A portfolio with 50 holdings looks diversified. A portfolio with 50 holdings where one name has 60% weight is not. The Herfindahl–Hirschman Index (HHI) and its reciprocal, the effective number of holdings, are the cleanest one-number summaries of true concentration. FM103's Concentration sub-pill tracks both over time.
Definition
For portfolio weights w1, w2, ..., wn summing to 1, the Herfindahl–Hirschman Index (Hirschman 1964) is:
It ranges from 1/n (perfect equal-weight across n names) to 1 (single-name portfolio). The reciprocal:
is the "effective number of holdings." A portfolio with 50 names where each has weight 0.02 has HHI = 50 · 0.02² = 0.02 and Neff = 50. A portfolio with 50 names where one has weight 0.5 and the rest split the other 0.5 equally has HHI ≈ 0.255 and Neff ≈ 3.9.
Why HHI beats "number of holdings"
Counting holdings ignores the size distribution. Market-cap-weighted strategies routinely have Neff < 10 even with 100+ holdings because the top few names absorb most of the weight (the S&P 500 itself has Neff ≈ 60 even though it holds 500 names). For a strategy claiming to be diversified, Neff is the honest number.
Thresholds
- Neff < 5: Concentrated. Returns determined by a handful of names.
- Neff 5–15: Moderate. Significant single-name risk but not dominant.
- Neff 15–30: Reasonably diversified for an active strategy.
- Neff > 30: Diversified; tracking-error from a benchmark dominated by macro factors.
HHI over time
The Concentration sub-pill plots HHI per rebalance period. Three patterns:
- Stable HHI. Healthy. Strategy maintains target diversification across regimes.
- Rising HHI. Concentration is creeping up — usually because winners are being held and not being trimmed. Mechanical issue, fixable with rebalancing rules.
- Falling HHI. Equal-weight drift — rebalancing is pulling toward equal even when conviction differs. Re-examine whether the rebalance rule matches intent.
Style drift as a related diagnostic
The Concentration sub-pill also computes style drift: how much the sector composition changes from period to period. Different from holdings change — you can swap names within the same sector and have zero style drift. High style drift with high HHI is the dangerous combination: small portfolio rotating between sectors.
The single-stock contribution check
The sub-pill also reports the largest single-stock weight at each rebalance. Some users cap single-stock weight at 5% or 10% as a portfolio rule. The plot reveals whether the cap is actually binding, and how often.
How concentration interacts with the Strategy Health Card
The Risk sub-score in the Strategy Health Card penalises high average HHI heavily. A strategy with strong factor return but persistent Neff < 5 will earn a high Factor sub-score and a low Risk sub-score — landing in MONITOR or REVIEW. The diagnosis is clear: "the factor works but the implementation is over-concentrated." Fix is straightforward: cap single-stock weight, or move to equal-weighted instead of cap-weighted within the top-N.
HHI for sectors, not just stocks
HHI also applies to sector exposure. A strategy with 30 names spread across 11 sectors but with 60% in tech has sector HHI ≈ 0.4 (Neff,sector ≈ 2.5). Stock-level concentration may look fine while sector concentration is severe. The sub-pill shows both.
Sharpe, drawdown, and VaR all scale with Neff in non-obvious ways. A portfolio with Neff = 5 has roughly the diversification benefit of a 5-stock portfolio — not the 50-stock portfolio it nominally is. Before drawing any conclusion from risk-adjusted return, verify Neff.
Further Reading
Foundational papers
- Hirschman, A. O. (1964). The Paternity of an Index. American Economic Review, 54(5), 761–762.
Textbook references
- Grinold, R. C. & Kahn, R. N. (1999). Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk (2nd ed.). McGraw-Hill.
- Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk (3rd ed.). McGraw-Hill.
Related QuanterLab articles
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For market-cap-weighted strategies, effective N is almost always lower than holdings count. Cap single-stock weight at 5% to enforce a meaningful diversification floor.