Why AMMs Create Impermanent Loss: The Constant Product and Arbitrage

The companion calculator computes impermanent loss from how far a price has moved. That loss can feel mysterious, why should providing liquidity leave you worse off than simply holding? The answer lies in how automated market makers actually work: a simple mathematical rule that rebalances the pool, enforced by arbitrage traders, that systematically has the pool sell whatever is rising and buy whatever is falling. Understanding the constant-product mechanism and the arbitrage behind it demystifies impermanent loss and shows it to be a structural feature, not an accident. This is educational background on how the mechanism works, not financial advice; cryptocurrency is highly volatile and risky, and any figures are illustrative.

The Automated Market Maker

Traditional exchanges match buyers and sellers through an order book. An automated market maker, or AMM, works differently: it holds a pool of two tokens and lets anyone trade against that pool at a price set by a formula, no counterparty needed. Liquidity providers supply the two tokens to the pool and earn a share of trading fees. The pool always offers a price, and that price is determined entirely by the ratio of the two tokens it holds. This design is what makes decentralized trading possible without an order book, and it is also the root of impermanent loss.

The Constant-Product Rule

The most common AMM keeps the product of the two token quantities constant: multiply the amount of token A by the amount of token B, and that number must stay fixed as people trade. To buy some of token A from the pool, you must add enough token B to keep the product unchanged, which is what sets the price and moves it as the pool's balance shifts.

Because the formula forces the pool to give up whichever token is becoming more valuable and accumulate whichever is becoming less valuable, the pool automatically sells winners and buys losers, the opposite of what a holder who simply sat tight would have done.

Arbitrage Is the Enforcer

What actually makes the pool rebalance is arbitrage. When an asset's price moves on the wider market, the AMM's formula-set price briefly lags, so arbitrage traders step in to buy the underpriced token from the pool or sell the overpriced one to it, pocketing the difference, until the pool's price matches the market again. This arbitrage is essential, it keeps the pool's prices honest, but it is precisely the process that drains the appreciating token out of the pool. The liquidity provider effectively pays the arbitrageurs, and that payment is impermanent loss. It is the cost of the pool always offering a slightly stale price that arbitrageurs correct.

Why "Impermanent," and When Fees Win

The loss is called impermanent because it only becomes real if you withdraw while prices have diverged, if the price ratio returns to where you started, the loss disappears. But if you exit after a large divergence, it is realized as a permanent shortfall versus holding. Crucially, impermanent loss is only half the ledger: liquidity providers earn trading fees, and in a high-volume pool those fees can offset or exceed the impermanent loss. Whether providing liquidity pays depends on the race between fees earned and impermanent loss incurred, which is why stable, correlated pairs, with little price divergence, are popular for liquidity provision.

Providing Liquidity With Eyes Open

Use the calculator to quantify the impermanent loss for a given price move, and understand it as the built-in cost of the AMM design: the constant-product rule makes the pool sell winners and buy losers, and arbitrage enforces it. Weigh that loss against the fees a pool earns, and recognize that volatile, diverging pairs carry the most impermanent loss while correlated pairs carry the least. The formula measures the loss; understanding the AMM is what tells you why it happens and when fees make it worthwhile.