# What is the effective number of bets? Definition and example Canonical URL: https://marketheist.io/portfolio/learn/independent-bets By Jeffrey Lin · Published 2026-10-25 · Updated 2026-10-25 · Data through Sep 2026 Category: Diversification ## Key takeaways - The effective number of bets counts independent sources of risk, not tickers: five funds that move together are closer to one bet than to five. - Every version takes risk shares that add up to one and computes exp(−Σ pᵢ ln pᵢ), which equals k when risk is split evenly across k sources and 1 when one source carries it all. - MarketHeist's co-movement groups measure first merges holdings whose average correlation is 0.5 or more (average-linkage clustering), then counts the groups by their risk shares. - Meucci's minimum-torsion effective number of bets instead splits risk across uncorrelated factors built to stay close to the holdings; it answers a different question and reads close to the holding count for an equal-weight basket of near-identical funds. - On an equal-weight basket of highly correlated US stock funds, the groups measure reads about one bet while minimum torsion reads about the number of funds, which is why MarketHeist headlines the groups measure. ## Definition The effective number of independent bets is a count of how many separate sources of risk a portfolio is really spread across: 1 when a single source drives everything, rising toward the number of holdings when risk is spread evenly over unrelated sources. It is computed as the exponential of the entropy of the risk shares, so holdings that move together count once, not many times. Also known as: effective number of bets, ENB, independent bets, number of independent risk sources, effective number of uncorrelated bets. ## How it works Counting holdings overstates diversification when the holdings move together. The effective number of bets replaces the count with a number that asks how many separate things the portfolio is actually exposed to, weighted by how much risk each one carries. Both measures on this page end with the same step. Take a list of risk shares that are not negative and add up to one, and compute the exponential of their entropy. This is the effective number ecologists use for species diversity (a Hill number of order one): a forest with ten equally common species scores ten, a forest that is nearly all one species scores close to one, however many rare species it has. The two measures differ in what counts as a source. MarketHeist's co-movement groups measure clusters the holdings by correlation. Clusters merge while the average correlation between them is at least 0.5, so a group's members have, on average, moved together. Each group's risk share is the sum of its members' risk contributions, and the entropy is taken over the groups. The question it answers is: how many clusters of holdings that tend to move together carry the risk? Meucci's minimum-torsion measure rebuilds the portfolio from uncorrelated factors, one per holding, chosen to stay as close as possible to the original holdings, and takes the entropy of each factor's contribution to variance (uncertainty). The question it answers is: across uncorrelated directions of risk tied to each holding, how evenly is the variance spread? Those factors always exist, even when two holdings are near copies; for near copies they are built from the small differences between the holdings. That is where the two disagree. When holdings are interchangeable and equally weighted, symmetry gives every minimum-torsion factor the same share, so the measure returns the holding count whatever the correlation. The groups measure merges them and reads one. Neither is an arithmetic error; they measure different things. MarketHeist headlines the groups measure because it is the one that matches the question most investors ask about overlap. ## Formula **Effective number of bets from risk shares** ``` N = \exp⁡( - \sum_{k = 1}^K p_k \ln⁡p_k), p_k \ge 0, \sum_{k = 1}^K p_k = 1 ``` where: - N: effective number of bets, between 1 and K - pₖ: share of risk carried by source k (a group or a factor) - K: number of sources (groups for the co-movement measure, holdings for minimum torsion) - ln: natural logarithm; a share of zero contributes nothing **Co-movement groups: merge rule and group shares** ``` \bar{\rho} (A,B) = \frac{1}{|A| |B|} \sum_{a \in A} \sum_{b \in B} \rho_{a b} \ge \rho^*, p_g = \frac{\max (0,\sum_{i \in g} RC_i)}{\sum_h \max (0,\sum_{i \in h} RC_i)} ``` where: - ρ̄(A, B): average correlation over every pair with one holding in cluster A and one in cluster B - ρₐᵦ: correlation of the monthly returns of holdings a and b - |A|: number of holdings in cluster A - ρ*: group threshold, 0.5 by default - g, h: co-movement groups - RCᵢ: risk contribution of holding i (its share of portfolio