Portfolio thinking: correlation and exposure
Picks that share a match, team, league, model or kickoff are not independent. Measure their correlation from a score model, avoid multiplying probabilities for same-game combinations, and cap exposure by what picks have in common.
Learning objectives
- Compute the joint probability and correlation of two same-match outcomes from a score grid.
- Show with numbers why multiplying probabilities misprices same-game combinations, in both directions.
- Estimate the effective number of independent picks for a set of correlated picks.
- Set exposure limits by match, team, model and kickoff for a simulated bankroll.
Why same-match outcomes move together
Match result, total goals and both-teams-to-score are all functions of one scoreline. Even if each team's goals are independent Poisson draws, events defined on the same score are linked: most home wins involve the home side scoring two or more, which also pushes the total up. Real scores add a further layer: allowing modest dependence between the two teams' goals improves the fit Maher (1982), and Dixon and Coles added a correction for low scores Dixon & Coles (1997).
Take a hypothetical match with expected goals λ = 1.6 (home) and μ = 1.0 (away), independent Poisson, no low-score correction. The resulting score grid gives: home win 51.34%, draw 25.00%, away win 23.66%, over 2.5 goals 48.16% and BTTS ‘Yes’ 50.45%.
ρ(A, B) = (P(A∩B) − P(A)·P(B)) / √(P(A)(1 − P(A)) · P(B)(1 − P(B)))| Pair | P(A) | P(B) | Joint P(A∩B) | Product P(A)·P(B) | Correlation |
|---|---|---|---|---|---|
| Home win & Over 2.5 | 51.34% | 48.16% | 29.95% | 24.73% | +0.21 |
| BTTS Yes & Over 2.5 | 50.45% | 48.16% | 38.57% | 24.30% | +0.57 |
| Draw & Under 2.5 | 25.00% | 51.84% | 19.31% | 12.96% | +0.29 |
| Away win & Over 2.5 | 23.66% | 48.16% | 12.52% | 11.39% | +0.05 |
| Home win & BTTS Yes | 51.34% | 50.45% | 21.98% | 25.90% | −0.16 |
Same-game combinations: why multiplication fails
Multiplying probabilities is only valid for independent events. For Home win & Over 2.5, multiplication gives 24.73% (fair odds 4.04), but the score grid gives 29.95% (fair odds 3.34). For BTTS & Over 2.5 the gap is larger: 24.30% (4.12) against 38.57% (2.59).
The error runs both ways. With positive correlation, multiplication understates the joint probability, so a correctly priced combination looks poor. With negative correlation — Home win & BTTS, ρ = −0.16 — multiplication overstates it: 25.90% (fair odds 3.86) instead of 21.98% (4.55). That is the dangerous direction, because it makes an over-priced combination look attractive.
Concentration beyond a single match
Correlation also comes from shared causes across matches:
- Same team: picks on one side's attack in several markets, or across consecutive fixtures, all depend on the same rating.
- Same league or round: a league-wide change in scoring rates moves every totals pick together.
- Same model: if your model overrates home attacks, every home-win and over pick is wrong at once. This is often the largest hidden correlation.
- Same kickoff: simultaneous picks settle together, leaving no chance to react between results.
N_eff = n / (1 + (n − 1) × ρ̄)Ten picks with ρ̄ = 0.21 behave like about 3.46 independent picks; with ρ̄ = 0.5, like 1.82. Twenty picks with ρ̄ = 0.10 are worth about 6.90. Correlation also fattens the tails: for unit picks at fair odds on Home win and Over 2.5, the standard deviation of combined profit is 1.56 instead of 1.42, and the chance that both lose is 30.45% rather than the 25.22% independence suggests. For BTTS and Over 2.5 it is 39.96% against 25.69%.
