Chapter 12 Exercises: Line Shopping and Odds Optimization

Instructions: Complete all exercises in the parts assigned by your instructor. Show all work for calculation problems. For programming challenges, include comments explaining your logic and provide sample output.

Part A: Line Shopping Fundamentals (10 exercises, 5 points each)

Exercise A.1 --- The Cost of Not Shopping

You plan to bet the Kansas City Chiefs moneyline in 200 games over an NFL season. Your average bet size is $200. Book A consistently offers KC at -145, while Book B offers KC at -135. Calculate:

(a) The profit per winning bet at each book.

(b) If you win 60% of your KC bets, the total season profit at each book.

(c) The cost of exclusively using Book A instead of Book B across the full season.

(d) Express this cost as a percentage of total amount wagered.

Exercise A.2 --- Odds Comparison Across Formats

Five sportsbooks offer the following lines on the same NBA game (Team A moneyline):

(a) Convert all odds to decimal format.

(b) Convert all odds to implied probability.

(c) Rank the books from best to worst for a bettor backing Team A.

(d) Calculate the implied probability difference between the best and worst book. How significant is this over 500 bets?

Exercise A.3 --- Half-Point Value Quantification

In NFL point spread betting, different books sometimes offer spreads that differ by half a point. For each pair below, calculate the difference in implied probability and the EV difference per $110 bet (assuming both are at -110):

(a) Team A -3 vs. Team A -2.5

(b) Team A -7 vs. Team A -6.5

(c) Team A -10 vs. Team A -9.5

(d) Team A -14 vs. Team A -13.5

Use the historical NFL margin distribution where approximately 15.5% of games land on 3, 9.1% on 7, 3.9% on 10, and 3.6% on 14.

Exercise A.4 --- Multi-Sport Line Shopping

A bettor has accounts at five sportsbooks. On a given day, they identify 10 bets they want to place. For each bet, the best available odds and the worst available odds among their five books are shown:

(a) Calculate the implied probability difference for each bet.

(b) Calculate the average implied probability savings from line shopping.

(c) If this bettor places 10 bets per day, 300 days per year, at $100 per bet, estimate the annual dollar value of line shopping.

(d) Which sport/bet type shows the largest line shopping opportunity? What structural factors might explain this?

Exercise A.5 --- Timing Your Bet

A bettor tracks the closing line for NFL spreads and records when they placed their bet relative to the closing line:

(a) Is there a clear pattern between timing and CLV? What explains this pattern?

(b) If the bettor sizes all bets equally at $100, calculate the total profit for each timing group (assume -110 odds).

(c) Why might a bettor choose to bet on Saturday instead of Tuesday despite the lower CLV?

(d) Design an optimal timing strategy that balances CLV with other factors. What constraints should the bettor consider?

Exercise A.6 --- Reduced Juice Shopping

Some sportsbooks offer reduced juice (e.g., -105 instead of -110). Compare the long-term impact:

(a) Calculate the break-even win rate at -110 vs. -105 vs. -102.

(b) A bettor with a true 53% win rate places 1,000 bets at $100. Calculate their profit at -110, -105, and -102 juice.

(c) Express the value of reduced juice in terms of equivalent CLV per bet.

(d) If Book A offers -110 lines but better odds selection, while Book B offers -105 lines but inferior selection, how should a bettor decide which to use? Frame this as a mathematical comparison.

Exercise A.7 --- Line Shopping Across Correlated Markets

A sportsbook offers the following on an NFL game: DAL -6 (-110) / NYG +6 (-110). Meanwhile, the moneyline is DAL -250 / NYG +210.

(a) Calculate the no-vig probability implied by the spread market.

(b) Calculate the no-vig probability implied by the moneyline market.

(c) If these differ, which market is pricing Dallas higher and by how much?

(d) Explain how a bettor could use discrepancies between correlated markets (spread, moneyline, totals, team totals) as a form of line shopping. What risks does this approach carry?

Exercise A.8 --- The Account Management Problem

A serious line shopper maintains accounts at 12 sportsbooks. Each requires a minimum deposit of $500 and the bettor allocates $50,000 total across all accounts.

(a) If the bettor distributes funds equally, how much is in each account? Is this sufficient for $500 average bet sizes?

(b) The bettor finds that 60% of their best-line bets come from just 3 books. Should they redistribute funds? Design an allocation strategy.

(c) After 6 months, the bettor is limited at 2 of their top 3 books (bet limits reduced from $500 to $50). How does this affect their line shopping edge? Quantify the impact.

