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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!

# Present Value of a Future Payoff Stream

Often in economics we want to evaluate the present value of a stream of payoffs. Among many examples:

• in finance, we may want to know how to value an investment with a stream of payoffs in the future
• in game theory, we may want to know the value of payoffs that occur in a multi-period game
• in industrial organization and antitrust, we may want to know the value of investing in a collusive relationship

Let’s look at how to evaluate all of these by first looking at one future payoff; then several; then an infinite number.

## Present value of a single payoff in the future

To think of the present value of a payoff in the future, think first of a concrete example: suppose you invested €1000 at 10% interest. After one year, it would grow to be worth $1.1 \times €1000 = €1100$; after two years, it would be worth $1.1 \times €1100 = 1210$; after three, $1.1 \times 1210 = €1331$; and so forth.

More generally, if you invested some present amount $v$ at an interest rate of $r$ (where $r = 0.1$ corresponds to a 10% interest rate), it would grow exponentially: \begin{aligned} \text{Initial value} &= v\\ \text{Value after one year } &= (1 + r)v\\ \text{Value after two years } &= (1+r)^2v\\ \text{Value after three years } &= (1+r)^3v\\ & \vdots\\ \text{Value after }t\text{ years } &= (1+r)^tv\\ \end{aligned} Now let’s flip the script, and think about how much you would need to invest to achieve some target amount in the future. Let’s define $PV$ as the amount of money we would need to invest in order to be worth some amount $x$ at some point $t$ years into the future. Then by the above equation we have $$(1+r)^t \times PV = x$$ solving for $PV$ by dividing both sides by $(1 + r)^t$ gives us $$PV(x,r,t) = {x \over (1 + r)^t}$$ This relationship is visualized in the graph below. Try adjusting $x$ and $t$ (by dragging the dot) and $r$ (by adjusting the slider) to see how the present value of the future payoff changes:

## Present value of a two-period payoff stream (now and in the future)

Now let’s think about a two-period income stream, comprised of present and future income. Let’s denote time in terms of periods into the future; so we’ll think of the “present” as being “period 0”, and the “future” as being “period 1;” so we can write our income stream as a payment of $x_0$ now and $x_1$ in the future. The interest rate between these two periods is $r$.

To calculate the present value of this income stream, imagine that we took our $x_0$ and invested it at an interest rate of $r$; in period 1, it would have grown to be worth $(1+r)x_0$. We would then receive our payment of $x_1$, meaning that our “future value” of this income stream after one period would be $$FV = (1 + r)x_0 + x_1$$ The present value would be the amount that, if invested in period 0, would grow to that future value after one period: \begin{aligned} (1 + r)PV &= FV\\ (1 + r)PV &= (1 + r)x_0 + x_1\\ PV &= x_0 + {x_1 \over 1 + r} \end{aligned} We can see this equivalence in the following graph: the $PV$ grows at interest rate $r$ to become the same amount as you would end up with if you had the income stream:

Note that this equation for the present value is the same one we derived when looking at the budget constraint of a two-period intertemporal consumption model.

## Present value of a multi-period (or infinite-period) stream of constant payments

We’ve established that the value of a payment of $x$ received $t$ years in the future (written $x_t$) at interest rate $r$ is given by $$PV(x_t) = {x \over (1+r)^t}$$ and that the value of receiving $x_0$ now and $x_1$ one period in the future is $$PV(x_0,x_1) = x_0 + {x_1 \over 1+r}$$ We can extend this to an arbitrary series of payments $(x_0, x_1, x_2, …, x_t)$ by just adding the value of the payment received in each period: $$PV(x_0,x_1,x_2,...,x_t) = x_0 + {x_1 \over 1 + r} + {x_2 \over (1 + r)^2} + \cdots + {x_t \over (1 + r)^t}$$ Now let’s think about the present value of a stream $n$ payments of $x$ dollars each, starting one period in the future: that is, $x_0 = 0$ and $x_1 = x_2 = \cdots = x_n = x$. The present value of this stream is $$PV = {x \over 1 + r} + {x \over (1 + r)^2} + \cdots + {x \over (1 + r)^t}$$ You can see this for various values of $n$ in the graph below. Try dragging $n$ from 1 to 5 to see how the present value changes:

It’s clear that as $n$ increases, the present value changes by a lot at first, then by less and less. The mathematical formula for the limit of this expression as $n$ approaches infinity is $$PV = {x \over r}$$ Indeed, if you check the “Show $PV$ for $n = \infty$” in the graph above, you can see this theoretical limit, which is pretty close to the $PV$ of $n = 5$ if $r$ is large!

There are two elegant ways of deriving this limit:

• The economic explanation is pretty simple: if you put away some value $PV$ and never take it out, and it pays you a certain amount each period, the amount it pays each period is just the interest rate times the principal you’ve invested. So $$x = r \times PV$$or$$PV = {x \over r}$$There’s actually a special name of this kind of financial instrument, which is a perpetuity – one can think of it as a bond which never comes due. For example, an asset that pays €10 per year “in perpetuity” is worth €40 if the interest rate is 25%, but only €20 if the interest rate is 50%.

• The mathematical proof is by recursion. The infinite sum may be written as $$PV = {x \over 1 + r} + {x \over (1 + r)^2} + {x \over (1 + r)^3}\cdots$$ Suppose we factor out $1/(1 + r)$ from every term in the right-hand side. This gives us $$PV = {1 \over 1 + r}\left[ x + {x \over 1 + r} + {x \over (1 + r)^2} + \cdots\right]$$ But notice that the bracketed term is just $x$ plus the infinite series which is equal to PV! So if we substitute $PV$ back in and solve, we get \begin{aligned} PV &= {1 \over 1 + r}\left[ x + PV\right]\\ (1+r)PV &= x + PV\\ r \times PV &= x\\ PV &= {x \over r} \end{aligned}

Copyright (c) Christopher Makler / econgraphs.org