Impermanent loss formula: how to calculate it, with a worked example

The impermanent loss formula

IL = 2 × √r ÷ (1 + r) − 1, with r = price ratio at the end ÷ price ratio at the start.

• r compares the two tokens, not one token with the dollar. If token A doubles against a stablecoin, r = 2. If both tokens double, r = 1 and there is no impermanent loss.
• 2√r ÷ (1 + r) is the value of the pool position divided by the value of holding. It is 1 at r = 1 and below 1 for every other r.
• IL is that ratio minus 1, so it is always zero or negative. A doubling costs 5.72%, a tripling 13.40% and a fivefold move 25.46%.

In Excel, with r in B2: =2*SQRT(B2)/(1+B2)-1. The formula assumes a 50/50 pool that keeps x · y = k constant, the design of the classic automated market makers, and it ignores trading fees.

Why the pool loses against holding

A constant-product pool always holds equal dollar values of both tokens. When token A rises, traders buy A from the pool until its price inside the pool matches the market, so the pool ends with less A and more of the stablecoin. You sold part of the winner on the way up. When A falls, the pool buys more of it on the way down. Either way you end with more of the token that did worse.

From the formula you can also derive the two values directly. For a deposit D split 50/50, value if held = D/2 × r + D/2 and value in the pool = D × √r. Dividing the second by the first gives 2√r ÷ (1 + r).

Worked example: $10,000 in a pool, token A doubles

You deposit $5,000 of token A and $5,000 of a stablecoin. Token A then doubles, so r = 2. Had you held the tokens in a wallet, the A half would be worth $10,000 and the stablecoin half still $5,000, a total of $15,000. In the pool the position is worth $10,000 × √2 = $14,142.14.

The impermanent loss is $14,142.14 − $15,000 = −$857.86, or −5.72% of the value if held. The position still gained $4,142.14 in dollars, which is why impermanent loss is easy to overlook: you made money, just less than holding would have made. To break even with holding, the fees earned over the period must reach $857.86, 8.58% of the original deposit.

Worked example: impermanent loss by price ratio and for a $10,000 deposit

The price ratios and the deposit are yellow inputs. Every other number is a formula. The tokens are invented.

Worked example: impermanent loss by price ratio and for a $10,000 deposit

	A	B	C	D
1	Price change of token A	Price ratio r	Pool ÷ holding	Impermanent loss
2	-75%	0.25	80.00%	-20.00%
3	-50%	0.50	94.28%	-5.72%
4	-25%	0.75	98.97%	-1.03%
5	0%	1.00	100.00%	0.00%
6	25%	1.25	99.38%	-0.62%
7	50%	1.50	97.98%	-2.02%
8	100%	2.00	94.28%	-5.72%
9	200%	3.00	86.60%	-13.40%
10	400%	5.00	74.54%	-25.46%
11
12	Dollar example: $10,000 deposited 50/50, token A doubles
13	Deposit D (half token A, half stablecoin), $			10,000
14	Price ratio r of token A			2.00
15	Value if held = D/2 × r + D/2			15,000.00
16	Value in the pool = D × √r (before fees)			14,142.14
17	Impermanent loss in dollars			-857.86
18	Impermanent loss in % = pool ÷ held − 1			-5.72%
19	Fees needed to break even, % of deposit			8.58%

Tokens, prices and the deposit are fictional and for teaching only. This is not investment advice.

Up or down: the loss is symmetric in the ratio, not in percent

The table shows that r = 2 and r = 0.5 give exactly the same loss, 5.72%, because the formula gives the same result for r and 1/r. A token that halves hurts as much as one that doubles. In percent terms that is not symmetric: a 50% fall matches a 100% rise, and a 75% fall (r = 0.25) costs 20%, the same as a fourfold rise.

Small moves cost little: a 25% rise costs 0.62% and a 25% fall 1.03%. This is why pools of two assets that move together, such as two stablecoins, have low impermanent loss, and why pools pairing a volatile token with a stablecoin need high fee income to compete with holding.

Why it is called impermanent, and when it becomes permanent

The loss depends only on the current price ratio. If the ratio returns to where it started, r goes back to 1 and the loss disappears, while the fees you collected stay. That is the reason for the name. When you withdraw while r is far from 1, the loss is locked in and is as permanent as any other realised loss.

Fees offset the loss, but they depend on trading volume, the fee tier and your share of the pool, and none of these is guaranteed. Concentrated-liquidity pools, where you provide liquidity only within a price range, earn more fees per dollar but suffer a larger impermanent loss inside that range, and pools with uneven weights such as 80/20 have a smaller one. The simple formula applies to the 50/50 full-range case only. Smart-contract risk and token risk come on top.

Common mistakes

• Using the dollar price change instead of the ratio between the two tokens. If both tokens rise 50%, r = 1 and there is no impermanent loss.
• Forgetting the −1. 2√r ÷ (1 + r) is the pool value relative to holding, about 94.3% at r = 2. The loss is that minus 1.
• Comparing the pool with the starting deposit. Impermanent loss compares the pool with holding the same tokens. The pool can be up in dollars and still show a loss.
• Assuming a fall is less harmful than a rise. A 50% fall (r = 0.5) costs exactly as much as a 100% rise (r = 2).
• Applying the 50/50 formula to concentrated or weighted pools. Range positions and 80/20 pools need their own formulas.
• Counting fees as certain. Fee income varies with volume and competition. Compare the loss with realistic, not best-case, fees.