Exposure limits
Exposure limits cap the total share of a simulated bankroll that rides on one shared cause. Sensible dimensions are per match, per team, per model and per kickoff window. Three picks of 2% each on one match — home win, over 2.5 and BTTS — are one 6% position on that scoreline. In the model above, all three lose together 18.57% of the time (the scores 0-0, 0-1 and 0-2) and all three win together 21.98% of the time.
A practical rule: size the per-match cap so that the joint loss of all picks on a match is no bigger than you would accept on a single pick. Then apply the same logic to team and model. Diversification only counts when the picks fail for different reasons.
Worked example
Worked example · Pricing a same-match pair at Kestrel Park v Dunmore
A score model gives Kestrel Park (home) λ = 1.6 and Dunmore μ = 1.0 expected goals, independent Poisson, no low-score correction. A hypothetical quote of 3.50 is offered on ‘Kestrel Park win and Over 2.5 goals’.
- Marginal probabilities
the home / draw / away probabilities and over/under probabilities on a Poisson score grid (λ = 1.6, μ = 1.0)Home win 51.34%; Over 2.5 48.16% - Naive product
0.5134 × 0.481624.73% → fair odds 4.04 - Joint from the score grid
Sum of P(x, y) over scores with x > y and x + y ≥ 329.95% → fair odds 3.34 - How far off is the product?
0.2995 / 0.2473Joint is 1.21× the product - Correlation
(0.2995 − 0.2473) / √(0.5134 × 0.4866 × 0.4816 × 0.5184)ρ = +0.21 - EV of the quote under each method
0.2473 × 3.50 − 1 vs 0.2995 × 3.50 − 1−13.5% (product) vs +4.8% (joint) - Effective size of the two legs
2 / (1 + 1 × 0.21)N_eff ≈ 1.65 picks, not 2
Multiplying probabilities flipped the sign of the EV. Even the +4.8% from the joint calculation rests entirely on λ, μ and the independence assumption; with an interval on those inputs it may not survive (see lesson 14). Either way, the two legs are one position on one scoreline.
Practice · BTTS and Over 2.5 on one match
Using the same model (λ = 1.6, μ = 1.0), compute for ‘BTTS Yes and Over 2.5’: the product of the two probabilities, the joint probability from the grid, the correlation and both fair odds. Then compute N_eff for six picks with average correlation 0.21, and the chance that separate BTTS and Over 2.5 picks both lose.
Show what you should find
P(BTTS) = 50.45% and P(Over 2.5) = 48.16%; the product is 24.30% (fair odds 4.12), while the joint is 38.57% (fair odds 2.59) — 1.59× the product, correlation +0.57. Six picks with ρ̄ = 0.21 give N_eff = 6 / (1 + 5 × 0.21) ≈ 2.93. Both picks lose together 39.96% of the time, against 25.69% if they were independent.
Common mistakes
- ✗ Multiplying probabilities for picks on the same match.Compute the joint probability from a score grid. For BTTS and Over 2.5 the product (24.30%) is far below the joint (38.57%).
- ✗ Assuming same-match correlation is always positive.Home win and BTTS have ρ = −0.16 here; multiplying then overstates the joint probability and makes an over-priced combination look attractive.
- ✗ Counting ten picks as ten independent pieces of evidence.With average correlation 0.21, ten picks carry about as much information as 3.46 independent ones.
- ✗ Diversifying across markets on the same match.Result, totals and BTTS picks on one match share one scoreline. Cap them together as one position.
When it fails
Supporting research
- Dixon & Coles (1997): Modelling Association Football Scores and Inefficiencies in the Football Betting MarketJournal of the Royal Statistical Society: Series C (Applied Statistics) · Peer-reviewed empiricalFootball scores are modelled with team strengths plus a low-score dependence correction, so markets on one match share a joint distribution.
- Maher (1982): Modelling association football scoresStatistica Neerlandica · Peer-reviewed empiricalAllowing modest dependence between the two teams' scores improves the fit of Poisson score models.
Check your understanding
Practice only: saved on this device, no XP and no account record.