Bet Sizing

Bet sizing

A probability is not a position, and the four steps between them each cost something measurable. Advances in Financial Machine Learning (Lopez de Prado 2018), chapter 10. The size curve, the averaging of concurrent bets, the discretisation, and the leverage budget — all four are usually written in one line each, and three of them are usually wrong.

Every number below is printed by scripts/bet_sizing.py (numpy + scipy, seed 0, 2.8 s).

⚠️ This chapter is the one place in this plugin with no runnable reference implementation. mlfinpy 0.1.2 ships no bet-sizing module (its packages are labeling, sample_weights, sampling, filters, structural_breaks, cross_validation, ensemble, data_structure, util, dataset), and 🔴 mlfinlab is not installable — ✅ checked 2026-09-09: the PyPI JSON API returns 404 and the simple index lists zero distribution files. The formulas below are transcribed from the published snippets; what is measured is every property they should have.

1. The size curve is not 2p - 1

z = (p - 1/n_classes) / sqrt(p (1 - p))        m = 2 * Phi(z) - 1        signal = side * m

⚠️ AFML Snippet 10.1 (page 142), getSignal / getBetSize. ✅ Measured against the obvious alternative:

✅ m(0.5) = 0.0 exactly, m(1) = 1.0000, m(0) = -1.0000. The curve sits below the linear size up to about p = 0.90 and above it after: it is sceptical near the coin flip and saturating at the extremes, which is the opposite shape to what "just use 2p-1" gives you. At p = 0.55 it asks for 80 % of the linear size; at p = 0.95, 107 %.

🚨 n_classes is not decoration. The pivot is 1/n_classes, not 0.5. ✅ Measured: m(0.40, n=3) = +0.1082 — a long — while m(0.40, n=2) = -0.1617, a short. Pass a three-class model's probability through the binary formula and every marginal call comes out with the wrong sign.

⚠️ The z-to-size map assumes z is approximately standard normal, which is a modelling convention rather than a derived fact. What it guarantees is monotonicity, the right sign, and saturation in [-1, 1]; it does not make m an optimal fraction of capital. That question is Kelly's, and ../../../fin-strategies/skills/position-sizing-kelly/SKILL.md owns it.

2. Averaging concurrent bets, not adding them

⚠️ AFML Snippet 10.2 (page 144), avgActiveSignals: at each bar, average the signals of the bets active at that bar. ✅ Measured on 297 overlapping bets over 5,000 bars (holding 5-60 bars, mean concurrency 1.95, max 8, 14.4 % of bars flat; calibration check: mean true q 0.662 against a realised hit rate of 0.653):

• 🚨 Adding the sizes gives you 3.2x leverage at the peak and puts the book above 1x on 11 % of bars — a leverage decision made by the arrival rate of signals, not by anyone.
• ✅ Averaging divides the peak by 3.2x and the mean by 2.0x, and halves the turnover.
• 🚨 It is not a rescaling. The two position series correlate 0.8662, because the divisor is the concurrency, which moves with time. A bet's contribution to the book shrinks when other bets arrive — that is a real, and deliberate, change to the return series, not a change of units.

3. 🚨 Discretisation: rounding alone barely reduces turnover

⚠️ AFML Snippet 10.3 (page 145): signal1 = (signal0 / stepSize).round() * stepSize, clipped to [-1, 1]. The stated purpose is to stop trading on trivial changes in the probability. ✅ Measured on the averaged signal, against a variant that only moves when the raw signal has drifted a full step from the level currently held:

🚨 At step 0.05 the rounded position trades MORE than the continuous one (100.3 %). A signal hovering at a grid boundary flips across it every bar, and each flip is a full step of trading. Rounding changes the set of values the position takes; it does not by itself change how often the position moves. Only the coarsest grid (step 1.00, i.e. {-1, 0, 1}) cuts turnover, and it costs a correlation of 0.73 with the continuous signal.

✅ Requiring a full step of drift before moving cuts turnover to 32 % at step 0.50 and costs tracking (correlation 0.69) rather than resolution. (At step 1.00 that rule never trades at all, because the averaged signal never reaches 1 — hence the blank row.)

