Present value: Difference between revisions
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''The REP salary of a single token with infinite lifetime is <math display="block">f_0^1\left(t\right)=\int_{0}^{t}{\frac{f^\prime\left(s\right)}{R\left(s\right)}ds}.</math>The present value of a single token in a DAO with income stream <math>f_0^\prime</math> is <math display="block">PVf_0^1=\int_{0}^{\infty}{e^{-rt}\frac{f^\prime\left(t\right)}{R\left(t\right)}dt}.</math>'' | ''The REP salary of a single token with infinite lifetime is <math display="block">f_0^1\left(t\right)=\int_{0}^{t}{\frac{f^\prime\left(s\right)}{R\left(s\right)}ds}.</math>The present value of a single token in a DAO with income stream <math>f_0^\prime</math> is <math display="block">PVf_0^1=\int_{0}^{\infty}{e^{-rt}\frac{f^\prime\left(t\right)}{R\left(t\right)}dt}.</math>'' | ||
== Code == | |||
== See Also == | |||
* [[Reputation Tokenomics]] | |||
== Notes == | == Notes == |
Revision as of 09:56, 7 March 2023
The present value (PV) is the value of an expected income stream determined as of the date of valuation. This formula is of primary importance in REP tokenomics when valuing a REP token based on its future income through the REP salary.
Derivation of present value formulas for REP tokens
Suppose you know that next year you are guaranteed to be given $100. What is that promise worth today? Like any financial or economic question, there are many assumptions that go into that question. To eliminate psychological variations between people, let’s abstract the situation. Assume for the moment you are a bank, not a person with varying idiosyncratic desires. And further, you are perfectly confident you will receive the money next year.
To answer the question, we need only know the interest rate to determine how much an investment today would return in 1 year. Then the present value of the $100 promise one year in the future can be found by solving the equation
For instance if the annual interest rate is 7% then substituting we calculate the promise of $100 one year in the future has the same value as $93.46 in your hand today. (Financial theorists justify this valuation with a technical argument that macroscopically, over many business deals, any other contract would lead to an arbitrage opportunity.) The term is called the discount factor.
In case we are going to invest and earn interest on this money for several periods, say years, then the compound interest formula is
Our main goal in REP tokenomics is to find the present value of a reputation token, which has one more complication. We will receive many, differently-sized future rewards due to the reputational salary which depends on the fees the DAO earns and the number of REP tokens that share in the salary. To calculate the present value of all these future rewards, we merely sum them up their individual present value. The analogous traditional terminology is an income stream for, say, a corporation or a retiree. To motivate the general formula which includes an integral, let’s first make the simplifying assumption that our fees arrive as constant discrete payments each year, forever. Suppose we get paid each year. The financial term for such a tool is an annuity, and in particular this is perpetuity since it theoretically never ends. In this case and the present value of the fees given for all eternity is
This illustrates an important intuition: even though the income stream is infinite as , nevertheless the present value of the perpetuity is finite. Further we see, e.g., that the present value is inversely related to the interest rate. If the interest rate is higher, then the future payments will be worth less, so the present value is lower.
Now if we allow the fees to vary in time, we simply adjust the formula[1] to
Proposition 1
The REP salary of a single token with infinite lifetime is
Code
See Also
Notes
- ↑ For the following general reasons, we ignore the obvious fact that the DAO will not receive fees for all eternity. The formula is still a good first approximation if we don’t have any information about when the DAO will collapse, because the convergence of the series makes the tail shrink to 0 as grows. However, this is not always insignificant. If is bounded the tail shrinks like . For instance, if the interest rate is 7% (the historical stock market return minus historical inflation) and we assume the DAO will last 50 years, then this factor is 3.4%. Not 0. The savvy investor will account for this when choosing their bids for REP tokens in a REP market.
- ↑ This generalizes the discrete case by using the formalism of tempered distributions, which include atomic distributions.