tab on the Risk page answers the question of how many mutually independent risks a portfolio contains and how this changes over time. It decomposes the correlation matrix of the positions into its eigenvalues, compares them with the bound for pure randomness, counts the effective bets at the portfolio's weights and tracks these measures on a rolling basis without look-ahead. This page describes what the code in core/dimensions.py actually calculates.

Data and estimation

The basis is the same data as on the rest of the Risk page: monthly returns of completed months in the base currency from the settings, for all positions over their common period, at the target weights of the selected portfolio (positive weights only, scaled to 100%). The period is the entire common history or the last 36, 60 or 120 months; at least 24 common months and at least two positions are required. A position that does not fluctuate at all during the period stops the calculation with a notice, because without fluctuation there is no correlation.

As in the risk decomposition, the code estimates the covariance matrix Σ as a sample covariance, as an EWMA with decay factor λ (default 0.97 per month, half-life 22.8 months) or with Ledoit-Wolf shrinkage towards a scaled identity target (Ledoit and Wolf 2004). From Σ it derives the correlation matrix C = D^−½ Σ D^−½, with the variances on the diagonal of D. A benchmark plays no role for the dimensions.

Spectrum and noise threshold

The eigenvalues λ₁ ≥ … ≥ λ_N of the correlation matrix sum to the number of positions N; λₖ/N is the share of total variance carried by the k-th dimension. Each dimension is an eigenvector, i.e. a direction in which the positions move together. The loadings table shows up to the first six dimensions; because the sign of an eigenvector is arbitrary, the code flips each one so that the sum of its loadings is positive, and names the three largest loadings in absolute terms for each dimension.

The yardstick is the band in which the eigenvalues of purely random correlations lie (Marchenko and Pastur (1967); for stocks Laloux et al. (1999)). Eigenvalues above the upper bound count as dimensions with signal (Marchenko-Pastur distribution).

The bound holds strictly for the sample correlation of many independent, identically distributed series; with few positions it is a rule of thumb. For EWMA, the code uses the effective number of months. This is an approximation. Ledoit-Wolf pulls the eigenvalues towards the middle, so the count tends to be conservative. With fewer than twice as many months as positions, a notice appears; with at most as many months as positions, the correlation matrix (except with Ledoit-Wolf) is not invertible, and at least N − T + 1 eigenvalues are zero.

Effective number of bets

Following Meucci (2009), the code decomposes the covariance matrix into its principal components Σ = E Λ Eᵀ and writes the portfolio as a mix of the principal component portfolios, w̃ = Eᵀw. Each contributes the share pₖ = λₖ w̃ₖ² / wᵀΣw to the variance; the shares are never negative and sum to 1. The effective number of bets is exp(−Σ pₖ ln pₖ): 1 if one component carries everything, N with N equal shares. If two eigenvalues of the covariance matrix are less than 1% apart, the direction of the components is barely determined; a notice then appears.

The minimum torsion bets following Meucci et al. (2013) look for uncorrelated factors F = tR that stay as similar as possible to the positions: t minimises Σₙ Var((Fₙ − Rₙ)/σₙ) subject to the factors being uncorrelated. Written in terms of the correlation, t = σ · diag(d) · O · C^−½ · σ⁻¹ with an orthogonal matrix O. Like Meucci's “exact” procedure, the code alternates between O = (dCd)^−½ · d · C^½ (the orthogonal part of the polar decomposition) and d = diag(O · C^½), starting with d = 1, until the relative change in d is at most 10⁻¹² (at most 10,000 steps). The shares pₙ = b̃ₙ² Var(Fₙ) / wᵀΣw with b̃ = t⁻ᵀw each belong to one position, are never negative and sum to 1; from them, the count is again exp(−Σ pₙ ln pₙ). If the smallest eigenvalue of the correlation matrix is not greater than 10⁻¹⁰, minimum torsion stays empty.

There is one peculiarity: if two positions move almost identically, each still keeps its own factor. Minimum torsion then splits the common risk between the two and counts them as separate bets. The shares are close to the Euler contributions. With two positions with a correlation of 0.999 and a third, independent one, all equally volatile and equally weighted, minimum torsion counts 2.87 bets and the principal components 1.65. From a correlation of 0.95 between two positions, a notice points this out.

Alongside are two measures without a decomposition: the effective number of weights 1 / Σwᵢ² (Herfindahl index) and the diversification ratio Σwᵢσᵢ / σ_p following Choueifaty and Coignard (2008), the same as in the Positions tab.

Absorption ratio and history

The absorption ratio following Kritzman et al. (2011) is the share of the k largest eigenvalues in the sum of all eigenvalues, with k = ⌈N/5⌉ and at least 1. Kritzman et al. use the covariance; Portfolio Lab uses the correlation so that highly volatile positions do not determine the measure on their own. The spectrum and the absorption ratio thus count every position equally, while the bets are weighted by the portfolio.

On a rolling basis, the code calculates the effective number of bets (both kinds), the dimensions with signal and the absorption ratio in windows of 24 to 120 months (in steps of 6, default 36) over the entire common history, with the same covariance estimate, today's target weights and, for each window, only the months up to its end. The history appears once at least the window length plus 12 months of data are available. If the window has at most as many months as positions, minimum torsion stays empty in the history.

Limitations

• Backward-looking: all measures are estimates from past monthly returns. Correlations change, and in crises they often rise at the same time.
• Monthly data: daily returns would provide more observations but would be offset for exchanges in different time zones; asset classes are only available monthly.
• Normal-distribution view: the spectrum and the bets are based on variances and covariances; they do not capture tail dependence.
• Twins: minimum torsion counts positions that move almost identically as separate bets; as a number of independent risks, the count by principal components is then closer.
• Not a return measure: more dimensions do not mean more return.

This page describes how Portfolio Lab calculates; past results are not a forecast. All terms in the glossary.