Impermanent Loss

This page provides examples exploring impermanent loss.

In these examples, we initially deposited $100 each of token AAAA at a value of $0.10 (1000 tokens), and token BBBB at a value of $1.00 (100 tokens). We therefore had a total of $200 in the liquidity pool, at a token ratio of 10:100.

Example 1

The value of token AAAA rose 100% to $0.20, and the value of token BBBB also rose by 100% to $2.00. This means the token ratio is now 20:200, which is the same as 10:100. Our tokens in the liquidity pool are now worth a total of $400 ($200 of each token), which is the same as they would be worth if we had simply held them in our wallet. Therefore, there is no impermanent loss.

If we were to rework this example, so that both tokens lost an equal percentage of value, we would find that no impermanent loss occurs. This goes to show that, with impermanent loss it is not the token values that matter: whether their values increase or decrease: rather what matters is the ratio between the two tokens values.

Example 2

The value of token AAAA rose 200% to $0.30, but the value of token BBBB remains unchanged at $1.00. This means the token ratio has narrowed to 30:100. With this shift in the token ratio, arbitrage rebalances the pool by selling some AAAA and buying up extra BBBB. Our tokens are rebalanced at 575.35 tokens of AAAA, and 173.2 tokens of BBBB. We now have a total of $346.40 in the liquidity pool ($173.20 of each token). However, if we had simply held the tokens in our wallet, we'd have $300 worth of AAAA and $100 worth of BBBB, for a total of $400. This means we are at 13.46% impermanent loss, but that only affects our profits since our total value has still increased.

Example 3

This time, the value of token AAAA rose 50% to $0.15, but the value of token BBBB rose even more significantly by 500% to $6.00. The token ratio has widened to 15:600, which is the same as 2.5:100. Arbitrage once again rebalances the pool, doubling our number of AAAA tokens to 2000, but halving our number of BBBB tokens to 50. We now have a total of $600 in the liquidity pool ($300 of each token). This is $150 less than if we had simply held our tokens in our wallet, as we would have had $150 of AAAA and $600 of BBBB, for a total of $750. Our impermanent loss is 20%, but as in the previous example this only affects our profits.

Example 4

Token AAAA has lost 10% of its value to $0.09, but token BBBB rose 200% to a value of $3.00. Once again the token ratio has widened, this time to 9:300, which can also be expressed as 3:100. Arbitrage rebalances the pool, so we now have 1825.74 of token AAAA, and 54.77 of token BBBB. Our tokens in the pool are worth a total of $328.63 ($164.31 of each token). Had we instead held those tokens in our wallet, we'd have $90 of AAAA and $300 of BBBB, for a total of $390. Missing out on that extra $61.37 means we have an impermanent loss of 15.73%. Luckily, once again this only affects our profits thanks to the gains of BBBB.

Example 5

The markets have turned bearish, and token AAAA has lost 20% of its value to $0.08, while token BBBB has lost 50% to $0.50. The token ratio has now narrowed to 8:50, which we can also express as 16:100. After arbitrage rebalances the pool, we have 790.57 of token AAAA, and 126.49 of token BBBB. Our total token value is $126.49, as we have $63.25 worth of each token. Had we simply held our tokens in our wallet, we'd have $80 of AAAA and $50 of BBBB, for a total of $130. Our impermanent loss is 2.7%, and this time it is eating into our principal investment.

Additional Resources

Whiteboard Crypto has a handy impermanent loss calculator, along with a detailed explanation on impermanent loss - Impermanent Loss Calculator (Examples + 3 Versions).