Chapter 14 Key Takeaways: Advanced Bankroll and Staking Strategies

Key Concepts

  1. The Kelly criterion maximizes long-term geometric growth by betting $f^* = (pb - q)/b$. This formula emerges from optimizing the expected logarithm of wealth. It inherently avoids ruin (never recommends betting everything), scales with bankroll (proportional betting), and produces zero recommendations for non-positive-EV bets. The derivation rests on the law of large numbers applied to the product of multiplicative returns.

  2. Fractional Kelly is essential in practice. Full Kelly assumes perfect knowledge of probabilities, which bettors never have. Quarter-Kelly (25% of the full Kelly fraction) captures approximately 75% of the theoretical growth rate while reducing variance by 75% and making the strategy robust to estimation errors. Most professional bettors use Kelly fractions between 0.20 and 0.33.

  3. Bet portfolios benefit from the same diversification principles as financial portfolios. The Markowitz mean-variance framework adapts naturally to sports betting. Allocating across independent bets improves the Sharpe ratio by $\sqrt{n}$. Correlated bets (e.g., same-game props) reduce but do not eliminate the diversification benefit. The optimal number of simultaneous bets is typically 5-15.

  4. Covariance between bets must be estimated and accounted for. Same-sport bets, same-game bets, and bets influenced by common factors (weather, sharp action) are positively correlated. Ignoring these correlations leads to over-allocation and underestimation of portfolio risk.

  5. Correlated parlays can be positive EV when books misprice the correlation. If two positively correlated outcomes are priced as independent in a parlay, the true joint probability exceeds the implied joint probability, creating extra value. Same-game parlays are the primary market where this occurs.

  6. Drawdowns are inevitable and quantifiable. The expected maximum drawdown grows logarithmically with the number of bets. A bettor with a 2-3% edge at quarter-Kelly should expect maximum drawdowns of 15-25% over a 500-bet season. Understanding this in advance is critical for psychological preparedness.

  7. Recovery time grows rapidly with drawdown depth. Recovering from a 10% drawdown takes roughly 100-150 bets at quarter-Kelly with a typical edge. A 30% drawdown takes roughly 400-600 bets. A 50% drawdown can take over 1,000 bets. Prevention is far more efficient than recovery.

  8. Pre-committed drawdown policies prevent emotional decision-making. A tiered policy (e.g., reduce bet size at 15%, review model at 25%, pause at 40%) removes the burden of real-time decision-making during the psychological stress of a drawdown. Write the plan when thinking clearly; execute it when you are not.

  9. Multi-account management is a practical necessity for serious bettors. Allocating bankroll across sportsbooks should consider juice efficiency, bet limits, restriction risk, promotional value, and withdrawal speed. Threshold-based rebalancing (not fixed-schedule) minimizes transaction costs while maintaining adequate allocation.

  10. Seasonal bankroll shifts are required because the US sports calendar creates natural cycles. October-November offers peak diversification across five major sports. July is the leanest month. Bankroll allocation should flex with the available opportunities, deploying more during high-diversification periods and building reserves during low-opportunity months.

Key Formulas

Formula Expression Usage
Kelly Criterion $f^* = (pb - q) / b$ Optimal bet fraction for single bets
Growth Rate at Kelly $G(f^*) = p\log(1+bf^*) + q\log(1-f^*)$ Theoretical long-run growth rate
Portfolio Expected Return $\mu_P = \sum_i f_i \mu_i$ Weighted expected return of bet portfolio
Portfolio Variance $\sigma_P^2 = \sum_i \sum_j f_i f_j \sigma_{ij}$ Risk of bet portfolio including covariance
Sharpe Ratio $S = \mu_P / \sigma_P$ Risk-adjusted return measure
Diversification Benefit Sharpe improves by $\sqrt{n}$ for $n$ independent bets Quantifies the value of adding bets
Joint Probability (correlated) $P(A \cap B) = p_A p_B + \rho\sqrt{p_Aq_Ap_Bq_B}$ True parlay hit rate with correlation
Recovery Time (approx.) $n \approx -\log(1-d) / G(f)$ Bets to recover from drawdown $d$
Variance Drain Log return $\approx \mu - \sigma^2/2$ Growth reduction from volatility

Quick-Reference Decision Framework

Step 1 --- Estimate Your Edge. Calculate $\mu = pb - q$ for each available bet. If $\mu \leq 0$, do not bet.

Step 2 --- Calculate the Kelly Fraction. Apply $f^* = \mu / b$ (simplified for decimal net odds). Multiply by your chosen Kelly multiplier (0.20 to 0.33 recommended).

Step 3 --- Construct the Portfolio. Estimate correlations between simultaneous bets. Apply mean-variance optimization with your risk aversion parameter, or simply scale down proportionally if total allocation exceeds your maximum (typically 15-20% of bankroll).

Step 4 --- Check Drawdown Status. Before placing bets, check your current drawdown level against your drawdown policy. Adjust bet sizing if any threshold is breached.

Step 5 --- Execute Across Best Lines. Place each bet at the sportsbook offering the best price. Track which accounts absorb each bet for rebalancing purposes.

Step 6 --- Record and Review. Log every bet with model probability, odds, book, and result. Weekly: review CLV and edge estimates. Monthly: rebalance accounts if thresholds are exceeded.

The core principle: Bankroll management is not a secondary concern -- it is the bridge between having a mathematical edge and actually realizing a profit. The best model in the world is useless without the discipline and framework to survive the inevitable drawdowns.

Ready for Chapter 15? Self-Assessment Checklist

Before moving on to Chapter 15 ("Modeling the NFL"), confirm that you can do the following:

  • [ ] Derive the Kelly criterion from the expected log wealth maximization problem
  • [ ] Calculate Kelly fractions for American odds, decimal odds, and multi-outcome bets
  • [ ] Explain why fractional Kelly is preferred and quantify the trade-off between growth and variance reduction
  • [ ] Apply Markowitz mean-variance optimization to a portfolio of sports bets
  • [ ] Estimate and incorporate covariance between correlated bets (e.g., same-game, same-sport)
  • [ ] Evaluate whether a parlay has positive expected value given leg correlations
  • [ ] Simulate maximum drawdown distributions and calculate recovery times
  • [ ] Design a three-tier drawdown management policy
  • [ ] Allocate bankroll across multiple sportsbook accounts using a scoring framework
  • [ ] Create a seasonal bankroll plan that accounts for the US sports calendar
  • [ ] Build and run a Monte Carlo simulation to project bankroll outcomes

If you can check every box with confidence, you are well prepared for Chapter 15. If any items feel uncertain, revisit the relevant sections of Chapter 14 or work through the corresponding exercises before proceeding.