🧮Total Returns

Before we derive the total return we must first state the formula for the value of the stake in the liquidity pool taking into account price variation and trading fees. That is combining the two formulas stated on Variation and Returns:

$V_t = 2L \cdot \sqrt{R}\cdot e^{\alpha t}$

Now, to measure the return from $t_0$ to $t_1$, we would use the following formula:

$R = \frac{V_1-V_0}{V_0}=\frac{2L \cdot \sqrt{R_1}\cdot e^{\alpha }-2L \cdot \sqrt{R_0}}{2L \cdot \sqrt{R_0}}$

Which after simplification becomes:

$R=\sqrt{\frac{p_1}{p_0}}\cdot e^\alpha-1$

If we define $P$ as the price ratio of p at any $t$ from $p$ at $t_0$, then we can generalize the return formula as follows for any time $t$

$R_t = \sqrt{P_t} \cdot e^{\alpha \cdot t}-1$

We can now plot the graphs of the total returns of the liquidity provider’s stake at different growth rates of the pool’s reserves from trading fees as price varies.