The Impact of Liquidation on Health Factor in Overcollateralized Loans

Why does the Health Factor of a position increase after a liquidation? If the liquidation mechanism provides liquidators with a collateral bonus, does this always improve the collateral-to-loan ratio?

In this note, we demonstrate that a liquidation generally improves the health factor of a position. We also examine the rare cases where the health factor might decrease, which can occur due to the bonus given to the liquidator.

Consider a loan smart contract, such as AAVE, that permits liquidations when the loan-to-collateral ratio exceeds a specific threshold. This condition is reflected in the Health Factor of the position, \(H\), defined as:

where \(C\) represents the collateral, \(L\) the loan amount, and \(\delta\) the liquidation threshold.

For simplicity, we assume that both currencies have the same price. The contract allows liquidations when:

During a liquidation, a liquidator repays a percentage of the debt \(L\) in exchange for collateral. We denote this percentage as \(k\) (where \(0

Let’s demonstrate how a liquidation affects the health factor of a position:

• After liquidation, since the liquidator repays \(kL\), the remaining loan is reduced to \((1-k)L\).
• The lender’s collateral is reduced by the value covered plus the bonus: \(C-(1-k)L-(1-k)Lb=C-(1-k)L(1+b)\)

The health factor after liquidation becomes:

Therefore, the health factor improves if:

As an example, consider a position with a loan-to-collateral ratio of \(0.9\). Assuming a liquidation threshold of \(0.8\), this implies that the health factor is below 1 and therefore eligible for liquidation. Let’s also consider a bonus rate of \(0.05\).

Plot showing the relationship between \(k\) and the health factor components. The blue curve represents \(\frac{1}{1-k}\) and the red line represents \((1+b)\frac{L}{C}\) where \(b=0.05\) and \(\frac{L}{C}=0.9\).

As shown in the plot, in this case the equation is satisfied for any close factor (\(k<1\)). For the equation to be violated, the close factor would need to be very small and the bonus rate very high.