Computing Impermanent Loss in Liquidity Pools

Liquidation is the biggest risk associated with leverage yield farming in DeFi. In a previous article we computed the liquidation threshold for a simple liquidity pool (LP) position. Impermanent loss (IL) is another major risk. IL occurs when our equity asset token price fluctuates.

To quantify IL, assume we have entered an X — Y token pair LP. Following the notation we used in this article, our initial position value in the LP can be written in USD ($) terms as:

We also previously computed the change in position value when the X token has a relative fluctuation value of δ:

Combining these two equations:

Now lets compute the equivalent position value had we held the tokens instead of entering them into the LP:

IL is defined as:

We have computed all the elements needed to compute this equation:

We can simplify further by using our definition of δ. Recall:

This yields the final expression for IL:

Below we plot this result:

Example: If the asset doubles, IL = (2*1.41)/3 -1 = -5.7%.

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