variance) - p_g: risk share of group g after clamping negative group totals to zero Clusters merge in order of the highest average correlation; the groups are the clusters that formed at or above the threshold. The group shares then go into the first formula. **Meucci minimum-torsion contributions (correlation form)** ``` ν = C^{1/2} (\sigma⊙w), p_i = \frac{ν_i^2}{\sum_jν_j^2} ``` where: - C: correlation matrix of the holdings' monthly returns - C^(1/2): symmetric square root of C (the same eigenvectors, square roots of the eigenvalues) - σ ⊙ w: each holding's weight times its volatility (volatility-scaled weights) - νᵢ²: variance contributed by minimum-torsion factor i; never negative - pᵢ: factor i's share of portfolio variance, which goes into the first formula With equal weights, equal volatilities and one common correlation, every νᵢ is the same by symmetry, so pᵢ = 1/n and the measure equals n for any correlation below one. ## Worked example Two portfolios measured over one common window, Dec 2004 to Sep 2026: the Golden Butterfly (equal parts US total market, small-cap value, long Treasuries, short Treasuries and gold) and an equal-weight basket of five US stock funds (SPY, VTI, VV, VB and VTV). The Golden Butterfly holds 5 funds. The co-movement groups measure finds 3 groups, and the largest one (IJS and VTI) carries 66.0% of the risk. Counted by the groups' risk shares, the portfolio holds 2.36 effective bets. Meucci's minimum-torsion measure reads 3.96 for the same portfolio, more bets than the groups measure finds. It gives VTI and IJS, and TLT and SHY, partly separate factors; the groups measure puts each of those pairs in one group, because the pair's correlation is at or above 0.5. The basket of five US stock funds shows the disagreement at its sharpest. The funds' monthly returns are so alike that the groups measure puts all 5 in a single group carrying 100.0% of the risk and reads 1.00 bets. Minimum torsion reads 5.00, close to the fund count, as if the basket were 5 separate bets. Both numbers are correct arithmetic. Minimum torsion reports how evenly variance spreads across decorrelated factors, and such factors exist even between funds that are near copies of each other. The groups measure reports how many clusters of co-moving funds carry the risk. For the question of whether five US stock funds diversify one another, the groups measure gives the answer an investor would recognise, and that is why it is the figure MarketHeist shows as Independent Bets. Data through Sep 2026. Reproduce id: 7fdf24a7d007b90a (this guide, its lab setup and the data month). ## How MarketHeist computes it - **Data**: Monthly total returns (dividends reinvested). Funds younger than the period shown are extended backward with older funds, spliced on returns, not prices. - **Window**: The worked example and the lab use the common window of the holdings, Dec 2004 to Sep 2026 (262 months), ending at the data month of this page. - **Group threshold**: Clusters merge into one co-movement group while their average cross correlation is at least 0.5. A single pair inside a group can sit below 0.5; the group's mean cannot. (`GROUP_CORRELATION` in packages/portfolio/engine/overlap.ts) - **Linkage**: Average linkage (UPGMA on the correlation itself): at each step the two clusters with the highest mean correlation over all cross pairs merge. Single linkage, used for the HRP tree, is not used here because it chains: a fund could join a group through one correlated member while being uncorrelated with the others. (`GROUP_LINKAGE` in packages/portfolio/engine/overlap.ts) - **Tree heights and the cut**: Merge heights are correlation distances d = √(0.5(1 − ρ)). The groups are the subtrees whose top merge sits at or below the distance of the threshold, which for 0.5 is d = 0.5. (`correlationDistance` in packages/portfolio/engine/hrp.ts) - **Group risk shares**: Each group's share is the sum of its members' risk contributions, computed at the target weights from the population covariance of monthly returns. A group whose total is negative (a net diversifier) is set to zero, and the remaining shares are rescaled to sum to 1 before the entropy. (`effectiveBetsFromGroups` in packages/portfolio/engine/overlap.ts) - **Minimum-torsion ENB**: Computed in correlation space: volatility-scaled weights σᵢwᵢ, the symmetric square root of the correlation matrix from a Jacobi eigen-decomposition (round-off negative eigenvalues set to zero), factor contributions νᵢ², then the exponential of their entropy. A single holding returns 1; a zero-variance portfolio returns the holding count. (`effectiveNumberOfBets` in packages/portfolio/engine/risk.ts) - **Near-duplicate pairs**: Separately, any pair of holdings with a correlation of 0.9 or more is listed as a near-duplicate. The flag does not change either count. (`REDUNDANT_CORRELATION` in