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# Equations

#### Staking

$pTHEO \ amount\ to \ unstake^‡ = THEO$
‡ At or beyond the full staking Time Period.
$pTHEO \ amount\ to\ unstake^¶ = (Principal \ Deposit - Penalty)THEO$
¶Before the full staking Time Period.
Reward rates
The reward rate for staking is variable, based on the APYVariable as follows:
$APY_{Variable}= APY_{Fixed} + SC_{RS} + SC_{YS}$
$SC_{RS} = D_{RS}ΔtokenPrice$
$SC_{YS} = D_{YS}ΔtreasuryYield$
Where
$D_{RS}$
is a discount rate applied to the fluctuation of the token price (ΔtokenPrice; the difference in token price between time intervals of 8 hours), as a percentage.
$D_{RS}$
$D_{YS}$
is the discount rate applied to ΔtreasuryYield, as a percentage. ΔtreasuryYield is the fluctuation of the treasury yield, measured in arrears from the prior month.
$D_{YS}$
is set manually and can be updated.
$APY_{Fixed}$
(Starting Percentage) for each tranche will unwind over time, eventually settling to a final value.
$APY_{Variable}$
is capped by a Max Percentage. The Starting Percentage, Max Percentage, and wind down structure for the staking tranches are shown in the table below.
Year
Starting Percentage, Locked Tranche
Max Percentage, Locked Tranche
Starting Percentage, Unlocked Tranche
Max Percentage, Unlocked Tranche
1
12%
18%
4%
6%
2
10.5%
15.75%
3.5%
5.25%
3
9%
13.5%
3%
4.5%
4
7.5%
11.25%
2.5%
3.75%
5
6%
9%
2%
3%
The reward rate (r) for the next 8-hour performance-update period is calculated using APYVariable as follows:
$r = ne^{1/n \ ln(APY_{variable} + 1)} - n$
Where n is 1095, the number of 8-hour performance-updates per 365 days.

### Bonding

Users can deposit a principal in exchange for a bond from a specified market.
Bonds can be redeemed after their vesting period, for a payout, in THEO or sTHEO (depending on the type of deposit available).
The payout amount is variable, depending on the current price (P) of the principal token in THEO terms, calculated as follows:
$P = CMV (1 - BondRate_{Variable})$
Where CMV is the current market value, obtained from Uniswap V3 price oracle data.
$BondRate_{Variable} = BondRate_{Fixed} + BC_{RB} + BC_{YB}$
$BondRate_{Fixed}$
is a fixed rate set when the bonding market is created.
$BC_{RB}$
is the Bond Control Return for Bonding, calculated as:
$BC_{RB} = D_{RB}ΔtokenPrice$
$D_{RB}$
(Discount Rate Return Bond) is the discount rate applied to the fluctuation of the token price (ΔtokenPrice; the difference in token price between time intervals of 8 hours), as a percentage.
$D_{RB}$
is set manually and can be updated.
$BC_{YB}$
is the Bond Control Treasury for Bonding, calculated as:
$BC_{YB} = D_{YB}ΔtreasuryYield$
$D_{YB}$
is the discount rate applied to the fluctuation of the treasury yield (ΔtreasuryYield; measured in arrears from the prior month), as a percentage.
$D_{YB}$
is set manually and can be updated.

#### Whitelisted Bonding

Addresses can deposit a principal in exchange for a bond that is redeemable for THEO after the vesting period. Bonds are priced using a fixed US Dollar rate per THEO (depending on the market; for example \$10 per THEO). The principal deposit amount is converted to the equivalent amount in US Dollars using Chainlink oracle data.