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Understanding Returns

Portfolio values

A portfolio value gives the value of a LP share for a given exchange rate. The portfolio value of an AMM can be derived from the trading invariant or vice-versa.
It has been shown that portfolio values of all constant function automated market makers are nondecreasing, non-negative, and concave. This means that without external rewards, providing liquidity is always worse than holding the underlying tokens.

PMMP portfolio values

The portfolio value of a PMMP liquidity position is shown below:
V(p)={2pp1p20pp1p12p>p1V(p)= \begin{cases} 2p*p_{1}-p^{2} & 0\le p\le p_{1}\\ p_{1}^{2} & p > p_{1} \end{cases}
The value of a Numoen LP position with p1 as 3