---
title: AMM Slippage & IL — Variables, Prices, r, and Slippage
tags: AMM, DeFi, Slippage, Impermanent Loss, Uniswap v2
---

[TOC]

# Contents
Compact mathematical reference for:
- **AMM state variables**
- **pool reserves and prices**
- **constant-product relations**
- **price-move factor** $r$ and **slippage formulas** (with/without fees),

# Variables & Notation

## AMM state (constant-product, Uniswap v2-style)
- $X$ — reserve of token **A** in the pool.
- $Y$ — reserve of token **B** in the pool.
- $k$ — invariant: $k = X\cdot Y$
- **Price** of token A in units of token B: $P = \dfrac{Y}{X}$

## Initial \& Final States
- Initial pool state (pre-trade, time $t = 0$):
$$\left(X_0, \ Y_0, \ P_0=\dfrac{Y_0}{X_0} \right) \ \mathrm{with} \ \ k=X_0\cdot Y_0$$
- Final pool state (post-trade, time $t = 1$):
$$ \left(X_1, \ Y_1, \ P_1=\dfrac{Y_1}{X_1}\right) \ \mathrm{with} \ k = X_1\cdot Y_1 = X_0\cdot Y_0$$
- **Price ratio** and spot return:
$$
r =\frac{P_1}{P_0} \ \ \ (\mathrm{spot \ return} = r-1)
$$
Note that $r-1$ is also the **average** price change along AMM curve for an **infinitessimally** small buy quantity.

---

# Constant-Product Relations

## Reserves at target price $P_1$
Using $k=X_0Y_0 = X_1Y_1$ and $P_1=\dfrac{Y_1}{X_1}$
$$
X_1=\sqrt{\frac{k}{P_1}}=\frac{X_0}{\sqrt{r}} \qquad
Y_1=\sqrt{k\,P_1}=Y_0\sqrt{r} \qquad r = \frac{P_1}{P_0}
$$

## Net token movements
**Buy A** (add B, remove A from pool):
$$
\Delta A = X_0 - X_1 = X_0\!\left(1-\frac{1}{\sqrt{r}}\right),\qquad
\Delta B = Y_1 - Y_0 = Y_0(\sqrt{r}-1).
$$

With **input fee** $\tau_{in}$, if trader **spends** $B_{\text{spent}}$:
$$
\Delta B \;=\; (1-\tau_{in})\, B_{\text{spent}}\,.
$$
Input fee is a fee taken from what the trader sends into the pool before it touches the reserves. Therefore, $\tau_{in}$ is the fraction of the input that does not reach the reserves (goes to protocol, LP, creator). On the other hand, the output fee, $\tau_{out}$ is a fee taken from what the trader receives from the pool after the on-curve output has been computed (going to creator, protocol). It does not affect the reserve, since $\Delta A$ is fixed by the invariant once $\Delta B$ is realized.

## Inverse Relations (see appendix at the end for derivations)
The following formulas recover $r = P_1/P_0$ from a fill.
For **net quote** added (**buy** token **A** with token **B**, adds **B** to the pool): $$r=\left(1+\frac{\Delta B}
{Y_0}\right)^{\!2}$$
From **net base** removed (**buy** token **A** with token **B**, removes **A** from the pool): $$r=\left(1-\frac{\Delta A}{X_0}\right)^{\!-2}$$

Note: when selling token **A**, $\Delta B < 0$ and $\Delta A > 0$ (since $\Delta A = X_0 - X_1$), and the same formulas hold with appropriate sign changes.

In the presense of fees, we need to adjust the formula for $r$. Specifically, the input amount (quote) that enters the pool, $\Delta B = Y_1 - Y_0$, **includes** LP fee. However, the output amount (base) should exclude the LP fee to walk the AMM curve: $$\Delta B_{eff} = (1 - f_{LP})\Delta B$$ since LP fee is added to the input reserve, but does not entitle extra output. Therefore, $X_1 = k/(Y_0 + \Delta B_{eff})$, and taking $P_1 = Y_1/X_1$ and $P_0 = Y_0/X_0$ we get:

$$
r = \frac{P_1}{P_0} = \left(1+\frac{\Delta B}{Y_0}\right)\cdot \left(1 + \frac{\Delta B_{eff}}{Y_0}\right)
$$

---

# Slippage on the AMM Curve

Note that the final spot price differs from the average execution price paid along the AMM curve.

