Equations

Staking

$$deposit = withdrawal$$

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 And the amount of sPB withdrawn from the staking contract will always result in the same amount of PB.

$$rebase = 1 - ( PBeposits /sPB Outstanding )$$

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

Minting

Minting happens by allowing users to purchase a bond. This bond price is the Mint price.

$$bond Price = 1 + Premium$$

PB has an intrinsic value of 1 MIM, which is roughly equivalent to $1. In order to make a profit from minting, PiggyBankDAO charges a premium for each minting action.

$$Premium = debt Ratio * BCV$$

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 PB is minted from the profit and subsequently distributed among all stakers.

$$debt Ratio = bondsOutstanding/PBupply$$

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

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

Bond payout determines the number of PB sold to a minter. For reserve mints, the market value of the assets supplied by the minter is used to determine the bond payout. For example, if a user supplies 1000 MIM and the mint price is 250 MIM, the user will be entitled 4 PB.

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

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

PB Supply

$$PB_{supplyGrowth} = PB_{stakers} + PB_{bonders} + PB_ {DAO}$$

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

• PB is minted and distributed to the stakers.

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

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

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

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

$$PB_{bonders} = bondPayout$$

Whenever someone purchases a bond, a set number of PB is minted. These PB will not be released to the minter 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 Minting section above to see how it is calculated.

$$PB_{DAO} = PB_{bonders}$$

The DAO receives the same amount of PB as the minter. This represents the DAO profit.

Backing per PB

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

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

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

The stablecoin balance in the treasury grows when bonds are sold. Backing per PB is calculated differently for different mints types.

$$BackingPerPB_{reserveBond} = assetSupplied$$

For reserve mints such as MIM minting, the Backing per PB simply equals to the amount of the underlying asset supplied by the minter.

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

For LP Mints such as PB-AVAX Minting, the RBacking Per PB is calculated differently because the protocol needs to mark down its value. Why? The LP token pair consists of PB, and each PB in circulation will be backed by these LP tokens - there is a cyclical dependency. To safely guarantee all circulating PB are backed, the protocol marks down the value of these LP tokens.