Chapter 13 Exercises: Finding and Quantifying Your Edge
These exercises progress from basic EV calculations through Kelly sizing, edge estimation, and portfolio-level thinking. Solutions are available in code/exercise-solutions.py.
Section A: Expected Value Calculations (Exercises 1-8)
Exercise 1: Basic EV Computation
A prediction market asks "Will the central bank raise interest rates at the next meeting?" and the YES contract is priced at $0.42. You estimate the true probability of a rate hike at 55%.
(a) Compute the EV of buying one YES contract. (b) Compute the EV of buying one NO contract. (c) Which side should you trade? (d) What is the EV as a percentage return on capital invested?
Exercise 2: Break-Even Probability
A YES contract for "Will Company X report earnings above $2.50 per share?" is trading at $0.68.
(a) What is the break-even probability -- the minimum true probability at which buying YES has non-negative EV? (b) If you estimate the true probability at 72%, what is the EV per contract? (c) If the platform charges a 2% fee on winnings, what is the adjusted break-even probability?
Exercise 3: Multi-Outcome EV
A prediction market offers contracts on which party will win an election: - Party A: $0.45 - Party B: $0.35 - Party C: $0.15 - Other: $0.05
Your probability estimates are: Party A: 50%, Party B: 30%, Party C: 12%, Other: 8%.
(a) Compute the EV for buying each contract. (b) Compute the EV for selling (shorting) each contract. (c) Which trades have positive EV? (d) If you could only make one trade, which would you choose and why?
Exercise 4: EV Scanner
You are scanning five markets with the following data:
| Market | Your Probability | Market Price (YES) |
|---|---|---|
| M1 | 0.75 | 0.62 |
| M2 | 0.30 | 0.40 |
| M3 | 0.55 | 0.53 |
| M4 | 0.10 | 0.18 |
| M5 | 0.85 | 0.80 |
(a) For each market, compute the EV of the best trade (YES or NO). (b) Rank the markets by EV percentage return on capital. (c) Which markets would you trade if your minimum EV threshold is 5%?
Exercise 5: EV with Transaction Costs
You want to buy YES contracts priced at $0.55. You estimate the true probability at 65%. The platform charges: - Entry fee: 1% of position value - Exit fee on winning trades: 2% of profit - No fee on losing trades
(a) Compute the gross EV per contract (ignoring fees). (b) Compute the net EV per contract after fees. (c) What minimum edge (in percentage points) do you need to overcome the fees?
Exercise 6: Conditional EV
Market A asks "Will Event X happen?" at $0.60. Market B asks "If Event X happens, will Event Y also happen?" at $0.50.
You believe: P(X) = 0.65, P(Y|X) = 0.70, P(Y|not X) = 0.20.
(a) What is the EV of buying YES on Market A? (b) What is the EV of buying YES on Market B? (c) If both events resolve YES, what is your combined profit from buying one contract in each market? What is the probability of this occurring?
Exercise 7: Time Value of EV
Two markets offer the same EV per contract: - Market A: EV = $0.05 per contract, resolves in 1 week - Market B: EV = $0.05 per contract, resolves in 6 months
(a) Compute the annualized EV return for each market, assuming the YES price is $0.50 for both. (b) Which market is more attractive? Why? (c) What other factors might make Market B preferable despite the lower annualized return?
Exercise 8: EV Under Parameter Uncertainty
You are estimating the probability that a candidate wins an election. Your base rate analysis suggests 55%, your polling model suggests 62%, and your qualitative assessment suggests 58%.
(a) If you use a simple average as your estimate, what is the EV of buying YES at $0.54? (b) If you use a weighted average (40% weight on polling model, 30% on each of the other two), what is the EV? (c) If your 80% confidence interval for the true probability is [52%, 66%], and the market is at $0.54, what is the EV using the lower bound? The upper bound? (d) Should you trade? Justify your answer.
Section B: Kelly Criterion (Exercises 9-16)
Exercise 9: Basic Kelly Calculation
You find a prediction market where you estimate the true probability at 70% and the YES price is $0.55.
(a) Compute the full Kelly fraction for buying YES. (b) If your bankroll is $5,000, how many dollars should you allocate at full Kelly? (c) How many YES contracts can you buy? (d) Compute the half-Kelly and quarter-Kelly allocations.
Exercise 10: Kelly for NO Bets
A market prices YES at $0.75, but you estimate the true probability at only 55%.
