# Liquidation Penalty

*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 $$max(cf)=0.9$$​

$$
p <\frac{1}{0.9\*lt} - 1
$$

&#x20;is then a safe condition for liquidation penalty for any asset.