# Minting and Redeeming
## Minting
All BD-Stable tokens are fungible with one another and entitled to the same proportion of collateral no matter what collateral ratio they were minted at. This system of equations describes the minting function of the Blindex Protocol:
$$BD = \overbrace{(Y*P\_y)}^{\text{collateral value}} + \overbrace{(Z*P\_z)}^{\text{BDX value}}$$
$$(1-C\_r)(Y*P\_y) = C\_r(Z*P\_z)$$
$$BD$$ is the units of newly minted BD-Stable
$$C\_r$$ is the collateral ratio
$$Y$$ is the units of collateral transferred to the system
$$P\_y$$ is the price in BD-Stable underlying fiat of $$Y$$ collateral
$$Z$$ is the units of BDX burned
$$P\_z$$ is the price in BD-Stable underlying fiat of BDX
#### **Example A: Minting BD-Stable (BDEU - EUR pegged stable) at a collateral ratio of 100% with 0.05 ETH (ETH/EUR = 4000)**
To be explicit, we can start by finding the BDX needed to mint BDEU with 0.05 ETH (`worth 200 EUR`) at a collateral ratio of 1.00
$$(1-1.00)(100*1.00) = 1.00(Z*P\_z)$$
$$0 = (Z \* P\_z)$$
Thus, we show that no BDX is needed to mint BDEU when the protocol collateral ratio is 100% (fully collateralized). Next, we solve for how much BDEU we will get with the 0.05 ETH worth 200 EUR.
$$BD = (200\*1.00) + (0)$$
$$BD = 200$$
`200 BDEU` are minted in this scenario. Notice how the entire value of BDEU is in euro value of the collateral when the ratio is at `100%`. Any amount of BDX attempting to be burned to mint BDEU is returned to the user because the second part of the equation cancels to `0` regardless of the value of $$Z$$ and $$P\_z$$.
#### **Example B: Minting BDEU at a collateral ratio of 80% with 0.03 ETH worth 120 EUR (ETH/EUR = 4000) and BDX/EUR =2.**
First, we need to figure out how much BDX we need to match the corresponding amount of ETH.
$$(1 - 0.8)(120 \* 1.00) = 0.8(Z\*2.00)$$
$$Z = 15$$
Thus, we need to deposit 15 BDX alongside 0.03 ETH (worth 120 EUR) under these conditions. Next, we compute how much BDEU we will get.
$$BD = (120*1.00) + (15*2.00)$$
$$BD = 150$$
`150 BDEU` are minted in this scenario. `120 BDEU` are backed by the value of ETH as collateral while the remaining `30 BDEU` are not backed by anything. Instead, BDX is burned and removed from circulation proportional to the value of minted algorithmic BDEU.
## Redeeming
Redeeming BD-Stable is done by rearranging the previous system of equations for simplicity, and solving for the units of collateral, $$Y$$, and the units of BDX, $$Z$$. However, there are 2 new components introduced. Effective Collateral Ratio and BDX Effective Coverage Ratio. Both described in the *Price Stability* section.
$$Y = \dfrac{BD\*(min(efC\_r,C\_r))}{P\_y}$$
$$Z = \dfrac{efBDXC\_r \* BD\*(1-min(efC\_r,C\_r))}{P\_z}$$
$$BD$$ are the units of BD-Stable redeemed
$$C\_r$$ is the collateral ratio
$$efC\_r$$ is the effective collateral ratio
$$efBDXC\_r$$ is the effective BDX coverage ratio
$$Y$$ are the units of collateral transferred to the user
$$P\_y$$ is the price in BD-Stable underlying fiat of $$Y$$ collateral
$$Z$$ are the units of BDX minted to the user
$$P\_z$$ is the price in BD-Stable underlying fiat of BDX
#### **Example D: Redeeming 170 BDEU at a collateral ratio of 65%, effective collateral ratio of 100%, effective BDX coverage ratio = 100%, ETH/EUR = 4000 and BDX/EUR = 3.75.**
$$Y = \dfrac{170\*(.65)}{4000}$$
$$Z = \dfrac{1 \* 170\*(.35)}{3.75}$$
Thus, $$Y = 0.027625$$ and $$Z = 15.867$$
Redeeming `170 BDEU` returns `170` EUR of value to the redeemer: `0.027625 ETH` (worth `110.5 EUR`) from the collateral pool and `15.867 of BDX` (worth `59.5 EUR`) from BDX reserves (stored in BDEU treasury) at the current BDX market price.
#### **Example E: Redeeming 170 BDEU at a collateral ratio of 65%, effective collateral ratio of 60%, effective BDX coverage ratio = 75%, ETH/EUR = 4000 and BDX/EUR = 3.75.**
$$Y = \dfrac{170\*(.6)}{4000}$$
$$Z = \dfrac{0.75 \* 170\*(.4)}{3.75}$$
Thus, $$Y = 110.5$$ and $$Z = 15.867$$
Redeeming `170 BDEU` returns `0.0255 ETH` (worth `102 EUR`) of value to the redeemer from the collateral pool and `13.6 of BDX` (worth `51 EUR`) from BDX reserves (stored in BDEU treasury) at the current BDX market price. Please note that `170 BDEU` were exchanged for the value of only `152 EUR`. This is an important modification to the original FRAX protocol. In times when collateralization is low, we prevent the early dumpers from benefitting from leaving the protocol early at the expense of faithful holders.
Additionally, there is a 1 block delay parameter (adjustable by governance) on withdrawing redeemed collateral to protect against flash loans.
#### NOTE: These examples do not account for the mint and redeem fees, which are set to 0.3%
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