Liquidation Penalty

Rationale for Liquidation Penalty values
Goals:  
 - liquidation is lucrative for liquidator,
 - liquidation improves health of the borrower;
Given
health=1−bcu/bchealth = 1 - bcu/bc\\health=1−bcu/bc
for
deposited asset A worth of d with collateral factor cfAcf_{A}cfA​
,
borrowed asset B worth of b with liquidation threshold ltBlt_{B}ltB​
​
Liquidation starts when
h0=1−(b/ltB)(d∗cfA)=1−bd∗cfA∗ltB=0,⇓d/b=1/(cfA∗ltB)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)h0​=1−(d∗cfA​)(b/ltB​)​=1−d∗cfA​∗ltB​b​=0,⇓d/b=1/(cfA​∗ltB​)
Upon liquidation of portion α\alphaα
 of B with penalty pAp_ApA​
, health must increase:
h1=1−(b(1−α))cfA∗ltB∗(d−α∗b(1+pB))>h0=0⇓b(1−α)<cfA∗ltB∗(d−α∗b(1+pB))⇓(1−α)<cfA∗ltB∗((d/b)−α∗(1+pB))1−α<1−α∗(1+pB)∗cfA∗ltBα>α∗(1+pB)∗cfA∗ltB(1+pB)∗cfA∗ltB<1(1+pB)<1cfA∗ltB⇓pB<1cfA∗ltB−1h_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} - 1h1​=1−cfA​∗ltB​∗(d−α∗b(1+pB​))(b(1−α))​>h0​=0⇓b(1−α)<cfA​∗ltB​∗(d−α∗b(1+pB​))⇓(1−α)<cfA​∗ltB​∗((d/b)−α∗(1+pB​))1−α<1−α∗(1+pB​)∗cfA​∗ltB​α>α∗(1+pB​)∗cfA​∗ltB​(1+pB​)∗cfA​∗ltB​<1(1+pB​)<cfA​∗ltB​1​⇓pB​<cfA​∗ltB​1​−1
is the condition for liquidation to increase health. Given max(cf)=0.9max(cf)=0.9max(cf)=0.9
​
p<10.9∗lt−1p <\frac{1}{0.9*lt} - 1p<0.9∗lt1​−1
 is then a safe condition for liquidation penalty for any asset.
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