Impermanent Loss: Mathematical Explanation and Practical Impacts

Detailed Instructions

Begin with the constant product formula for a Uniswap V2-style AMM: x * y = k, where x and y are the reserves of two assets (e.g., ETH and USDC) and k is a constant. Define the initial state: let P be the initial price of asset X in terms of Y, so P = y / x. This implies the initial reserves are x and y = Px. The liquidity provider's initial portfolio value in terms of asset Y is V_pool,initial = Px + y = 2Px.

• Sub-step 1: Write the invariant: x * y = k.
• Sub-step 2: Express price P as the ratio of reserves: P = y / x.
• Sub-step 3: Calculate initial portfolio value by converting all assets to a common denomination (Y).

pythonCopy
# Initial conditions
x_initial = 10  # e.g., 10 ETH
y_initial = 40000  # e.g., 40,000 USDC
k_initial = x_initial * y_initial  # k = 400,000
P_initial = y_initial / x_initial  # Price = 4000 USDC/ETH

Tip: The constant k only changes when liquidity is added or removed (fee-free), not during swaps.

Detailed Instructions

Assume the price of asset X changes to a new price P' = r * P, where r is the price ratio (P' / P). For the HODL portfolio, the value remains in the original quantities x and y. Its new value in terms of Y is V_hold = P' * x + y = rPx + Px = Px(r + 1).

For the pool portfolio, after arbitrage restores the pool price to P', the new reserves (x', y') must satisfy two conditions: x' * y' = k and P' = y' / x'. Solve this system.

• Sub-step 1: Define the price change ratio r = P' / P.
• Sub-step 2: Calculate the HODL value: V_hold = Px(r + 1).
• Sub-step 3: Solve for new pool reserves: x' = x / sqrt(r) and y' = y * sqrt(r).

pythonCopy
import math
r = 2  # Price of ETH doubles to 8000 USDC/ETH
# New pool reserves after arbitrage
x_new = x_initial / math.sqrt(r)  # ~7.071 ETH
y_new = y_initial * math.sqrt(r)  # ~56568.54 USDC
V_pool_new = P_new * x_new + y_new  # Price is now P' = r*P_initial

Tip: The pool's value becomes V_pool = 2 * sqrt(r) * Px. The sqrt(r) term is the key to impermanent loss.

Detailed Instructions

With V_hold = Px(r + 1) and V_pool = 2 * sqrt(r) * Px, calculate the impermanent loss (IL) as the percentage difference: IL = (V_pool - V_hold) / V_hold. Substitute the expressions and simplify.

• Sub-step 1: Write the difference: V_pool - V_hold = 2sqrt(r)Px - (r+1)Px = Px(2sqrt(r) - (r+1)).
• Sub-step 2: Divide by V_hold = Px(r+1).
• Sub-step 3: Cancel Px to get the core IL formula: IL = [2 * sqrt(r) / (r + 1)] - 1.

This shows IL is purely a function of r, the ratio of the new price to the old price. The result is always <= 0, indicating a loss relative to holding.

pythonCopy
# Calculate IL percentage for a given price ratio r
def impermanent_loss(r):
    return (2 * math.sqrt(r) / (r + 1)) - 1

r_values = [0.5, 1, 2, 4]
for r in r_values:
    il_pct = impermanent_loss(r)
    print(f"Price ratio {r}: IL = {il_pct:.2%}")
# Output: 0.5: -5.72%, 1: 0.00%, 2: -5.72%, 4: -20.00%

Tip: The loss is symmetric in log price space; a 2x price increase or 0.5x decrease yields the same -5.72% IL.

Detailed Instructions

Analyze the function f(r) = 2*sqrt(r)/(1+r). Take its derivative to find critical points and understand where impermanent loss is minimized or maximized.

• Sub-step 1: Observe that f(r) = f(1/r), proving symmetry around r=1 (no price change).
• Sub-step 2: Calculate the derivative f'(r) = (1 - sqrt(r)) / (sqrt(r)*(1+r)^2).
• Sub-step 3: Set f'(r) = 0. The numerator zero gives sqrt(r) = 1, so r=1 is a critical point. The second derivative confirms it's a maximum.

Thus, f(r) has a maximum value of 1 at r=1 (0% IL). The impermanent loss increases as r deviates from 1. As r -> 0 or r -> infinity, f(r) -> 0, meaning IL -> -100% (total loss relative to holding).

pythonCopy
# Visualizing the IL curve
import numpy as np
r_range = np.logspace(-2, 2, 100)  # r from 0.01 to 100
il_values = (2 * np.sqrt(r_range) / (r_range + 1)) - 1
# The plot shows a symmetric, concave-down curve crossing 0 at r=1.

Tip: The convexity of the payoff shows LPs are short a gamma option, earning fees as compensation for this negative convexity.

Detailed Instructions

The two-asset, constant-product derivation is a specific case. For a generalized Constant Function Market Maker (CFMM), the trading function is Ψ(R) = k, where R is the vector of reserves. Impermanent loss arises from any divergence between the external market price vector and the pool's implied marginal prices (∇Ψ).

For a 3-asset pool with weights w_i and constant product, the invariant is (R1)^w1 * (R2)^w2 * (R3)^w3 = k. The derivation follows similar steps but uses weighted geometric means.

• Sub-step 1: Define the generalized invariant Ψ(R).
• Sub-step 2: Calculate marginal price of asset i as the partial derivative ∂Ψ/∂R_i.
• Sub-step 3: Compare the value of the LP share after a price change to its HODL value.

For Curve-style stableswap pools, which blend constant-sum and constant-product invariants, impermanent loss is minimized for small price deviations (e.g., between stablecoins) but follows the constant-product curve for large deviations.

solidityCopy
// Simplified view of a stableswap invariant (Curve)
// D is the invariant representing the total coins when balanced.
function get_D(uint256[3] memory xp, uint256 amp) internal pure returns (uint256) {
    // Sum S = xp[0] + xp[1] + xp[2];
    // Invariant D is solved via Newton's method from: amp * S + D = amp * D * (xp[0]/D * xp[1]/D * xp[2]/D) + D
}

Tip: The amplification coefficient (A) in Curve pools controls the "flatness" of the bonding curve, directly affecting the IL profile for correlated assets.