# Uniswap V3 Explained - Concentrated Liquidity, Impermanent Loss, Slippage

Uniswap protocol is an ETH native smart contract system which enables swapping of pairs of ERC20<>ERC20 and ERC20<>ETH.

Uniswap uses automated market maker (AMM) algorithm to execute trades. Users provide liquidity in pairs of tokens to create a liquidity pool. Trades are executed by depositing the offered token in the pool and withdrawing the asked token from the pool. A swap fee is applied to the amount of ask token which is distributed to the liquidity providers (LPs).

Uniswap V3 is the latest version of the protocol which has introduced concentrated liquidity and many other concepts. In V3 there are several fee tiers available based on the risk of providing liquidity. The fees is collected in the 2 tokens of the pool and is not invested back into the pool.

UNI is a governance token for the Uniswap protocol. UNI token holders might be eligible for protocol fee in future. The current protocol fee is 0%. UNI token holders can change the protocol fee.

## Concentrated Liquidity

Uniswap V3 uses concentrated liquidity market maker (CLMM) which is much efficient market marking algorithm than standard constant product market maker (CPMM) algorithm.

There are 2 tokens in a pool token0 and token1. The price (P) of token0 is expressed in terms of token1. For example 100UNI per 1ETH in a pool of UNI<>ETH.

In CLMM the LPs have to choose a range of price between which they are providing liquidity. If the price P moves outside the range of a pool, it gets inactive and the swap is performed using the next available pool in the changed price range.

In CLMM the pool tracks the square root of the price (P) and the liquidity (L) in the pool. The amount of the tokens in the pool are not needed to calculate the amount of tokens received in a swap.

Below are the formulas which define the relationship between amount of tokens, price and liquidity.

```
# x is the amount of token0, y is the amount of token1
# price of token0 in terms of token1
P = y / x
# liquidity is the geometric mean of the amount of tokens
L = sqrt(x*y)
```

In V3 the liquidity is defined as the change in amount of token1 for a given change in square root P. Based on this concept the below V3 formulas are used to calculate the amount of tokens you can get.

```
Δy = Δ(√P) * L
Δx = Δ(1/√P) * L
```

The above formulas is used for movement of price per adjacent tick. A tick is an integer which represents the price using the below formula.

```
P = 1.0001^i
sqrt(P) = 1.0001^(i/2)
i = log(sqrt(P)) * 2 / log(1.0001)
```

Each tick is 0.1% away from the adjacent one. If the price movement for the complete swap is beyond the adjacent tick then swap is performed in step functions moving from one tick to another until all the tokens are swapped.

CLMM follows the constant product formula for the price movement within 2 adjacent ticks. CLMM is a variation of the constant product formula.

Below is a script I wrote to emulate a swap using concentrated liquidity formula. I have ignored applying fee in the swap. Only the swap for token1 to token0 is implemented.

In V3 the liquidity pools are represented as NFTs since each pool is distinct from each other. A single swap might move from pool to pool based on the price impact of the swap.

Concentrated liquidity is highly efficient when compared to the standard constant product algorithm. CLMM uses the full liquidity in the pool within the price range of the pool. But CPMM spreads the liquidity over 0 to infinity. This is because CLMM has different formulas to calculate the new state of the pool.

### Price Impact

When a swap is made agains a pool the ratio of the tokens in the pool changes. The ratio of tokens in the pool is the price (P) of the token0 in terms of token1.

At the at the beginning of the swap the ratio of the poll is 100UNI : 1ETH. But you will not get 100UNI when swapping with 1ETH because the ratio of the pool changes. This is called the price impact on the swap.

Let us take an example of UNI<>ETH liquidity pool. With current ratio 100UNI per 1 ETH. We will be using V2 formula of CPMM because the calculations are much easy but the concept still the same for V3.

```
# x and y are number of tokens
# x_uni = 10000, y_eth = 100
x_uni * y_eth = k
(x_uni - recieve) * (y_eth + deposit) = k
(10000 - receive) * (100 + 1) = 10000 * 100
receive = 10000 - (10000 * 100 / 101)
receive = 99.0099
```

In the above calculation you see that for 1 ETH you get 99.0099 UNI tokens. The ratio of tokens in the pool has changed but the product of the amount of tokens is still the same.

### Slippage

Transactions with higher gas can be executed before transactions with lower fee. It is not possible to predict at which point in time will the transaction execute. The state of the pool might have changed between the transaction broadcast and execution. The changed state of the pool might result in a very different price for the swap than predicted. This change in price is considered as slippage.

### Impermanent loss

Liquidity providers are taking risk by providing liquidity. The ratio of tokens in the pool will keep on changing based on the current market price. Arbitrageurs will trade with pool to match the pool token ratio (price) with that of the larger market. This rebalancing of the portfolio is risky for the LPs because when they decide to withdraw the funds the ratio might be very skewed in the direction of token which has lost value.

Lets us take an example to see this. The below example uses V2 CPMM because it has a simple formula but the concept is same for V3 as well.

Alice and Bob decide the fund the BTC<>ETH pool. We will see the state of the liquidity pool at different times. The state of the pool is calculated using 2 equations.

```
# token_x and token_y are number of tokens
# k is the constant product and r is the ratio of tokens
token_x * token_y = k
token_x / token_y = r
# substituting the value of token_y
token_x^2 / r = k
token_x = √(k*r)
token_y = √(k/r)
```

token_x = BTC, token_y = ETH

**At T0**

r = 1/10

Initial pool state = 900 BTC + 9000 ETH

Alice deposits 100 BTC + 1000 ETH

Final pool state = 1000 BTC + 10000 ETH

Alice is 10% owner of the pool

**At T1**

r = 1/8

Initial pool state = 1118 BTC + 8944 ETH

Bob deposits 80 BTC + 640 ETH

Final pool state = 1198 BTC + 9584 ETH

Bob owns 6.67% of the total pool

Alice now owns 9.33% of the pool

**At T2**

r = 1/5

Initial pool state = 1515.36 BTC + 7576.8 ETH

Alice decides to withdraw from the pool

Alice will get 9.33% of the pool which is 141.38 BTC + 706.91 ETH. Which at current rate is worth 282.76 BTC.

If Alice would have held the tokens instead of adding then to the pool she would have 100 BTC + 1000 ETH which is worth 300 BTC at current rates. So Alice lost 17.24 worth BTC in her holding.

Final pool state = 1373.98 BTC + 6869.89 ETH

Bob now owns 7.356% of the pool and decides to keep his funds in the pool.

**At T3**

r = 1:8

Initial pool state = 1086.22 BTC + 8689.76 ETH

Bob decides to withdraw his funds from the pool

Bob will get 7.356% of the pool which is 79.9 BTC + 639.218 ETH. Which at the current rate is worth 159.8 BTC (consider this 160 because of decimal errors it is coming 159.8). If Bob would have not deposited in the pool he would have 80 BTC + 640 ETH which at the current rate is worth 160 BTC.

Here we see that Bob did not lose any value because the ratio of the pool is same as when he deposited his tokens.

This is the reason it is called impermanent loss. If the ratio of the pool is same as when you deposited the token there is no loss.

LPs get trading fees for every trade. If the trading fees collected by the LP is greater than the impermanent loss then the LP can withdraw the funds from the pool with a profit.