Editing 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:''' A''ssume 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.