(a) Compute the full Kelly fraction for buying NO. (b) If your bankroll is $8,000, what is the half-Kelly allocation? (c) What happens to the Kelly fraction as your estimate decreases from 55% to 50% to 45%?
Exercise 11: Kelly with Small Edge
You estimate a true probability of 52% for an event priced at 48%.
(a) Compute the full Kelly fraction. (b) With a $10,000 bankroll, how much should you bet at full Kelly? At half Kelly? (c) If you make this exact bet 100 times (each time with the same edge and bankroll fraction), what is the expected total profit at full Kelly? What is the standard deviation? (d) Explain why small-edge situations require careful consideration of transaction costs.
Exercise 12: Over-Betting Danger
Suppose the true probability of an event is 60% and the market price is $0.45.
(a) Compute the full Kelly fraction. (b) Compute the expected growth rate (log-growth) at full Kelly. (c) Compute the expected growth rate at 2x Kelly. (d) Compute the expected growth rate at 3x Kelly. (e) What fraction of your bankroll would you need to bet for the expected growth rate to be zero?
Exercise 13: Kelly with Multiple Bets
You have a $20,000 bankroll and find three independent positive EV opportunities:
| Market | Your Prob | Market Price | Side |
|---|---|---|---|
| M1 | 0.65 | 0.50 | YES |
| M2 | 0.40 | 0.55 | NO |
| M3 | 0.80 | 0.70 | YES |
(a) Compute the full Kelly fraction for each bet independently. (b) Compute the half-Kelly dollar allocation for each bet. (c) What is the total allocation as a fraction of your bankroll? Is this reasonable? (d) If you impose a maximum total allocation of 40% of bankroll, how would you allocate among the three bets?
Exercise 14: Kelly with Estimation Uncertainty
You estimate the probability of an event at 68% with an 80% confidence interval of [58%, 78%]. The market price is $0.55.
(a) Compute the Kelly fraction using your point estimate (68%). (b) Compute the Kelly fraction using the lower bound (58%). (c) Compute the Kelly fraction using the upper bound (78%). (d) What Kelly fraction would you use in practice? Justify your answer. (e) At what confidence level (lower bound) does the Kelly fraction become zero?
Exercise 15: Ruin Probability
You use full Kelly betting on a series of independent bets, each with probability 60% and market price $0.45.
(a) What is the probability that your bankroll ever drops to 50% of its starting value? (b) What is the probability for half Kelly? (c) What is the probability for quarter Kelly? (d) If you switch from full Kelly to half Kelly, how much long-run growth rate do you sacrifice (expressed as a percentage of full Kelly's growth rate)?
Exercise 16: Kelly Criterion Simulation
Using the code in code/example-02-kelly-criterion.py or your own implementation:
(a) Simulate 1,000 bets with true probability 60%, market price $0.45, starting bankroll $10,000, using full Kelly. Plot 10 sample paths. Report the median and mean final bankroll. (b) Repeat with half Kelly. Compare the median, mean, and 5th percentile final bankrolls. (c) Repeat with quarter Kelly. (d) Find the Kelly fraction that minimizes the probability of the bankroll ever dropping below $5,000 (50% drawdown) while maintaining at least 50% of full Kelly's median growth.
Section C: Edge Estimation and Decomposition (Exercises 17-24)
Exercise 17: Base Rate Analysis
You are estimating the probability that a newly launched tech product will still be on the market one year after launch.
(a) What reference class would you use? Suggest at least three possible reference classes from broad to narrow. (b) Suppose historical data shows: All consumer products: 45% survive one year. Tech products: 35%. Products in the same category launched in the last 5 years: 40%. How would you combine these estimates? (c) The product has received unusually positive reviews and strong pre-orders. How would you adjust your estimate, and by how much?
Exercise 18: Calibration Assessment
Over your last 100 trades, your probability estimates and outcomes were:
| Your Estimate Range | Number of Trades | Actual YES Outcomes |
|---|---|---|
| 0.10 - 0.20 | 15 | 1 |
| 0.20 - 0.30 | 10 | 4 |
| 0.30 - 0.40 | 12 | 5 |
| 0.40 - 0.50 | 8 | 4 |
| 0.50 - 0.60 | 18 | 12 |
| 0.60 - 0.70 | 15 | 9 |
| 0.70 - 0.80 | 12 | 10 |
| 0.80 - 0.90 | 10 | 7 |
(a) Compute the actual frequency for each bin. (b) Plot (or describe) a calibration curve. Is it above or below the diagonal? (c) Compute the Brier score for your estimates (use the bin midpoint as your estimate). (d) Are you overconfident or underconfident? In which regions? (e) How would you adjust your future estimates based on this analysis?
