Liquidity Pool Calculator

Providing liquidity is a trade of one exposure for another: the provider gives up the payoff of simply holding two assets and receives a stream of trading fees plus a payoff that is short volatility. Fee income is mechanical. Total fees equal daily volume times the fee rate, and a provider earns them in proportion to pool share, which is deposit divided by total value locked after the deposit. This is why the ratio of volume to TVL, not TVL alone, drives yield: a pool with 5 million USD locked and 5 million USD of daily volume at 0.30 percent returns roughly 109 percent annualized before losses, while the same fee rate on a pool turning over one tenth of its depth per day returns about 11 percent. The offsetting term comes from the constant product invariant. Because the pool must always satisfy x times y equal to k, arbitrageurs rebalance it toward the market price, leaving the provider systematically heavier in the asset that fell. The resulting divergence loss, universally called impermanent loss, is 2 times the square root of r divided by 1 plus r, minus 1, where r is the price ratio change. It depends only on that ratio and is convex: about 0.6 percent at a 1.25x move, 2.0 percent at 1.5x, 5.7 percent at 2x, 20.0 percent at 4x and 25.5 percent at 5x. The label is misleading, since the loss only reverses if prices return to the entry ratio. Note also that this model prices a V2-style full-range position; a concentrated V3 range earns more fees per unit of capital but suffers considerably worse divergence loss and stops earning entirely once price leaves the range.

Automated market making on-chain was proposed before it was practical. Bancor described a constant reserve ratio formula in its 2017 whitepaper and raised roughly 153 million US dollars in June 2017 to build it, but the design that won was the simpler constant product rule sketched in Vitalik Buterin posts around 2016 and 2017 and implemented by Hayden Adams in Uniswap V1, deployed in November 2018. The underlying mathematics is older still: the constant product form is a Cobb-Douglas trading function, and logarithmic market scoring rules for prediction markets, formalized by Robin Hanson in 2003, are a direct intellectual ancestor. The cost side of liquidity provision was quantified publicly in mid-2019, when an analyst writing under the pseudonym Pintail published a Medium study asking whether Uniswap was a good deal for liquidity providers, deriving the divergence formula and popularizing the term impermanent loss. Curve, launched by Michael Egorov in January 2020, attacked the problem from the other direction with a stableswap invariant that flattens the curve near parity so that pegged assets suffer almost no divergence, which is why stablecoin pools can run 0.01 to 0.05 percent fees profitably. Uniswap V3 in May 2021 introduced concentrated liquidity, turning passive provision into an active position-management problem. Later academic work reframed the loss as loss-versus-rebalancing, arguing it is better understood as value paid to arbitrageurs than as an accounting artifact.