Detailed Instructions

At the heart of the most common AMMs, like Uniswap V2, lies the Constant Product Formula: x * y = k. This invariant ensures that the product of the quantities of two tokens in a liquidity pool (x and y) must always remain constant (k), regardless of trades. This simple rule automatically determines prices based on the available supply of each asset.

• Sub-step 1: Define the variables: Let x be the reserve of Token A (e.g., ETH) and y be the reserve of Token B (e.g., DAI) in the pool.
• Sub-step 2: Establish the constant: The product k = x * y is calculated after the initial liquidity deposit and must be preserved after every trade.
• Sub-step 3: Observe price impact: The price of Token A in terms of Token B is given by Price_A = y / x. As x increases (someone sells A), y must decrease to keep k constant, making A cheaper.

Tip: This model creates slippage: larger trades execute at worse average prices because they move the ratio x/y more significantly.

Detailed Instructions

To perform a swap, a user specifies an input amount of one token, and the AMM calculates the output of the other token using the constant product rule. The formula to calculate the output amount Δy when inputting Δx (with a small fee φ, typically 0.3%) is derived from (x + Δx * (1 - φ)) * (y - Δy) = k.

• Sub-step 1: Account for the fee: First, deduct the protocol fee from the input. For a 0.3% fee, the effective input to the pool is Δx_effective = Δx * 0.997.
• Sub-step 2: Calculate the output: Solve the invariant for the new y' reserve: y' = k / (x + Δx_effective). The output amount is Δy = y - y'.
• Sub-step 3: Update the reserves: After the trade, the new pool state becomes (x_new, y_new) = (x + Δx_effective, y - Δy).

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// Example: Pool has 100 ETH (x) and 400,000 DAI (y). k = 40,000,000.
// User swaps 10 ETH for DAI with 0.3% fee.
let effectiveInput = 10 * 0.997; // = 9.97 ETH
let newX = 100 + 9.97; // = 109.97 ETH
let newY = 40000000 / newX; // ≈ 363,735.66 DAI
let outputDAI = 400000 - 363735.66; // ≈ 36,264.34 DAI

Tip: The user receives approximately 36,264 DAI. Notice the price per ETH paid is ~3,626 DAI, higher than the initial 4,000 DAI/ETH price due to slippage.

Detailed Instructions

Liquidity providers (LPs) deposit an equivalent value of both tokens into the pool. They must deposit in a ratio equal to the current reserve ratio to avoid arbitrage and preserve the price. In return, they receive LP tokens, which represent their share of the pool and accrue trading fees.

• Sub-step 1: Determine deposit amounts: If the pool has reserves (x, y), and you want to deposit based on an amount A of Token A, you must also deposit B = (y / x) * A of Token B.
• Sub-step 2: Mint LP tokens: The number of LP tokens minted is proportional to your share of the pool. If totalSupply is the current LP token supply, you receive minted = totalSupply * (A / x).
• Sub-step 3: Track your share: Your LP tokens entitle you to a proportional claim on the entire pool's reserves and the accumulated fees (which are automatically reinvested by increasing k).

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// Example: Adding liquidity to the 100 ETH & 400,000 DAI pool.
// You deposit 1 ETH. Required DAI deposit = (400,000 / 100) * 1 = 4,000 DAI.
// If total LP supply is 1000 tokens, you get minted = 1000 * (1 / 100) = 10 LP tokens.
// You now own 1% of the pool (10/1000).

Tip: Always deposit using a front-end interface like the Uniswap UI at app.uniswap.org or via a contract call to the addLiquidity function to ensure correct ratios.

Detailed Instructions

Impermanent Loss (IL) is not a direct loss of funds but an opportunity cost incurred by LPs when the price ratio of the pooled assets changes compared to when they were deposited. It occurs because the AMM automatically rebalances the pool, selling the appreciating asset and buying the depreciating one to maintain x * y = k.

• Sub-step 1: Calculate value if held: Determine the value of your initial deposit if you had simply held the tokens in your wallet, based on the new market prices.
• Sub-step 2: Calculate value in the pool: Determine the value of your LP share, which is your portion of the new pool reserves (x_new, y_new) at the new market prices.
• Sub-step 3: Quantify the difference: Impermanent Loss = (Value in Pool - Value if Held) / Value if Held. The loss is greatest when prices diverge significantly.

