# Liquidation Free Loans

cDOT is issued to crowdloan participants as proof of crowdloan contribution when the contributed project wins the parachain slot. For each DOT that the user contributed to the crowdloan, one cDOT is minted and issued back to the crowdloan contributor, such that he/she can redeem the cDOT back to DOT at a one-to-one ratio when the corresponding parachain lease is up (i.e. 96 weeks after parachain auction is won). Therefore, from a financial standpoint, cDOT is a 96-week zero-coupon bond on DOT, and the price relationship between DOT and cDOT is expressed below:

Where Px is the price of asset X, T is the remaining term-to-maturity of the cDOT and r is the implied yield of the cDOT. Here the implied yield represents the market expected APR of DOT through term T.

Example: cDOT is traded at 0.8 DOT, and the cDOT will mature in 1.5 years. Then implied yield r can be derived as . This essentially means by purchasing the cDOT now and holding it through to its maturity and redeeming it 1:1 to DOT, we are making 16.04% DOT-denominated APR.

When a person put two assets of equal value to a liquidity pool, when the relative price of the two assets changes, the total value of the two assets in the LP position becomes lower than that if the liquidity provider just held the two assets. The difference between the two is referred to as impermanent loss. The term “impermanent” refers to the fact that the loss from liquidity provision depends on the relative price of the two assets in the pool, which is ever-changing.

However, in the case of cDOT-DOT liquidity providing, since cDOT vs DOT price is certain at the maturity of the cDOT (i.e. cDOT is redeemable 1:1 to DOT at the maturity of the cDOT), the impermanent loss becomes permanent. Mathematically speaking, since our DEX uses the most widely adopted constant-product AMM mechanism (i.e. x*y=k model), when the LP position is opened, the constant product k is determined. Then at at maturity of the cDOT, the number of cDOT x and number of DOT y in the LP will be equal (as cDOT is redeemable 1:1 to DOT). Therefore at maturity, the LP position consists of √k

**cDOT and √k DOT, which in turn can be redeemed into 2√k DOT.****In other words, the DOT-denominated value of LP position at the time of cDOT maturity is determined when the LP position is opened.****Same example from the previous section:**cDOT is traded at 0.8 DOT, and the cDOT will mature in 1.5 years. Now Jane deposits 100 cDOT and 80 DOT into the liquidity pool for yield farming (i.e. k=100*80=8000). We know Jane will be able to withdraw

**√**k=89.44 cDOT and 89.44 DOT from her LP position and redeem the 89.44 cDOT into 89.44 DOT to get in total 178.88 DOT. If she held onto the 100 cDOT and 80 DOT for the same period, she would have gotten 180 DOT back, which implies her (im)permanent loss is 0.62%.

**As a matter of fact, given that today similar liquid crowdloan products are trading at ~0.7 DOT, for an LP to provide liquidity in the cDOT-DOT pool, the permanent loss would be merely 1.6%**when the cDOT matures after roughly 2 years. Given that the loss of cDOT-DOT liquidity provision is certain and trivial if held until the maturity of cDOT, providing liquidity is a great long-term value-add for cDOT holders.

Following the trail of permanent loss of providing liquidity in CDOT-DOT pool – as the number of DOT that the LP position will be redeemable to is certain at the maturity of the cDOT,

**we can provide liquidation-free DOT loan to borrowers with the cDOT-DOT LP token as collateral, as long as the maturity value of the LP token will cover the principal and interest of the borrowed DOT.**To put such relationship in mathematical term: assume price of cDOT is p per DOT,

**and each cDOT-DOT LP token consists of 1 DOT and 1/p cDOT, the maturity value of one LP token (without considering any swap fee reward) is:**Where k is the constant product of the LP, which equals to 1/p in this case.

The maximum loan amount at the maturity of the cDOT-DOT LP token is:

where CF is the collateral factor denoted as the max value of DOT that can be borrowed as a percentage of the value of the collateralised LP token, and is the borrow rate cap of DOT in the money market and T is the remaining term to maturity of the cDOT.

Subsequently, the range of CF can be calibrated per the inequality:

In other word, as long as the above inequality holds, the loan never needs to be liquidated. Because even if the collateral value (i.e. LP token value) is temporarily less than that of the loan, it is certain that later when cDOT matures, the value of the collateralised LP token will rise back up and be able to cover the loan amount even if the loan keeps accruing interest at the maximum borrowing interest rate. See figure below for demonstration.

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Therefore, the maximum liquidation-free loan collateral factor that can be offered to the borrower with cDOT-DOT LP token as collateral can be derived as a function of p, and T.

Example: Current cDOT price is 0.8 DOT, and the cDOT will mature in 1.5 years. Also, the borrowing rate cap of DOT in the money market is 50%. Assume each LP token consists of 1 DOT and 1.25 cDOT (i.e. total value equals 2 DOT). For each LP token as collateral, we can provide collateral factor or 1.22 DOT liquidation-free loan to the borrower.

Note the LP token is only enabled as collateral in E-mode, and the borrowing of DOT will be liquidation-free by default. Also, given that of is static at 30%, ideally CF should be calculated dynamically based on the spot of p and T, but for launch of V1, the maximum CF is set at 65%, which is calibrated based on T being 96 weeks and conservative level of p.