Editing Fixed-period BOND (section)

=== Fixed-period BOND contracts under finite-lifetime REP ===
Next, we consider the situation when there is a finite lifetime ''<math>L<\infty</math>'' on a DAO's REP tokens. We make the further assumptions of constant minting ratio ''<math>m</math>'' and constant fees ''<math>f'_0</math>''. We assume the BOND tokens are minted after the system reaches equilibrium. In this case, there are always ''<math>R_{\infty}=mf'_0L</math>'' of the REP tokens in the system. Then diluting the system with ''<math>q</math>'' artificially minted BOND tokens at time ''<math>t_0=0</math>'' which have the same lifetime ''<math>L_B=L</math>'' gives''<math display="block">f_B^1 (t)=f_0^1 (t)= \begin{cases}t \frac{f'_0}{q+mf'_0L} & \text{if }t\leq L \\ \frac{f'_0L}{q+mf'_0L} & \text{if }t\geq L \end{cases}</math>''

So<math display="block">PVf_B^1=\int_0^\infty e^{-rt} \frac{d}{dt} f_0^1 (t)dt=\frac{f'_0}{q+mf'_0L}\int_0^L e^{-rt}dt=\frac{f'_0}{q+mf'_0L}\left ( \frac{1-e^{-rL}}{r}\right ).</math>Therefore we solve the equation <math>qPVf_B^1=b</math> for <math>q</math>  to get

'''Proposition 6:''' A''ssume the rate of fees <math>f'_0</math>'' ''is constant, and the lifetime of all tokens (REP and BONDs) is <math>L<\infty</math>. To pay a bounty with present value worth <math>$b</math> a DAO can mint <math>q</math> BOND tokens where<math display="block">q= \frac{brmf'_0L}{f'_0(1-e^{-rL})-rb}</math>'' 

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.

Notice Proposition 6 gives a bound on the value ''<math>$b</math>'' of BOND tokens that can be minted based on the amount of fees ''<math>f'_0</math>'' the DAO is earning ''<math display="inline">b<f'_0(1-e^{-rL})/r</math>''.

The major problem with these formulas is that the assumption that the rate of fees ''<math>f'_0</math>'' is constant is false and will often be very inaccurate, especially when a DAO is small. The above solutions make BONDs a gamble for both the developer and the DAO. If the rate of fees ''<math>f'</math>'' increases during the lifetime ''<math>L_B</math>'' then the reward’s value will be greater than ''<math>$b</math>'', and if the rate of fees decreases it will be worth less. However, as mentioned above, Jensen’s inequality gives us a bound, showing these results are conservative. Specifically, if the fees’ rate is not constant, but that the fees merely have expected value ''<math>f'_0</math>'' then these formulas will be generous to the BOND holder. If however, the actual values of the fees have an average less than this expected value, the BOND holders can still end with less than ''<math>$b</math>'' remuneration in present value.

Such uncertainties can be eliminated by more complicated contracts which have [[Riskless BONDs|variable lifetimes]].