Why can a liquidity provider end up worse off than someone who just held?

This is the most misunderstood arithmetic in DeFi, and the misunderstanding is expensive. The claim is not that liquidity providers lose money. The claim is sharper and stranger: an LP can finish with less value than if they had done nothing at all — and the size of that shortfall is a fixed function of one variable, computable in advance.

Where it comes from

A pool must keep x × y = k. When the external price of X rises, the pool is briefly cheap, so arbitrageurs buy X out of it until the ratio matches. That is the mechanism working correctly — and it means the pool automatically sells the asset that is going up and buys the one that is going down. The LP does not choose this. It is the invariant.

Hold both tokens in a wallet instead and you keep every unit of the riser. That difference is impermanent loss, also called divergence loss — a better name, because there is nothing temporary about it once you withdraw.

Worked: a 2× move, in full

Deposit 10 ETH and 20,000 USDC at $2,000 per ETH.

Initial value = (10 × 2,000) + 20,000 = $40,000 k = 200,000

ETH doubles to $4,000. Arbitrage rebalances the pool until y/x = 4,000, subject to x·y = 200,000:

4,000x² = 200,000 → x² = 50 → x = 7.0711 ETH y = 28,284.27 USDC

(Check: 7.0711 × 28,284.27 = 200,000. ✓)

Pool value = (7.0711 × 4,000) + 28,284.27 = 28,284.27 + 28,284.27 = $56,568.54

Had you simply held = (10 × 4,000) + 20,000 = $60,000

Shortfall = $3,431.46, which is 3,431.46 ÷ 60,000 = 5.72% of the hold value.

The LP is up $16,568 in absolute terms and down 5.72% against doing nothing. Both statements are true simultaneously, and marketing material tends to feature only the first.

The closed form

Let r be the price ratio (new ÷ old). Writing V₁ = 2√(k·p₁) for the pool and V_hold = y₀(1+r) for the wallet, the constants cancel and you get:

IL = 2√r / (1 + r) − 1

At r = 2: 2 × 1.414214 ÷ 3 − 1 = 0.942809 − 1 = −5.72% — the figure derived above, with no reference to the deposit size. It depends on nothing but the price ratio: not on how much you deposited, not on the fee tier, not on how long you stayed.

Two properties worth internalising. It is symmetric: r = 0.5 gives exactly the same −5.72% as r = 2 (run the halving case numerically — the pool ends at 14.1421 ETH and 14,142.14 USDC, worth $28,284.27 against $30,000 held). And it is convex: it accelerates. Doubling costs 5.72%; a tenfold move costs 42.50%.

The four sentences people get wrong

1. "It's impermanent, so it reverses." Only if the price ratio returns to exactly where you entered. Withdraw at any other ratio and it is realised, permanently.
2. "It only matters if the price falls." No — it is symmetric, and a rise produces exactly as much divergence loss as the equivalent fall.
3. "Stablecoin pairs have none." They have none while the peg holds. When one side depegs, the pool has already sold you into the failing asset — the mechanism works against you precisely when it matters.
4. "The APY covers it." Sometimes. That comparison is arithmetic, not assumption, and it is the whole of the next lesson.

Nothing here suggests providing liquidity, and nothing here suggests avoiding it. It states what the invariant does to a position, which is a fact about the maths.

Try it now

1. A year of each asset is below. Measure the first to get r — the ratio of end price to start price — then compute IL from the closed form. Rebuild it the long way as well (rebalanced reserves, pool value, hold value) and confirm the two agree.

2. Now Measure the second chart over the identical window and repeat. Which pair would have produced more divergence loss? The answer is purely about the ratio between the two moves, not about either one on its own — check that your two numbers say so.
3. Compute IL for a pair where both assets doubled. Show that it is zero, and explain in one sentence why correlated moves cost a liquidity provider nothing at all.