Chapter 4 Exercises: Bankroll Management Fundamentals

Part A: Conceptual Questions

A.1 Explain the Kelly Criterion in your own words. Why does it maximize the long-run geometric growth rate of a bankroll rather than the arithmetic expected value? What is the fundamental insight that distinguishes Kelly betting from simply maximizing expected profit on each bet?

A.2 A bettor argues: "If I have a 5% edge on every bet, I should bet as much as possible to maximize my profits." Explain why this reasoning is flawed. Use the concept of "gambler's ruin" and the difference between additive and multiplicative processes to support your answer. What happens to a bettor who consistently overbets relative to their edge?

A.3 Discuss the philosophical difference between "unit sizing" (flat betting a fixed percentage) and "proportional sizing" (Kelly-based staking). Under what circumstances might a sophisticated bettor prefer flat betting despite knowing the Kelly Criterion? Consider factors such as edge estimation uncertainty, psychological comfort, and practical constraints.

A.4 The concept of "risk of ruin" assumes a bettor will continue betting indefinitely. In practice, bettors have finite time horizons. How does a finite time horizon change the risk-of-ruin calculation? Does it increase or decrease the practical risk a bettor faces? Explain your reasoning with reference to both the mathematical formula and real-world betting behavior.

A.5 Explain why fractional Kelly (e.g., half Kelly) is widely recommended by professional bettors despite sacrificing expected growth rate. Address the following in your answer: (a) the shape of the growth rate curve near the Kelly optimal point, (b) the impact of edge estimation errors, (c) the psychological tolerance for drawdowns, and (d) the asymmetry between the cost of overbetting vs. underbetting.

A.6 Two bettors each have a $10,000 bankroll. Bettor A bets 1 unit ($100) on every wager regardless of perceived edge. Bettor B sizes each bet according to the Kelly Criterion based on their estimated edge. Assuming both bettors have the same average edge of 3% and make 1,000 bets per year, discuss the likely differences in their outcomes after 5 years. Consider variance, drawdowns, final bankroll, and the practical challenges each bettor faces.

A.7 A bettor discovers that the Kelly Criterion recommends betting 8% of their bankroll on a single NFL game. They feel uncomfortable risking $800 of their $10,000 bankroll on one game. Is this discomfort rational? Discuss the tension between mathematically optimal behavior and psychological sustainability in bankroll management. What practical steps can the bettor take to resolve this tension?

A.8 Explain the concept of "bet correlation" and why it matters for bankroll management. If a bettor places five bets on different NFL games in the same week, are these bets truly independent? How should correlated bets affect (a) individual bet sizing, (b) total exposure limits, and (c) risk-of-ruin calculations? Provide specific examples of bet types that are likely to be correlated.

Part B: Calculation Problems

B.1 A bettor estimates a 55% probability of winning a bet offered at American odds of +100 (even money, decimal 2.00).

(a) Calculate the full Kelly fraction for this bet. (b) Calculate the half Kelly and quarter Kelly fractions. (c) If the bettor's bankroll is $5,000, what is the recommended bet size under each strategy? (d) Calculate the expected growth rate (log growth) per bet under each strategy.

B.2 A bettor has identified a bet with the following parameters: - True probability of winning: 62% - Decimal odds offered: 1.80 (implied probability 55.6%) - Current bankroll: $20,000

(a) Calculate the Kelly fraction. (b) What is the expected edge (as a percentage)? (c) Calculate the optimal bet size in dollars. (d) If the bettor uses 40% Kelly (f* = 0.40 * Kelly), what is the bet size? (e) Compare the expected log growth rate for full Kelly vs. 40% Kelly.

B.3 Calculate the exact risk of ruin for each of the following scenarios using the formula R = ((1-p)/p)^(B/u), where p is win probability, B is bankroll, and u is unit size (assuming even-money bets):

(a) p = 0.52, bankroll = 100 units (b) p = 0.52, bankroll = 200 units (c) p = 0.55, bankroll = 100 units (d) p = 0.55, bankroll = 50 units (e) p = 0.60, bankroll = 100 units

Compare the results and explain what they reveal about the relationship between edge, bankroll size, and ruin probability.