(d) Describe three strategies for managing sportsbook accounts to minimize the risk of being limited.

Exercise A.9 --- Live Betting Line Shopping

Live (in-play) betting markets often have wider discrepancies between books due to varying model speeds. A bettor tracks live NFL line discrepancies:

(a) Calculate the average line discrepancy. How does this compare to pre-game discrepancies (typically 0.5-1.0 points)?

(b) Why are live betting discrepancies larger than pre-game discrepancies?

(c) What practical challenges make live line shopping more difficult than pre-game shopping?

(d) If a bettor could capture 50% of the average live discrepancy as CLV on 200 live bets per season, estimate the profit at $100 per bet.

Exercise A.10 --- Building a Line Shopping Workflow

Design a complete line shopping workflow for a bettor who places 5-10 bets per day across NFL, NBA, and MLB. The workflow should specify:

(a) Pre-market preparation: What tools, data sources, and accounts should be set up before lines are released?

(b) Market opening: What actions should the bettor take within the first 30 minutes of lines being posted?

(c) Mid-week monitoring: How should the bettor track line movements and identify the optimal entry point?

(d) Pre-game execution: What is the final decision process 1-2 hours before game time?

Part B: CLV and Odds Optimization (10 exercises, 5 points each)

Exercise B.1 --- CLV from Line Shopping

A bettor places two identical bets (same game, same side) at two different books:

• Bet 1: Team A -3 at Book X (-110). Closing line: Team A -4 (-110).
• Bet 2: Team A -3 at Book Y (-105). Closing line: Team A -4 (-110).

(a) Calculate the CLV in points for each bet.

(b) Calculate the CLV in implied probability for each bet.

(c) Explain why the juice difference matters even though the spread is the same.

(d) Over 200 bets where the bettor consistently gets -105 instead of -110, quantify the cumulative advantage.

Exercise B.2 --- Optimizing Parlay Prices

A bettor wants to place a 3-leg parlay. Each leg is available at different prices across three books:

(a) Calculate the parlay payout at each book if all three legs are placed at the same book.

(b) Calculate the "optimal parlay" payout using the best odds for each leg.

(c) What is the percentage improvement from optimizing each leg?

(d) If parlay rules at a single book require all legs to be at the same book, what is the bettor's best single-book option?

Exercise B.3 --- Season-Long Shopping Value

A bettor projects the following for their NFL season:

• 250 spread bets at $200 each
• 50 moneyline bets at $200 each
• 30 total bets at $200 each

Without line shopping, they expect an average of -110 on all bets. With line shopping across 6 books, they expect:

• Spreads: average -107
• Moneylines: average 2.5% better implied probability
• Totals: average -106

(a) Calculate the break-even win rate improvement for each bet type.

(b) For a bettor with a true 53% win rate on spreads, calculate the season profit with and without line shopping.

(c) Calculate the total dollar value of line shopping across all 330 bets.

(d) If maintaining 6 sportsbook accounts costs the bettor 3 hours per week in account management, calculate the effective hourly rate earned from line shopping.

Exercise B.4 --- Alternative Line Markets

Some sportsbooks offer alternative lines (buying or selling points). Compare:

• Standard: Team A -3 (-110)
• Alt line: Team A -2.5 (-125)
• Alt line: Team A -3.5 (-100)

Another book offers: Team A -2.5 (-115)

(a) For each option, calculate the implied probability.

(b) Which option offers the best value for a bettor who believes Team A has a 55% chance of winning by 3+?

(c) Is buying the half-point at -125 better or worse than the second book's -2.5 at -115?

(d) When is it mathematically justified to "buy" points through alternative lines versus simply shopping for a better number?

Exercise B.5 --- Arbitrage from Line Shopping

Two books post the following on an NBA game:

• Book A: Team X -3 (-105) / Team Y +3 (-115)
• Book B: Team X -2.5 (-115) / Team Y +2.5 (-105)

(a) Is there a pure arbitrage opportunity here? Calculate.

(b) If not a pure arbitrage, is there a "middle" opportunity? What would need to happen for the middle to hit?

(c) Calculate the expected value of a middle play (bet Team X -2.5 at Book B and Team Y +3 at Book A) if the probability of the game landing on exactly 3 is 5%.

(d) What are the risks of middle plays that pure arbitrage does not carry?

