Perpetuities: Proofs & Stuff
Annuity and perpetuities can feel a little abstract. In general, both concepts are just the present value of a timeline of payments. Perpetuities are an infinite timeline of payments \(C\), whereas annuities are a finite timeline of payments \(C\).
In this article, I’ll discuss perpetuities and prove the formula for perpetuities. Take a look at the timeline below, where our first payment \(C\) is in Period 1. The timeline below is the exact definition of a perpetuity:
Note that perpetuities have this EXACT timeline. Specifically, the first payment happens in Period 1, and we get the present value one period before this in Period 0.
If you recall, the basic way we value payments throughout time is via the risk-free interest rate \( r \). We divide \( C \) by \( (1+r) \) to move backwards one period, and multiply \( C \) by \( (1+r) \) to move forwards one period. For instance, the value of \( C \) in period 3 will be: $$ \frac{C}{(1+r)^3}$$
This means we can write down some expression for the present value of this timeline: $$ PV = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \dots \quad (1) $$
Here I've just added dots to show that the sum keeps going on infinitely. On that note, we should ask ourselves why this sum is valued at anything other than infinity? You might suggest that if we have infinite payments then our present value should be infinite. However, as we move forward period by period, the value of \( C \) is discounted more and more. This means that our present value eventually converges to some finite value, provided \( r \) is positive.
If each payment \( C \) is 1 and our \( r \) is \( \frac{1}{2} \), then a visual representation of why this is finite might look something like this where:
$$ PV = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 1. $$
You can see in the empty white area where this pattern would keep continuing infinitely, and not get bigger than the overall square which sums to 1.
That’s pretty cool, but before I show you the proof - perhaps our original perpetuity formula was a little basic. What happens if our perpetuity grows period by period by some rate g? Well, then our timeline would look like this:
In every period, \( C \) grows by \( g \). It’s important to note that our first payment \( C \) is as is – growth is only applied from Period 2 onwards. Hence, in period 2, the payment has grown by \( C\times G\) and hence \(C+C\times G=C (1 + g) \). For instance, to find the present value of Period 2 we will divide by \(1+r\) such that: \[ \frac{C(1 + g)}{(1+r)^2}\] Let’s find the present value of this timeline:
\[\quad \text{PV} = \frac{C}{(1+r)} + \frac{C(1+g)}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3} + \dots\]
Okay, we’ve made it complicated enough – how do we get the formula for it? I’ll give you a little reveal first – the formula for a perpetuity (with this exact timing of payments and growth) is \[ \text{PV} = \frac{C}{r - g} \] That means that if you ever want to find the value of an infinite stream of payments where the first payment is in one period, and growth is applied from the second period onwards – you can use this formula!
In the proof below, the important bit is that you understand that our starting point is just the PV of the timeline above.
Let’s label that expression as \((1)\): \[ \quad \text{PV} = \frac{C}{(1+r)} + \frac{C(1+g)}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3} + \dots \quad (1) \] Now multiply \((1)\) on both sides by \(\frac{1+g}{1+r}\) to get \((2)\): \[ \frac{(1+g)\,\text{PV}}{1+r}= \frac{C(1+g)}{(1+r)^2} + \frac{C(1+g)^2}{(1+r)^3} + \frac{C(1+g)^3}{(1+r)^4} + \dots \quad (2) \]
Subtract \((2)\) from \((1)\) such that the equality still holds: \[ \text{PV} - \frac{(1+g)\,\text{PV}}{1+r} \;=\; \frac{C}{(1+r)} - \frac{C(1+g)}{(1+r)^2} \;+\; \frac{C(1+g)}{(1+r)^2} - \frac{C(1+g)^2}{(1+r)^3} \;+\; \frac{C(1+g)^2}{(1+r)^3} \dots \] Notice that all terms cancel out on the right handside except \(\frac{C}{1+r}\). Finally, rearrange to solve for \(\text{PV}\):
\[ PV \left( 1 - \frac{1+g}{1+r} \right) = \frac{C}{1+r} \]
\[ PV \left( \frac{1+r}{1+r} - \frac{1+g}{1+r} \right) = \frac{C}{1+r} \]
\[ \frac{PV (r - g)}{1+r} = \frac{C}{1+r} \]
\[ PV = \frac{C(1+r)}{(r - g)(1+r)} \]
Finally, you get: \[ \text{PV} \;=\; \frac{C}{r - g} \] Just note that it must be discounted by more than the growth rate every period, so the PV isn’t infinite. Hence, \(r > g\) and definitely not equal—otherwise the PV would be undefined. And this is how your perpetuity formula is derived!