Fixed-period BOND

A Fixed-period BOND is a BOND token which expires after a fixed lifetime $L_{B}$. This page details the design of the contract which governs these tokens so their valuation is fair. We use the results and notation from the reputation tokenomics page.

Overview

Suppose we pay a developer with $q$ newly minted BOND tokens, which dilutes the total REP in the DAO as fees are now shared with the $q+R$ tokens. The number $q$ of tokens determines the rate of payout, as a larger $q$ means a larger share of the REP salary. The exact rate of payout for $q$ BOND token is proportional to the incoming fees as $f'q/(q+R)$.

Depending on whether the REP token design of the DAO has REP tokens with finite or infinite lifetime we get different formulas for how long the lifetime $L_{B}$ should be set for a fixed-period BOND.

Fixed-period BOND contracts under infinite-lifetime REP

Under the assumption that a DAO's REP lifetime is unbounded, $L=\infty$, we wish to find the lifetime $L_{B}$ we should set for a BOND so that it will have a particular value. In this case the income stream $f_{B}^{1}(t)$ of one BOND token is the same as one REP token $f_{0}^{1}(t)$

$$f_{B}^{1}(t)=\int _{0}^{t}{\frac {f'(s)}{q(s)+R(s)}}ds$$

until the BOND expires. In general, we have ${\textstyle R(t)=R_{0}+\int _{0}^{t}m*f'(s)ds.}$ The term $q(s)$ represents all the BONDs which are active at time $t=s$.

Now suppose you want to find the lifetime $L_{B}$ of a BOND that will give an expected payout of $\$b$ for $q$ tokens. We therefore hope to solve the equation

$$qf_{B}^{1}(L_{B})=b$$

However, the terms $L_{B}$ and $f'$ are complicated. $L_{B}$ is a stopping time, and $f'$ is a stochastic process. Nevertheless, these terms behave relatively well, since we know the expected value $E[f_{B}^{1}(t)]$ will be an increasing function of $t$, because $f_{B}^{1}(t)$ is increasing, because $f'(s)\geq 0$. Simply taking the expected value

$$E_{0}\left[f_{B}^{1}(t)\right]=\int _{0}^{t}E_{0}\left[{\frac {f'(s)}{q(0)+R_{0}+\int _{0}^{s}m*f'(u)du}}\right]ds$$

setting it equal to $b$ and solving for $t=t^{*}$ gives you the desired stopping time $t^{*}=E_{0}[L_{B}]$. 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 $f'(s)=f'_{0}$ and no further BOND tokens are minted during the lifetime $L_{B}$. Then there is a minor change in the previous formulas

$$f_{B}^{1}(t)=\int _{0}^{t}{\frac {f'_{0}}{q_{0}+R_{0}+smf'_{0}}}ds={\frac {1}{m}}ln\left({\frac {q+R_{0}+mtf'_{0}}{q+R_{0}}}\right)$$

Therefore, to pay a developer $q$ tokens that will have payout with value $\$b$ at time $L_{B}$ we solve the equation ${\textstyle f_{B}^{1}(L_{B})=b}$ for $L_{B}$. We get

$$L_{B}=(q+R_{0}){\frac {e^{mb}-1}{mf'_{0}}}$$

Further, we can find the present value of a single BOND token with arbitrary lifetime  under the assumption of constant fees is

$$PVf_{B}^{1}=\int _{0}^{L_{B}}e^{-rt}{\frac {d}{dt}}f_{B}^{1}(t)dt={\frac {1}{m}}exp\left({\frac {r(q+R_{0})}{mf'_{0}}}\right)\int _{\frac {r(q+R_{0})}{mf'_{0}}}^{L_{B}}{\frac {e^{-s}}{s}}ds$$

Therefore we have the following solutions:

Proposition 5: Assume the rate of fees is constant $f'_{0}$ and that no further BOND tokens are minted during the lifetime $L_{B}$.

Then $q$ BOND tokens will have payout $\$b$ by time $L_{B}$ at a payout rate of $qf'_{0}$ by setting the lifetime of a BOND to be

$$L_{B}=(q+R_{0}){\frac {e^{mb}-1}{mf'_{0}}}$$

Again, assuming constant rate of fees, if you wish to pay a developer $q$ BOND tokens with arbitrary predetermined lifetime $0\leq L_{B}\leq \infty$ that will have expected present value worth $\$b$ solve the following equation for $q$

$$b=q{\frac {1}{m}}exp\left({\frac {r(q+R_{0})}{mf'_{0}}}\right)\int _{\frac {r(q+R_{0})}{mf'_{0}}}^{L_{B}}{\frac {e^{-s}}{s}}ds\qquad \quad {\scriptstyle {\text{(Equation 7)}}}$$

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.

Fixed-period BOND contracts under finite-lifetime REP

Next, we consider the situation when there is a finite lifetime $L<\infty$ on a DAO's REP tokens. We make the further assumptions of constant minting ratio $m$ and constant fees $f'_{0}$. We assume the BOND tokens are minted after the system reaches equilibrium. In this case, there are always $R_{\infty }=mf'_{0}L$ of the REP tokens in the system. Then diluting the system with $q$ artificially minted BOND tokens at time $t_{0}=0$ which have the same lifetime $L_{B}=L$ 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.