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// Example: You deposited 1 ETH & 4,000 DAI (Price: 1 ETH = 4,000 DAI).
// Later, market price shifts to 1 ETH = 8,000 DAI.
// Value if Held: 1*8000 + 4000 = 12,000 DAI.
// Pool rebalances. New reserves can be calculated. Your 1% share might be worth ~11,314 DAI.
// IL = (11314 - 12000) / 12000 ≈ -5.7%

Tip: IL is 'impermanent' because if the price returns to its original ratio, the loss vanishes. It becomes permanent upon withdrawing at the changed price. Trading fees are meant to compensate for this risk.

Detailed Instructions

An Automated Market Maker (AMM) is a decentralized exchange protocol that uses mathematical formulas to price assets. Liquidity providers (LPs) deposit equal value of two tokens (e.g., ETH and USDC) into a liquidity pool. In return, the AMM mints and issues LP tokens (Liquidity Provider tokens) to the depositor's wallet address. These tokens are a receipt and represent your proportional share of the entire pool.

• Sub-step 1: Choose a Pool: Select a trading pair like ETH/USDC on a platform like Uniswap V3. The pool address might be 0x8ad599c3A0ff1De082011EFDDc58f1908eb6e6D8.
• Sub-step 2: Deposit Equal Value: You must deposit assets in a 50/50 value ratio. To provide $10,000 of liquidity, deposit ~$5,000 worth of ETH and ~$5,000 worth of USDC.
• Sub-step 3: Receive LP Tokens: Upon deposit, the contract mints LP tokens to your address. You can view them in your wallet; their quantity defines your pool share.

Tip: The value of your LP tokens fluctuates with the total value of the pool and is subject to impermanent loss.

Detailed Instructions

LPs earn a percentage of every trade that occurs in their pool, typically 0.01% to 1% per swap. However, they are exposed to impermanent loss (IL), which is the loss compared to simply holding the assets, occurring when the price ratio of the pooled tokens changes. The loss is 'impermanent' because it can reverse if prices return to the original ratio.

• Sub-step 1: Fee Calculation: For a pool with a 0.3% fee, a $10,000 trade generates $30 in fees, distributed proportionally to all LPs.
• Sub-step 2: IL Simulation: Use a calculator. If ETH price doubles relative to USDC after you deposit, your LP position might be worth ~$13,400 versus $14,100 if you had just held, resulting in an IL of ~5%.
• Sub-step 3: Net Profit Analysis: Your real yield is Fees Earned - Impermanent Loss. Consistently high volume can offset moderate IL.

Tip: Concentrated liquidity models (like Uniswap V3) allow you to set custom price ranges to earn higher fees but require active management.

Detailed Instructions

Interacting directly with the AMM's smart contract requires precise function calls. The primary function is often addLiquidity, which handles the token approvals and deposit. Always verify contract addresses on the project's official documentation or a block explorer.

• Sub-step 1: Approve Tokens: First, approve the AMM router to spend your tokens. For USDC on Ethereum, you might call:

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usdcContract.approve(routerAddress, amount)

• Sub-step 2: Call Add Liquidity: Then, call the router's function. For a Uniswap V2-style router:

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router.addLiquidity(
  address(tokenA),
  address(tokenB),
  amountADesired,
  amountBDesired,
  amountAMin,
  amountBMin,
  to,
  deadline
)

• Sub-step 3: Set Parameters: amountAMin and amountBMin are slippage tolerances (e.g., 0.5%). The deadline is a future block timestamp (e.g., block.timestamp + 1200 for 20 minutes).

Tip: Using a front-end UI (like app.uniswap.org) simplifies this process but understanding the underlying calls is crucial for safety.

Detailed Instructions

Continuous monitoring is essential. You must track your accumulated fees (often automatically added to the pool, increasing the value of your LP tokens) and the current pool prices. To exit, you burn your LP tokens to withdraw your underlying assets, which will include your original deposit plus any accrued fees, minus any impermanent loss realized upon withdrawal.

• Sub-step 1: Track Your Position: Use a dashboard like DeBank or Zapper. Input your wallet address 0xYourAddressHere to see the value of your LP tokens and estimated fees.
• Sub-step 2: Remove Liquidity: Call the removeLiquidity function on the router. You must specify the amount of LP tokens to burn and minimum amounts of each token you will accept.

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router.removeLiquidity(
  address(tokenA),
  address(tokenB),
  liquidity, // amount of LP tokens to burn
  amountAMin,
  amountBMin,
  to,
  deadline
)

• Sub-step 3: Claim Rewards (if any): Some protocols offer additional liquidity mining rewards in a governance token (e.g., UNI). You may need to claim these from a separate staking contract.

Tip: The withdrawal transaction will finalize any impermanent loss. Always check that the current pool composition aligns with your expectations before withdrawing.