Fixed-period BOND
A Fixed-period BOND is a BOND token which expires after a fixed lifetime .
Suppose we pay a developer with newly minted BOND tokens, which dilutes the total REP in the DAO as fees are now shared with the tokens. The number of tokens determines the rate of payout, as a larger means a larger share of the REP salary. The exact rate of payout for BOND token is proportional to the incoming fees as
1.1.1 Infinite lifetime
Under the assumption that REP lifetime is unbounded, , then the income stream of one BOND token is the same as a REP token
In general, we have The term represents all the BONDs which are active at time .
Now suppose you want to find the lifetime of a BOND that will give an expected payout of for tokens. We therefore hope to solve the equation
However, the terms and are complicated. is a stopping time, and is a stochastic process. Nevertheless, these terms behave relatively well, since we know the expected value of will be an increasing function of , because is increasing, because . Simply taking the expected value
setting it equal to and solving for gives you the desired stopping time . There is no simple formula for the expected values of fractions, so we make no further elaboration of the general case in this presentation. However, Jensen’s inequality allows us to give an estimate of the general case, and the following calculations give an upper limit.
Now assume the rate of fees is constant and no further BOND tokens are minted during the lifetime . Then there is a minor change in the previous formulas
Therefore, to pay a developer tokens that will have payout with value at time we solve the equation
for . We get
Further, we can find the present value of a single BOND token with arbitrary lifetime under the assumption of constant fees is
Therefore we have the following solutions:
Proposition 5: Assume the rate of fees is constant and that no further BOND tokens are minted during the lifetime .
Then BOND tokens will have payout by time at a payout rate of by setting the lifetime of a BOND to be
To pay a developer BOND tokens with arbitrary predetermined lifetime that will have present value worth solve the equation for
Equation 7
Equation 7 has no elementary solution, but admits efficient solutions through standard numerical algorithms. However, next we assume the REP tokens have finite lifetimes which guarantees explicit elementary solutions.
1.1.2 Finite lifetime
Next, we consider the situation when there is a finite lifetime on all tokens. We make the further assumptions of constant minting ratio and constant fees . We assume the BOND tokens are minted after the system reaches equilibrium. In this case, there are always of the REP tokens in the system. Then diluting the system with artificially minted BOND tokens at time which have the same lifetime gives
So
Therefore we solve the equation
for to get
Proposition 6: Assume the rate of fees is constant and the lifetime of all tokens is . To pay a bounty with present value worth a DAO can mint BOND tokens where
Similar calculations can be made to get the formula for the number of BOND tokens when we choose the lifetime independently of the lifetime of the normal REP tokens.
The major problem with these formulas is that the assumption that the rate of fees is constant is usually false. The above solutions make BONDs a gamble for both the developer and the DAO. If the rate of fees increases during the lifetime then the reward’s value will be greater than , and if the rate of fees decreases it will be worth less. However, as mentioned above, Jensen’s inequality gives us a bound, showing these results are conservative. Specifically, if the fees’ rate is not constant, but that the fees merely have expected value then these formulas will be generous to the BOND holder. If however, the actual values of the fees have an average less than this expected value, the BOND holders can still end with less than remuneration in present value.