Loan: Payback

Borrowers are incentivized to repay loaned synthetics and withdraw collateral when the market price is below the protocol price. This demand exists until the market price is again at equilibrium with the protocol price thus restoring the peg. Withdrawal of collateral is enabled when the collateralization ratio of the loan is above the borrowing ratio.

$$if \ Collateralization \ Ratio > Borrowing \ Ratio \ then \ Withdrawal \ enabled$$

Borrowers may redeem the synthetic asset for the collateral of the loan until all of the collateral has been withdrawn. Borrowers are incentivized to repay loans and withdraw collateral because the repayment value is less than the withdrawn collateral value.

The example below shows withdrawal of collateral from a sample loan. The initial collateralization ratio is below the borrowing ratio. Withdrawal is not enabled until the collateralization ratio goes above the borrowing ratio.

Parameters
Borrowing Ratio	150%
Collateral Price	$10,000/ETH
Synth Price	$100/ETH

Accounts Before Payback

Loan		Borrower
Collateral	14 ETH	Account	0 ETH
Liability	1000 Synths		800 Synths
Collateralization Ratio	140%

Suppose the Borrower repays 80 synth by calling the payback function

$$Collateral \ Withdrawn = Collateral(Before) - Collateral(After)$$

$$Collateral(After) = \frac{Liability(After) * Price(Synth) * Borrowing  \ Ratio}{Price(Collateral)}$$

$$Liability(After) = 1000 \ Synths - 800 \ Synths = 200 \ Synths$$

Collateral(After) = \frac{200 \ Synths * $100 / Synth * 150\%}{$10,000/ETH}

Accounts After Payback

Loan		Borrower
Collateral	3 ETH	Account	9 ETH
Liability	200 Synths		0 Synths
Collateralization Ratio	150%

This shows that the Borrower is incentivized to pay back the loan. The 800 Synths paid back by the Borrower redeemed 11 ETH while only costing 8 ETH. Also, Collateralization Ratio of the loan is again restored to the Borrowing Ratio.