# Liquidation Penalty

Rationale for Liquidation Penalty values
Goals: - liquidation is lucrative for liquidator, - liquidation improves health of the borrower;
Given
$health = 1 - bcu/bc\\$
for
• deposited asset A worth of d with collateral factor
$cf_{A}$
,
• borrowed asset B worth of b with liquidation threshold
$lt_{B}$
Liquidation starts when
$h_0 = 1 - \frac{(b/lt_B)}{(d*cf_A)} = 1 - \frac{b}{d*cf_A*lt_B} = 0,\\ \Downarrow\\ d/b = 1/(cf_A*lt_B)$
Upon liquidation of portion
$\alpha$
of B with penalty
$p_A$
, health must increase:
$h_1 = 1 - \frac{(b(1-\alpha))}{cf_A*lt_B*(d-\alpha * b (1 + p_B))} > h_0 = 0\\ \Downarrow\\ b(1-\alpha) < cf_A*lt_B*(d-\alpha * b (1 + p_B))\\ \Downarrow\\ (1-\alpha) < cf_A*lt_B*((d/b)-\alpha * (1 + p_B))\\ 1 - \alpha < 1- \alpha * (1 + p_B) * cf_A*lt_B \\ \alpha > \alpha * (1 + p_B) * cf_A*lt_B\\ (1 + p_B) * cf_A*lt_B < 1\\ (1 + p_B) < \frac{1}{cf_A*lt_B}\\ \Downarrow\\ p_B <\frac{1}{cf_A*lt_B} - 1$
is the condition for liquidation to increase health. Given
$max(cf)=0.9$
$p <\frac{1}{0.9*lt} - 1$
is then a safe condition for liquidation penalty for any asset.