Liquidation math

The narrative — why liquidation is partial, why it pauses while the market is closed — is on liquidations. This page is the arithmetic, in the order the contract runs it (LendingPoolRouter).

Eligibility

After accruing interest, the position must satisfy

$$BV > M_{\text{LT}}$$

strictly, with both sides as on position health; otherwise the call reverts AssetNotLiquidatable. Also required: the borrower has shares, the liquidator asks to repay more than zero, both prices are Fresh when requireFreshForLiquidation is set (it is, on every market), and the collateral token's issuer has not paused transfers.

The liquidator chooses how many shares $s_r$ to repay. The contract caps that at the borrower's $s$ and then at the close factor below, so liquidation(borrower) — which asks for everything — repays as much as the close factor allows, not necessarily the whole debt.

Close factor

One liquidation takes what is needed to make the position healthy again, with a margin, and no more. Write the health factor and the market's bad-debt line as

$$hf = \frac{M_{\text{LT}}}{BV}, \qquad k = \text{LT}\,(1 + b)$$

Repaying a fraction $f$ of the debt removes $f \cdot BV$ of debt and $f \cdot BV(1 + b)$ of collateral value, so the health factor afterwards is

$$hf' = \frac{hf - f\,k}{1 - f}$$

It reaches the target $T = 1.05$ at the restoring fraction

$$f = \frac{T - hf}{T - k} = \frac{1.05 - hf}{1.05 - \text{LT}\,(1 + b)}$$

computed as a WAD and rounded up. The most shares one liquidation may repay is (IsHealthy · 0x9259…F9a1, a view; the router applies the same rule to every liquidation through the shared CloseFactor library, LendingPoolRouter):

A request above $s_{\max}$ is capped to it, the same way a request above $s$ is: the call does not revert. Rounding up to a whole share means the position ends at 1.05, not one wei below it.

Why a target. A target is a state, not a share of the debt. After a first, smaller liquidation, the allowance is exactly what is still needed to reach 1.05, so splitting a liquidation into several calls never repays more than one call would. In practice it repays less: each slice raises the health factor, and once it is back above 1 the position is not liquidatable at all.

Why 1.05. Exactly 1 would leave the borrower one tick from the next liquidation. 1.05 is a 5% margin, small enough that the borrower keeps most of the position.

Dust

A partial liquidation — one that leaves shares ($s_{\text{rep}} < s$) and does not take the whole Position ($\sigma < 10^{18}$) — must leave at least $\text{minBase} / 2$ of debt and of collateral:

$$\left\lfloor \frac{BV\,(s - s_{\text{rep}})}{s} \right\rfloor \ge \frac{\text{minBase}}{2} \quad\text{and}\quad \left\lfloor \frac{CV'\,(10^{18} - \sigma)}{10^{18}} \right\rfloor \ge \frac{\text{minBase}}{2}$$

otherwise the call reverts MustNotLeaveDust(leftoverDebt, leftoverCollateral, minLeftover). $\text{minBase}$ is IsHealthy.closeFactorMinBase(), one value for every market: $10 on mainnet (read on 27 October 2026), so the minimum left behind is $5. The owner can change it.

If even $\lceil s \cdot f \rceil$ would leave less than that, the whole position may go — that is the third row of the table — so an unhealthy position can always be liquidated by someone. "Small" is judged by what reaching the target would leave, never by the position's size now, which is what keeps the rule split-proof. Repaying exactly $s_{\max}$ never reverts as dust; only a smaller partial can.

A liquidation the Position cannot cover (the bad-debt case below) takes all of the collateral and is never refused as dust.

Live markets

Mainnet's parameters, read at block 94915008; MSFT/USDG and AAPL/USDG at block 99393260, QQQ/USDG, GOOGL/USDG and AMD/USDG at block 113204091. Every $k$ is below 0.95, so a position in bad debt ($hf < k$) can always be liquidated in full.

