Present value: Difference between revisions

Revision as of 07:06, 6 March 2023

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 adjust the formula^[1]

The more general case[2] allows for continuous compounding which is derived from the formula

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

which gives the present value of a reward promised in years in the future as

Then the income stream leads to the desired present value formula is given by taking the limit of Riemann sums

This formula follows from the idea that the reward you will get during time period will be the amount your total fees increased . We fix regular time intervals . Taking the limit gives the foundational formulas:

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

Equation 1

The present value of a single token in a DAO with income stream is

Equation 2

[2] This generalizes the discrete case by using the formalism of tempered distributions, which include atomic distributions.

↑ 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 ${\textstyle \sum _{i=n+1}^{\infty }{\frac {f^{\prime }\left(i\right)}{\left(1+r\right)^{i}}}}$ shrink to 0 as $n$ grows. However, this is not always insignificant. If $f^{\prime }$ is bounded the tail shrinks like ${\textstyle {\frac {1}{\left(1+r\right)^{n}}}}$ . 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.

[1] 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 ${\textstyle \sum _{i=n+1}^{\infty }{\frac {f^{\prime }\left(i\right)}{\left(1+r\right)^{i}}}}$ shrink to 0 as $n$ grows. However, this is not always insignificant. If $f^{\prime }$ is bounded the tail shrinks like ${\textstyle {\frac {1}{\left(1+r\right)^{n}}}}$ . 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.

[1]

@@ Line 12: / Line 12: @@
 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 <math>f_0^\prime</math> 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 <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, that 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 adjust the formula[1]
+Now if we allow the fees to vary in time, we 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.</ref>
 The more general case[2] allows for continuous compounding which is derived from the formula
@@ Line 35: / Line 35: @@
 Equation 2
-----[1] We are ignoring the obvious fact that the DAO will not receive fees for all eternity for the following general reasons. 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.
 [2] This generalizes the discrete case by using the formalism of tempered distributions, which include atomic distributions.

Present value: Difference between revisions

Revision as of 07:06, 6 March 2023

Derivation of present value formula for time series

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