B.4 A bettor makes even-money bets with a 53% win rate. Their current bankroll is $8,000 and they bet $200 per game (2.5% of bankroll).

(a) Calculate the risk of ruin for this bettor. (b) How large would their bankroll need to be (keeping the $200 bet size) to reduce risk of ruin below 1%? (c) Alternatively, if they keep the $8,000 bankroll, what maximum bet size gives a risk of ruin below 1%? (d) If they switch to Kelly staking, what would their initial bet size be? How does this compare to their current approach?

B.5 A bettor has three simultaneous betting opportunities:

(a) Calculate the individual Kelly fraction for each bet. (b) If all three bets are placed simultaneously and are independent, should the bettor simply add the Kelly fractions? Why or why not? (c) Calculate the total Kelly exposure across all three bets. If this exceeds a comfortable threshold (say 15% of bankroll), how should the bettor adjust? (d) Rank the three bets by Kelly fraction. Does this ranking match the ranking by edge percentage? Explain any discrepancies.

B.6 A bettor starts with a $10,000 bankroll and uses Kelly staking on even-money bets where their true win probability is 54%.

(a) Calculate the Kelly fraction. (b) After winning their first bet, what is their new bankroll and new bet size? (c) After losing their second bet (from the bankroll in part b), what is their new bankroll and bet size? (d) Trace the bankroll through the sequence: W, W, L, W, L, L, W, W, W, L (where W=win, L=loss). (e) Compare the final bankroll to what it would have been under flat $400 betting (4% of initial bankroll) with the same sequence of outcomes.

B.7 Fractional Kelly comparison problem. A bettor has an edge of 4% on even-money bets and a starting bankroll of $25,000.

(a) Calculate the expected log growth rate per bet for Kelly fractions of: 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, and 2.00 times the optimal Kelly fraction. (b) At what multiple of Kelly does the expected growth rate become zero? (c) At what multiple of Kelly does the expected growth rate become negative (i.e., the bettor expects to lose money in the long run despite having a positive edge)? (d) Plot or sketch the growth rate as a function of Kelly multiple. What is the shape of this curve and what does it tell us about the risks of overbetting?

Part C: Programming Exercises

C.1 — Kelly Criterion Calculator

Build a comprehensive Kelly Criterion calculator in Python that handles the following:

1. Even-money bets: Given win probability p, compute the Kelly fraction f* = 2p - 1.
2. General odds: Given win probability p and decimal odds d, compute f = (pd - 1) / (d - 1).
3. American odds input: Convert American odds to decimal odds, then compute Kelly.
4. Fractional Kelly: Accept a Kelly fraction multiplier (e.g., 0.5 for half Kelly) and compute the adjusted bet size.
5. Multiple simultaneous bets: Given a portfolio of bets with probabilities and odds, compute individual Kelly fractions and total exposure.
6. Edge sensitivity analysis: For a given bet, show how the Kelly fraction changes as the estimated probability varies by +/- 1%, 2%, and 5%.

Include input validation (probabilities in [0,1], positive odds, etc.) and clear output formatting. Demonstrate your calculator with at least five different betting scenarios.

C.2 — Bankroll Simulator

Write a Python class BankrollSimulator that simulates bankroll evolution over a series of bets. Your simulator should:

1. Accept parameters: initial bankroll, win probability, odds, number of bets, and staking strategy.
2. Implement at least four staking strategies: - Flat betting (fixed dollar amount) - Fixed percentage of current bankroll - Full Kelly - Fractional Kelly (user-specified fraction)
3. Run N Monte Carlo simulations (default N=1000) for each strategy.
4. Track and report: final bankroll distribution, maximum drawdown distribution, risk of ruin (percentage of simulations where bankroll dropped below a threshold), median and mean final bankroll.
5. Generate plots comparing strategies: bankroll paths (overlay 20 sample paths per strategy), distribution of final bankrolls (histogram or box plot), drawdown analysis.

Test your simulator with: p=0.54, decimal odds=2.00, 500 bets, initial bankroll=$10,000.