Exercise 19: Edge Decomposition
You have been trading political prediction markets for six months. Your records show:
- Average edge (your_prob - market_price, signed by trade direction): +4.2 percentage points
- When you remove your private information sources, your average edge drops to +1.8 percentage points
- When you use a naive model (just the market price) instead of your model, your edge drops to +2.5 percentage points
- Your average entry price is 1.3 percentage points better than the VWAP
(a) Estimate your information edge. (b) Estimate your model edge. (c) Estimate your timing edge. (d) What is the residual edge? (e) Which component should you focus on improving?
Exercise 20: Measuring Realized Edge
You made 50 trades on YES contracts. Of these, 30 events occurred (you won) and 20 did not (you lost). Your average purchase price was $0.55. The average market price at the time of your trades (for the same events) was also $0.55 (you traded at market price).
(a) What is your win rate? (b) What is your average profit per trade? (c) What is your total P&L? (d) Is your edge statistically significant? (Hint: under the null hypothesis that the market price is correct, what is the expected win rate and its standard deviation?)
Exercise 21: Brier Score Comparison
For a set of 20 predictions, you and the market gave the following probability estimates (all for YES), and the actual outcomes are shown:
| # | Your Estimate | Market Price | Outcome |
|---|---|---|---|
| 1 | 0.70 | 0.65 | 1 |
| 2 | 0.30 | 0.35 | 0 |
| 3 | 0.55 | 0.50 | 1 |
| 4 | 0.80 | 0.75 | 1 |
| 5 | 0.20 | 0.25 | 0 |
| 6 | 0.60 | 0.55 | 0 |
| 7 | 0.45 | 0.50 | 1 |
| 8 | 0.90 | 0.85 | 1 |
| 9 | 0.15 | 0.20 | 0 |
| 10 | 0.65 | 0.60 | 1 |
| 11 | 0.40 | 0.45 | 0 |
| 12 | 0.75 | 0.70 | 1 |
| 13 | 0.35 | 0.40 | 1 |
| 14 | 0.85 | 0.80 | 0 |
| 15 | 0.50 | 0.50 | 1 |
| 16 | 0.25 | 0.30 | 0 |
| 17 | 0.70 | 0.65 | 1 |
| 18 | 0.55 | 0.50 | 0 |
| 19 | 0.80 | 0.75 | 1 |
| 20 | 0.10 | 0.15 | 0 |
(a) Compute the Brier score for your estimates. (b) Compute the Brier score for the market. (c) Who is better calibrated? (d) Compute the log-loss for both. Who is better by this metric?
Exercise 22: Edge Decay Analysis
You have been using the same trading strategy for 12 months. Your monthly average edge (in percentage points) has been:
| Month | Avg Edge |
|---|---|
| 1 | 8.5 |
| 2 | 7.2 |
| 3 | 6.8 |
| 4 | 6.1 |
| 5 | 5.5 |
| 6 | 4.9 |
| 7 | 4.2 |
| 8 | 3.8 |
| 9 | 3.1 |
| 10 | 2.7 |
| 11 | 2.3 |
| 12 | 1.9 |
(a) Plot (or describe) the edge decay curve. (b) Fit an exponential decay model: $E(t) = E_0 \cdot e^{-\lambda t}$. Estimate $E_0$ and $\lambda$. (c) At what month will the edge drop below 1 percentage point? (d) If your trading costs (fees + opportunity cost) are 1.5 percentage points per trade, at what month should you stop trading this strategy? (e) What steps might you take to slow the decay?
Exercise 23: Combining Probability Estimates
You are estimating the probability that a new regulation will be implemented by year-end. You have four estimates:
- Base rate from similar regulations: 40%
- Expert panel consensus: 55%
- Your quantitative model: 48%
- Prediction market price: 52%
(a) Compute the simple average. (b) Compute a weighted average with weights: base rate 20%, expert 30%, model 30%, market 20%. (c) Apply extremization with parameter $a = 2.0$ to the simple average. What is the extremized estimate? (d) If the market price is 52% and your extremized estimate is your final answer, what is your edge? (e) Is this edge large enough to trade, given typical transaction costs of 1-2%?
