The Kelly Criterion is a mathematically grounded way to size bets when you know (or estimate) your edge. Applied sensibly, it helps grow a bankroll while controlling risk — but it requires honest estimates of win probability and payout, and extra care with crypto (price swings, fees, liquidity). This article explains the Kelly formula, shows clear step-by-step calculations, gives conservative fractional-Kelly practical rules, and covers crypto-specific adjustments so you can use Kelly responsibly.
Important disclaimer: This is educational content, not financial or legal advice. Betting involves risk. If you are unsure, consult a licensed professional. Never bet money you cannot afford to lose.
1) What the Kelly Criterion does (plain)
Kelly gives the fraction of your bankroll to wager on a repeated bet to maximize the long-term growth rate of your money if your probability and payout estimates are correct. It balances reward and risk: bet more when your edge is bigger, bet nothing when you have no edge.
The classical Kelly formula for a single binary bet with decimal odds O and win probability p is:
Where f∗ is the fraction of bankroll to stake. If f∗ is negative → don’t bet (negative expected value).
2) How to estimate the inputs (p and O) for crypto bets
Kelly is only as good as your inputs. For casino/sports/crypto markets:
- O (decimal odds / payout multiple): use the net payout multiple you’ll receive on a win. For a double-up bet that returns 2.00 (your stake + equal win), O=2.00 and b=1.00. For an auto-cashout at 1.5× you receive 1.5 times stake on success → O=1.5, b=0.5.
- p (win probability): estimate using reliable data:
- For a matched-betting or trading edge, use historical hit rates from many rounds (empirical).
- For sports: combine models, stats and market-implied probabilities (but beware of overfitting).
- For casino promotions: estimate the effective p by factoring house edge and bonus conditions.
- Adjust for fees & slippage: subtract transaction fees, exchange spreads, bookmaker margins from payouts before computing O. Use worst-case settlement costs for conservative sizing.
Rule: If p or O are uncertain, use fractional Kelly (see section 5).
3) Step-by-step Kelly calculation — worked example (digit-by-digit)
Scenario A — a positive-edge bet (illustrative):
You believe a crypto sports market has positive edge: your estimated probability to win p=0.55 and decimal payout O=2.00 (even money).
Compute b=O−1.
- O−1=2.00−1.00.
- Step: subtract 1.00 from 2.00.
- Result: b=1.00.
Compute q=1−p.
- 1.00−0.55.
- Step: subtract 0.55 from 1.00.
- Result: q=0.45.
Compute numerator bp−q.
- bp=1.00×0.55.
- Step: multiply 1.00 by 0.55.
- Result: 0.55.
- bp−q=0.55−0.45.
- Step: subtract 0.45 from 0.55.
- Result: 0.10.
Compute f∗=(bp−q)/b.
- (bp−q)/b=0.10/1.00.
- Step: divide 0.10 by 1.00.
- Result: f∗= 0.10 → 10% of bankroll.
Interpretation: Full Kelly says stake 10% of your bankroll each independent repetition. That’s aggressive in practice; most people use a fraction (e.g., 25%–50% of Kelly) to reduce volatility.
4) Example showing why negative Kelly means “don’t bet”
Scenario B — casino-style negative EV (illustrative):
A dice-style bet pays 1.95× on a success probability of p=0.51 (house edge in payout).
Compute b=O−1=1.95−1.00.
- 1.95−1.00=0.95.
- So b=0.95.
Compute q=1−p=1.00−0.51.
- 1.00−0.51=0.49.
- So q=0.49.
Compute bp=0.95×0.51.
- Multiply: 0.95×0.51.
- 0.95 × 0.51 = 0.4845. (Compute digit by digit: 95×51=4845 then scale by 4 decimal places → 0.4845.)
Compute numerator bp−q=0.4845−0.49.
- 0.4845−0.49=−0.0055.
- Result is negative.
Compute f∗=−0.0055/0.95.
- Division yields a negative number ≈ −0.005789…(approx).
- Negative Kelly means no stake; the bet has negative expected value even if your p > 0.5 because payout is worse than fair.
Practical rule: If you get a negative Kelly, don’t place the bet — look for better odds or a different market.
5) Fractional Kelly — practical risk management
Full Kelly maximizes long-term growth but has very large volatility and drawdowns. Most practitioners use fractional Kelly:
Common choices:
- Half-Kelly (k = 0.5): reduces volatility substantially while retaining much growth.
- Quarter-Kelly (k = 0.25): safer; good for noisy p estimates.
- Rule of thumb: if your estimate of p is uncertain, prefer k≤0.25.
Worked fractional example (continuing Scenario A):
Full f∗=0.10(10%). Half-Kelly = 0.5×0.10=0.05 → stake 5% of bankroll.
Digit-by-digit:
- 0.5×0.10=0.05.
- Multiply 0.10 by 0.5.
6) Applying Kelly with cryptocurrency-specific issues
A — convert to a stable, consistent bankroll unit
Cryptocurrency prices move independently of your bets. Two ways to handle this:
- Use stablecoins (USDT/USDC) for play bankroll: compute Kelly stakes in stablecoin units to avoid BTC/ETH price noise during sessions.
