# Equations

## Staking

$deposit = withdrawal$
Swaps between VSQ and sVSQ during staking and unstaking are always honored 1:1. The amount of VSQ deposited into the staking contract will always result in the same amount of sVSQ. And the amount of sVSQ withdrawn from the staking contract will always result in the same amount of VSQ.
$rebase = 1 - ( vsqDeposits / sVSQOutstanding )$
The treasury deposits VSQ into the distributor. The distributor then deposits VSQ into the staking contract, creating an imbalance between VSQ and sVSQ. sVSQ is rebased to correct this imbalance between VSQ deposited and sVSQ outstanding. The rebase brings sVSQ outstanding back up to parity so that 1 sVSQ equals 1 staked VSQ.

## Bonding

$bond Price = 1 + Premium$
VSQ has an intrinsic value of 1 DAI, which is roughly equivalent to \$1. In order to make a profit from bonding, Olympus 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 VSQ is minted from the profit and subsequently distributed among all stakers.
$debt Ratio = bondsOutstanding/vsqSupply$
The debt ratio is the total of all VSQ promised to bonders divided by the total supply of VSQ. This allows us to measure the debt of the system.
$bondPayout_{reserveBond} = marketValue_{asset}\ /\ bondPrice$
Bond payout determines the number of VSQ 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 DAI and the bond price is 250 DAI, the user will be entitled 4 VSQ.
$bondPayout_{lpBond} = marketValue_{lpToken}\ /\ bondPrice$
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 VSQ-DAI LP token which is valued at 1000 DAI at the time of bonding, and the bond price is 250 DAI, the user will be entitled 4 VSQ.

## VSQ Supply

$VSQ_{supplyGrowth} = VSQ_{stakers} + VSQ_{bonders} + VSQ_{DAO} + VSQ_{pvsqExercise}$
VSQ supply does not have a hard cap. Its supply increases when:
• VSQ is minted and distributed to the stakers.
• VSQ is minted for the bonder. This happens whenever someone purchases a bond.
• VSQ is minted for the DAO. This happens whenever someone purchases a bond. The DAO gets the same number of VSQ as the bonder.
• VSQ is minted for the team, investors, advisors, or the DAO. This happens whenever
the aforementioned party exercises their pVSQ.
$VSQ_{stakers} = VSQ_{totalSupply} * rewardRate$
At the end of each epoch, the treasury mints VSQ at a set reward rate. These VSQ will be distributed to all the stakers in the protocol. You can track the latest reward rate on the VESQ Policy dashboard.
$VSQ_{bonders} = bondPayout$
Whenever someone purchases a bond, a set number of VSQ is minted. These VSQ 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.
$VSQ_{DAO} = VSQ_{bonders}$
The DAO receives the same amount of VSQ as the bonder. This represents the DAO profit.
$VSQ_{pvsqExercise} = pVSQ + DAI$
The individual would supply 1 pVSQ along with 1 DAI to mint 1 VSQ. The pVSQ is subsequently burned. Read this Medium article for more information on pVSQ.

## Backing per VSQ

$VSQ_{backing} = treasuryBalance_{stablecoin} + treasuryBalance_{otherAssets}$
Every VSQ in circulation is backed by the VESQ 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 DAI bond and FRAX 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 VSQ-DAI bond and VSQ-FRAX bond, the RFV is calculated differently because the protocol needs to mark down its value. Why? The LP token pair consists of VSQ, and each VSQ in circulation will be backed by these LP tokens - there is a cyclical dependency. To safely guarantee all circulating VSQ are backed, the protocol marks down the value of these LP tokens, hence the name risk-free value (RFV).