Chapter 12 Exercises: Box Plus-Minus (BPM) and VORP

Overview

These exercises cover BPM methodology, OBPM and DBPM calculations, VORP interpretation, and practical applications of these metrics. Problems range from basic calculations to advanced analytical challenges.

Section A: Foundational Concepts (Exercises 1-8)

Exercise 1: True Shooting Percentage

Calculate True Shooting Percentage for each player:

a) Calculate TS% for each player. b) Rank players by shooting efficiency. c) If league average TS% is 56.5%, which players are above average?

Exercise 2: Replacement Level Understanding

A team has the following players:

a) Which players are above replacement level (-2.0)? b) Calculate the difference between each player's BPM and replacement level. c) If Deep Bench is replaced by a replacement-level player, what is the expected impact?

Exercise 3: Position Estimation

Given the following player profiles, estimate their position (1-5 scale):

a) Use the position estimation formula to calculate each player's position. b) What traditional position label would you assign to each?

Exercise 4: BPM Interpretation

Classify each BPM value according to the standard interpretation scale:

a) +11.5 b) +6.2 c) +3.8 d) +0.5 e) -1.5 f) -3.0

Provide the category (MVP, All-NBA, All-Star, etc.) for each.

Exercise 5: Component Analysis

A player has the following BPM breakdown: - OBPM: +6.5 - DBPM: +0.8 - Total BPM: +7.3

a) What percentage of this player's value comes from offense? b) Describe the player archetype this profile suggests. c) Compare to a player with OBPM: +3.0 and DBPM: +3.5.

Exercise 6: VORP Basics

Calculate VORP for the following scenarios (assume 82-game season):

a) Calculate VORP for each player using the formula: VORP = (BPM + 2.0) * (MP / 3936) b) Rank players by VORP. c) Explain why the VORP ranking might differ from the BPM ranking.

Exercise 7: Rate vs. Volume

Two players have the following statistics:

Player X: - BPM: +9.0 - Minutes: 1,800

Player Y: - BPM: +5.5 - Minutes: 3,100

a) Calculate VORP for both players. b) Which player provided more total value? c) If both players could play 3,000 minutes, who would have higher projected VORP? d) Discuss the tradeoffs between these player profiles.

Exercise 8: Historical Context

Using the historical BPM leaders table from the chapter:

a) What is the highest single-season BPM ever recorded? b) Which player appears most frequently in the top 10 single-season list? c) What is the typical BPM range for MVP-caliber seasons? d) Calculate the average OBPM and DBPM for the top 5 seasons.

Section B: OBPM Calculations (Exercises 9-15)

Exercise 9: Scoring Efficiency Component

Calculate the scoring efficiency component of OBPM for players with:

Use the formula: Scoring = 0.5 * (TS% - LeagueTS%) * sqrt(USG%) * 10

a) Calculate the scoring component for each player. b) Explain why Player D might have negative scoring contribution despite 61% TS%. c) Which player has the most valuable scoring profile?

Exercise 10: Playmaking Component

Calculate playmaking contributions:

Use: Playmaking = 0.02 * AST% + 0.015 * (AST% - TOV%)

a) Calculate the playmaking component for each player. b) Which player is the most efficient playmaker (considering turnovers)? c) What does a negative (AST% - TOV%) difference indicate?

Exercise 11: Three-Point Spacing

Given the following three-point profiles:

Using spacing coefficient of 0.003:

a) Calculate the spacing component for each player. b) Explain why three-point volume matters for OBPM. c) How might the spacing component undervalue certain players?

Exercise 12: Complete OBPM

Calculate complete OBPM for Player A with these statistics:

• TS%: 0.620, League TS%: 0.565
• USG%: 30
• AST%: 32
• TOV%: 14
• ORB%: 3.5
• 3PAr: 0.38
• Position: 2 (Shooting Guard)

Use coefficients from the chapter and show all work.

Exercise 13: Position Adjustment Effects

Compare OBPM for two players with identical box scores but different positions:

Both players: - TS%: 0.590, USG%: 25 - AST%: 30, TOV%: 12 - ORB%: 4.0, 3PAr: 0.35

Player A: Position = 1 (Point Guard) Player B: Position = 4 (Power Forward)

a) Explain why Player B would receive a larger position adjustment bonus. b) Estimate the OBPM difference between them. c) Is this adjustment justified? Why or why not?

Exercise 14: High-Usage Efficiency Tradeoff

Analyze these three high-usage players:

a) Calculate the scoring efficiency component for each. b) Calculate the playmaking component for each. c) Which player likely has the highest OBPM? d) Discuss the tradeoff between usage and efficiency.

Exercise 15: OBPM Case Comparison

Compare the OBPM profiles of these player archetypes:

Floor General: - PTS/100: 20, TS%: 0.560 - AST%: 45, TOV%: 16 - USG%: 22, 3PAr: 0.30

Volume Scorer: - PTS/100: 35, TS%: 0.590 - AST%: 15, TOV%: 10 - USG%: 34, 3PAr: 0.40

a) Calculate approximate OBPM for each. b) Which has higher offensive value? c) What roster construction would each player fit best?

