Exercise 2: Mathematical Calculations and Analysis

⏰ Time Investment: 45-60 minutes 🎯 Goal: Master the mathematical foundations of AMMs through hands-on calculations

📚 Required Reading Integration 📖 Primary: Lesson 2: The Mathematics of Liquidity Provision 📖 Supporting: Lesson 1: Understanding AMM Fundamentals

🔢 Phase 1: Formula Mastery (15 minutes)

Fee vs Gas Break-Even Calculator

Formula Mastery Checklist

Constant Product Formula Practice

Exercise 1: Complete the table

ETH Reserves	USDC Reserves	k (constant)	Price (USDC/ETH)
10	20,000	_____	_____
11	_____	200,000	_____
_____	18,181.82	200,000	2,000
9	_____	200,000	_____

Price Impact Calculation

Exercise 2: Calculate price impact for different trade sizes

Pool: 100 ETH, 200,000 USDC (k = 20,000,000)

Trade 1: Buy 0.1 ETH

• New ETH: _____

• New USDC: _____

• Price impact: _____%

Trade 2: Buy 1 ETH

• New ETH: _____

• New USDC: _____

• Price impact: _____%

Trade 3: Buy 10 ETH

• New ETH: _____

• New USDC: _____

• Price impact: _____%

Observation: As trade size increases, price impact _____ (increases/decreases).

📊 Phase 2: Fee Mathematics (15 minutes)

Fee Distribution Exercise

Scenario:

• Pool TVL: $1,000,000

• Daily Volume: $500,000

• Fee Rate: 0.3%

• Your Capital: $10,000

Calculate:

Step 1: Your pool share

• Share = $10,000 ÷ $1,000,000 = _____%

Step 2: Daily fees generated

• Daily fees = $500,000 × 0.003 = $_____

Step 3: Your daily fee share

• Your fees = $_____ × % = $

Step 4: Monthly fees

• Monthly = $_____ × 30 = $_____

Step 5: Annual fees (before IL)

• Annual = $_____ × 12 = $_____

Step 6: APY calculation

• APY = ($_____ ÷ $10,000) × 100% = _____%

Fee vs. Gas Analysis

Scenario: You're considering providing liquidity on Ethereum L1

Setup:

• Position: $5,000

• Expected monthly fees: $50

• Gas costs: $30 (deposit) + $20 (withdraw) = $50 total

Analysis:

• Net monthly return: $50 - $50 = $_____

• Is this profitable? _____ (Yes/No)

• Minimum position size for profitability: $_____ (assuming same fees)

Conclusion: For L1, you need positions >$_____ to be profitable.

🧮 Phase 3: Advanced Calculations (20 minutes)

Liquidity Depth Comparison

Exercise 3: Compare three pools

Pool A: 20 ETH, 40,000 USDC Pool B: 100 ETH, 200,000 USDC Pool C: 500 ETH, 1,000,000 USDC

Calculate depth:

• Depth_A = √(20 × 40,000) = √(_____) = _____

• Depth_B = √(_____ × ) = √() = _____

• Depth_C = √(_____ × ) = √() = _____

Which pool can handle a $100,000 trade with least impact?

• Answer: Pool _____

Multi-Trade Analysis

Exercise 4: Calculate cumulative price impact

Pool: 100 ETH, 200,000 USDC (k = 20,000,000) Starting Price: 2,000 USDC/ETH

Trade Sequence:

1. Buy 1 ETH → New price: _____

2. Buy 1 ETH → New price: _____

3. Buy 1 ETH → New price: _____

Compare to single 3 ETH trade:

• Single trade price: _____

• Cumulative price: _____

• Difference: _____

Observation: Splitting trades _____ (reduces/increases) price impact.

📈 Phase 4: Real-World Scenario (10 minutes)

Complete Pool Analysis

Scenario: You're analyzing an ETH/USDC pool on Arbitrum

Pool Data:

• TVL: $5,000,000

• Daily Volume: $1,000,000

• Fee Rate: 0.05%

• Your Capital: $25,000

• Network: Arbitrum (L2)

Calculate Expected Returns:

Daily Analysis:

• Your share: _____%

• Daily fees: $_____

• Daily gas (amortized): $_____

• Net daily: $_____

Monthly Analysis:

• Monthly fees: $_____

• Monthly gas: $_____

• Net monthly: $_____

Annual Analysis:

• Annual fees: $_____

• Annual gas: $_____

• Net annual: $_____

• APY: _____% (before IL)

Risk Assessment:

• Minimum viable position: $_____ (based on gas costs)

• Your position is _____ (viable/not viable) for active management

🎯 Phase 5: Mastery Check (10 minutes)

Self-Evaluation

Rate your confidence (1-10):

• Calculating swap amounts: _____/10

• Understanding price impact: _____/10

• Fee distribution math: _____/10

• Liquidity depth concepts: _____/10

• Gas economics: _____/10

Total: _____/50

Key Formulas Mastered

Write from memory:

Constant Product Formula:

• Formula: _____

• What does k represent? _____

Price Calculation:

• Formula: _____

• How does it change with trades? _____

Fee Share:

• Formula: _____

• What factors affect your share? _____

📚 Next Steps

• Practice calculations with different pool sizes

• Use online calculators to verify your work

• Explore Uniswap interface to see real calculations

• Move to Lesson 3 to understand IL (the hidden cost)

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← Back to Lesson 2 | Next: Exercise 3 →