And whether any of it is worth doing is a cost question, not an accuracy question. ✅ The same three position series scored at four cost levels:

🚨 At zero cost discretisation strictly loses (0.90 → 0.84 → 0.68); at 20 bp it is the only version still above water. Choose the step from your cost curve (../../../fin-core/skills/execution-cost-analysis/SKILL.md owns that), not from a default, and count the choice as a trial.

4. 🚨 The concurrency budget divides by a number you have not seen yet

The natural way to bound leverage is to divide the summed signal by the maximum number of simultaneous bets. ✅ Measured:

• 🚨 The full-sample maximum is not known until bar 1,236 of 5,000 — a quarter of the way through. Everything before that is sized with information from the future.
• 🚨 The causal version exceeds the full-sample version's own peak on 0.7 % of bars: that is leverage the look-ahead hid, and it is exactly the tail where it matters.
• 🚨 Over the first quarter of the sample the causal budget runs at 1.58x the full-sample one, because its divisor has not yet met the crowded stretch.

A backtest that divides by a full-sample maximum reports a smoother book than you could have run, and it will not warn you. Use an expanding maximum, a fixed budget declared in advance, or a quantile of trailing concurrency — and check the whole position series with ../../../fin-core/skills/signal-construction/scripts/assert_causal.py.

5. Traps

• 🚨 2p - 1. Section 1. Wrong shape at both ends, and it ignores n_classes.
• 🚨 A binary size formula on a three-class model. Section 1: m(0.40, n=3) and m(0.40, n=2) have opposite signs.
• 🚨 Adding concurrent positions. Section 2: 3.2x leverage nobody chose.
• 🚨 Assuming discretisation cuts turnover. Section 3: at step 0.05 it increases it.
• 🚨 A full-sample concurrency divisor. Section 4.
• 🚨 An uncalibrated probability. All of section 1 assumes p means what it says. A model whose reported 0.8 is right 60 % of the time will be sized as if it were right 80 % of the time. Check calibration (predicted probability against realised frequency, by bucket) before sizing, and note that a probability from an unpurged fold is not calibrated for the future (../../../fin-libraries/skills/lib-purgedcv/SKILL.md).
• 🚨 The size at t must use p known at t. A probability re-reported mid-bet is fine; a probability computed with the bet's own outcome is not.
• ⚠️ m is a relative size, not a capital fraction. Multiplying it by the full account is a Kelly-scale decision — ../../../fin-strategies/skills/position-sizing-kelly/SKILL.md — and volatility targeting sits on top of that (../../../fin-core/skills/portfolio-and-risk/SKILL.md).
• ⚠️ Both discretise variants use half-to-even rounding (np.round, like pandas' .round()): a size of exactly 0.25 at step 0.5 becomes 0.0, not 0.5. It only matters on exact ties, and exact ties happen more often than you expect once a signal is itself discretised.
• 🚨 The step size is a trial. Six steps times two rules times four cost levels is 48 numbers, and the best of them is not an estimate of anything. ../../../fin-core/skills/backtest-validation/scripts/trial_ledger.py.

6. Scripts

• scripts/bet_sizing.py — the size curve and its linear alternative, a calibrated overlapping-bet simulator whose probability is re-reported every bar, averaging versus summing active signals, both discretisation rules, turnover and Sharpe at four cost levels, and the full-sample versus expanding concurrency budget. numpy + scipy only. Seed 0, 2.8 s.

7. Where this sits

../meta-labeling/SKILL.md produces the probability this skill consumes, and stops at the threshold. ../../../fin-strategies/skills/position-sizing-kelly/SKILL.md owns the Kelly fraction and how much capital the whole book gets — this skill deliberately does not repeat it. ../../../fin-core/skills/portfolio-and-risk/SKILL.md owns volatility targeting and the risk budget. ../../../fin-core/skills/execution-cost-analysis/SKILL.md owns the cost curve that decides the step size in section 3, and ../../../fin-strategies/skills/execution-algorithms/SKILL.md owns getting the resulting size into the market. ../sample-weights-and-uniqueness/SKILL.md owns the concurrency series that section 4 divides by. ../../../fin-core/skills/signal-construction/SKILL.md owns the causality check on the position series.