Section C: DBPM Calculations (Exercises 16-20)

Exercise 16: Defensive Statistics

Calculate the defensive contribution for these players:

Using coefficients: STL (0.15), BLK (0.10), DRB (0.008)

a) Calculate raw defensive contribution for each. b) Apply position adjustments for Players B and D (assume Position = 5). c) Apply the 30% defensive regression.

Exercise 17: Position-Based Expectations

For a center (Position = 5), expected defensive stats are approximately: - Expected BLK%: 4.5 - Expected STL%: 1.3 - Expected DRB%: 22

Calculate the position adjustment for:

a) A rim protector with BLK% = 7.0, STL% = 1.0, DRB% = 25 b) A mobile center with BLK% = 3.0, STL% = 2.5, DRB% = 18 c) Which player receives a larger positive adjustment? Why?

Exercise 18: Defensive Limitations Analysis

Consider two players with identical DBPM (+1.5):

Player A (Shot Blocker): - BLK%: 5.5, STL%: 1.0, DRB%: 23 - Poor help defense, limited lateral movement

Player B (Help Defender): - BLK%: 1.5, STL%: 2.0, DRB%: 18 - Excellent rotations, strong communication

a) Explain why these players might have similar DBPM despite different skill sets. b) Which player is likely undervalued by DBPM? c) What additional data would help evaluate their defense?

Exercise 19: Team Defensive Context

A team has the following defensive context: - Team Defensive Rating: 105.0 (league average: 110.0) - Total team minutes: 19,680 (82 games * 48 min * 5 players)

Players on this team:

a) Calculate raw DBPM for each player. b) How should the team's defensive overperformance be distributed? c) What challenges arise in attributing team defense to individuals?

Exercise 20: Complete DBPM

Calculate complete DBPM for a power forward (Position = 4) with: - STL%: 1.8 - BLK%: 2.5 - DRB%: 19 - Height: 6'9"

Include: a) Base defensive contribution b) Position adjustment c) Defensive regression d) Final DBPM estimate

Section D: VORP Analysis (Exercises 21-28)

Exercise 21: VORP Calculation

Calculate VORP for the following players:

a) Calculate VORP for each player using the full season denominator (3,936). b) Which player accumulated the most value? c) How does missed playing time affect VORP rankings?

Exercise 22: Replacement Level Sensitivity

Re-calculate VORP using different replacement level thresholds:

Player: BPM = +5.0, Minutes = 2,500

a) VORP with replacement level = -2.0 (standard) b) VORP with replacement level = -1.5 c) VORP with replacement level = -2.5 d) How sensitive is VORP to the replacement level assumption?

Exercise 23: Seasonal Projection

A player has the following mid-season statistics (45 games played): - BPM: +6.2 - Minutes: 1,620 (36 MPG) - Team games remaining: 37

Assuming consistent performance: a) Calculate current VORP. b) Project end-of-season VORP. c) At what pace would they finish with 8.0 VORP?

Exercise 24: Roster VORP Analysis

Analyze this team's VORP distribution:

a) Calculate VORP for each player. b) Calculate total team VORP. c) What percentage of team VORP comes from the top 3 players? d) Identify roster weaknesses based on VORP.

Exercise 25: Trade Analysis Using VORP

Evaluate this proposed trade using VORP:

Team A Receives: - Player X: BPM +5.5, expected 2,200 minutes - Player Y: BPM +1.0, expected 1,500 minutes

Team B Receives: - Player Z: BPM +4.0, expected 2,800 minutes

a) Calculate expected VORP for each player. b) Which team wins the trade on VORP basis? c) What factors beyond VORP should influence the trade decision?

Exercise 26: Career VORP Projection

A 24-year-old player has accumulated: - Career BPM: +5.5 - Career VORP: 18.5 (4 seasons) - Average minutes/season: 2,400

Using typical aging curves: a) Estimate BPM for ages 25-30 (assuming peak at 27). b) Project cumulative VORP by age 30. c) What career VORP total might this player achieve?

Exercise 27: VORP vs. Wins

Using the approximation that ~30 points of differential = 1 win:

Player: VORP = 6.5

a) How many "points added" does this represent? b) Convert to approximate wins added. c) If a win is worth $3.5 million in the current NBA, estimate this player's value.

Exercise 28: Replacement Level by Position

Consider that replacement level might vary by position: - PG replacement level: -1.8 - SG replacement level: -1.5 - SF replacement level: -1.8 - PF replacement level: -2.2 - C replacement level: -2.5

Using these position-specific thresholds, recalculate VORP:

a) Calculate VORP with standard -2.0 threshold. b) Calculate VORP with position-specific thresholds. c) How does this affect relative value assessment?

Section E: Comparative Analysis (Exercises 29-35)

Exercise 29: BPM vs. RAPM Comparison

Given these players with both BPM and RAPM data:

a) Calculate the BPM-RAPM difference for each player. b) Which player is most overrated by BPM? c) Which player is most underrated by BPM? d) What might explain these discrepancies?

