Editing Variable-period BOND (section)

== Mechanism design ==
Suppose we wish to pay a developer a bounty worth <math>$b</math>. We can give them <math>q</math> BOND tokens which each pay out the same as a REP token by participating in the REP salary. Our goal is to find the variable stopping time formula for the expiration date <math>L_B</math> given the rate of payout and the value <math>$b</math>.

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>.

The income stream for a BOND token is therefore<math display="block">f_B^1(t)=\int_{0}^{t} \frac{f'(s)}{q(s)+R(s)}ds</math>

for time <math>0 \leq t \leq L_B</math>. As before, <math display="inline">R(t)=R_0+ \int_{0}^{t} m*f'(s)ds</math> for infinite lifetime REP tokens or ''<math display="inline">R(t)=\int_{t-L}^t m*f' (s)ds</math>'' if the lifetime <math>L</math> is finite.The present value of a single BOND token is<math display="block">PVf_B^1=\int_0^{L_B}e^{-rt}\frac{d}{dt}f_B^1(t)dt</math>

We assume <math>q_0</math> tokens are minted to make a reward of value <math>$b</math>. We use <math>q(t)</math> to denote the total number of BOND tokens that are earning reputational salaries in the DAO at any given time <math>t</math>, which includes <math>q_0</math> and any other BOND tokens that have been minted during the relevant time period. Combining these facts gives the following result.

'''Proposition 7''' ''To pay a bounty of <math>q_0</math> BOND tokens minted to have exact initial present value <math>$b</math> make the stopping time <math>L_B</math> the random variable that satisfies the formula<math display="block">q_0\int_{0}^{L_B} \frac{e^{-rt}f'(t)}{q(t)+R(t)}dt=b</math>''



This formula works under general assumptions, as long as ''<math>f'</math>'' grows fast enough for the left-hand side of the equation to grow past the right-hand side for some ''<math>L_B< \infty</math>''. We discuss this constraint in the [[Variable-period BOND#Constraint on BOND issuance|next section]].

The result is not as deep as all the technical terminology might make it seem. The basic idea is simple. First, keep track of the random processes given by the fees ''<math>f'</math>'', any new BOND tokens added ''<math>q(t)</math>'', and the interest rate ''<math>r(t)</math>''. That simply means we record the history of their values. Then the stopping time ''<math>L_B</math>'' is reached at the first time ''<math>t^*=L_B</math>'' that the above equation is satisfied. The stopping time is merely the moment we end the fees paid to the ''<math>q_0</math>'' BOND tokens. When programming the smart contract which controls this financial device, the integral simply becomes a sum, and the stopping condition is given by an IF THEN statement. The only variable that poses any difficulty in decentralized environments is the interest rate ''<math>r(t)</math>'', which requires an oracle, since the other two variables, ''<math>f(t)</math>'' and ''<math>q(t)</math>'', are automatically recorded.

Proposition 7 gives a means for calculating the expected value and variance of the stopping time <math>L_B</math> under various assumptions on the parameters  ''<math>r</math>'', ''<math>f</math>'', ''<math>m</math>'' and ''<math>q</math>''.

=== Constraint on BOND issuance ===
In order for a BOND contract to be fully paid, the DAO must remain solvent, meaning the fees it earns must be great enough for the present value to be eventually realized. In the case of riskless BONDs the fees must satisfy the constraint''<math display="block">\int_{0}^{\infty} \frac{e^{-rt}f'(t)}{q(t)+R(t)}dt\geq \frac{b}{q_0}  \qquad  \quad  {\scriptstyle\text{(Equation 8)}}</math>''

This gives a limit for how many bounties can be proposed, lest the BONDs cannot be paid if ''<math>f'</math>'' is too small.  

For example let us assume that the lifetime of REP tokens is ''<math>L<\infty</math>'' and all the parameters are constant, such as the rate fees ''<math>f'_0</math>''.  Assuming the group has reached REP equilibrium, we have '' <math>R_{\infty}=mf'_0L</math>.'' Further simplify by assuming the only outstanding BONDs ''<math>q(t)</math>'' are the ''<math>q_0</math>'' that are currently under consideration. Then the integral in Equation 8 may be solved to get''<math display="block">\frac{q_0f'_0}{r(q_0+mf'_0L)}\geq b</math>''

A DAO cannot repay a bounty ''<math>$b</math>'' that doesn’t satisfy this constraint. The limit is''<math display="block">b\leq\frac{f'_0}{r}</math>''

which is seen by letting ''<math>q_0\rightarrow \infty</math>'', because, in that case, the DAO will use all its fees to pay back the ''<math>q_0</math>'' BONDs for eternity.

'''Proposition 8.''' ''A DAO cannot mint bonds of value in excess of'' ''<math display="inline">f'_0/r</math>''.

Conversely, solving the above constraint for ''<math display="inline">f'_0</math>'' shows the rate of fees must be large enough to satisfy''<math display="block">f'_0>\frac{brq_0}{q_0-brmL}>br</math>''

or else the DAO cannot ever repay the bounty, no matter how large the stopping time. Therefore, a good rule of thumb is not to seek a bounty ''<math display="inline">$b</math>'' from a DAO unless you can expect their fee rate to eventually far exceed ''<math display="inline">br</math>''.