# ♾️3.5 Liquidations

## 3.5.1 Automatic Collateral Assignment

The FROB has a cross-collateral system, meaning each loan is backed by an amount of collateral that is automatically computed when a "liquidability check" is performed.

The criteria that is used is the pro rata collateral assignment: since each DebtPosition represents a percentage of the total borrower debt, that loan is backed by an equivalent percentage of the borrower collateral.

To be able to compute the amount of auto-assigned collateral dynamically in a scalable way, the protocol tracks both

total borrower debt and

total borrower collateral

so the automatically assigned collateral is

$C_{L_{B}}(t) = \frac{D_{L_{B}}(t)}{D_{B}(t)} C_{B}(t)$

with

$C_{B}(t)$ the total collateral of the borrower $B$ at the time $t$

$D_{B}(t)$ the total debt of the borrower $B$ at the time $t$

$D_{L_{B}}(t)$ the total outstanding debt of the loan $L_{B}$ belonging to the borrower $B$ at time $t$ (amount due to the lenders)

$C_{L_{B}}(t)$ the collateral automatically assigned to the loan $L$ belonging to the borrower $B$ at time $t$

The notation stresses how all of these quantities change over time as the result of the various actions the borrower and lender can take, which are

$D_{L_{B}}(t)$ can be partially reduced both by the borrower via partial repayment and the lender via self-liquidation (when it is allowed)

$D_{B}(t)$ the total debt of the borrower changes as new loans are opened and closed

$C_{B}(t)$ the total collateral of the borrower changes as he deposits and withdraws it or liquidations and self-liquidations happen

## 3.5.2 Eligibility for Liquidation

Each loan has a collateral ratio computed as

$\rho_{L_{B}}(t) = \frac{C_{L_{B}}(t)}{D_{L_{B}}(t)}$

The protocol has 2 collateral thresholds that are checked

$\rho_{o}$ the minimum collateral ratio for opening a new loan and

$\rho_{l} < \rho_{o}$ the minimum collateral ratio for making a loan eligible for a standard liquidation

The initial values are set to

$\rho_{o} = 150\%$

$\rho_{l} = 130\%$

Overdue loans are also eligible for liquidation, regardless of the borrower's collateral ratio.

## 3.5.3 Liquidation Mechanisms

**3.5.3.1 Standard Liquidation**

**3.5.3.1 Standard Liquidation**

When a loan is eligible for standard liquidation, the liquidator is supposed to cover all the $D_{L_{B}}$ with an equivalent amount of USDC

Since

$D_{L_{B}} = V_{L_{B}}$

with

$V_{L_{B}}$ the face value of the loan, so the sum of all the related lender(s) credits

In exchange, the liquidator gets an equivalent amount of collateral computed using our oracle price $P$ at the time of liquidation

If $\rho_{L_{B}} > 100\%$ it means the liquidation is profitable, and there is an incentive reward to be paid to liquidators. The liquidator gets up to a fixed 5% reward on the loan's face value. Then, the collateral remainder is split as follows:

This is split as follows

$\nu_{b}$ collateral remainder percentage back to the liquidated borrower

$\nu_{p}$ collateral remainder percentage to the protocol

**3.5.3.2 Self Liquidation**

**3.5.3.2 Self Liquidation**

When $\rho_{L_{B}} < 100\%$ it means the loan is undercollateralized.

Running a standard liquidation on such a loan would mean the liquidator suffers a net loss. This is a negative incentive for any 3rd party liquidator, so the only actors likely to perform a standard liquidation on an undercollateralized loan is one of Size's bots, with the purpose of taking a loss to make the lender whole.

This kind of liquidation will be subsidized by means of an insurance fund. However, it is not possible to assume that insurance capacity will be sufficient to make all the lenders whole in every potential scenario. In this case, lenders can also perform a "self-liquidation".

When this happens, the lender cancels an amount of debt in the DebtPosition equivalent to their credit and, in exchange, they get an amount of collateral equivalent to the percentage of the total credit of that loan, so its face value $V_{L_{B}}$, from the borrower reserves.

