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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
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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:

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:

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