C.3 — Risk of Ruin Monte Carlo

Implement a Monte Carlo risk-of-ruin analyzer that:

1. Simulates a bettor's bankroll path over a specified number of bets.
2. Tracks whether the bankroll ever drops below a ruin threshold (default: 0% of initial bankroll, i.e., going broke; also support custom thresholds like 50% or 25% of initial).
3. Runs at least 10,000 simulations per parameter set.
4. Compares Monte Carlo estimates to the analytical risk-of-ruin formula for even-money bets.
5. Generates a heatmap showing risk of ruin as a function of (a) edge (x-axis: 1% to 10%) and (b) bet size as percentage of bankroll (y-axis: 1% to 20%).
6. Identifies the "safe zone" where risk of ruin is below 5% and the "danger zone" where it exceeds 25%.

Include proper progress reporting for the simulations and visualization of results.

C.4 — Drawdown Analyzer

Create a drawdown analysis tool that:

1. Takes a bankroll history (list of bankroll values over time) and computes: maximum drawdown (peak-to-trough), maximum drawdown duration (time from peak to recovery), all drawdowns exceeding a specified threshold, current drawdown (from most recent peak).
2. Generates a "drawdown chart" showing the bankroll path on top and the drawdown percentage below (like a financial underwater equity curve).
3. Computes expected maximum drawdown statistics from Monte Carlo simulation given: win probability, odds, bet size, and number of bets.
4. Answers the question: "Given my edge and staking approach, what is the probability of experiencing a drawdown of X% or worse over Y bets?"

Demonstrate with a simulated bankroll path of 2,000 bets at 53% win rate, even-money odds, and 2% Kelly.

C.5 — Staking Strategy Comparator

Build a tool that compares the performance of multiple staking strategies head-to-head:

1. Implement at least six staking strategies: - Flat betting (fixed amount) - Percentage of bankroll - Full Kelly - Half Kelly - Quarter Kelly - Proportional to edge (variable Kelly based on per-bet edge estimates)
2. Simulate all strategies on the same sequence of bet outcomes (same random seed) so comparisons are fair.
3. Generate a comprehensive comparison report including: final bankroll (mean, median, 5th percentile, 95th percentile), maximum drawdown statistics, risk of ruin, Sharpe ratio of returns, growth rate (geometric mean return per bet).
4. Create visualizations: overlay bankroll paths, box plot of final bankrolls, risk-return scatter plot (x=volatility, y=growth rate) for each strategy.
5. Allow the user to specify a "variable edge" scenario where bet edges are drawn from a distribution (e.g., uniform between 1% and 8%) to simulate realistic betting.

Part D: Analysis Questions

D.1 You are given the following data from a bettor's last 200 even-money bets: - Wins: 112, Losses: 88 - Starting bankroll: $5,000 - Ending bankroll: $7,400 (using flat $250 bets)

(a) Estimate the bettor's win rate and edge. (b) Construct a 95% confidence interval for the true win rate. (c) Based on the confidence interval, calculate the range of Kelly fractions that could be appropriate. (d) What Kelly fraction would you recommend given the uncertainty? Justify your choice. (e) If the bettor had used Kelly staking from the start (based on the point estimate of 56% win rate), simulate what the bankroll path might have looked like. How does it compare to the flat betting result?

D.2 Analyze the following three staking strategies applied to the same sequence of 1,000 bets (55% win rate, even-money odds, $10,000 starting bankroll):

(a) Why is the mean final bankroll for Full Kelly so much higher than the median? (b) Which strategy would you recommend for a recreational bettor with low risk tolerance? For a professional with high risk tolerance? Justify both answers. (c) The Full Kelly strategy has a 0.3% ruin rate despite having a 10% edge. Explain how this is possible. (d) Calculate the coefficient of variation (CV = Std Dev / Mean) for each strategy. What does this metric tell us about the risk-adjusted performance? (e) If you could only choose one metric to evaluate a staking strategy, which would you choose and why?

D.3 A professional sports bettor shares their results from one NFL season (September through February):

• Total bets placed: 340
• Record: 184-156 (54.1% win rate)
• Average odds: -108 (decimal ~1.926)
• Starting bankroll: $50,000
• Maximum bankroll during season: $68,500
• Minimum bankroll during season: $41,200
• Ending bankroll: $62,300
• Staking method: 2% of current bankroll per bet

Analyze this bettor's season: (a) Calculate their ROI and profit. (b) Estimate their true edge at the average odds. (c) Was their staking strategy appropriate? Calculate what the Kelly fraction would have recommended. (d) Calculate their maximum drawdown in both dollar and percentage terms. (e) Using their estimated edge and staking approach, what is the probability they experience a losing season (ending with less than they started)? (f) What adjustments, if any, would you recommend for next season?

