Editing Present value (section)

== 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 <math display="inline">PV</math> of the $100 promise one year in the future can be found by solving the equation <math display="block">PV\left(1+r\right)=100</math>since the interest on <math display="inline">PV</math> after one period (one year in the present example) is <math display="inline">PV(1+r)</math> so you would have the original <math display="inline">PV</math> plus the interest, which is <math display="inline">rPV</math>. Here $100 is the '''future value''' <math display="inline">FV</math> of the present value . So the formula for determining present value from the future value given by<math display="block">PV\left(1+r\right)=FV.</math>

For instance if the annual interest rate is 7% then substituting <math display="inline">r=.07</math> 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 <math display="inline">r</math> is called the '''discount factor'''.

In case we are going to invest and earn interest on this money for several periods, say <math>n</math> years, then the compound interest formula is <math display="block">PV\left(1+r\right)^n=FV</math>assuming interest is compounded annually, so<math display="block">PV=\frac{FV}{\left(1+r\right)^n}.</math>


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 [[Reputation#REP salary mechanism|reputational salary]] which depends on the fees the DAO earns and the [[Reputation#REP Token Minting Mechanism|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 <math>f_0^\prime</math> each year. The financial term for such a tool is an [[wikipedia:Annuity|annuity]], and in particular this is [[wikipedia:Perpetuity|perpetuity]] since it theoretically never ends. In this case <math>f\left(n\right)=nf_0^\prime</math> and the present value of the fees given for all eternity is<math display="block">PVf=\frac{f_0^\prime}{\left(1+r\right)^0}+\frac{f_0^\prime}{\left(1+r\right)^1}+\frac{f_0^\prime}{\left(1+r\right)^2}+\ldots=\sum_{i=0}^{\infty}\frac{f_0^\prime}{\left(1+r\right)^i}=\left(1+\frac{1}{r}\right)f_0^\prime</math>using the geometric series formula. 

This illustrates an important intuition: even though the income stream is infinite <math>f\left(n\right)\rightarrow\infty</math> as <math>n\rightarrow\infty</math>, 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 <math>r</math> 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<ref>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 <math display="inline">\sum_{i=n+1}^{\infty}\frac{f^\prime\left(i\right)}{\left(1+r\right)^i}</math> shrink to 0 as <math>n</math> grows. However, this is not always insignificant. If <math>f^\prime</math> is bounded the tail shrinks like <math display="inline">\frac{1}{\left(1+r\right)^n}</math>. 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]].</ref> to<math display="block">PVf=\sum_{i=0}^{\infty}\frac{f^\prime\left(i\right)}{\left(1+r\right)^i}</math>The more general case<ref>This generalizes the discrete case by using the formalism of tempered distributions, which include atomic distributions.</ref> allows for continuous compounding, which is derived from the formula

<math display="block">V\left(1+\frac{r}{n}\right)^{tn}=FV</math>which describes how interest accumulates over a period of <math>t</math> years if the compounding periods are broken into <math>n</math> periods per year. The continuously compounding formula is then

<math display="block">\lim_{n \to \infty}{{PV\left(1+\frac{r}{n}\right)}^{tn}}=FV</math>which gives the present value <math>PV</math> of a reward <math>FV</math> promised <math>t</math> years in the future as 

<math display="block">PV=FVe^{-rt}.</math>Then the income stream <math display="inline">f_0^1\left(t\right)</math> leads to the desired present value formula by taking the limit of Riemann sums:

<math display="block">PVf_0^1\approx\frac{f_0^1\left(t_1\right)-f_0^1\left(t_0\right)}{t_1-t_0}e^{-rt_1}\Delta t+\frac{f_0^1\left(t_2\right)-f_0^1\left(t_1\right)}{{\ t}_2-t_1}e^{-rt_2}\Delta t+...+\frac{f_0^1\left(t_n\right)-f_0^1\left(t_{n-1}\right)}{{\ t}_n-t_{n-1}}e^{-rt_n}\Delta t
</math>

<math display="block">\approx{\frac{d}{dt}}f_0^1\left(t_1\right)e^{-rt_1}\Delta t+{\frac{d}{dt}}f_0^1\left(t_2\right)e^{-rt_2}\Delta t+...+{\frac{d}{dt}}f_0^1\left(t_n\right)e^{-rt_n}\Delta t
</math>This formula follows from the idea that the reward you will get during time period <math display="inline">\left[t_1,{\ t}_2\right]</math> will be the amount your total fees increased <math>f_0^1\left(t_2\right)-f_0^1\left(t_1\right)</math>. We fix regular time intervals <math display="inline">\Delta t={\ t}_{i+1}-t_i</math>. Taking the limit gives the foundational formulas:

'''Proposition 1''' <math display="block">PVf_0^1=\int_{0}^{\infty}{e^{-rt}{\frac{d}{dt}}f_0^1\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>''