Editing Fixed-period BOND (section)

=== Fixed-period BOND contracts under infinite-lifetime REP ===
Under the assumption that a DAO's REP lifetime is unbounded, <math>L= \infty</math>, we wish to find the lifetime <math>L_B</math> we should set for a BOND so that it will have a particular value. In this case the income stream <math>f_B^1(t)</math> of one BOND token is the same as one REP token <math>f_0^1(t)</math>

<math display="block">f_B^1(t)=\int_{0}^{t} \frac{f'(s)}{q(s)+R(s)}ds</math>

until the BOND expires. In general, we have <math display="inline">R(t)=R_0+ \int_{0}^{t} m*f'(s)ds.</math> The term <math>q(s)</math> represents all the BONDs which are active at time <math>t=s</math>.

Now suppose you want to find the lifetime <math>L_B</math> of a BOND that will give an expected payout of <math>$b</math> for <math>q</math> tokens. We therefore hope to solve the equation<math display="block">qf_B^1(L_B)=b</math>

However, the terms <math>L_B</math> and <math>f'</math> are complicated. <math>L_B</math> is a stopping time, and <math>f'</math> is a stochastic process. Nevertheless, these terms behave relatively well, since we know the expected value <math>E[f_B^1(t)]</math> will be an increasing function of <math>t</math>, because <math>f_B^1(t)</math> is increasing, because <math>f'(s)\geq0</math>. Simply taking the expected value

<math display="block">E_0 \left[f_B^1(t) \right ]=\int_{0}^{t}E_0 \left [ \frac{f'(s)}{q(0)+R_0+\int_{0}^sm*f'(u)du} \right ] ds</math>setting it equal to <math>b</math> and solving for <math>t=t^*</math> gives you the desired stopping time <math>t^*=E_0[L_B]</math>. 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 <math>f'(s)=f'_0</math> and no further BOND tokens are minted during the lifetime <math>L_B</math>. Then there is a minor change in the previous formulas<math display="block">f_B^1(t)=\int_{0}^{t} \frac{f'_0}{q_0+R_0+smf'_0}ds=\frac{1}{m}ln\left ( \frac{q+R_0+mtf'_0}{q+R_0}\right )</math>Therefore, to pay a developer <math>q</math> tokens that will have payout with value <math>$b</math> at time <math>L_B</math> we solve the equation <math display="inline">f_B^1(L_B)=b</math> for <math>L_B</math>''.'' We get <math display="block">L_B=(q+R_0) \frac{e^{mb}-1}{mf'_0}</math>

Further, we can find the present value of a single BOND token with arbitrary lifetime <math>L_B</math> under the assumption of constant fees is<math display="block">PVf_B^1=\int_0^{L_B}e^{-rt}\frac{d}{dt}f_B^1(t)dt=\frac{1}{m}exp\left ( \frac{r(q+R_0)}{mf'_0} \right)\int_{\frac{r(q+R_0)}{mf'_0}}^{L_B}\frac{e^{-s}}{s}ds</math> 

Therefore we have the following solutions:

'''Proposition 5:''' ''Assume the rate of fees is constant <math>f'_0</math> and that no further'' BOND ''tokens are minted during the lifetime <math>L_B</math>.''

''Then <math>q</math> ''BOND ''tokens will have payout <math>$b</math> by time <math>L_B</math> at a payout rate of <math>qf'_0</math> by setting the lifetime of a'' BOND ''to be<math display="block">L_B=(q+R_0) \frac{e^{mb}-1}{mf'_0}</math>''

''Again, assuming constant rate of fees, if you wish to pay a developer <math>q</math> ''BOND ''tokens with arbitrary predetermined lifetime <math>0 \leq L_B \leq \infty</math> that will have expected present value worth <math>$b</math> solve the following equation for <math>q</math><math display="block">b=q\frac{1}{m}exp\left ( \frac{r(q+R_0)}{mf'_0} \right)\int_{\frac{r(q+R_0)}{mf'_0}}^{L_B}\frac{e^{-s}}{s}ds  \qquad  \quad  {\scriptstyle\text{(Equation 7)}}</math>''

Equation 7 has no elementary solution, but admits efficient solutions through standard numerical approximation algorithms. However, next we assume the REP tokens have finite lifetimes, which guarantees explicit elementary solutions.