## Average execution slippage (no fees)
Average paid price for a buy over the path is the **geometric mean** of starting and ending prices (see exact derivation in the appendix at the end):
$$
P_{\text{avg}}=\sqrt{P_0 P_1}=P_0\sqrt{r}.
$$
Price-relative average execution slippage for **buy**:
$$
\Sigma= \frac{P_{\text{avg}}}{P_0}-1=\sqrt{r}-1 = \frac{\Delta B}{Y_0}
$$
For sell, simply reverse the sign $\Sigma \rightarrow -\Sigma$

## Average execution slippage (with input fees)
If fees are charged on input before hitting the curve, the **price-plus-fee uplift** (buy) is:
$$
\frac{P_{\text{avg}}}{P_0}\cdot\frac{1}{\,1-\tau\,}-1
\;=\;
\frac{\sqrt{r}}{\,1-\tau\,}-1
$$

Rule of thumb for **cost per \$ traded** for **buy** (expanding above in Taylor series):
$$
\Gamma \approx\; (\sqrt{r}-1) \;+\; \tau
$$

With LP fee $f$ retained in the pool we have (for small trades):
$$
\sqrt{r} - 1 \approx \frac{\Delta B + \Delta B_{eff}}{2Y_0} \approx \left(1-\frac{f}{2}\right)\frac{\Delta B}{Y_0}
$$
Note that LP fee increases the constant $k$ in the constant product formula:
$$
k' = k\cdot \left(1+\frac{\Delta B}{Y_0}\right)\left(1+\frac{\Delta B_{eff}}{Y_0}\right) > k
$$

For small trades:
$$
\frac{k'}{k} \approx 1 + f\cdot \frac{\Delta B}{Y_0}
$$

---

# Impermanent Loss (see derivation in appendix)
For a 50/50 constant-product pool (ignoring fees) and price ratio $r=P_1/P_0$:
$$
\boxed{\;\text{IL}(r)\;=\;\frac{2\sqrt{r}}{1+r}-1\;}
$$
Note: negative for $r\neq 1$. Fees earned can offset IL if volume is sufficient.

---

# Appendix — AMM mathematics

### Recovering $r$ from **net quote added** $\Delta B$ (buy $A$ with $B$)

- Net change in B reserves: $Y_1 = Y_0 + \Delta B$
- Constant product: $X_1 = \dfrac{k}{Y_1} = \dfrac{X_0Y_0}{Y_0 + \Delta B}$

Substitution yields:
$$
\begin{aligned}
r \;=\; \frac{P_1}{P_0}
&= \frac{Y_1/X_1}{Y_0/X_0}
= \frac{Y_1 X_0}{X_1 Y_0}
= \frac{Y_1 X_0}{\big(\frac{X_0Y_0}{Y_1}\big) Y_0}
= \frac{Y_1^2}{Y_0^2}
= \left(\frac{Y_0+\Delta B}{Y_0}\right)^2
= \left(1+\frac{\Delta B}{Y_0}\right)^{\!2}
\end{aligned}
$$
- When **selling** $A$ for $B$, $\Delta B<0$, implying $r<1$.

---

### Recovering $r$ from **net base removed** $\Delta A$ (buy $A$ with $B$ )

Use the **“removed from pool”** sign convention:
$$
\Delta A \;\equiv\; X_0 - X_1 \quad\Rightarrow\quad
\begin{cases}
\Delta A > 0 & \text{buy (A leaves pool)}\\
\Delta A < 0 & \text{sell (A enters pool)}
\end{cases}
$$

- $X_1 = X_0 - \Delta A$
- $Y_1 = \dfrac{k}{X_1} = \dfrac{X_0Y_0}{X_0-\Delta A}$

Substitution yields:
$$
\begin{aligned}
r \;=\; \frac{Y_1/X_1}{Y_0/X_0}
&= \frac{Y_1 X_0}{X_1 Y_0}
= \frac{\big(\frac{X_0Y_0}{X_0-\Delta A}\big) X_0}{(X_0-\Delta A) Y_0}
= \frac{X_0^2}{(X_0-\Delta A)^2}
= \left(1-\frac{\Delta A}{X_0}\right)^{-2}
\end{aligned}
$$
- When **selling**, $\Delta A<0$ so the denominator is greater than 1 and $r<1$.

### Average execution price over path derivation

**Spot price along the path**
Since $k=XY$ is constant,
$$
Y=\frac{k}{X}\;\;\Rightarrow\;\;P(X)=\frac{Y}{X}=\frac{k}{X^2}.
$$

**Infinitesimal trade relation**
Removing an infinitesimal $dA>0$ from the pool corresponds to $dX=-dA$. Keeping $k$ constant:
$$
(X-dX)(Y+dB)=XY
\;\Rightarrow\;
-\,Y\,dX + X\,dB = 0
\;\Rightarrow\;
dB=\frac{Y}{X}\,dA=P(X)\,dA.
$$

**Integrate totals**
$$
\Delta A=\int dA = X_0 - X_1
$$

$$
\Delta B=\int dB=\int_{X_0}^{X_1}\frac{k}{X^2}(-dX)
= -k\!\int_{X_0}^{X_1}\frac{dX}{X^2}
= k\!\left(\frac{1}{X_1}-\frac{1}{X_0}\right).
$$

