Borrowing

Borrowing transaction
Let xxx
 be the decrease in the XXX
 pool. This decrease is due to the borrower withdrawing xxx
 assets from the pool.
​yyy
 be the increase in the YYY
 pool.
​zzz
 be the increase in the ZZZ
 pool.
The value of xxx
, yyy
, and zzz
 is calculated from maintaining constant K.K.K.
 
​(X−x)×(Y+y)×(Z+z)=K(X-x)\times(Y+y)\times(Z+z)=K(X−x)×(Y+y)×(Z+z)=K
 
So as yyy
 increases, zzz
 has to decrease ; when zz z
increases, yy y
has to decrease.
The maximum ofyyy
 is when z=0z=0z=0
 :
​(X−x)×(Y+ymax)×(Z+0)=K(X-x)\times(Y+y_{max})\times(Z+0)=K(X−x)×(Y+ymax​)×(Z+0)=K
 
The maximum of zzz
 is when y=0:y=0:y=0:
 
​(X−x)×(Y+0)×(Z+zmax)=K(X-x)\times(Y+0)\times(Z+z_{max})=K(X−x)×(Y+0)×(Z+zmax​)=K
 
Let d be time duration from the time of user transaction to maturity time of pool in seconds
The borrower receives xxx
 assets from the pool.
The Debt that a borrower will have to repay before maturity is x+dyx+dyx+dy
 with a constraint that there is a minimum interest debt to be paid by borrowers such that y>=ymax16y>=\frac{y_{max}}{16}y>=16ymax​​
 
The Collateral (w) that a borrower will have to lock is z_{max}+zd/2^25  
For the borrowers, the more collateral they lock, the lower interest they have to pay. This reflects the fact that the higher the collateral locked, the less is the probability for the borrower to default at the maturity date. Thus, borrowers have the flexibility to decide their opportunity cost vs cost of loan profile at the time of borrowing assets from the pool.
The amount of collateral the borrower has to stake has a minimum value of zmax z_{max}zmax​
 when z=0z=0z=0
 is chosen so that enough collateral will always cover the amount borrowed xxx
. Similarly the debt has a minimum value when y=ymax/16y={y_{max}}/{16}y=ymax​/16
 such that there is always a minimum interest payable by a borrower
Deep-dive into a Borrowing transaction
Bob wants to borrow DAI using his collateral ETH. Suppose a LP initiates the DAI-ETH pool with similar parameters as mentioned in the previous lending transaction such that:
​X=10,000X = 10,000X=10,000
 
​Y=0.0000475Y=0.0000475Y=0.0000475
 
​Z=4.16Z = 4.16Z=4.16
​
​K=1.979K=1.979K=1.979
 
Now Bob wants to borrow 1000 DAI by depositing collateral in the form of ETH. So as per the AMM equation for a borrowing transaction:
​(10000−1000)×(0.0000475+y)×(4.16+z)=1.979(10000-1000)\times(0.0000475+y)\times(4.16+z)=1.979(10000−1000)×(0.0000475+y)×(4.16+z)=1.979
 
Solving for ymaxy_{max}ymax​
,
​(10000−1000)×(0.0000475+ymax)×(4.16+0)=1.979(10000-1000) \times(0.0000475+y_{max}) \times(4.16+0) =1.979(10000−1000)×(0.0000475+ymax​)×(4.16+0)=1.979
 
​ymax=0.0000053y_{max}=0.0000053ymax​=0.0000053
​
Solving for zmaxz_{max}zmax​
 ,
​(10000−1000)×(0.0000475+0)×(4.16+zmax)=1.979(10000-1000)\times(0.0000475+0)\times(4.16+z_{max}) =1.979(10000−1000)×(0.0000475+0)×(4.16+zmax​)=1.979
 
​zmax=0.469z_{max}=0.469zmax​=0.469
​
Let d=d=d=
 2592000 (duration of 30 days from the time of tx in seconds)
The annual interest rate that can be selected by Alice will have a range of (ymax16)×dyearx(\frac{y_{max}}{16})\times \frac{d_{year}}{x} (16ymax​​)×xdyear​​
 to ymax×dyearxy_{max}\times \frac{d_{year}}{x}ymax​×xdyear​​
 (where dyeard_{year}dyear​
 is no. of sec in a year = 31556926). 
In this case, Bob can select APR from 1% to 16%
Assuming Bob chose to pay annual interest of 10%
​=>=>=>
 Interest value per second =y=1000×0.131556926=0.00000316=y=1000\times \frac{0.1}{31556926} =0.00000316=y=1000×315569260.1​=0.00000316
 
Debt to be repaid by Bob= x+dy=1000+2592000×0.00000316=1008.2x+dy=1000+2592000\times 0.00000316=1008.2x+dy=1000+2592000×0.00000316=1008.2
 
Solving for zzz
,
​(10000−1000)×(0.0000475+0.00000316)×(4.16+z)=1.979(10000-1000)\times(0.0000475+0.00000316)\times(4.16+z)=1.979(10000−1000)×(0.0000475+0.00000316)×(4.16+z)=1.979
 
​z=0.18z=0.18z=0.18
  

 =>=>=>
 Collateral to be locked z_{max}+zd/2^25 = 0.469+0.18*2592000/33554432 = 0.4829 ETH
Bob now has the optionality to pay back his debt of 1008.2 DAI before maturity in order to unlock his locked collateral of 0.4829 ETH. In case Bob decides to default, his collateral will be automatically forfeited at maturity
Summarising:
Bob borrowed 1000 DAI from the pool before 1 month of pool expiry by depositing 0.4829 ETH as collateral. When Bob repays 1008.2 DAI before maturity of the pool, he will receive his collateral back. If Bob defaults on the borrowing, his locked collateral of 0.4829 ETH will be proportionately distributed to the lenders based on the Insurance tokens lenders hold.
Last modified 1yr ago