# Staking & Impermanent Loss

One of the main issues with liquidity providing on an LP is impermanent loss. IL is the loss incurred by a market making position vs. keeping the initially allocated amounts fixed. Another approach of IL could be referred as unrealized profit or loss (**uPNL**): this a more accurate way to call such profit or loss, as it only becomes realized if one chooses to sell out of a position which has changed in value.

Given an initial price $$P\_0$$, the value of $$P\_0$$ of a 2-asset portfolio is initially given by:

$$V = V\_0 + P\_0 \cdot y$$

On the other hand, a new value $$V\_1$$ after the first transactions $$P\_1$$ is given by:

$$V\_1 = x' + P\_1 \cdot y'$$

Given a new value and new prices, the equation will still be the following:

$$x+ P\_1 \cdot y$$

The **IL** or **uIL** is the delta between the portfolio change of the market making portfolio and the change in value of a portfolio of assets with fixed quantities. This is the loss on top of a mark to market move of an equivalent fixed-quantity portfolio:

$$x' + P\_1 \cdot y' - (x + P\_0 \cdot y) - (x + P\_1 \cdot y - (x + P\_0 \cdot y))$$

Which simplifies to:

$$x' - x + P\_1 \cdot y' - P\_0 \cdot y - (P\_1 - P\_0) \cdot y$$

Also, we will call $$R$$ the ratio between the current and previous price given a fixed-period of time $$t$$: 

$$R = \frac{P\_1}{P\_0}$$

Hence, unrealized impermanent loss $$\epsilon$$ will be given by the following equation:

$$\epsilon = \frac{uPNL}{V\_0} = \sqrt{R} - \frac{1}{2} \cdot (R+1)$$


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