Relative Positional Value Change in Liquidity Pools
An important question arises when we yield farm with liquidity pools:
Question: How does our position in a liquidity pool change when one asset price fluctuates by δ %?
Answer: The relative positional value change is:
Proof:
Consider an arbitrary crypto token: X. For example X = ETH. Denote the quantity of X tokens we hold at time t as:
The value of the token X relative to USD ($) at time t is governed by a conversion factor:
This puts a $ value on our holding:
Now consider a liquidity pool (LP) in the token pair: X — Y. This LP is governed by two rules:
- Constant product: at any time t we have:
2. Balanced liquidity: at any time t, the total $ value of LP tokens is balanced:
Substituting the second rule into the first rule gives a simple rule for the quantities of the tokens in the pool:
Here we defined the conversion factor from X to Y as:
which can be interpreted as the conversion of X to $ followed by the conversion of $ to Y.
Now assume that at time t=1 we decide to enter a position into the X — Y LP. We provide a balanced (i.e. 50:50) set of tokens:
Our position value at time t=1 is simply:
The LP will typically have many more token than what we provide it and our position value will be a fraction of the total LP value. We can compute this value using the rules of the LP:
where we used the balanced liquidity rule of the LP to reach the final equation.
Now consider the LP at time t=2 and construct the ratio of position values:
Notice how this ratio is independent of our fractional ownership and also independent of the LP liquidity k.
Assume now that the value of the Y token remains fixed between t=1 and t=2, while the value of the X token fluctuates as:
This will simplify the positional ratio we computed before:
Stated differently, the relative positional value change is:
This completes the proof.
Caveat: This is an approximation as we didn’t model the nieces of a LP such as transaction fees.
Bonus: The model above is most accurate when the change in the asset is small. In this scenario we can use the binomial approximation to reduce further:
Which gives us a simple rule of thumb: relative position value change is about half the change in the asset.