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:
∑ivi=1   vi≥0\sum_i v_i = 1 \ \ \ v_i \geq 0i∑​vi​=1   vi​≥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
∑ipiqi−∑ipi′qi′∑ipi′qi′\frac{\sum_i p_iq_i - \sum_i p'_iq'_i}{\sum_i p'_iq'_i}∑i​pi′​qi′​∑i​pi​qi​−∑i​pi′​qi′​​
This measures how much the portfolio increased in $ terms since yesterday.
Daily Crypto Basis Returns
∑ipiqi−∑ipiqi′∑ipiqi′\frac{\sum_i p_iq_i - \sum_i p_iq'_i}{\sum_i p_iq'_i}∑i​pi​qi′​∑i​pi​qi​−∑i​pi​qi′​​
This value will be negative if we would prefer yesterday’s portfolio to today’s portfolio at today’s prices.
Rebalanced Portfolio Benchmark
∑i(pipi′)vi−1\sum_i \left(\frac{p_i}{p'_i}\right) v_i - 1i∑​(pi′​pi​​)vi​−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:
∏qivi=C∑ivilog⁡qi=log⁡C\begin{align*} 
\prod q_i^{v_i} & = C \\
\sum_i v_i \log q_i & = \log C
\end{align*}∏qivi​​i∑​vi​logqi​​=C=logC​
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:
∏i(pipi′)vi−∑i(pipi′)vi∑i(pipi′)vi\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}∑i​(pi′​pi​​)vi​∏i​(pi′​pi​​)vi​−∑i​(pi′​pi​​)vi​​
This value is at most 0 when the prices are the same. Clipper’s crypto performance (loss or gain) should substantially outperform this benchmark.
Last modified 1mo ago