Present value

The present value (PV) is the value of an expected income stream determined as of the date of valuation.

Derivation of present value formula for time series

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 ${\textstyle PV}$ of the $100 promise one year in the future can be found by solving the equation

$$PV\left(1+r\right)=100$$

since the interest on ${\textstyle PV}$ after one period (one year in the present example) is ${\textstyle PV(1+r)}$ so you would have the original ${\textstyle PV}$ plus the interest, which is ${\textstyle rPV}$. Here $100 is the future value ${\textstyle FV}$ of the present value . So the formula for determining present value from the future value given by

$$PV\left(1+r\right)=FV.$$

For instance if the annual interest rate is 7% then substituting ${\textstyle r=.07}$ 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 ${\textstyle r}$ is called the discount factor.

In case we are going to invest and earn interest on this money for several periods, say $n$ years, then the compound interest formula is

$$PV\left(1+r\right)^{n}=FV$$

assuming interest is compounded annually, so

$$PV={\frac {FV}{\left(1+r\right)^{n}}}.$$

Our main question on the present value of a reputation token is a bit more complicated. We will receive many, differently-sized future rewards due to the reputational salary. To calculate these future rewards’ present value we merely sum them up. 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 $f_{0}^{\prime }$ 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 $f\left(n\right)=nf_{0}^{\prime }$ and the present value of the fees given for all eternity is

$$PVf={\frac {f_{0}^{\prime }}{\left(1+r\right)^{0}}}+{\frac {f_{0}^{\prime }}{\left(1+r\right)^{1}}}+{\frac {f_{0}^{\prime }}{\left(1+r\right)^{2}}}+\ldots =\sum _{i=0}^{\infty }{\frac {f_{0}^{\prime }}{\left(1+r\right)^{i}}}=\left(1+{\frac {1}{r}}\right)f_{0}^{\prime }$$

using the geometric series formula. This illustrates, that even though the income stream is infinite $f\left(n\right)\rightarrow \infty$ as $n\rightarrow \infty$, 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 $r$ 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 to

$$PVf=\sum _{i=0}^{\infty }{\frac {f^{\prime }\left(i\right)}{\left(1+r\right)^{i}}}$$

The more general case allows for continuous compounding which is derived from the formula

$$V\left(1+{\frac {r}{n}}\right)^{tn}=FV$$

which describes how interest accumulates over a period of $t$ years if the compounding periods are broken into $n$ periods per year. The continuously compounding formula is then

$$\lim _{n\to \infty }{{PV\left(1+{\frac {r}{n}}\right)}^{tn}}=FV$$

which gives the present value $PV$ of a reward $FV$ promised $t$ years in the future as

$$PV=FVe^{-rt}.$$

Then the income stream ${\textstyle f_{0}^{1}\left(t\right)}$ leads to the desired present value formula is given by taking the limit of Riemann sums

$$PVf_{0}^{1}\approx {\frac {f_{0}^{1}\left(t_{1}\right)-f_{0}^{1}\left(t_{0}\right)}{t_{1}-t_{0}}}e^{-rt_{1}}\Delta t+{\frac {f_{0}^{1}\left(t_{2}\right)-f_{0}^{1}\left(t_{1}\right)}{{\ t}_{2}-t_{1}}}e^{-rt_{2}}\Delta t+...+{\frac {f_{0}^{1}\left(t_{n}\right)-f_{0}^{1}\left(t_{n-1}\right)}{{\ t}_{n}-t_{n-1}}}e^{-rt_{n}}\Delta t$$

$$\approx {\frac {d}{dt}}f_{0}^{1}\left(t_{1}\right)e^{-rt_{1}}\Delta t+{\frac {d}{dt}}f_{0}^{1}\left(t_{2}\right)e^{-rt_{2}}\Delta t+...+{\frac {d}{dt}}f_{0}^{1}\left(t_{n}\right)e^{-rt_{n}}\Delta t$$

This formula follows from the idea that the reward you will get during time period ${\textstyle \left[t_{1},{\ t}_{2}\right]}$ will be the amount your total fees increased $f_{0}^{1}\left(t_{2}\right)-f_{0}^{1}\left(t_{1}\right)$. We fix regular time intervals ${\textstyle \Delta t={\ t}_{i+1}-t_{i}}$. Taking the limit gives the foundational formulas:

Proposition 1 The reputational salary of a single token with infinite lifetime is

$$f_{0}^{1}\left(t\right)=\int _{0}^{t}{{\frac {f^{\prime }\left(s\right)}{R\left(s\right)}}ds}$$

The present value of a single token in a DAO with income stream $f_{0}^{\prime }$ is $f_{0}^{\prime }$

Equation 2