**Average execution price**
$$
P_{\text{avg}}=\frac{\Delta B}{\Delta A}
= \frac{k\left(\frac{1}{X_1}-\frac{1}{X_0}\right)}{X_0 - X_1}.
$$
Using $k=X_0Y_0=X_0^2P_0$:
$$
P_{\text{avg}}
= \frac{X_0^2P_0\left(\frac{X_0-X_1}{X_0X_1}\right)}{X_0-X_1}
= P_0\,\frac{X_0}{X_1}.
$$
Also, from $P=\dfrac{k}{X^2}$:
$$
P_1=\frac{k}{X_1^2}=\frac{X_0^2P_0}{X_1^2}
\rightarrow\;
\frac{X_0}{X_1}=\sqrt{\frac{P_1}{P_0}}.
$$
Therefore,
$$
\boxed{\,P_{\text{avg}}=P_0\sqrt{\frac{P_1}{P_0}}=\sqrt{P_0P_1}\, }
$$

**Interpretation**
The average paid price over a constant-product path equals the **geometric mean** of the start and end spot prices. Hence the (fee-excluded) average execution slippage for a buy is
$$
\frac{P_{\text{avg}}}{P_0}-1=\sqrt{\frac{P_1}{P_0}}-1=\sqrt{r}-1,\quad r=\frac{P_1}{P_0}.
$$

### Impermanent loss formula derivation

**Goal.** Show that for a price move factor $r = \dfrac{P_1}{P_0}$ (price of A in units of B), an LP’s
impermanent loss relative to just holding is:
$$
\boxed{\;\text{IL}(r)\;=\;\frac{2\sqrt{r}}{1+r}\;-\;1\;}
$$
Assume that LP owns $0 < s \le 1$ fraction of the pool, and consistent with V2, 50/50 deposit at entry is mandated.

Using $r = P_1/P_0$, $k=X_0Y_0$ and $P_1 = Y_1/X_1$ we get:

$$
X_1 = \sqrt{\frac{k}{P_1}} = \frac{X_0}{\sqrt{r}},
\qquad
Y_1 \;=\; \sqrt{k\,P_1} \;=\; Y_0\,\sqrt{r}.
$$

**LP value vs holding value at $P_1$:**

Take **B** as the numeraire (values in units of token B).

- **LP value at exit** (holding a fraction $s$ of pool):
$$
V_{\text{LP}}(P_1)
= s\Big(X_1\,P_1 + Y_1\Big).
$$

- **Holding value at $P_1$** (if the LP had simply kept their entry amounts $sX_0,\,sY_0$):
$$
V_{H}(P_1)
= s\Big(X_0\,P_1 + Y_0\Big).
$$

Simplify $V_{\text{LP}}$ using $P_1 = rP_0$ and $P_0=Y_0/X_0$, we compute the term $X_1 P_1$:
$$
X_1 P_1
= \left(\frac{X_0}{\sqrt{r}}\right)\!\cdot\!P_1
= \left(\frac{X_0}{\sqrt{r}}\right)\!\cdot\!(r P_0)
= \left(\frac{X_0}{\sqrt{r}}\right)\!\cdot\!\left(r \frac{Y_0}{X_0}\right)
= \sqrt{r}\,Y_0.
$$
Hence
$$
V_{\text{LP}}(P_1)
= s\Big(\underbrace{\sqrt{r}\,Y_0}_{X_1P_1} + \underbrace{Y_0\sqrt{r}}_{Y_1}\Big)
= s\cdot 2Y_0\sqrt{r}.
$$

Similarly, the holding value:
$$
V_{H}(P_1)
= s\Big(X_0\,P_1 + Y_0\Big)
= s\Big(X_0(rP_0) + Y_0\Big)
= s\Big(rY_0 + Y_0\Big)
= s\cdot Y_0(1+r).
$$

**Impermanent Loss as a relative shortfall:**
By definition (relative performance of LP vs holding):
$$
\text{IL}(r)
= \frac{V_{\text{LP}}(P_1)}{V_{H}(P_1)} - 1
= \frac{s\cdot 2Y_0\sqrt{r}}{s\cdot Y_0(1+r)} - 1
= \frac{2\sqrt{r}}{1+r} - 1.
$$

This completes the derivation.

**Properties of IL:**
- $\text{IL}(1)=0$ (no price change → no divergence).
- Symmetry: $\text{IL}(r)=\text{IL}(1/r)$.
- Always non-positive for $r\neq 1$ (LP underperforms holding on pure price divergence).
- Fees accrued from trading can offset IL in practice, but are **not** part of the formula above.