Editing Variable-period BOND

A variable-period BOND is a [[BOND tokens|BOND token]] which expires after a variable lifetime <math>L_B</math> which depends on the variable payout rate. This expiration date<ref>Expiration date, lifetime, and period are all interchangeable terms for the amount of time a BOND is active, denoted <math>L_B</math>.</ref> <math>L_B</math> will change as the fees the DAO earns changes. This page details the design of the contracts which govern these tokens so their valuation is fair. The variables of payout rate, the expiration time, and the value of the BOND are dependent on each other. A DAO can choose two of the variables at will, then solve for the third to determine a fair contract using REP tokenomics theory.

The results and notation from the [[Reputation tokenomics|reputation tokenomics page]] are used here derive the formulas under the assumptions that a DAO's underlying REP tokens have infinite lifetime and when they have finite lifetime. We give estimates for <math>L_B</math> under the further assumptions that the fees the DAO earns are constant.
==Overview==
Suppose we wish to pay a developer a bounty worth <math>$b</math>. We can give them BOND tokens which pay out the same as a REP token would by participating in the REP salary. Our goal is to find formulas determining how large we should set the expiration date given the rate of payout and the value.

When we pay the developer with <math>q</math> newly minted BOND tokens, which dilutes the total REP in the DAO as fees are now shared with the <math>q+R</math> tokens. The number <math>q</math> of tokens determines the rate of payout, as a larger <math>q</math> means a larger share of the REP salary. The exact rate of payout for <math>q</math> BOND token is proportional to the incoming fees as <math>f'q/(q+R)</math>.

Depending on whether the REP token design of the DAO has REP tokens with finite or infinite lifetime we get different formulas for how to program the lifetime <math>L_B</math> for a variable-period BOND.