Automated market makers and impermanent loss, with the arithmetic
How a constant-product pool prices a trade, why a liquidity provider ends up short of simply holding, and the exact size of the shortfall.
An automated market maker replaces the order book with a formula: a pool holds two assets, and the ratio in which it holds them determines the price at which it will trade. Nobody quotes, nobody cancels, and the pool will always take the other side of a trade, which is the source of both its usefulness and the losses its depositors absorb. The loss has a closed form, and knowing it is the difference between measuring a position and guessing at it.
How the formula prices a trade
The most common design is the constant product formula, written x * y = k, where x and y are the quantities of the two assets held by the liquidity pool. A trade may change the composition of the pool but not the product, so buying x means supplying enough y to keep k intact. The marginal price implied by the pool is simply y / x, and the two reserves at any price p are x = sqrt(k/p) and y = sqrt(k*p).
Because the product is fixed, a large trade walks the price along a curve rather than executing at a single number. The gap between the marginal price before the trade and the average price actually paid is price impact, and it grows with trade size relative to the pool. This is why a quoted price on a decentralized exchange is only meaningful next to a size, and why slippage limits are part of every transaction. A pool never runs out of either asset in the arithmetic sense, since the price approaches infinity as one reserve approaches zero, but it becomes unusable long before that.
The shortfall against holding, computed exactly
A liquidity provider deposits both assets and receives a claim on a share of the pool. If the external price of one asset moves, arbitrage traders trade against the pool until its implied price matches the outside market. They buy the asset the pool is now underpricing, which means the pool ends up holding less of whichever asset rose and more of whichever fell. That mechanical rebalancing is not a mistake; it is how the price gets corrected, and the provider pays for it.
Let r be the ratio of the new price to the old price. The pool value relative to simply holding the original two amounts is
value ratio = 2 * sqrt(r) / (1 + r)
and impermanent loss is that ratio minus one. It is always negative for any r other than one, and it is symmetric: a halving and a doubling produce the same shortfall.
| Price ratio r | Value if simply held | Shortfall |
|---|---|---|
| 0.80 or 1.25 | 0.9938 | -0.62% |
| 0.67 or 1.50 | 0.9798 | -2.02% |
| 0.50 or 2.00 | 0.9428 | -5.72% |
| 0.33 or 3.00 | 0.8660 | -13.40% |
| 0.25 or 4.00 | 0.8000 | -20.00% |
| 0.20 or 5.00 | 0.7454 | -25.46% |
| 0.10 or 10.00 | 0.5750 | -42.50% |
Worked through with quantities: a pool holds 10 units of a volatile asset and 20,000 units of a stablecoin, so k = 200,000 and the implied price is 2,000. The price doubles to 4,000. The new reserves are sqrt(200,000 / 4,000) = 7.0711 units and sqrt(200,000 x 4,000) = 28,284.27 stablecoins, worth 56,568.54 in total. Holding the original 10 units and 20,000 stablecoins would have been worth 60,000. The gap is 3,431.46, or 5.72 percent, exactly as the formula says.
The name is misleading and the better term is divergence loss. Nothing about it is temporary once a position is withdrawn; it only reverses if the price ratio returns to where it started while the provider is still in the pool. The economics are those of a short volatility position: the provider is systematically selling what is rising and buying what is falling, and the trading fees are the premium collected for doing so.
Fees, and the harder benchmark
The provider's return is fee income minus divergence loss, and fee income scales with volume rather than with price movement. The ratio to watch is therefore turnover against the size of the pool: DEX volume relative to value locked says how hard each unit of deposited capital is working, and a pool with a high fee tier but no flow earns nothing. Comparing daily DEX volume and total value locked across venues is the same comparison at the protocol level.
Concentrated liquidity lets a provider allocate capital only within a chosen price range, which multiplies both fee income and divergence loss within that range and pays nothing at all once the price leaves it. It converts a passive position into an active one that requires a view on where the price will trade, and it makes the simple two-asset formula above a lower bound on the loss rather than the answer.
There is also a benchmark stricter than holding. Divergence loss compares the pool against a passive holder, but the deeper cost is that the pool quotes a stale price between trades and informed traders take the difference. Measured against a portfolio that continuously rebalanced at market prices, the shortfall is larger, and it is a transfer to arbitrage traders rather than a market movement. This is closely related to MEV: the right to be the first to correct a stale pool price is valuable, and it is auctioned in the block-building market rather than earned by the pool.
What the design does well
Constant product pools are not a poor imitation of an order book; they solve a different problem. They provide a price for assets too small or too new for a professional market maker to quote, they need no active management to stay live, and their state is public, which makes routing and aggregation trivial. Pools for assets expected to trade near a fixed ratio, such as two stablecoins, use curves that keep the price flat over a wide band and only steepen at the edges, which is why depth for those pairs can be very good and why the same pools deteriorate suddenly during a depeg.
Fee income and the shortfall against holding are both measurable after the fact, which is more than can be said for most trading strategies. The DEX activity pages carry venue-level volume and turnover, the value locked pages show where the deposits sit, and the yield article next in this track separates trading fees from token incentives, which are frequently reported together as a single number.