Tools
Liquidity Provision Fee Break-even Calculator
How much fee income does a pooled two-asset position have to earn before it matches simply holding the two amounts, once the price ratio between them has moved?
Supplying two assets to a pool is not the same as holding them. The pool is rebalanced continuously by whoever trades against it, so it is always selling whichever asset is rising — and over any move in the ratio between the two, that ends up worth less than simply having held them. Fee income is what closes the gap. This calculator says how much fee income closes it exactly, and how far the ratio can move before a fee income you state stops covering it.
What this tool does not decide
- Whether to provide liquidity. A break-even is a price, not a verdict. Fee income above it does not make a position worth holding and fee income below it does not make it a mistake.
- How likely any move is. No probability appears anywhere on this page. You state where the ratio ended; the model says what that cost.
- What any venue pays. This tool ships no pool, protocol, token or published yield, names none, and fetches none. Both figures are yours — which is also why nothing on this page can quietly go out of date.
- What the position actually cost. Gas, slippage, deposit and withdrawal cost, fee tiers, concentrated ranges and incentives are all outside the arithmetic. The break-even is a floor.
- Whether either asset is worth holding at all. The comparison is between two ways of holding the same two assets, so a view about the assets themselves is not in it.
- Anything about tax, in any jurisdiction.
Your figures
Your position
Two numbers, both yours. Nothing is sent anywhere, nothing is stored, and no rate, price or venue is looked up.
These entries need attention before the calculation can run
A factor, not a percentage. Enter 1 if the two assets have moved together and the ratio is unchanged. Enter 2 if one has doubled against the other, or 0.5 if one has halved against it — both give the same answer, because the arithmetic is symmetric.
Result
Jump to resultEnter your two figures above and select Calculate. Nothing is sent anywhere: the calculation runs in this browser, and no value is stored, shared or placed in the address bar.
- Value of the pooled position, against simply holding
- Not yet calculated
- Divergence before any fee income
- Not yet calculated
- Fee income you entered
- Not yet calculated
- Where that leaves you against simply holding
- Not yet calculated
- Result
- Not yet calculated
This is arithmetic on two numbers you enter, not an assessment of a position. It compares the value of a rebalanced pooled position against simply holding the two amounts, and it cannot see the risks that decide whether the position is a sensible one at all.
A break-even is not a verdict. Fee income above it does not make a position worth holding and fee income below it does not make it a mistake — what the position risks, and what it would have cost to do something else, are not quantities on this page.
How this is worked out
What is being compared
Two ways of holding the same two assets through the same price move. Every figure this tool reports is the difference between them.
| Option | What it is | What it is worth at the end |
|---|---|---|
| Holding | The two original amounts, kept as they are | Their value at the end is whatever the two assets are then worth |
| The pooled position | The same two amounts, supplied to a pool that rebalances continuously | Its value at the end is set by the ratio, plus whatever fee income it earned |
Why a pooled position falls behind at all
It is not a fee and not a penalty. A pool is rebalanced by whoever trades against it, which means it is continuously selling whichever asset is rising and buying whichever is falling. Over any move that ends somewhere other than where it started, that is a worse outcome than having held the two amounts — and it is worse for every move, in either direction, with the two exactly equal only where the ratio is unchanged.
The shortfall is symmetric: a ratio that ends at four and a ratio that ends at a quarter produce the same figure, because the identity does not know which of the two assets a reader thinks of as having moved.
The identity
Writing k for the factor the price ratio ended at, the pooled position is worth this share of simply holding, before any fee income:
pooled ÷ held = 2√k ÷ (1 + k)That quantity is at most one for every positive k, and exactly one only atk = 1. Requiring fee income to close the gap gives the figure this page reports:
break-even fee income = (1 + k) ÷ (2√k) − 1Read the other way round, a stated fee income covers a band of ratios rather than a single one. The band's two endpoints multiply to exactly one, which is the same symmetry arriving from a different direction: whatever ratio is the bottom of the band, its reciprocal is the top.
What you enter
- Price ratio
- A factor. 1 is unmoved; 2 is one asset having doubled against the other
- Fee income
- A share of the position, over the same period as the move
What is assumed
- The position starts with equal value in each of the two assets.
- The pool rebalances continuously under a constant product, which is what the closed form describes.
- Only where the ratio ended matters. The path it took between the two points does not enter the arithmetic.
- Fee income is stated as a share of the position over the same period, and is treated as earned rather than projected.
- Both assets continue to trade. An asset that becomes worthless is a boundary rather than a ratio this identity has content at.
What is not modelled
Each of these moves the answer, and none of them is in the arithmetic. A break-even computed without them is a floor rather than a full account of what the position costs.
- the price path between the start and the end — only where the ratio ended matters here
- gas, slippage, deposit and withdrawal cost, and any transaction fee
- a concentrated range, a fee tier, or any venue-specific rule
- rewards, incentives or emissions paid in a third asset
- the possibility that either asset becomes worthless or ceases to trade
- tax, in any jurisdiction
The nearest thing this site publishes on a headline yield that survives less than it appears to is the staking rewards, fees and dilution explainer. It is a different mechanism and the same shape of question.
Calculation model and corrections
- Calculation model
- Liquidity Position Break-even v1.0
- Last reviewed
- The comparison
- Against holding the two original amounts, unpooled, through the same price move
- Starting position
- Equal value in each of the two assets, which is what the identity assumes
- Price ratio
- Where the ratio ended, as a factor. 1 is unmoved; 2 is one asset having doubled against the other
- Direction
- Symmetric — a ratio of 4 and a ratio of 0.25 give the same divergence
- Fee income
- A share of the position, over the same period, entered by you
- Rebalancing
- Continuous, under a constant product, which is what the closed form describes
- Break-even
- The fee income at which the two are exactly equal, reported before any cost of transacting
- Excluded
- Gas, slippage, deposit and withdrawal cost, fee tiers, concentrated ranges, incentives and tax
- Rounding
- Display only; intermediate values remain unrounded