Exercise 24: Confidence Interval Impact on Sizing
For the same market priced at $0.50, consider three scenarios:
- Scenario A: Your estimate is 60% with 80% CI [55%, 65%]
- Scenario B: Your estimate is 60% with 80% CI [45%, 75%]
- Scenario C: Your estimate is 60% with 80% CI [35%, 85%]
(a) Compute the Kelly fraction using the point estimate for all three. (b) Compute the Kelly fraction using the lower bound of the CI for each. (c) Apply half-Kelly to the lower-bound Kelly for each. What are your final position sizes (as fraction of bankroll)? (d) Discuss how estimation uncertainty should affect your trading decision.
Section D: Portfolio and Strategy Problems (Exercises 25-30)
Exercise 25: Portfolio EV
You have a $50,000 bankroll and the following open positions:
| Market | Side | Contracts | Price Paid | Your Prob | Current Price |
|---|---|---|---|---|---|
| M1 | YES | 500 | $0.40 | 0.55 | $0.48 | ||
| M2 | NO | 300 | $0.45 | 0.40 | $0.52 | ||
| M3 | YES | 200 | $0.70 | 0.80 | $0.72 |
(a) Compute the total capital deployed. (b) Compute the unrealized P&L based on current prices. (c) Compute the expected P&L based on your probabilities (if you hold to resolution). (d) If you had to liquidate one position immediately at current prices, which would you close? Why?
Exercise 26: Strategy Comparison
Two traders use different strategies over 200 trades each:
Trader A: Takes large positions on high-edge opportunities. Average edge: 8%, win rate: 62%, average position size: 15% of bankroll, Sharpe ratio: 0.8.
Trader B: Takes many small positions on moderate-edge opportunities. Average edge: 3%, win rate: 58%, average position size: 4% of bankroll, Sharpe ratio: 1.4.
(a) Which trader has higher expected total return? (b) Which trader has a lower risk of ruin? (c) If you had to choose one strategy, which would you prefer and why? (d) How would you combine elements of both strategies?
Exercise 27: Fee Impact Analysis
You are considering two platforms for the same trades:
Platform A: 5% fee on net profits, no per-trade fees, minimum withdrawal $50. Platform B: $0.01 per contract traded (both entry and exit), no fee on profits.
Your typical trade: buy 100 YES contracts at $0.50, hold to resolution.
(a) If the event occurs (you win), what is your total fee on each platform? (b) If the event does not occur (you lose), what is your total fee on each platform? (c) Given a true probability of 55% for your typical trade, what is the net EV per trade on each platform? (d) At what probability level are the two platforms equally expensive?
Exercise 28: Edge Tracking Dashboard
Design (on paper or in pseudocode) an edge tracking dashboard that displays:
(a) A rolling 30-day P&L chart (b) Calibration curve updated after each resolved trade (c) Edge decomposition pie chart (d) Current open positions with real-time EV estimates (e) An alert system that flags when your rolling edge drops below a threshold
Describe the data inputs needed for each component and how often each should be updated.
Exercise 29: Market Regime Analysis
Your trading history shows different performance in different market "regimes":
| Regime | Trades | Win Rate | Avg Edge | Avg P&L/Trade |
|---|---|---|---|---|
| Low volatility | 80 | 62% | +5.2% | +$32 |
| High volatility | 45 | 48% | +1.1% | -$8 |
| Trending | 55 | 67% | +7.8% | +$51 |
| Mean-reverting | 20 | 55% | +3.0% | +$15 |
(a) In which regime are you most profitable? (b) In which regime should you reduce or eliminate trading? (c) If you could predict the regime in advance, how would you adjust your Kelly fraction for each? (d) Compute the overall average P&L per trade across all regimes.
Exercise 30: Comprehensive Trading Plan
Design a complete trading plan for prediction markets, incorporating the concepts from this chapter. Your plan should include:
(a) Market selection criteria: Which markets will you trade and why? (b) Edge estimation process: Step-by-step procedure for estimating your probability for each market. (c) Position sizing rules: Kelly fraction, maximum position size, portfolio limits. (d) Entry and exit criteria: When to enter, when to exit before resolution, when to add to positions. (e) Risk management: Drawdown limits, loss limits, correlation limits. (f) Tracking and review: What will you log, how often will you review, what triggers strategy modification? (g) Edge classification: How will you categorize your edge for each trade? (h) Retirement criteria: Under what conditions will you stop trading altogether?
This exercise has no single right answer. The goal is to produce a coherent, complete plan that you could actually follow.