packages/portfolio/engine/overlap.ts) - **Where each measure appears**: The app's Independent Bets figure, the Balance-risk table, the risk guides and the MCP tools all report the co-movement groups measure over the full window. Minimum torsion appears on this page only, by name. (`portfolioEffectiveBets` in packages/portfolio/engine/overlap.ts) - **Engine functions**: `exposureOverlap` (packages/portfolio/engine/overlap.ts), `effectiveBetsFromGroups` (packages/portfolio/engine/overlap.ts), `effectiveNumberOfBets` (packages/portfolio/engine/risk.ts) ## Effective number of independent bets compared with its neighbours Ways to count bets, and what each reads for five near-identical funds held in equal weights | Measure | What counts as a source | Five near-identical funds | Main weakness | | --- | --- | --- | --- | | Holding count | Each ticker | Five | Ignores weights and correlation | | Effective number of holdings, 1 / Σwᵢ² | Each ticker, by money weight | Five | Ignores volatility and correlation | | Co-movement groups (MarketHeist) | Clusters whose average correlation is at least 0.5 | About one | A hard threshold: a pair just under it counts as two groups | | Minimum-torsion ENB (Meucci, Santangelo and Deguest) | Uncorrelated factors kept close to the holdings | About five | Reads near the holding count on interchangeable holdings | | Principal-component ENB (Meucci 2009) | Principal components of the covariance | About one | Components are hard to interpret and arbitrary when eigenvalues tie | ## Limitations - Both measures come from one historical window of monthly returns. Correlations rise in some crises, so a portfolio can hold fewer independent bets in a crash than its long-run figure suggests. - The groups measure uses a hard threshold. Two funds with a correlation just under 0.5 count as separate groups, and just over it as one, so the count can step when the data move slightly. - Minimum torsion is smooth and needs no threshold, but it does not merge near-identical holdings, which is the case most investors care about when they ask about overlap. - Neither measure looks inside the funds. They see only how returns moved, not what the funds hold, so two funds with different holdings that happened to move together are treated as one source. - A higher count describes how risk was spread in the past. It is not a forecast of returns and not a rating of a portfolio. ## Frequently asked questions **What does an effective number of bets of 1 mean?** That one source drove essentially all of the portfolio's risk. A portfolio of several funds can still read close to 1 when they all moved together, as broad US stock funds do. **Why does MarketHeist show fewer bets than Meucci's ENB?** MarketHeist merges holdings that move together into co-movement groups before counting, so near-identical funds count once. Meucci's minimum-torsion measure gives each holding its own uncorrelated factor and does not merge them, so it reads close to the holding count for an equal-weight basket of similar funds. **Does adding more funds increase the number of independent bets?** Only when the new fund moves differently from what the portfolio already holds and takes a meaningful share of the risk. Adding another broad US stock fund to a US stock portfolio leaves the groups measure where it was. **Is the effective number of bets the same as the effective number of holdings?** No. The effective number of holdings, 1 divided by the sum of squared weights, looks only at how the money is split. The effective number of bets looks at how the risk is split, after accounting for volatility and correlation. **Why is the number of bets not a whole number?** It is the exponential of an entropy, so it measures evenness as well as count. Two groups with equal risk read 2; two groups where one carries most of the risk read somewhere between 1 and 2. ## References 1. Meucci, A. (2009) Managing Diversification . Risk (2009). Introduces the effective number of bets on principal components. 2. Meucci, A., Santangelo, A. and Deguest, R. (2015) Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors . Risk (2015), pp. 70-75; SSRN working paper 2013. The minimum-torsion factors. 3. Hill, M. O. (1973) Diversity and Evenness: A Unifying Notation and Its Consequences . Ecology 54(2). The exponential of entropy as an effective number. 4. Sokal, R. R. and Michener, C. D. (1958) A Statistical Method for Evaluating Systematic Relationships. University of Kansas Science Bulletin 38. Average linkage (UPGMA). ## Try it App: https://marketheist.io/portfolio/p/golden-butterfly MCP prompt (MarketHeist MCP server, https://marketheist.io/mcp): ``` Using MarketHeist, how many independent bets does my portfolio hold, and which holdings move together? 25% SPY, 25% QQQ, 25% VTI, 25% BND ``` Historical simulation, not advice.