RESEARCH NOTE R4 · Research notes · complete derivations
Distinguish fee income from trading edge.
Connect average execution, path-dependent rebalancing costs and fee demand without double-counting losses.
Two pools with the same volume can deliver different LP results. The difference can come from when trades arrive, which direction inventory moves, and how much value is transferred while quotes catch up. A professional desk evaluates the quality of flow, not just its size.
Jump to the interactive experiment ↓Marginal price is not average execution
In a fee-free full-range pool starting at x = 100 ETH and y = 270,000 USDC, spot is y/x = 2,700. A trader buying 10 ETH leaves 90 ETH, so the new USDC reserve is 27,000,000/90 = 300,000. The trader pays 30,000: average execution 3,000, ending marginal price 3,333.33. Selling 10 ETH instead pays out 24,545.45 USDC, averaging 2,454.55, ending at 2,231.40. These are separate trades from the same start. Price impact is curve movement; slippage tolerance is a transaction constraint, not a fee.
The benchmark behind LVR
Loss-versus-rebalancing compares AMM principal with a hypothetical self-financing portfolio that follows its inventory but trades at external market prices. It isolates an adverse-selection cost under its modeling assumptions. It is not the initial-token holding benchmark used by IL. The research model assumes idealized trading and arbitrage; an actual hedge pays costs and cannot trade continuously. Use the distinction as an analytical lens, not a promise of achievable benchmark returns.
Same ending price, different path cost
For a discrete toy comparison, let xᵢ be LP ETH inventory at Pᵢ. A comparator starting with the same capital first earns xᵢ₋₁(Pᵢ−Pᵢ₋₁), then rebalances at Pᵢ without fees. Its final value minus LP principal is the illustrated rebalancing gap. A round trip can return the LP to its initial inventory, giving zero endpoint IL, while the comparator ends ahead. This discrete calculation is not an estimator from live swap logs, and finite step sizes matter.
Local curvature explains the direction
For a small step away from boundaries, xδP − δV ≈ −½Γ(δP)², which is nonnegative for a standard LP. The continuous, frictionless analogue integrates curvature against price quadratic variation. This connects short-convexity exposure to path cost; it does not say realized fee income is fixed. Fees, inventory returns, execution quality, capital costs and any hedges must all be evaluated on a consistent basis.
Do not subtract IL and LVR twice
If you already value every asset and reconcile contributions, withdrawals and fees, trading losses are already embedded in actual P&L. IL and LVR are alternative decompositions against different benchmarks, not extra cash expenses to subtract again. For ADPP research, test whether a fee or bound change improves net results and execution quality out of sample. Raising fees may reduce demand; a boundary may reject trades but leave inventory exposed. Policy protection at the Router concerns consistency between quote and execution, not elimination of LP adverse selection.
TRY IT YOURSELF
Make the mechanics visible.
Compare a flat path with a price round trip ending at the same 3,000. Follow the inventory-matched comparator step by step. Add a hypothetical net fee amount and distinguish fee-free divergence, rebalancing gap and actual illustrative P&L.
| Step | ETH inventory | LP principal | Comparator value | Gap · USDC |
|---|---|---|---|---|
| 0 | 1.50715 | 10,000.00 | 10,000.00 | 0.00 |
| 1 | 0.70215 | 10,328.52 | 10,452.14 | 123.63 |
| 2 | 1.50715 | 10,000.00 | 10,241.50 | 241.50 |
Rᵢ = Rᵢ₋₁ + xᵢ₋₁(Pᵢ−Pᵢ₋₁). The comparator then rebalances to xᵢ at Pᵢ. Each path step assumes arbitrage has occurred; external prices alone do not move reserves. This is a discrete illustration, not continuous-time LVR or a backtest. Fees are independently assumed, not generated by the path.
CHECK YOUR UNDERSTANDING
What does this mean for you?
Would a fee increase still improve your market if eligible volume halved? What evidence distinguishes informed flow from uninformed flow?
CONNECT IT TO AMM MARKETS
From understanding to a real decision.
Read AMM Markets depth, price bounds and directional fees together. These tools describe constraints and capacity; assessing a policy requires historical evidence and a clear counterfactual, not just high fee APR.
Inspect market policy and depth ↗Opens a live service. Exploring is separate from submitting a transaction.References and scope
- Milionis et al. · Loss-Versus-Rebalancing (2024 version) ↗
- Uniswap v3 whitepaper · §§2–2.1 ↗
- Uniswap v4 whitepaper · hooks and custom accounting ↗
Course edition: October 2026. Distinguish protocol principles from product instructions; actual market parameters come from the live interface and on-chain state.