Exercise B.6 --- Odds Aggregation Methods

Given closing odds from 8 sportsbooks for Team A moneyline:

Book 1: -142, Book 2: -145, Book 3: -140, Book 4: -150, Book 5: -138, Book 6: -148, Book 7: -143, Book 8: -155

(a) Calculate the mean, median, and mode of the implied probabilities.

(b) Calculate the "market consensus" using the no-vig midpoint of the sharpest book (Book 5 at -138/+118).

(c) Which aggregation method would you recommend as the "true" probability and why?

(d) If your model gives Team A a 56% chance, which books offer value and which do not?

Exercise B.7 --- Dynamic Line Shopping

A bettor wants to bet NFL Over 44.5. Current lines:

By Wednesday:

(a) What was the best time and book to place the Over bet on Monday?

(b) What is the CLV of betting at Book A on Monday vs. the Wednesday consensus?

(c) If the bettor waited until Wednesday, which book is now the best option?

(d) Develop a decision rule: when should a bettor "lock in" an early line versus waiting for potential improvement?

Exercise B.8 --- Cross-Market Line Shopping

A bettor notices the following on an NFL game:

• Book A: Team total Over 24.5 (-110)
• Book B: Game total Over 48 (-110), Spread: Team -3.5 (-110)

Using the relationship: Team Total = (Game Total + Spread) / 2:

(a) Calculate the implied team total from Book B's game total and spread.

(b) Is there a discrepancy between Book A's team total and the derived team total?

(c) If the discrepancy represents value, how would you exploit it?

(d) What are the limitations of using cross-market relationships for line shopping?

Exercise B.9 --- Historical Line Shopping Analysis

You have a dataset of 1,000 bets with the following columns: best_available_odds, worst_available_odds, odds_used, outcome.

(a) Design a metric to measure "line shopping efficiency" --- the percentage of the available odds improvement that the bettor actually captured.

(b) If the bettor captured 70% of the available improvement on average, estimate the profit difference versus a bettor who always used the worst line.

(c) What factors would cause a bettor's shopping efficiency to be below 100%?

(d) Propose three improvements the bettor could make to increase their shopping efficiency.

Exercise B.10 --- The Limits of Line Shopping

(a) A bettor finds that their line shopping edge averages +1.5% in implied probability per bet. If they place 500 bets per year at $200 each, calculate their expected annual profit from line shopping alone (assuming they have no model edge).

(b) Is line shopping alone sufficient for long-term profitability? Explain using the relationship between shopping edge and the vig.

(c) If a sportsbook limits the bettor to $25 maximum bet, calculate how many accounts they need to maintain their $200 average bet size.

(d) At what point does the cost of maintaining accounts (deposits, time, fees) exceed the value of line shopping? Create a break-even analysis.

Part C: Applied Analysis and Programming (10 exercises, 6 points each)

Exercise C.1 --- Odds Comparison Engine

Build a Python class OddsComparator that accepts odds from multiple sportsbooks for the same event and identifies the best available odds.

Exercise C.2 --- Line Shopping Profit Calculator

Write a function that takes a bettor's historical odds versus the best available odds and calculates the total profit lost or gained from line shopping behavior.

Exercise C.3 --- Automated Odds Monitor

Build a system that monitors a dictionary of odds across multiple books and generates alerts when a new best price appears or when odds diverge beyond a threshold.

Exercise C.4 --- Timing Optimization Analysis

Write a simulation that models optimal bet timing by generating opening lines, intermediate prices, and closing lines, then measuring CLV for bets placed at different times.

Exercise C.5 --- Line Shopping ROI Calculator

Create a comprehensive calculator that takes a bettor's season of bets and computes the ROI attributable specifically to line shopping (versus the ROI from their model edge alone).

Exercise C.6 --- Half-Point Value Calculator

Implement a function that calculates the exact value of each half-point in NFL spread betting using historical margin distributions.

Exercise C.7 --- Arbitrage Detector

Write a function that scans odds across multiple books for the same event and identifies pure arbitrage opportunities, calculating the guaranteed profit percentage.

Exercise C.8 --- Account Balance Optimizer

Build a simulation that optimizes the allocation of bankroll across multiple sportsbook accounts based on historical patterns of which books offer the best odds.

Exercise C.9 --- Shopping Efficiency Tracker

Create a tracking system that records the best available odds, the odds actually used, and computes a rolling "shopping efficiency" score.

Exercise C.10 --- Complete Line Shopping Pipeline

Combine the tools from C.1-C.9 into a unified pipeline that loads multi-book odds data, identifies best prices, computes shopping value, and generates a daily report.

Scoring Summary