# Adaptive Fee Model

A model designed to free users from hefty fees while transacting.

Considering the CPF ${R_A}\times{R_B}=k$, liquidity providers always incur a loss named **impermanent loss**, due to a state deviation of reserves. The loss will disappear if the state returns to the initial state. Without loss of generality, let’s assume that $B$ outperforms $A$ after a period. Clearly, arbitrageurs will sell $A$ to get $B$ because the current pool is maintaining an undervalued price of $B$.

(24) $(R_A,R_B)\xrightarrow{0<\alpha\leq1}(\frac{1}{\alpha}R_A,{\alpha}R_B)$.

Thus, the price is transforming from $p=R_B/R_A$ to $p'=\alpha^2R_B/R_A$. Compared with a *HODL* strategy, adding liquidity to a pool seems less profitable (without a fee model).

(25) $loss_{no-fee}=(p'R_A+R_B)-(p'\frac{1}{\alpha}R_A+\alpha{R_B})$

(26) $=(1-\alpha)^2R_B.$

Most AMMs use a fee model typically of 0.25% to cover the loss. The mistake here is the fee is fixed while the loss is varied. It’s better when the fee is adaptive, that means the fee will be large if the loss is large, and vice versa.

To create a more effective fee model, the AMM develops a function $\gamma=f(\alpha)$, where $0<\gamma\leq1$, respects to $\alpha$ to adapt the change. Because $\alpha$ describes how the pool state deviates from the current state, it’s reasonable for the fee function to rely on $\alpha$.

**Definition 2. **The impermanent loss is zero when the adaptive fee is following:

(27) $\gamma=\frac{1-\alpha}{2-\alpha}$.

*Proof. *Recall that a trader will get $(1-\alpha)R_B$ in a no-fee model. When the AMM applies the adaptive fee model, the trader will receive $(1 − \gamma)(1 − \alpha)R_B$. In summary,

(28) $(R_A,R_B)\xrightarrow{0<\alpha,\gamma\leq1}(\frac{1}{\alpha}R_A,{\alpha}R_B+\gamma(1-\alpha)R_B)$.

With $p'=\alpha(\alpha+\gamma(1-\alpha))R_B/R_A$, the loss now is

(29 )$loss_{adaptive-fee}=(p'R_A+R_B)-(p'\frac{1}{\alpha}R_A+\alpha+\gamma(1-\alpha))R_B)$

(30) $=((\alpha-2)(\alpha+\gamma(1-\alpha))+1)R_B$.

Because of the zero loss, we have an equation, $loss_{adaptive-fee}=0$. Hence, $\gamma=(1-\alpha)/(2-\alpha).$

**Corollary 1. **Applying the adaptive fee, the pool’s state will follow:

(31) $(R_A,R_B)\xrightarrow{0<\alpha\leq1}(\frac{1}{\alpha}R_A,\frac{1}{2-\alpha}R_B).$

Because the pricing curve is the same as the 2D CPF, the impermanent loss (IL) is obvious to Sentre’s AMM too. However, employing a variant of the adaptive fee model can cover the IL** **while also benefit LPs.

(32) $Protocol\ Fee=Adaptive\ Fee+0.05\%$

The approach can secure income for LPs. By flexibility, the fee is low corresponding to small price changes, and gets larger along with the price change.

Furthermore, the AMM collects a fractional fee as a tax to maintain the core engine at the beginning. In the future, this tax can be opened as a grant for community projects (see Fig. 4).

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