Equations

Staking

$$deposit = withdrawal$$

Swaps between IMMO and sIMMO during staking and unstaking are always honored 1:1. The amount of IMMO deposited into the staking contract will always result in the same amount of sIMMO. And the amount of sIMMO withdrawn from the staking contract will always result in the same amount of IMMO.

$$rebase = 1 - ( IMMODeposits / IMMOOutstanding)$$

The treasury deposits IMMO into the distributor. The distributor then deposits IMMO into the staking contract, creating an imbalance between IMMO and sIMMO. sIMMO is rebased to correct this imbalance between IMMO deposited and sIMMO outstanding. The rebase brings sIMMO outstanding back up to parity so that 1 sIMMO equals 1 staked IMMO.

Bonding

$$bond Price = 1 + Premium$$

IMMO has an intrinsic value of 1 cUSD, which is almost equivalent to $1. In order to make a profit from bonding, Immortal charges a premium for each bond.

$$Premium = debt Ratio * BCV$$

The premium is derived from the debt ratio of the system and a scaling variable called BCV. BCV allows us to control the rate at which bond prices increase.

The premium determines profit due to the protocol and in turn, stakers. This is because new IMMO is minted from the profit and subsequently distributed among all stakers.

$$debt Ratio = bondsOutstanding/IMMOSupply$$

The debt ratio is the total of all IMMO promised to bonders divided by the total supply of IMMO. This allows us to measure the debt of the system.

$$bondPayout_{reserveBond} = marketValue_{asset}\ /\ bondPrice$$

Bond payout determines the number of IMMO sold to a bonder. For reserve bonds, the market value of the assets supplied by the bonder is used to determine the bond payout. For example, if a user supplies 1000 cUSD and the bond price is 250 cUSD, the user will be entitled 4 IMMO.

$$bondPayout_{lpBond} = marketValue_{lpToken}\ /\ bondPrice$$

For liquidity bonds, the market value of the LP tokens supplied by the bonder is used to determine the bond payout. For example, if a user supplies 0.001 IMMO-CUSD LP token which is valued at 1000 cUSD at the time of bonding, and the bond price is 250 cUSD, the user will be entitled 4 IMMO.

IMMO Supply

$$IMMO_{supplyGrowth} = IMMO_{stakers} + IMMO_{bonders} + IMMO_{DAO} + IMMO_{pIMMOExercise}$$

IMMO supply does not have a hard cap. Its supply increases when:

• IMMO is minted and distributed to the stakers.

• IMMO is minted for the bonder. This happens whenever someone purchases a bond.

• IMMO is minted for the DAO. This happens whenever someone purchases a bond. The DAO gets the same number of IMMO as the bonder.

• IMMO is minted for the team, investors, advisors, or the DAO. This happens whenever

  the aforementioned party exercises their pIMMO.

$$IMMO_{stakers} = IMMO_{totalSupply} * rewardRate$$

At the end of each epoch, the treasury mints IMMO at a set reward rate. These IMMO will be distributed to all the stakers in the protocol.

$$IMMO_{bonders} = bondPayout$$

Whenever someone purchases a bond, a set number of IMMO is minted. These IMMO will not be released to the bonder all at once - they are vested to the bonder linearly over time. The bond payout uses a different formula for different types of bonds. Check the bonding section above to see how it is calculated.

$$IMMO_{DAO} = IMMO_{bonders}$$

The DAO receives the same amount of IMMO as the bonder. This represents the DAO profit.

$$IMMO_{pIMMOExercise} = pIMMO + cUSD$$

The individual would supply 1 pIMMO along with 1 cUSD to mint 1 IMMO. The pIMMO is subsequently burned. Read this medium article for more information on pIMMO.

Backing per IMMO

$$IMMO_{backing} = treasuryBalance_{stablecoin} + treasuryBalance_{otherAssets}$$

Every IMMO in circulation is backed by the Immortal treasury. The assets in the treasury can be divided into two categories: stablecoin and non-stablecoin.

$$treasuryBalance_{stablecoin} = RFV_{reserveBond} + RFV_{lpBond}$$

The stablecoin balance in the treasury grows when bonds are sold. RFV is calculated differently for different bond types.

$$RFV_{reserveBond} = assetSupplied$$

For reserve bonds such as cUSD bond, the RFV simply equals to the amount of the underlying asset supplied by the bonder.

$$RFV_{lpBond} = 2sqrt(constantProduct) * (\%\ ownership\ of\ the\ pool)$$

For LP bonds such as IMMO-cUSD bond, the RFV is calculated differently because the protocol needs to mark down its value. Why? The LP token pair consists of IMMO, and each IMMO in circulation will be backed by these LP tokens - there is a cyclical dependency. To safely guarantee all circulating IMMO are backed, the protocol marks down the value of these LP tokens, hence the name risk-free value (RFV).