RESEARCH NOTE R2 · Research notes · complete derivations
Impermanent loss: relative to what?
Separate absolute P&L, divergence from holding, and success against an investment mandate.
“LPs always lose” confuses a comparison with a result. “IL does not matter if I want the asset” makes the opposite mistake. A useful analysis identifies the benchmark first, then shows inventory, fees and costs separately. Your objective determines the decision; it does not change the arithmetic.
Jump to the interactive experiment ↓Define the comparison before the trade
Let P be USDC per ETH. Holding the initial token quantities gives H(P) = x₀P + y₀. Let V(P) be LP principal excluding fees. Define the signed divergence D = V − H, or IL% = V/H − 1. We report a negative number for underperformance; some sources quote its positive magnitude as “loss”. Compare identical starting capital, valuation time and currency. All-ETH, all-cash and an execution schedule are different benchmarks, not alternative definitions of the same IL.
Why a standard fee-free curve falls below holding
For a full-range constant-product position initially worth C, let r = P/P₀. With equal initial token values, V = C√r and H = C(1+r)/2, so IL% = 2√r/(1+r) − 1. This is nonpositive because the LP value curve is concave and the holding line is its tangent at entry. At r = 2 and C = 10,000, principal is 14,142.14 versus holding at 15,000: a 4,142.14 absolute gain AND an 857.86 holding shortfall. Concentrated positions require their own piecewise inventory formula, not this full-range shortcut.
- VLP: fee-free principal value; Vhold: value of the original tokens at the same price.
- r: ending price / entry price. Multiply the result by 100 for percent.
AssumptionsThe second identity assumes full-range constant product, fixed liquidity and initially equal value weights. Concentrated ranges require their own amounts.
Fees can change the total outcome
For an unchanged position with no interim capital flows, total result is V + F − costs − C; net excess versus holding is V + F − costs − H. Here F is all LP fees valued once at the evaluation price, not a promised yield. In the doubling example, hypothetical fees of 1,000 and costs of 50 give net profit 5,092.14 and a 92.14 advantage over holding. Actual fees depend on flow, competition, time in range and policy. If there are deposits or withdrawals, reconcile dated cash flows first; do not treat every withdrawal as profit.
“Impermanent” is a condition, not a promise
In the ideal fixed-liquidity, fee-free model, returning to the entry relative price restores the original inventory and removes endpoint divergence. The price may never return. Withdrawing or moving the range crystallizes that comparison at the time; gas, hedges and adverse execution do not disappear on a rebound. An endpoint returning to zero divergence also says nothing about the cumulative opportunity cost along the path.
Judge the mandate without hiding the shortfall
An ETH accumulator may prefer receiving ETH on a decline to staying entirely in USDC. Record ETH acquired, average acquisition cost, remaining budget and drawdown, alongside the cash/holding comparison. A staged seller may accept limited upside to reduce exposure. A neutral desk may benchmark against a hedged strategy instead. Set these objectives before entry; changing the benchmark after a loss is not risk management. Wanting inventory can reduce a hedging requirement, but cannot make its market risk or capital opportunity cost vanish.
TRY IT YOURSELF
Make the mechanics visible.
Compare the same concentrated position with initial-token holding, cash or all ETH. Change the end price and independently assumed fees/costs. The signed IL stays tied to initial-token holding even when you switch the decision benchmark.
P₀ = 3,000 · [a,b] = [2,400, 3,600] · C = 10,000 USDC
Fee income is an independent assumption, held at the same USDC value across the chart. It is not inferred from the price path. No interim flows, compounding or benchmark trading costs; all values use the same ending price. Cash and all-ETH alternatives change initial allocation. Only the entry-token comparison is labeled IL.
CHECK YOUR UNDERSTANDING
What does this mean for you?
Before adding capital, would you write your objective in ETH units, USDC value, a target inventory mix, or hedged P&L? What opportunity cost will you still report?
CONNECT IT TO AMM MARKETS
From understanding to a real decision.
Use AMM Markets Portfolio to distinguish principal, unclaimed fees and settled history. Agree the objective before choosing a market or discussing a bespoke policy. This experiment is not a live portfolio return calculator.
Compare with the Liquidity Lab ↗Educational simulation, no real funds.References and scope
Course edition: October 2026. Distinguish protocol principles from product instructions; actual market parameters come from the live interface and on-chain state.