# Liquidation Math

How we calculate the *health factor*:
$$ {\sum \text{collateral in ETH}_i \times lendFactor_i \over \sum \text{borrow in ETH}_i \times borrowFactor_i} = healthFactor$$

### Liquidation scenerio

Specs:

liquidator discount = 1.05e18 == 0.5%.

assetA:
    price: 1 ETH.
    lendFactor: 0.5e18.
    borrowFactor: 0.
    
assetB:
    price: 2 ETH.
    lendFactor: 0.
    borrowFactor: 1e18.

1. User provides `1e18` of `assetA` as collateral.
2. User borrows `0.25e18` of `assetB`. Now user's health factor is `1e18`.
3. The price of assetA drops to `0.9e18`. User's health factor becomes `0.9e18` and is now able to be liquidated.

---

How do we create a general formula for the liquidator to calculate the optimal `repayAmount` that will leave the user with a health factor of `1.25e18`?

$$
{totalCollateral - (x \times ratio \times \text{collateralAsset in ETH} \times lendFactor) \over totalBorrowed - (x \times \text{borrowAsset in ETH} \times borrowFactor)} = 1.25e18
$$

Let:
$a = ratio$
$b = \text{borrowAsset in ETH}$
$c = \text{collateralAsset in ETH}$
$d = borrowFactor$
$e = lendFactor$

1.
$$
{totalCollateral - (xace) \over totalBorrowed - (xbd)} = 1.25e18
$$

2.
$$
totalCollateral - (xace) = 1.25e18 \times (totalBorrowed - xbd)
$$

3.
$$
(totalCollateral \times 100) - (xace \times 100) = 125e18 \times (totalBorrowed - xbd)
$$

4.
$$
(totalCollateral \times 100) - (xace \times 100) = (125e18 \times totalBorrowed) - (125e18 \times xbd)
$$

5.
$$
(-xace \times 100) = (125e18 \times totalBorrowed) - (125e18 \times xbd) - (totalCollateral \times 100)
$$

6.
$$
(-xace \times 100) + (125e18 \times xbd) = (125e18 \times totalBorrowed) - (totalCollateral \times 100)
$$

7.
$$
25e18x(-4e18eac + 5e18bd) = (125e18 \times totalBorrowed) - (totalCollateral \times 100)
$$

**Solution**:

$$
x = {(5e18 \times totalBorrowed) - (4e18 \times totalCollateral) \over (5e18 \times \text{borrowAsset in ETH} \times borrowFactor) - (4e18 \times \text{ratio} \times \text{borrowAsset in ETH} \times borrowFactor)}
$$