- If you want BTC exposure: size the Kelly fraction on the fiat-equivalent bankroll and stake the corresponding BTC amount at time of bet. Recalculate stake if BTC price moves significantly before execution.
Practical step: choose a “bankroll currency” (USD stablecoin or BTC) and stick to it for sizing clarity.
B — include fees, gas & slippage in O
Before computing O reduce the payout by:
- Exchange/deposit/withdrawal fees,
- On-chain gas (if it affects the net you realize),
- Bookmaker commission or spread.
For micro-scalps the fee drag can turn a positive Kelly into negative — always model net payout.
C — bet sizing limits & house constraints
Many sportsbooks enforce bet limits. If full Kelly suggests a stake exceeding book limits, time-slice the desired exposure or use different providers.
D — tax & legal considerations
Gains on crypto bets may be taxable. Factor expected tax liability into your sizing if material — e.g., adjust the effective payout downward.
7) Practical workflow — from idea to executed Kelly bet
- Estimate p empirically: base this on historical data or a model (e.g., 1,000+ prior analog events if possible).
- Compute net payout O after fees.
- Compute full Kelly f∗ using formula in section 3.
- Choose fraction k (e.g., 0.25 or 0.5) based on confidence in p.
- Cap stake by absolute max percent (e.g., don’t exceed 5% of bankroll even if Kelly is higher).
- Place bet in your chosen bankroll currency.
- Log result, update bankroll, recompute next stake using new bankroll and updated p if you re-estimate.
Automation note: bots can auto-size by Kelly but only if your data feed and execution are reliable. Test extensively in dry-runs.
8) Example full practical plan (numeric)
- Bankroll (stablecoin): 1,000 USDT.
- Bet opportunity: model gives p=0.60.
- Market payout (net after fees): O=1.8 (you receive 1.8× stake on win).
- Compute b=O−1=0.8.
- 1.8−1.0=0.8.
- Compute q=1−p=1.0−0.60=0.40.
- Compute bp=0.8×0.60.
- 0.8×0.6=0.48.
- Compute numerator bp−q=0.48−0.40=0.08.
- Compute full Kelly f∗=0.08/0.8=0.10 → 10% full Kelly.
- Divide 0.08 by 0.8 = 0.10.
Conservative application: choose quarter-Kelly k=0.25.
- Stake fraction = 0.25×0.10=0.025 → 2.5% of bankroll.
- Numeric stake = 1,000×0.025
- 1,000×0.025= 25 USDT stake.
So place a 25 USDT bet (adjust if min/max limits or if you prefer half-Kelly).
9) Risk controls & sanity checks
- Cap maximum percent (e.g., never bet > 5% of bankroll regardless of Kelly).
- Use fractional Kelly for noisy or model-derived p.
- Recompute p regularly — stale edges evaporate quickly in efficient markets.
- Stress test worst-case scenarios: simulate sequences of losses to estimate drawdown risk.
- Keep emergency reserve outside Kelly bankroll (cold storage) to avoid whole-account ruin.
10) When Kelly fails — warning signs
- Your observed win rate drifts far from the model’s p.
- Market odds compress (your edge shrinks).
- Fees, limits or latency make theoretical staking impossible.
- You’re emotionally unable to follow stake changes (Kelly requires discipline).
If any of the above happens, stop and reassess. Reduce k or pause betting until you verify inputs.
11) Simple Kelly calculator (manual formula)
You can compute Kelly in a spreadsheet with these cells:
- A1 = Bankroll (numeric)
- A2 = Decimal payout OO (numeric)
- A3 = Estimated probability pp (decimal)
- A4 = b=A2−1
- A5 = q=1−A3
- A6 = num=A4×A3−A5
- A7 = f∗= num / A4 (if A4 = 0, no bet)
- A8 = Fraction k (e.g., 0.25)
- A9 = Stake = A1×A7×A8 (cap as needed)
12) Quick FAQ
Q: If the bet is casino negative EV, can Kelly find a stake anyway?
A: No. Kelly returns a negative value. Negative Kelly means don’t bet unless you can improve payout or estimate p differently with trustworthy data.
Q: How conservative is 1/4 Kelly vs half-Kelly?
A: 1/4 Kelly roughly halves the volatility of half-Kelly and reduces drawdowns significantly while sacrificing some growth. It’s often a good balance for noisy estimates.
Q: Do I recompute Kelly after each round?
A: Yes — Kelly is dynamic: update bankroll and re-estimate p and O if you have new information. Use percent staking (based on current bankroll) so your sizes evolve.
13) Final practical checklist — before you stake
- Choose a bankroll currency (stablecoin or BTC) and stick to it.
- Estimate p from reliable historical data or a validated model.
- Compute net payout O after all fees.
- Calculate full Kelly f∗f^* and then choose fraction k (e.g., 0.25).
- Apply a hard cap on % of bankroll (e.g., 5%).
- Log every bet and re-estimate p monthly or after significant new data.
- Use simulations (Monte Carlo) if stakes are large to understand drawdown dynamics.