D.4 Compare the bankroll management approaches suitable for three different bettor profiles:

• Profile A: Recreational bettor, $2,000 bankroll, bets 5 games per week during NFL season, estimated edge of 2-3%.
• Profile B: Semi-professional bettor, $25,000 bankroll, bets 15-20 games per week across multiple sports, estimated edge of 3-5%.
• Profile C: Professional bettor, $200,000 bankroll, bets 40+ games per week across all sports, estimated edge of 2-4% with high volume.

For each profile, recommend: (a) A staking strategy and bet sizing approach. (b) A ruin threshold and acceptable risk level. (c) Guidelines for when to increase or decrease bet sizes. (d) A maximum single-bet and maximum daily exposure limit. (e) A plan for handling losing streaks of 10+, 20+, and 30+ bets.

D.5 You have access to the following Monte Carlo simulation results for a bettor with a 3% edge on even-money bets over 1,000 bets:

(a) At what bet size does the median final bankroll peak? How does this compare to the full Kelly fraction? (b) Notice that at 15% and 20% bet sizes, the median final bankroll is below the starting bankroll despite a 3% edge. Explain this phenomenon. (c) Identify the "sweet spot" range of bet sizes that balances growth potential with acceptable risk. Justify your choice. (d) A bettor argues that 10% bet sizing is acceptable because the risk of ruin is "only" 9.1%. Construct a counterargument. (e) Using this data, explain the concept of "over-betting" to someone who has never heard of the Kelly Criterion.

Part E: Research Questions

E.1 Read John Larry Kelly Jr.'s original 1956 paper, "A New Interpretation of Information Rate" (Bell System Technical Journal, Vol. 35, pp. 917-926). Write a 500-word summary addressing: (a) What problem was Kelly originally trying to solve? (b) How does information theory connect to gambling? (c) What assumptions does Kelly make, and which of these assumptions are violated in sports betting? (d) How has the Kelly Criterion been adapted for use in financial markets and sports betting since 1956?

E.2 Research the debate between "Kelly bettors" and "fractional Kelly advocates" in both the gambling and finance literature. Find at least three published arguments in favor of fractional Kelly and three arguments for full Kelly. Summarize each argument and evaluate which side has stronger evidence for sports betting specifically. Consider the work of Edward Thorp, William Poundstone, and the academic literature on growth-optimal portfolios.

E.3 Investigate how three different professional sports bettors or betting syndicates approach bankroll management. Sources may include interviews, books, podcasts, or published articles. Compare their approaches on the following dimensions: (a) bet sizing methodology, (b) maximum single-bet exposure, (c) bankroll segregation (separate bankrolls for different sports or bet types), (d) rules for increasing/decreasing bet sizes, and (e) risk-of-ruin tolerance. What common principles emerge across all three approaches?

E.4 The Kelly Criterion was independently discovered by Daniel Bernoulli in 1738 in the context of the St. Petersburg Paradox (he proposed maximizing expected logarithmic utility). Research the connection between Bernoulli's work and Kelly's. Address: (a) How are the two formulations mathematically equivalent? (b) Why did it take over 200 years for Kelly to "rediscover" the result? (c) What role did Claude Shannon play in the development and popularization of the Kelly Criterion? (d) How did Edward Thorp apply Kelly's work to blackjack and later to financial markets?

E.5 Find and review at least three academic papers or book chapters that present empirical evidence on bankroll management in sports betting. For each source: (a) Summarize the key findings. (b) Describe the methodology used (simulation, empirical data, analytical). (c) What staking strategy did the authors find to be optimal or recommend? (d) What limitations or caveats did the authors identify? Compile your findings into a comparative table and identify areas where the literature agrees and disagrees.

These exercises are designed to build progressively from conceptual understanding through mathematical application to practical implementation. Students are encouraged to complete Parts A and B before attempting the programming and analysis sections.