> For the complete documentation index, see [llms.txt](https://artichoke.gitbook.io/abstract/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://artichoke.gitbook.io/abstract/single-sided-lp-mechanics/total-returns.md).
# 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](/abstract/single-sided-lp-mechanics/variation-and-returns.md):
$$V\_t = 2L \cdot \sqrt{R}\cdot e^{\alpha t}$$
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.
---
# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.
## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.
Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:
```
GET https://artichoke.gitbook.io/abstract/single-sided-lp-mechanics/total-returns.md?ask=&goal=
```
`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.
Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.