# Appendix: Math

## Setup

Each day we have five relevant values:

* Yesterday’s asset prices in dollars, p’
* Today’s asset prices in dollars, p
* Yesterday’s (human) quantity vector, q’ on a *per LP token* basis
* Today’s (human) quantity vector, q on a *per LP token* basis
* Our target wealths vector, v, which we take to be wealth fractions:

$$
\sum\_i v\_i = 1 \ \ \ v\_i \geq 0
$$

Observe that v should *tend to* be proportional to the actual wealth (q\_i\*p\_i) on each asset i.

### Calculating Daily Returns & Benchmarks

All of these benchmarks are designed so that 0 represents no change. Since the values are expected to be small, they may not need to have log taken for additive basis points.

**Daily $ Returns**

$$
\frac{\sum\_i p\_iq\_i - \sum\_i p'\_iq'\_i}{\sum\_i p'\_iq'\_i}
$$

This measures how much the portfolio increased in $ terms since yesterday.

### Daily Crypto Basis Returns

$$
\frac{\sum\_i p\_iq\_i - \sum\_i p\_iq'\_i}{\sum\_i p\_iq'\_i}
$$

This value will be negative if we would prefer yesterday’s portfolio to today’s portfolio at today’s prices.

### Rebalanced Portfolio Benchmark

$$
\sum\_i \left(\frac{p\_i}{p'\_i}\right) v\_i - 1
$$

This is our target benchmark for Clipper: our daily dollar gain should track it closely and ideally outperform.

Observe that a rebalanced portfolio can *outperform* a static (”held”) portfolio consisting of any (!) initial combination of the assets themselves. For instance, imagine the two-asset case consisting of a risky asset and a stablecoin. On day one, the risky asset doubles in price. On day two the risky asset returns to its original value. It can be verified that a 50-50 rebalanced portfolio will return 1.125x, vs. 1x for simply holding any combination of assets for the two days.

### CPMM Loss Benchmark

In the CPMM, we have that:

$$
\begin{align\*}
\prod q\_i^{v\_i} & = C \\
\sum\_i v\_i \log q\_i & = \log C
\end{align\*}
$$

Day-to-day, the portfolio of the CPMM changes to have wealth proportional to the new set of prices. By working out this constant log-sum, we get the following expression for how much “impermanent” loss a CPMM mechanism takes on:

$$
\frac{\prod\_i \left( \frac{p\_i}{p'\_i} \right)^{v\_i} - \sum\_i \left( \frac{p\_i}{p'\_i} \right)v\_i}{\sum\_i \left( \frac{p\_i}{p'\_i} \right)v\_i}
$$

This value is at most 0 when the prices are the same. Clipper’s crypto performance (loss or gain) should substantially outperform this benchmark.


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