If a lender $E$ owns the credit $\gamma_{E, L_{B}}$ on the loan $L_{B}$ then what happens is

the new debt of the loan is $D_{L_{B}}' = D_{L_{B}} - \gamma_{E, L_{B}}$

the amount $C_{L_{B}} \frac{\gamma_{E, L_{B}}}{V_{L_{B}}}$ of collateral is transferred from the borrower $B$ reserves to the lender $E$ reserves

Using this mechanism, a proactive lender can limit the loss when the price of the collateral drops quickly. This mechanism is designed to leave it as an individual choice of each lender whether to self-liquidate as a stop loss and carry some inventory risk (or swap the ETH back into stablecoins) or keep the loss unrealized hoping the collateral ratio goes up again.

This operation does not change the collateralization of the borrower:

**Proof**

To simplify the notation let's redefine symbols as follows

$C$ : auto-assigned collateral (previously known as $C_{L_{B}}$)

$D$ : total loan debt (previously known as $D_{L_{B}}$)

$x \le D$ : amount of credit canceled in the self-liquidation process

To recap, the self-liquidation mechanism works as follows

lender forgives $x$ amount of credit and therefore the new debt is $D' = D - x$

in exchange, he receives collateral pro rata so the new collateral is $C' = C - C \frac{x}{D}$ so $C' = C(1 - \frac{x}{D})$

Collateral Ratio before

$\rho_{0} = \frac{C}{D}$

Collateral Ratio after

$\rho_{1} = \frac{C'}{D'}$

Let's compute the difference in CR so

$\Delta \rho = \rho_{0} - \rho_{1}$

$\frac{C}{D} - \frac{C(1 - \frac{x}{D})}{D - x} = \frac{CD - xC - CD + xC}{D(D-x)} = 0$

Let's double-check computing also the aggregated CR

CR Before $\frac{C_{0} + C_{1}}{D_{0} + D_{1}}$ where $C_{0}$ is the auto-assigned collateral to that loan and $D_{0}$ is the debt of that loan

CR After $\frac{C_{1} + C_{0}(1 - \frac{x}{D_{0}})}{D_{1} + D_{0} - x}$

Let's compute the delta CR and for the sake of simplicity let's skip the denominator since it is not relevant

$C_{0}(D_{0} - x) + C_{0}D_{1} + C_{1}(D_{0} - x) + C_{1}D_{1} - C_{0}(1 - \frac{x}{D_{0}})(D_{0} + D_{1}) - C_{1}(D_{0}+D_{1})$

Skipping the simplifications we get that this is equal to zero if

$x(C_{0} \frac{D_{1}}{D_{0}} - C_{1}) = 0$

which is true if $\frac{C_{0}}{D_{0}} = \frac{C_{1}}{D_{1}}$ which means that a specific loan needs to have the same collateral ratio as all the other loans and we know this is always true by construction.

**3.5.3.3 Liquidation with replacement**

**3.5.3.3 Liquidation with replacement**

This mechanism consists of a liquidator performing a standard liquidation, but the repaid proceeds do not become immediately available for the lender(s) to be claimed but are used to fill a borrow market order instead, making sure that the due date remains the same as the one of the liquidated lender so the lender(s) can claim when they expect to

Letting aside protocol fees that have already been taken on the liquidated loan during the standard liquidation process, at the due date the new borrower must return the face value of the liquidated loan $V_{L_{B}}$ plus the new protocol fees

If the liquidation happens at time $\tau$ and the due date is at $T$ then the new loan needs to have a duration of $T-\tau$ and the corresponding rate is obtained by applying this term to the yield curve $\Psi_{B'}$ of the new borrower $B'$ so that we obtain the new rate $r' = \Psi_{B'}(T - \tau)$ which allows the computation of the actual amount to lend out aka issuance value $I_{L_{B'}} = \frac{V_{L_{B}}}{r'}$ of the new loan that replaces the old one.

Since $r' > 0$ always then $V_{L_{b}} > I_{L_{B'}}$ and this determines a net profit $V_{L_{b}} - I_{L_{B'}}$ that is earned in cash by the protocol

**3.5.3.4 Overdue Liquidation**

**3.5.3.4 Overdue Liquidation**

When the loan is overdue, it is eligible for liquidation in the same way as underwater loans. However, overdue loans with healthy collateral ratios have reduced penalties for the borrower.

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