Mathematical Basis of the DAO

# Staking

$aEGIS = EGIS$
$aEGIS will always be redeemable for$EGIS as a 1:1 swap since the DAO will always convert bond sale profits to EGIS rewards.
$Rebase = 1 - (EGIS_{deposited}/aEGIS_{outstanding})$
The treasury deposits EGIS into the distributor. The distributor then deposits EGIS into the staking contract, creating an imbalance between EGIS and aEGIS. aEGIS is therefore rebased to correct this imbalance between EGIS deposited and aEGIS outstanding. The rebase brings aEGIS outstanding back up to parity with aEGIS so that 1 aEGIS always equals 1 staked EGIS. This rebasing mechanism facilitates the aEGIS = EGIS equation above.

# Bonding

$Bond Price = 1 + Market Premium$
The price of a bond is governed by the interaction between FTM and the market sentiment of EGIS. The sentiment of the market determines the premium applied to bond prices.
$MarketPremium = DebtRatio * BCV$
The Market Premium is derived from the debt ratio of the system and a scaling variable called BCV, which is in turn governed by market sentiment.
$DebtRatio = BondsOutstanding / EGIS_{supply}$
The debt ratio is a variable which allows us to measure the debt of the Protocol to Bonders, allowing for a dynamic Market Premium.
$BondPayout\scriptsize ReserveBond \normalsize = MarketValue \scriptsize Asset \normalsize / BondPrice$
Bond payout determines the number of EGIS 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 200 FTM and the bond price is 50 FTM, the user will be entitled 4 EGIS since 200/50 = 4.
$BondPayout \scriptsize lpBond \normalsize = MarketValue \scriptsize lpToken \normalsize / BondPrice$
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 EGIS-FTM LP token which is valued at 1000 FTM at the time of bonding, and the bond price is 250 DAI, the user will be entitled to 4 EGIS.

# Asset Backing

$EGIS \scriptsize Backing \normalsize = TreasuryBalance$
Every circulating EGIS token is backed by the treasury in either stablecoins or non-stablecoins.
$TreasuryBalance \scriptsize stablecoins \normalsize = RFV \scriptsize ReserveBond \normalsize + RFV \scriptsize lpBond$
When bonds are sold, the stablecoin reserve of the treasury grows. RFV (risk-free value) is calculated differently based upon the bond type.
$RFV \scriptsize ReserveBond \normalsize = AssetSupplied$
Reserve bonds in stablecoins like DAI, the RFV is equal to the value of the asset suppled by the bonder.
$RFV\scriptsize lpBond \normalsize = 2 \sqrt ConstantProduct * \% Ownership\scriptsize Pool$
For LP bonds such as EGIS-DAI bond, the RFV is calculated differently because the protocol needs to mark down its value.
Why? The LP token pair consists of EGIS, and each EGIS in circulation will be backed by these LP tokens - there is a cyclical dependency. To safely guarantee all circulating EGIS are backed, the protocol marks down the value of these LP tokens, hence the name risk-free value (RFV).