The Price of Impermanent Loss in DeFi AMM Protocols

5 min readDec 1, 2020

While DeFi protocols like Uniswap have grown dramatically over the past few months, there is almost nothing published about the valuation implications of impermanent loss (see references below). The following post attempts to address this by introducing a new concept to the DeFi market: Net APY = Nominal APY - Impermanent Loss Rate

(Note: The term impermanent loss rate here is used to mean the annualized drag on returns associated with the non-linearity of liquidity pool returns relative to returns of the underlying LP tokens. In Wall Street parlance, this is known as the option cost, since the non-linear nature of impermanent loss in liquidity pools resembles the non-linear payoff function of ordinary call and put options.)

Say we have a liquidity pool that employs the Automated Market Making (AMM) protocol used on Uniswap. Let m be the quantity of one token in the pool and n be the quantity of the second token.

At t=0,

Moreover, it can be shown¹ that

Here, π — the relative price of the two tokens — is assumed to be exogenous e.g., π is the pair price on centralized exchanges.

Per the constant product formula of the AMM protocol, at t=1 we have

The zero arbitrage quantities of each token after the price move follow the following equation

By substitution, we have

and

Now, the total value of the liquidity pool (denominated by the first token m) at time 0 is

At t=1, the value of the pool is

However, if one simply held the two tokens, the value of the portfolio would have been

Now, let’s assume markets are “complete” and interest rates are 0. In academic literature, completeness means that tradeable instruments exist to fully hedge out the risk e.g., Binance futures. When markets are complete, one can show that derivatives can be priced by assuming that the underlying asset (discounted by the risk-free rate) is a zero-drift stochastic process i.e.,

If we invest in a portfolio of the two tokens, we have

which is the result we would expect. In contrast, the liquidity pool’s present value is

Now, let’s assume π is lognormally distributed per the Black Scholes model. It can be shown (see references) that

Finally, we can write

In other words, if a liquidity pool pays no fees, then it always destroys value for any value of σ>0.

If σ represents annualized volatility and t is assumed to be 1 year, then we have

For small values of σ, note that

Therefore,

If we assume that daily price returns are independent and identically distributed, then we can estimate annualized volatility as follows

and we can use the Stats 101 variance estimator formula on daily price returns i.e.,

where x is the daily price return, x-bar is the sample mean of daily price returns, and n is the number of days in the sample period.

As an example, Figure 1 shows the impermanent loss rate for the ETH-USDC pair. Sigma is the annualized standard deviation of returns over a rolling 31 day period.

As of 12/1/2020, the nominal APY of USDC-ETH LP on Uniswap is ~25%. Per the above graph, the current impermanent loss rate is ~10%. Hence, the Net APY is ~15%, which is still far above market rates of return. In other words, while the impermanent loss rate isn’t insignificant, there is still plenty of value left on the table for liquidity pool investors.

References:

http://www.its.caltech.edu/~cvitanic/PAPERS/MOOC%20problems.pdf