Folio Lab/Documentation

Effective Number of Constituents

FolioLab's effective_n metric is the inverse Herfindahl concentration measure . It answers a narrow but useful question: how many equally weighted holdings would create the same weight concentration as the current portfolio?

Overview

Counting line items is a poor measure of diversification. A portfolio with 100 tickers can still be highly concentrated if a few names dominate the weights. The inverse-HHI effective number fixes that by translating weight concentration into an equivalent count of equally weighted holdings.

This page keeps the legacy route slug /docs/metrics/effective-number-bets, but the implemented metric is not Meucci's Effective Number of Bets on uncorrelated factors. It is a weight-only effective number of constituents. Correlations, PCA structure, and independent risk bets are outside this formula.

Mathematical Formulation

Core Formula

The implemented metric is the reciprocal of the Herfindahl-Hirschman Index of portfolio weights:

where is the weight of asset and is the number of holdings. The denominator is the HHI, which ranges from under equal weights to 1 for a single-position portfolio.

Diversification Ratio

The effective number of constituents can be scaled by the actual holding count to obtain a simple diversification ratio:

This ratio ranges from for a single dominant position to for equal weighting.

Properties

  • Range: . The minimum occurs when one asset carries all the weight; the maximum occurs under equal weights.
  • Equal weights: If for all , then .
  • Monotonicity: Shifting weight from a larger position to a smaller one increases .
  • Weight-only: The formula depends only on . Two portfolios with identical weights but very different correlation structures will have the same value.

Worked Example: Equal Weights

Consider a portfolio of assets, each with weight :

An equally weighted portfolio of 10 assets has an effective number of 10 constituents and a diversification ratio of 100%.

Worked Example: Concentrated Portfolio

Now consider a portfolio of assets with one dominant position: and the remaining nine positions at :

Despite holding 10 positions, the portfolio's weight concentration is equivalent to only 3.6 equally weighted holdings.

Advantages & Limitations

Advantages

  • Intuitive: It directly translates concentration into an equivalent count of equally weighted holdings.
  • Concentration detection: It reveals hidden weight concentration that raw holding counts miss.
  • Simple: It requires only the portfolio weights.
  • Operationally useful: It is convenient for monitoring portfolio concentration limits over time.

Limitations

  • Not true independent bets: It ignores correlations, so it should not be read as a PCA-based or factor-based diversification count.
  • Weight-only view: It does not account for volatility, covariance, or risk contribution.
  • Short positions: Interpretation becomes less clean for highly levered long-short portfolios.
  • No return content: It says nothing about expected return or portfolio quality.

References

  1. Herfindahl, O. C. (1950). Concentration in the U.S. Steel Industry. PhD dissertation, Columbia University.
  2. Hirschman, A. O. (1964). "The Paternity of an Index." American Economic Review, 54(5), 761-762.
  3. Strongin, S., Petsch, M., & Sharenow, G. (2000). "Beating Benchmarks." The Journal of Portfolio Management, 26(4), 11-27.
  4. Meucci, A. (2009). "Managing Diversification." Risk, 22(5), 74-79. Related background on a different, PCA-based Effective Number of Bets.