Example

1 NVDA against 100 USDG of debt on NVDA/USDG. NVDA falls to $150:

$$hf = \frac{0.65 \times 150}{100} = 0.975, \qquad f = \frac{1.05 - 0.975}{1.05 - 0.702} = \frac{0.075}{0.348} = 21.55\%$$

Without the close factor, the same liquidation could have repaid all 100 USDG and taken $108 of NVDA.

The same position at a tenth of the size — 0.06 NVDA ($9) against 6 USDG, also at $hf = 0.975$ — would keep only 4.71 USDG of debt after reaching 1.05, below the $5 minimum. So it may be liquidated in full (6 USDG repaid, $6.48 of NVDA seized), and a request to repay 3 USDG reverts MustNotLeaveDust.

Seize fraction

The contract works out what fraction $\sigma$ of the Position to hand over, as a WAD (LendingPoolRouter), on the shares left after the cap:

$$CV' = \left\lfloor \frac{M_{\text{LT}} \cdot 10^{18}}{\text{LT}} \right\rfloor, \qquad SV = \left\lfloor \left\lfloor \frac{BV \cdot s_r}{s} \right\rfloor \cdot \frac{10^{18} + b}{10^{18}} \right\rfloor$$ $$\text{if } SV < CV':\quad \sigma = \left\lfloor \frac{SV \cdot 10^{18}}{CV'} \right\rfloor,\quad s_{\text{rep}} = s_r$$ $$\text{otherwise}:\quad \sigma = 10^{18},\quad s_{\text{rep}} = \min\left(\left\lceil \frac{s \cdot \lfloor CV' \cdot 10^{18} / (10^{18} + b) \rfloor}{BV} \right\rceil,\ s\right)$$

Where:

• $CV'$ — the Position's value rebuilt from $M_{\text{LT}}$ (at most one unit below $CV$).
• $SV$ — the value to seize: the repaid share of the debt value, plus the bonus $b$.
• $s_{\text{rep}}$ — the shares actually repaid. When the Position cannot cover $\text{repaid} \times (1 + b)$, the liquidator takes all of it and repays only what it is worth at the bonus.

The liquidator then pays, rounded up in the lenders' favour:

$$a_{\text{rep}} = \min\left(\left\lceil \frac{s_{\text{rep}} \cdot \text{TBA}}{\text{TBS}} \right\rceil,\ \text{TBA}\right)$$

Protocol split

A share $\pi$ of the bonus — never of the repaid value — can go to the protocol (LendingPoolRouter):

$$\sigma_p = \left\lfloor \left\lfloor \frac{\sigma\,b}{10^{18} + b} \right\rfloor \cdot \frac{\pi}{10^{18}} \right\rfloor, \qquad \sigma_l = \sigma - \sigma_p$$

$\pi$ is liquidationProtocolShare, at most 100% (IsHealthy · 0x9259…F9a1), and 0% on every market. The Position then transfers $\lfloor q\,\sigma_l / 10^{18} \rfloor$ to the liquidator and $\lfloor q\,\sigma_p / 10^{18} \rfloor$ to the protocol (Position).

Bad debt

If the whole Position was taken ($\sigma = 10^{18}$) and the borrower still has shares, the rest of their debt is written off in the same transaction (LendingPoolRouter):

$$D_{\text{bad}} = \min\left(\left\lceil \frac{s_{\text{left}} \cdot \text{TBA}}{\text{TBS}} \right\rceil,\ \text{TBA}\right)$$

Supply absorbs it first, $S \mathrel{-}= \min(D_{\text{bad}}, S)$; anything beyond supply is charged to reserves, capped at what they hold; a shortfall beyond both is not recorded anywhere. Since a market's supply is the Earn vault's claim, the loss lands on every peUSDG holder.

Where it starts. A liquidation leaves bad debt as soon as the collateral cannot pay the debt plus the bonus:

$$CV < BV\,(1 + b) \quad\Longleftrightarrow\quad HF < \text{LT}\,(1 + b)$$

not when collateral falls below the debt. The liquidator is still paid their bonus on whatever is there, so lenders lose

$$D_{\text{bad}} \approx BV - \frac{CV'}{1 + b}$$

— which can be large even while $CV > BV$. Mainnet thresholds: 0.702 for NVDA/USDG, MSFT/USDG, AAPL/USDG and GOOGL/USDG, 0.749 for QQQ/USDG, 0.795 for SPY/USDG, 0.840 for WETH/USDG, 0.550 for SPCX/USDG, TSLA/USDG and AMD/USDG. Market creation requires $\text{LT}(1 + b) < 1$ (LendingPoolFactory · 0x7a2b…b1C6), so there is always a band of health factors in which liquidation clears a position without loss.

