Impermanent loss, explained arithmetically

Impermanent loss is the shortfall a liquidity provider suffers when the prices of the two assets in an automated market maker pool diverge, compared with simply holding those assets. It is not a fee and not a hack — it is a direct, calculable consequence of the constant-product formula. This article derives it from first principles with exact numbers.

The name is slightly misleading: the loss is “impermanent” only in the sense that it shrinks back toward zero if prices return to where they started. If you withdraw while prices have diverged, the loss is entirely real. We will build the arithmetic step by step, show the standard closed-form formula, and tabulate it — mechanics only, no view on whether providing liquidity is worthwhile.

The setup: a constant-product pool

Recall the constant-product rule from how automated market makers work: a pool holds reserves x and y of two assets, and every trade keeps

x × y = k

The pool’s price of asset X in terms of Y is the reserve ratio, P = y / x. Two facts follow that we will use repeatedly. Given a price P and the constant k, the reserves are pinned down:

• x = √(k / P)
• y = √(k × P)

You can verify these satisfy both x × y = k and y / x = P. The important consequence: as the price rises, the pool automatically holds less of the asset that went up and more of the one that went down. That automatic rebalancing is the seed of impermanent loss.

A worked example, step by step

Take a pool of 10 ETH and 10,000 USDC, with USDC as the unit of account.

• k = 10 × 10,000 = 100,000
• Starting price P₀ = 10,000 / 10 = 1,000 USDC per ETH
• Starting pool value = 10 × 1,000 + 10,000 = 20,000 USDC

Now suppose the market price of ETH doubles to 2,000 USDC. Arbitrageurs will trade against the pool until its internal price also equals 2,000. Using the formulas above with P₁ = 2,000:

• New ETH reserve: x = √(100,000 / 2,000) = √50 = 7.0711 ETH
• New USDC reserve: y = √(100,000 × 2,000) = √200,000,000 = 14,142.14 USDC

Check: 7.0711 × 14,142.14 = 100,000 ✓ and 14,142.14 / 7.0711 = 2,000 ✓. The pool sold ETH into the rally, exactly as the formula forces.

Now value the two outcomes at the new price of 2,000:

The liquidity provider ends with 28,284.27 versus 30,000 for someone who did nothing. The difference is the impermanent loss:

28,284.27 / 30,000 − 1 = −0.0572 = −5.72%

The pool position is worth about 5.72% less than simply holding. Note it did not fall in absolute terms — it grew from 20,000 to 28,284 — but it grew less than holding would have, because the pool kept selling ETH as it rose. Impermanent loss is always relative to holding, never a claim that the position lost nominal value.

The general formula

You do not have to redo the reserves every time. Let r be the price ratio, r = P₁ / P₀ (here r = 2,000 / 1,000 = 2). The impermanent loss for a 50/50 constant-product pool is:

IL(r) = 2√r / (1 + r) − 1

Plugging in r = 2: 2√2 / (1 + 2) − 1 = 2 × 1.41421 / 3 − 1 = 2.82843 / 3 − 1 = 0.94281 − 1 = −5.72% — exactly matching the worked example. The formula is standard and appears throughout the AMM literature, including analyses building on the Uniswap v2 whitepaper.

Two properties are worth internalising. First, IL is always negative (or zero) for any r ≠ 1 — divergence in either direction hurts relative to holding. Second, it is symmetric in the ratio: a halving of price (r = 0.5) produces the same −5.72% as a doubling, because 2√0.5 / 1.5 − 1 also equals −5.72%.

Impermanent loss at a glance

Evaluating the formula at several price ratios gives the reference table every liquidity provider should know:

These are illustrative outputs of the formula, not observed market data. The shape is the lesson: loss is mild for small moves and accelerates as divergence grows. A pool of two assets that stay close in price (say two dollar-pegged stablecoins) suffers almost no impermanent loss, which is why stable-pair pools exist; a pool of two assets whose ratio can swing wildly carries far more.

Where the formula comes from

The closed form is not magic; it drops out of the two facts we already established. Assume you provide the whole pool. Using x = √(k / P) and y = √(k × P), the value of the pool position at any price P is:

V_pool = x × P + y = √(k / P) × P + √(k × P) = 2√(k × P)

So the pool’s value scales with the square root of the price — it rises when the price rises, but only at a square-root rate. The HODL value, by contrast, holds the original reserves x₀ and y₀ and scales linearly with price. Divide one by the other, substitute the price ratio r = P₁ / P₀, and the k terms and starting reserves cancel to leave exactly:

V_pool / V_hodl = 2√r / (1 + r)

The impermanent loss is that ratio minus one. The square-root growth of the pool versus the linear growth of holding is the whole story: because the pool sheds the winning asset as it climbs, its value can never keep pace with simply holding, and the gap widens as the price ratio moves away from one in either direction.

Fees: the other side of the ledger

Impermanent loss is only half of a liquidity provider’s economics. Against it sits the trading-fee income earned on every swap the pool processes. Whether providing liquidity comes out ahead depends on whether accumulated fees exceed the impermanent loss realised when the position is closed — and that depends on volume, volatility, and how far prices ultimately diverge. This article deliberately does not tell you when that trade is worth making; it only equips you to compute the loss side exactly.

Common mistakes

• Thinking “impermanent” means “not real.” If you withdraw while prices have diverged, you lock in the loss. It is only unrealised until then.
• Confusing it with a nominal loss. IL is measured against holding, not against your starting value. Your position can grow and still exhibit impermanent loss.
• Ignoring the acceleration. The loss is trivial for small moves but grows steeply — a 4× move already costs 20% relative to holding.
• Forgetting the fee offset. Quoting impermanent loss without the fees earned tells only half the story.

What this means

Impermanent loss falls straight out of x × y = k: to keep the product constant, the pool must sell whatever rises and buy whatever falls, so a provider always underperforms holding when prices diverge — by 5.72% for a 2× move, symmetric up or down, and worse as the gap widens. The arithmetic is exact and the formula is short: IL(r) = 2√r / (1 + r) − 1. Learn to compute it, treat it as the cost side against fee income, and the economics of being a liquidity provider stop being mysterious.