Variable-period BOND

A variable-period BOND is a BOND token which expires after a variable lifetime $L_{B}$ which depends on the variable payout rate. This expiration date $L_{B}$ will change as the rate at which the DAO earns fees changes. This page details the design of the contracts which govern these tokens so their valuation is fair.

The three variables of payout rate, expiration time, and the value of the BOND are dependent on each other. A DAO can choose two of the variables at will, then solve for the third to determine a fair contract using REP tokenomics theory. A variable-period BOND is called a riskless BOND when the period is chosen so that the token expires once the stated value is paid out. In this page we detail the expiration formulas $L_{B}$ for paying out a fixed amount $\$b$ or after the present value of $\$b$ is reached.

The results and notation from the reputation tokenomics page are used here derive the formulas under the assumptions that a DAO's underlying REP tokens have infinite lifetime and when they have finite lifetime. The page on fixed-period BONDs estimates for the expected value of $L_{B}$ under the further assumptions that the fees the DAO earns are constant.

Overview

Normal bonds issued by a government may be considered risk-free in the sense that they are guaranteed by force of law to pay out as they are advertised. DAOs can issue BOND tokens with similar contracts guaranteeing any fixed payout, which can be assured by self-executing smart contracts, as long as the DAO remains solvent. But contracts based on the REP salary, with pre-determined end dates (so-called fixed-period BONDs) and values don't guarantee a specific value will be paid out.

One way to eliminate this risk is to change the expiration date into a variable, so that it doesn't expire until the specified value is paid through the REP salary. However, even this contract still carries risk (as do normal government bonds) in the sense that the interest rate may increase during the tenor of the contract, so the fixed return on the BOND may ultimately have a lower present value than expected.

We can eliminate even this type of risk, using the basic REP tokenomics equations. We simply make the lifetime of a BOND token into a variable which is dependent on the actual fees the DAO earns, instead of the expected fees as before. The basic idea is that the variable lifetime $L_{B}$ of a riskless BOND will grow if the fees shrink or the interest rate increases, and the lifetime will shrink if the fees grow or the interest rate decreases. Technically, a riskless contract makes the lifetime of the BOND $L_{B}$ a stopping time of the stochastic process given by the fees and the interest rate $r$. The smart contract governing the BOND tokens uses the record of fees to determine the expiration date dynamically, which guarantees the present value of the  BOND tokens at the time of issuance will be precisely the value advertised.

Suppose we wish to pay a developer a bounty worth $\$b$. We can give them $q$BOND tokens which pay out the same as a REP token would by participating in the REP salary. Our goal is to find formulas determining how large we should set the expiration date $L_{B}$ given the rate of payout and the value $\$b$.

When we pay the 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 $f'q/(q+R)$.

Mechanism design

Suppose we wish to pay a developer a bounty worth $\$b$. We can give them $q$ 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 $L_{B}$ given the rate of payout and the value $\$b$.

When we pay the 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 $f'q/(q+R)$.

The income stream for a BOND token is therefore

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

for time $0\leq t\leq L_{B}$. As before, ${\textstyle R(t)=R_{0}+\int _{0}^{t}m*f'(s)ds}$ for infinite lifetime REP tokens or ${\textstyle R(t)=\int _{t-L}^{t}m*f'(s)ds}$ if the lifetime $L$ is finite.The present value of a single BOND token is

$$PVf_{B}^{1}=\int _{0}^{L_{B}}e^{-rt}{\frac {d}{dt}}f_{B}^{1}(t)dt$$

We assume $q_{0}$ tokens are minted to make a reward of value $\$b$. We use $q(t)$ to denote the total number of BOND tokens that are earning reputational salaries in the DAO at any given time $t$, which includes $q_{0}$ 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 $q_{0}$ BOND tokens minted to have exact initial present value $\$b$ make the stopping time $L_{B}$ the random variable that satisfies the formula

$$q_{0}\int _{0}^{L_{B}}{\frac {e^{-rt}f'(t)}{q(t)+R(t)}}dt=b$$

This formula works under general assumptions, as long as $f'$ grows fast enough for the left-hand side of the equation to grow past the right-hand side for some $L_{B}<\infty$. We discuss this constraint in the 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 $f'$, any new BOND tokens added $q(t)$, and the interest rate $r(t)$. That simply means we record the history of their values. Then the stopping time $L_{B}$ is reached at the first time $t^{*}=L_{B}$ that the above equation is satisfied. The stopping time is merely the moment we end the fees paid to the $q_{0}$ 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 $r(t)$, which requires an oracle, since the other two variables, $f(t)$ and $q(t)$, are automatically recorded.

Proposition 7 gives a means for calculating the expected value and variance of the stopping time $L_{B}$ under various assumptions on the parameters $r$, $f$, $m$ and $q$.

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

$$\int _{0}^{\infty }{\frac {e^{-rt}f'(t)}{q(t)+R(t)}}dt\geq {\frac {b}{q_{0}}}\qquad \quad {\scriptstyle {\text{(Equation 8)}}}$$

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

For example let us assume that the lifetime of REP tokens is $L<\infty$ and all the parameters are constant, such as the rate fees $f'_{0}$.  Assuming the group has reached REP equilibrium, we have  $R_{\infty }=mf'_{0}L$. Further simplify by assuming the only outstanding BONDs $q(t)$ are the $q_{0}$ that are currently under consideration. Then the integral in Equation 8 may be solved to get

$${\frac {q_{0}f'_{0}}{r(q_{0}+mf'_{0}L)}}\geq b$$

A DAO cannot repay a bounty $\$b$ that doesn’t satisfy this constraint. The limit is

$$b\leq {\frac {f'_{0}}{r}}$$

which is seen by letting $q_{0}\rightarrow \infty$, because, in that case, the DAO will use all its fees to pay back the $q_{0}$ BONDs for eternity.

Proposition 8. A DAO cannot mint bonds of value in excess of ${\textstyle f'_{0}/r}$.

Conversely, solving the above constraint for ${\textstyle f'_{0}}$ shows the rate of fees must be large enough to satisfy

$$f'_{0}>{\frac {brq_{0}}{q_{0}-brmL}}>br$$

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 ${\textstyle \$b}$ from a DAO unless you can expect their fee rate to eventually far exceed ${\textstyle br}$.

See Also

• Fixed-period BOND