# Calculation Methodologies

## **Exchange Rate Calculation**

We use the formula below to calculate the exchange rate:

`rcv = a * B / (A + a)` 

`rcv_net = rcv * (1 - fee)` 

where `a` is the amount of token X sent by swapper and `A` is the total amount of tokens X in the pool before the swap

Then, the exchange rate is simply `lim a -> 0 (rcv / a)` which equates to `B/A`. 

In terms of how much a user shall receive `b` given `a` supplied:

`b = a * B / (A + a) * (1 - fee)` 

## Asset Price Calculation

Internally, a fixed pricing algorithm is used to calculate the spot $USD price of assets swapped via the Pact AMM pools. To achieve this the algorithm tracks every SWAP pool call, distinguishing between swaps with: 

* Stablecoin Swaps (USDT or USDC) and;
* General Swaps (all other assets)

#### Stablecoin Swaps

For swaps with either USDT or USDC (let’s call them STABLE), it assumes the stablecoin has price of $1 and uses the following formula:

$$
price\_{asset} = \frac{price\_{STABLE} \* total\_{STABLE} \* 10 ^{decimals\_{asset}}}{total\_{STABLE}  \*10 ^{decimals\_{STABLE}} }
$$

$$price\_{STABLE}$$ is the price of STABLE (1$) 

$$total\_{STABLE}$$ is a total number of STABLE in pool after SWAP

$$total\_{asset}$$ is a total number of asset in pool after SWAP

$$decimals\_{asset}$$ asset’s decimals 

$$decimals\_{STABLE}$$ STABLE’s decimals

#### General Swaps

For swaps between other assets, the price will only be calculated if there has been a prior SWAP app call to the `STABLE/A` or `STABLE/B` pool, because in this case `A/B` already have an implicit price calculated. From here, the formula is similar:

$$
price\_A=\frac{price\_B\*total\_B \* 10^{decimals\_A}}{total\_A \* 10^{decimals\_B}}
$$

$$price\_B$$ is the price of `B` calculated from SWAP app call in `STABLE/B` pool

$$total\_A$$ is a total number of `A` in pool after SWAP

$$total\_B$$ is a total number of `B` in pool after SWAP

$$decimals\_A$$is `A`’s decimals

$$decimals\_B$$is `B`’s decimals

Note: If there are no `STABLE/A` or `STABLE/B` pools, then the price won't be calculated and the value will be set to "-".


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