Free lesson · DeFi models

LP value versus holding the original tokens

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A liquidity provider can make money in dollars while underperforming a passive holding of the initial tokens. The benchmark must be named.

Start with equal-value base and quote reserves in a no-fee constant-product pool. After a price ratio change r, pool value grows with the square root of r while the original holdings have one asset growing with r.

The commonly called impermanent-loss ratio compares these two terminal values. It excludes fees and is not loss-versus-rebalancing, which uses a dynamically rebalanced benchmark. The underperformance can become realized when the position closes.

Apply it in a strategy

• Compare LP terminal value with a transparent buy-and-hold benchmark before adding fee income.
• Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
• Stress this failure condition: This no-fee full-range ratio cannot be applied unchanged to concentrated liquidity, rebalancing or path-dependent fee income.

Research deliverable

Build and explain a lp value versus holding the original tokens worksheet. Compare LP terminal value with a transparent buy-and-hold benchmark before adding fee income.

Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
from math import sqrt
def holding_difference(price_ratio):
    if price_ratio<=0: raise ValueError("Positive price ratio required")
    return 2*sqrt(price_ratio)/(1+price_ratio)-1

print(holding_difference(2))