Partial liquidation

Repaying a fraction $f$ of the debt removes $f \cdot BV$ of debt and $f \cdot BV (1 + b)$ of collateral value. The health factor afterwards is

$$HF' = \frac{\text{LT}\,\big(CV - f\,BV(1 + b)\big)}{BV\,(1 - f)}$$

and $HF' > HF$ exactly when $CV > BV(1 + b)$, i.e. $HF > \text{LT}(1 + b)$. Above that line, a partial liquidation improves the position, and the close factor stops it at 1.05. Below it, each partial liquidation makes the health factor worse, and the only clean outcome is a full one — which the close factor always allows there.

The chart shows a full liquidation, the only kind allowed at a health factor of 0.95 or below.

Full liquidation outcome by health factor, per 1 USDG of debt

Market

NVDASPYWETHSPCXTSLAMSFTAAPLQQQGOOGLAMD

Protocol share of the bonus

π = 0% (live)π = 20%

To liquidatorBorrower keepsBad debt (lenders lose)

Health factor

0.600

Collateral / debt

0.923

To liquidator

0.923

Borrower keeps

0.000

Bad debt

0.145

Show as a table

HF	Collateral	Liquidator	Protocol	Borrower keeps	Bad debt
0.400	0.615	0.615	0.000	0.000	0.430
0.500	0.769	0.769	0.000	0.000	0.288
0.600	0.923	0.923	0.000	0.000	0.145
0.702	1.080	1.080	0.000	0.000	0.000
0.800	1.231	1.080	0.000	0.151	0.000
0.900	1.385	1.080	0.000	0.305	0.000
0.950	1.462	1.080	0.000	0.382	0.000

Worked example

The illustrative Position from position health: 20 NVDA, 2400000000 borrow shares, debt 2402294128 (2,402.294128 USDG) after accrual, with $\text{TBA} =$ 20019117741, $\text{TBS} =$ 20000000000, $\text{LT} = 65\%$, $b = 8\%$. The prices below are hypothetical too.

At $180, capped by the close factor. $CV = \$3{,}600$, $M_{\text{LT}} = \$2{,}340 < BV = \$2{,}402.294128$: $HF = 0.97407$, liquidatable, and above 0.95. The liquidator asks for every share; the close factor allows

$$f = \left\lceil \frac{(1.05 - 0.974068900525572945) \cdot 10^{18}}{1.05 - 0.702} \right\rceil = 21.8193\%, \qquad s_{\max} = \lceil 2400000000 \times f \rceil = 523662755$$

Both sides stay far above the $5 minimum, so the cap is the restoring fraction. A second liquidation right after reverts AssetNotLiquidatable: the position is healthy.

Same, with $\pi = 20\%$: $\sigma_p =$ 2329614743312597 (0.23296%), so 0.046592 NVDA goes to the protocol and 3.098388 NVDA to the liquidator. The liquidator still repays the same 524.163318.

At $180, split in two. A first liquidation of 240000000 shares (10%) already lifts the health factor above 1, so the second is refused: 240,000,000 shares in total against 523,662,755 for one call. Asking for half the shares (1200000000) is capped to 523662755, like asking for all of them.

At $120, full liquidation. $CV = \$2{,}400$ — just 2.29 below the debt — and $HF = 0.649$: at or below 0.95, so the close factor allows every share, and below 0.702. $SV = \$2{,}594.48 \ge CV'$, so everything is seized:

The collateral was short of the debt by 2.29 USDG, but lenders lose 180.07: the 2.29 shortfall plus the liquidator's 8% bonus on $2,222.22 of repayment (177.78).