Exercise 30: Archetype Analysis

Classify these players by archetype and predict BPM accuracy:

Player A: 25 PPG, 45% from 3, excellent on-ball defense Player B: 12 PPG, 10 RPG, 4 BPG, rim protection specialist Player C: 8 PPG, 6 APG, floor general with limited scoring Player D: 15 PPG, 40% 3PT, switches 1-4 on defense

a) Identify each player's archetype. b) Predict whether BPM likely overrates or underrates each. c) Explain your reasoning.

Exercise 31: Era Comparison Challenge

Compare these players from different eras:

1990s Player: - BPM: +6.0 - League pace: 92 possessions/game - League TS%: 0.520

2020s Player: - BPM: +6.0 - League pace: 100 possessions/game - League TS%: 0.570

a) Are these performances directly comparable? b) What adjustments might be needed? c) Which player might be more valuable in the other's era?

Exercise 32: Stability Analysis

A player has the following season-by-season BPM:

a) Calculate the mean and standard deviation of BPM. b) Is this player's BPM stable or volatile? c) What true talent level would you estimate?

Exercise 33: Offensive vs. Defensive Value

Rank these players by overall value:

a) All have +6.0 BPM. Are they equally valuable? b) Consider playoff implications for each profile. c) Which player might be most underrated by BPM?

Exercise 34: Sample Size Analysis

A player has the following game log:

a) How does BPM stabilize as the sample grows? b) At what point would you consider the BPM reliable? c) What causes early-season BPM volatility?

Exercise 35: Multi-Metric Comparison

A player has the following advanced metrics:

• BPM: +5.5
• RAPM (1-year): +3.0
• RAPM (3-year): +4.5
• PER: 22.0
• WS/48: 0.180

a) What does the BPM-RAPM discrepancy suggest? b) Do the other metrics support BPM or RAPM? c) What is your best estimate of true impact?

Section F: Advanced Applications (Exercises 36-40)

Exercise 36: Draft Evaluation

Use BPM benchmarks for draft analysis:

Rookie BPM Expectations by Pick: - Picks 1-5: Average BPM of +1.0 - Picks 6-10: Average BPM of -0.5 - Picks 11-14: Average BPM of -1.5

A team has the 7th pick. Their preferred prospects: - Player A: Projected BPM +2.0 - Player B: Projected BPM +1.0 - Player C: Projected BPM +3.0 (but injury risk)

a) Calculate expected value above draft position for each. b) How should injury risk factor into the analysis? c) What BPM would a player need to justify trading up?

Exercise 37: Contract Valuation

Using VORP to evaluate contracts:

Player Contract: - Salary: $25 million/year - Expected BPM: +5.0 - Expected minutes: 2,500

Market Assumptions: - Win value: $3.5 million - Points per win: 30

a) Calculate expected VORP. b) Convert VORP to win equivalent. c) Calculate dollar value of production. d) Is this contract positive or negative value?

Exercise 38: Team Building Analysis

A team has a salary cap of $130 million and needs to build a roster.

Available Player Categories: | Category | Avg BPM | Avg Minutes | Avg Salary | |----------|---------|-------------|------------| | Star | +7.0 | 2,800 | $35M | | 2nd Star | +4.5 | 2,600 | $25M | | Quality Starter | +2.0 | 2,200 | $12M | | Rotation | +0.5 | 1,500 | $6M | | Minimum | -1.5 | 800 | $2M |

Design two roster constructions: a) Superstar-focused (2 stars) b) Depth-focused (no stars, many quality starters)

Calculate total VORP for each approach. Which maximizes value?

Exercise 39: Playoff Performance Analysis

Compare regular season and playoff BPM:

a) Which players show playoff improvement? b) What factors might cause BPM to change in playoffs? c) How should playoff BPM affect player evaluation?

Exercise 40: Historical Analysis Project

Using the historical BPM leaders:

a) Calculate the average BPM for the top 10 all-time seasons. b) Identify which decade produced the most top-20 seasons. c) Analyze the OBPM/DBPM ratio for historically great seasons. d) What BPM would a player need to rank in the top 20 all-time?

Answer Key Hints

Selected Answers:

Exercise 1a: TS% = PTS / (2 * (FGA + 0.44 * FTA)) - Player A: 25 / (2 * (18 + 2.64)) = 25 / 41.28 = 0.606 - Player B: 18 / (2 * (15 + 1.32)) = 18 / 32.64 = 0.551

Exercise 6a: VORP = (BPM + 2.0) * (MP / 3936) - Player A: (8.0 + 2.0) * (3000 / 3936) = 10.0 * 0.762 = 7.62 - Player B: (5.0 + 2.0) * (2500 / 3936) = 7.0 * 0.635 = 4.45

Exercise 21: Using the same VORP formula with appropriate minute inputs.

Exercise 27: - VORP of 6.5 represents 6.5 points per 100 possessions above replacement - Approximate wins: 6.5 * (1/30) * (playing time factor) ~ 5-6 wins

Full solutions available in the instructor's manual.