Detailed Instructions

Constant Product Market Maker (CPMM) is the core mechanism for many decentralized exchanges like Uniswap V2. The rule is simple: the product of the quantities of two tokens in a pool must remain constant after any trade. For a pool with tokens X and Y, the formula is x * y = k, where k is the constant. This creates the price relationship: Price of X in terms of Y = y / x. For example, if an ETH/USDC pool starts with 10 ETH and 20,000 USDC, k = 10 * 20,000 = 200,000. If a trader buys 1 ETH, the new pool state must satisfy (10 - 1) * new_usdc = 200,000, so new_usdc = 200,000 / 9 ≈ 22,222.22. The trader pays ~2,222.22 USDC for that 1 ETH.

• Sub-step 1: Identify the pool reserves: Check the current reserves of the two tokens, often via a blockchain explorer or the DApp interface.
• Sub-step 2: Calculate the constant k: Multiply the two reserve amounts together.
• Sub-step 3: Apply a trade scenario: Modify one reserve according to a hypothetical trade and solve for the new reserve of the other token using k.

Tip: The constant k only changes when liquidity is added or removed, not during swaps. This invariant is what leads to price slippage.

Detailed Instructions

Impermanent Loss (IL) quantifies the opportunity cost of providing liquidity compared to holding the assets. It's calculated as IL = (Value in Pool - Value if Held) / Value if Held. To compute this, you need two prices: the initial price when you deposited and the new, changed price. Assume you deposit into a 50/50 ETH/USDC pool when 1 ETH = 2,000 USDC. You deposit 1 ETH and 2,000 USDC (total value $4,000). If the external market price of ETH rises to 4,000 USDC, you must calculate two values. First, Value if Held: You would have 1 ETH * 4,000 + 2,000 USDC = $6,000. Second, Value in Pool: The pool's new reserves adjust per the constant product formula. With k = 1 * 2,000 = 2,000 and new price P = 4,000, the new reserves are sqrt(k/P) for ETH and sqrt(k*P) for USDC.

• Sub-step 1: Determine your pool share: If you provided 10% of the liquidity, your share is 10% of the reserves.
• Sub-step 2: Compute new pool reserves: Use the formulas new_x = sqrt(k / P) and new_y = sqrt(k * P) for a 50/50 pool.
• Sub-step 3: Calculate your pool value: (your_share_of_new_x * P) + your_share_of_new_y in USDC terms.

Tip: Impermanent loss is 'impermanent' because if the price returns to your entry point, the loss vanishes. It becomes permanent only upon withdrawal at a different price.

Detailed Instructions

The generalized IL formula for a 50/50 pool is IL = 2 * sqrt(price_ratio) / (1 + price_ratio) - 1, where price_ratio = new_price / initial_price. This formula shows IL is always zero when price_ratio = 1 (no change) and negative for any other ratio. For instance, if ETH doubles in price (price_ratio = 2), plug into the formula: IL = 2 * sqrt(2) / (1 + 2) - 1 = (2 * 1.4142) / 3 - 1 ≈ 0.9428 - 1 = -0.0572 or -5.72%. This means your pool value is 5.72% less than your hold value. You can compute this in code for analysis.

pythonCopy
def impermanent_loss(price_ratio):
    """Calculate IL percentage for a 50/50 pool."""
    return (2 * (price_ratio ** 0.5)) / (1 + price_ratio) - 1

# Example: Price increases 3x
loss = impermanent_loss(3)
print(f"IL: {loss:.2%}")  # Output: IL: -13.40%

• Sub-step 1: Calculate the price ratio: Divide the new token price by the old price for the asset you are pricing in the quote currency.
• Sub-step 2: Apply the formula: Use the mathematical expression or the code function above.
• Sub-step 3: Interpret the result: A negative percentage indicates the magnitude of the loss relative to holding.

Tip: The loss is symmetric; a price drop to 1/2 (price_ratio=0.5) also results in ~-5.72% IL. The worst IL occurs at large divergences.

Detailed Instructions

Fee income is the reward for providing liquidity and can compensate for impermanent loss. Your net profit/loss is Impermanent Loss + Fees Earned. Fees are typically a percentage (e.g., 0.3% on Uniswap V2) of the trade volume that passes through your share of the pool. High volume in a stable price range can make liquidity provision profitable despite IL. To estimate, you need to know your pool's total value locked (TVL), your share, and the daily volume. For example, in a pool with $1M TVL and 1% daily volume ($10,000 traded), the daily fees at 0.3% are $30. If your share is 5%, you earn $1.5 per day. Over 30 days, that's $45. If your IL from a price move is -$30 (on a $1,000 deposit), your net gain is +$15.

• Sub-step 1: Determine fee structure: Check the pool's swap fee percentage (e.g., 0.003 for 0.3%).
• Sub-step 2: Estimate volume through your share: Use analytics sites like Dune Analytics or The Graph to query historical volume for the pool address (e.g., 0x...).
• Sub-step 3: Calculate daily fee earnings: Your Fee = (Your Pool Share) * (Total Daily Volume) * (Fee Percentage).
• Sub-step 4: Compare to IL: Subtract the dollar amount of IL from your cumulative fee earnings over the period.

Tip: In highly volatile markets, IL can outpace fees. In stable, high-volume pairs (like stablecoin pairs), fees often dominate, making IL negligible.