Constant Mean Formula (Balancer Pool)

The formula is expressed as ∏ (B_k)^{w_k} = k, where B_k is the balance of token k and w_k is its normalized weight (summing to 1). This invariant must hold for all trades, ensuring the weighted geometric mean of the token balances remains constant. It is a superset of other AMM models: a 50/50 two-token pool is the classic Constant Product Market Maker (CPMM), while a pool with equal weights for all tokens is a Constant Mean Market Maker (CMMM).

Unlike fixed 50/50 pools, each asset in a Balancer pool is assigned a target weight (e.g., 80/20, 33/33/33). This allows pools to be optimized for specific use cases:

• 80/20 Pools: For governance token/stablecoin pairs, where liquidity providers (LPs) maintain a higher exposure to the volatile asset.
• Index Pools: To create token baskets that automatically rebalance as users trade against the pool, acting as a passive portfolio manager. The pool's smart contract enforces these weights through arbitrage incentives.

The formula natively supports pools with 2 to 8 assets. This enables complex, capital-efficient liquidity environments where a single pool can facilitate trades between multiple token pairs. For example, a pool containing WETH, USDC, and BAL allows direct swaps between any pairing (WETH/USDC, WETH/BAL, BAL/USDC) without routing through intermediate pools. This reduces slippage and fee costs for traders interacting with the pool's constituent assets.

The spot price for a swap is derived from the ratio of the tokens' balances and their weights. The price impact of a trade is determined by how much it moves the pool away from its target weights. Key mechanics:

• Arbitrage: Drives the pool back to its equilibrium weights, which align with external market prices.
• Swap Fee: A percentage (e.g., 0.05%) is taken from the input amount of each trade and added to the pool's reserves, accruing value for LPs.
• Formulaic Slippage: Larger trades cause greater price impact as they distort the invariant more significantly.

Liquidity providers deposit assets in precise proportion to the pool's target weights. In return, they receive Balancer Pool Tokens (BPT), which represent their share of the pool. Fees are automatically reinvested into the pool, increasing the value of each BPT. The customizable weights allow LPs to express a specific investment thesis, such as holding a higher percentage of a governance token while earning fees on its trading volume.

The Constant Mean Formula is part of a family of AMM invariants:

• Constant Product (x*y=k): Used by Uniswap V2. A special case of Balancer's formula with two assets at 50/50 weights.
• Constant Sum (x+y=k): Used by stablecoin pools (like Curve's StableSwap). Provides zero slippage at pegs but can be drained if the peg breaks.
• Hybrid Invariants: Protocols like Curve combine constant sum and constant product for low-slippage stablecoin swaps, which is a different approach than Balancer's generalized mean.

The Constant Mean Formula is the mathematical rule that defines the relationship between token balances in a Balancer pool. Unlike Uniswap's constant product formula (x*y=k), it generalizes to any number of tokens with arbitrary weights. The invariant is: ∏ (Balanceᵢ)^Weightᵢ = k, where the product of each token's balance raised to its weight remains constant. This allows for pools with more than two assets and non-50/50 weightings.

A key innovation is the ability to assign a weight (e.g., 80/20, 50/30/20) to each token in the pool. This weight determines the pool's target composition and influences price impact. For example, an 80/20 WBTC/WETH pool will be less sensitive to trades in the higher-weighted asset (WBTC), creating a different trading curve than a standard 50/50 pool. Weights are set by the pool creator and can be static or managed by smart contracts.

The formula natively supports pools with 2 to 8 tokens. This enables complex, capital-efficient liquidity provision for correlated assets, such as a stablecoin pool (DAI, USDC, USDT) or an index-like pool of DeFi governance tokens. Liquidity providers (LPs) deposit all tokens in the pool's target weight proportions, earning fees from all trading pairs within the single pool.

