Bankroll Management with Kelly Criterion: The Math of Position Sizing

Finding a positive-edge bet is the first problem in value betting. Sizing the bet correctly is the second — and it's the one that determines whether your edge translates into actual long-run growth.

Get the sizing wrong and you can have genuine edge and still blow up your bankroll. The Kelly Criterion is the mathematical framework that solves this problem.

The Kelly formula

John L. Kelly Jr., a researcher at Bell Labs, derived this formula in 1956. The Kelly Criterion answers: given a known edge over the market, what fraction of your bankroll should you bet to maximise long-run geometric growth?

f = (b × p − q) / b

Where:

• f = fraction of bankroll to bet
• b = net decimal odds minus 1 (so at 2.0 decimal, b = 1.0)
• p = probability of winning
• q = probability of losing = 1 − p

Equivalently, for decimal odds:

f = (decimal_odds × win_probability − 1) / (decimal_odds − 1)

Example

You have a bet at decimal odds of 2.20. BetEdge's model says the fair probability of winning is 52% (implied by Pinnacle's de-vigged odds).

f = (2.20 × 0.52 − 1) / (2.20 − 1)
  = (1.144 − 1) / 1.20
  = 0.144 / 1.20
  = 0.12 = 12%

Full Kelly says bet 12% of your bankroll.

On a $10,000 bankroll, that's $1,200.

Why the Kelly fraction maximises growth

Kelly's proof is elegant: any fraction above full Kelly reduces long-run geometric mean growth. Any fraction below full Kelly also reduces growth, but less severely, and with dramatically lower variance.

The Kelly formula maximises the expected logarithm of wealth — which is equivalent to maximising long-run compounded growth, not individual bet expectation.

This matters. A bettor maximising individual bet expectation would bet everything on any positive-edge bet. That leads to ruin the moment they lose. Maximising log-wealth (Kelly) explicitly penalises overbetting.

The variance problem with full Kelly

Here's why almost no serious bettor uses full Kelly:

Assumption 1: The formula assumes your edge estimate is perfectly accurate.

In practice, your edge estimate is a model output with uncertainty. If you're estimating 52% win probability and the true figure is 49%, your Kelly fraction is inflated — you're effectively overbetting.

Assumption 2: Kelly assumes unlimited bet divisibility and no correlation between bets.

Real bettors often have correlated bets (e.g., multiple selections from the same match) and minimum stake constraints.

The drawdown reality: At full Kelly with typical betting edges (2–8%), expected drawdowns of 30–50% are mathematically normal. Most bettors cannot sustain these psychologically — and react emotionally by abandoning the strategy at the worst moment.

Fractional Kelly: the practical solution

Fractional Kelly bets a fixed proportion of the full Kelly fraction.

| Kelly Multiple | Growth Retained | Drawdown Reduction | |---------------|----------------|-------------------| | Full (1.0×) | 100% | 0% | | Half (0.5×) | 75% | ~50% | | Quarter (0.25×) | 44% | ~75% |

The loss in long-run growth from using Quarter-Kelly instead of Full Kelly is surprisingly modest — you sacrifice 56% of the theoretical maximum growth, but you also cut expected drawdowns by ~75% and dramatically reduce the risk of catastrophic early losses.

BetEdge publishes Quarter-Kelly stake recommendations alongside every pick. This is our default for three reasons:

1. Uncertainty in edge estimates — our models have confidence intervals, not point estimates
2. Operator limits — many books restrict stakes before the full Kelly size can be placed
3. Psychological sustainability — smaller swings mean bettors stay rational longer

Working through an example

Your bankroll: €5,000 Pick: Arsenal to win at decimal odds 2.30 BetEdge edge estimate: 54% win probability (post Pinnacle de-vig + edge filter)

Full Kelly:

f = (2.30 × 0.54 − 1) / (2.30 − 1)
  = (1.242 − 1) / 1.30
  = 0.242 / 1.30
  = 0.186 → 18.6% of bankroll
  = €930

Quarter-Kelly: €930 × 0.25 = €232.50

If Arsenal win:

• Profit: €232.50 × (2.30 − 1) = €232.50 × 1.30 = €302.25
• New bankroll: €5,302.25

If Arsenal lose:

• Loss: −€232.50
• New bankroll: €4,767.50

The asymmetry means wins compound faster than losses erode.

When to deviate from Kelly

Kelly assumes each bet is independent. Sports betting violates this when:

Correlated bets: Multiple picks from the same match or same night of games. If you have 3 picks on a Champions League night and they're all from heavy favourites in European leagues, a "bad night" will hit all 3. Reduce fractional Kelly proportionally.

Bankroll scale: As your bankroll grows larger, your ability to get full bet on at your target odds may diminish. Kelly assumes you can always place the fraction. In reality, limit-friendly books (Pinnacle, exchanges) are the only venues where this holds above modest stakes.

Roll-up after losses: Some bettors reduce Kelly fraction further during cold streaks to slow bankroll erosion. This is psychologically sensible but not mathematically necessary — Kelly already accounts for variance if your edge estimate is correct.

The relationship between Kelly and CLV

Kelly sizing and CLV measurement are complementary systems.

Kelly answers: "How much to bet given this edge?" CLV answers: "Was the edge real?"

Together, they form a complete framework:

1. Use Pinnacle's de-vigged fair odds to estimate edge
2. Size via Quarter-Kelly based on that edge
3. After settlement, compute CLV to verify the edge estimate was accurate
4. Adjust future Kelly fractions if your CLV systematically differs from predicted edge

If your CLV is consistently +4% but your edge estimate was +6%, your Kelly fractions are inflated — scale them down proportionally.

Building the Kelly habit

The core habit is to pre-commit. Before placing a bet:

1. Estimate your win probability (use Pinnacle's line as the reference)
2. Compute the edge: edge = (bookmaker_odds × your_win_prob) − 1
3. If edge > 0, compute Kelly fraction
4. If edge is < 2% after margin, pass — noise is too high relative to margin cost
5. Size at Quarter-Kelly or half the Kelly fraction at most

Log every bet with these inputs. After 300 bets, compute average CLV. That number is your real performance metric.

BetEdge's Kelly calculator computes Quarter-Kelly stake recommendations for every published pick and lets you override with your preferred fraction in the Bankroll Tracker. Explore the compounded equity curve in the What-If Simulator.