# Leverage Max Amount Derivation

If we have a starting amount of collateral `S`, the Minimum Collateralization Ratio (MCR) of the collateral limits how much additional collateral `L` we can leverage:

After leveraging, we will have a total of `S+L` of collateral in our vault. The protocol enforces that we can borrow a maximum of `(S+L)/MCR` collateral's worth of PAR from our vault.

We will use this PAR to swap and repay the loan at the end of the leverage transaction, so the amount of PAR we withdraw must also be worth at least `L` collateral, neglecting flashloan fees (otherwise, we won't have enough to repay the loan)

In other words, when we leverage the maximum amount of collateral, the amount of PAR we exchange must be simultaneously worth at most `(S+L)/MCR` of collateral (as dictated by the MIMO protocol) and at least `L` of collateral (as dictated by the flashloan protocol). Thus, we can set both expressions equal to each other to get the equation:

`(L+S)/MCR = L`

Solving the above equation for `L` gives us the maximum amount of additional collateral `L` we can leverage:

`L = S/(MCR - 1)`

We can additionally account for any flashloan fees by further dividing `L` by `1 + (flashloan fees)`.

For example, if we assume:

* We start out with 1 ETH; i.e. `S = 1`
* We want to leverage an additional 1 ETH; i.e. `L = 1`
* The PAR/USD exchange rate is `1 PAR = 1.1 USD`
* The ETH/USD exchange rate is `1 ETH = 3000 USD`
* The MCR for ETH is 1.3; i.e. `MCR = 1.3`

The protocol enforces that we can borrow a maximum of `(S+L)/MCR`, or `(1+1)/(1.3)`, or `1.54` ETH's worth of PAR from our vault. That works out to `(1.54 ETH) * (3000 USD / 1 ETH) * (1 PAR / 1.1 USD)` or `4200` PAR.

We will also need to have at least `1 ETH`'s worth of PAR to repay the flashloan. `1 ETH` works out to `(1 ETH) * ( 3000 USD / 1 ETH ) * (1 PAR / 1.1 USD)` or \~ 2727.27 PAR.

Since the amount of PAR we can withdraw from our vault is higher than the amount of PAR we need to repay our loan, we can leverage this amount.

Assuming the above numbers, the maximum amount of additional ETH we could have leveraged was `S/(MCR-1)` or `(1 ETH)/(1.3 - 1)` or \~ 3.33 ETH.


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