The spot price between any two tokens in the pool is derived from their balances and weights. Price impact for a trade is calculated based on how much the trade moves the balances away from their weighted targets. The formula ensures that the pool always has liquidity for all its tokens, but trades that significantly imbalance the pool face higher slippage, incentivizing arbitrageurs to restore the target weights.

Balancer pools can charge a swap fee (e.g., 0.05% to 1%) on every trade, which is distributed proportionally to LPs. Additionally, protocol fees can be enabled, diverting a portion of the swap fee to the Balancer DAO treasury. For Boosted Pools that hold yield-bearing tokens (like Aave aTokens), the formula interacts with the underlying yield, allowing LPs to earn both trading fees and underlying yield.

Trades are executed via the Balancer Vault, which holds all pool assets. The Smart Order Router splits a single trade across multiple pools to find the best execution price. For instance, a swap from DAI to BAL might route through a DAI/WETH pool and then a WETH/BAL pool, or directly through a multi-token pool containing all three assets, optimizing for lowest price impact.

Unlike a Constant Product Market Maker (CPMM) where IL is symmetrical, a Balancer pool's IL profile is weighted by token proportions. In an 80/20 pool, the token with the higher weight experiences amplified IL relative to the lower-weight token during price divergence. This creates a non-linear risk profile that LPs must model.

• Example: In an 80/20 ETH/DAI pool, a 2x ETH price increase causes ~3.6% IL for the ETH side but only ~0.9% for the DAI side.

The entity deploying a Balancer pool controls critical parameters that define its security and incentive structure:

• Swap Fee: Set by the creator, this percentage is taken from each trade and distributed to LPs. It must balance attracting volume with LP profitability.
• Protocol vs. Pool Fees: Balancer v2 introduces a protocol fee (set by governance) that can be taken from the swap fee, creating a two-tier fee structure.
• Weight Updates: Creators can propose gradual weight updates (subject to timelocks) to rebalance the pool's strategy, which is a governance and economic action.

Custom weightings can make pools targets for specific economic attacks.

• Weight Manipulation: An attacker could execute large swaps to temporarily skew weights far from their equilibrium, creating arbitrage opportunities at the expense of LPs, especially in low-liquidity pools.
• Flash Loan Exploits: The formula's predictability can be combined with flash loans to manipulate prices across multiple pools in a single transaction, although Balancer v2's asset management reduces this risk.
• Pool Drain via Donation: As with any AMM, donating large amounts of a single token to skew the ratio can be used to drain value (see donation attacks).

The ability to create pools with high concentrations (e.g., 95/5) introduces concentrated risk:

• Capital Efficiency vs. Risk: High-weight assets are extremely capital efficient for swaps but expose LPs to massive IL if that asset's price moves.
• Oracle Reliability: Pools often serve as price oracles. A pool with 95% weight in one asset provides a less robust, more manipulable price feed for the minority asset.
• Exit Liquidity: For the low-weight token, the pool provides very little exit liquidity, making large sells highly impactful.

Balancer's implementation adds layers of complexity that must be audited and trusted.

• Vault Architecture (v2): All pool assets are held in a single, central Vault contract. A critical bug in this contract jeopardizes all pools.
• Governance Control: Key parameters, including protocol fees and the whitelist for liquidity gauges (for token emissions), are controlled by Balancer Governance (BAL) token holders.
• Timelocks & Pauses: Administrative functions are typically behind timelocks, and pools can be paused by governance in an emergency.

The economic model for LP profitability depends on multiple, sometimes competing, flows:

• Swap Fees: The primary source of yield, proportional to trading volume and the pool's fee setting.
• BAL Liquidity Mining: Pools may be eligible for BAL token emissions, distributing governance tokens as incentive. Emissions are directed via liquidity gauges.
• Protocol Fee Drag: The protocol fee (if activated) is a direct deduction from LP swap fee earnings.
• Tokenomics of Underlying Assets: In pools containing rebasing or fee-generating tokens (e.g., staked assets), LPs may also accrue additional native rewards.