In this article, we’re going to explore what the Present Value of a Perpetuity is, including:

- what it means
- what the Present Value of a Perpetuity formula is
- how to calculate the Present Value of a Perpetuity

Let’s get into it.

## What is Present Value of a Perpetuity

The Present Value of a Perpetuity is the value of a *Perpetuity* expressed in today’s terms. Essentially, there are 2 parts to this concept, including:

- the Present Value (PV), and
- a Perpetuity

Let’s consider what both these are individually first, and then we’ll look at how the two interact to make up the Present Value of a Perpetuity.

### What is Present Value?

The Present Value is the value of future cash flows expressed in today’s terms.

Money loses value over time because of the Time Value of Money. Thus, if we’re looking at *anything* involving money, it’s important to incorporate the Time Value of Money.

And as a result, it’s important to express future cash flows in today’s terms, or in *present* terms (hence the term “Present Value”).

### What is a Perpetuity?

A Perpetuity can be described as a constant stream of cash flows for an *infinite period of time*.

In other words, it’s anything that gives you the *same* *amount of cash* (equal cash flows) at the same pre-defined intervals (periodic payment), forever (i.e., indefinitely).

Examples of perpetuities in the real world are quite scarce. The British Government did issue ‘Consols’ (perpetual bonds), but these were sort of cancelled in the 2010s.

It’s understandable that there are few – if any – examples of perpetuities. Would you be willing to pay someone a fixed amount of money every period *forever*? Probably not.

But just because there aren’t explicit real-world examples of perpetuities doesn’t mean the concept is pointless.

Far from it.

The Present Value of a Perpetuity is used extensively in *any* (and every) discounted cash flow (DCF) valuation model in the estimation of what’s called the Terminal Value.

It’s also the fundamental basis for the Dividend Discount Model (DDM) for stock valuation.

Thus, you can’t really value a company – using a DCF valuation model – without knowing how to deal with perpetuities.

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### The Intuition Behind the Present Value of a Perpetuity

So why is it necessary to think about something that happens “forever”? Does anything really happen forever?

For humans individually, probably not. Every human dies at some point. But for companies, things are a tad bit different.

Companies can (theoretically) last/exist forever, indefinitely. This is ultimately because there is *separation of ownership and management *in companies.

Put simply, companies can outlive humans because the company is a separate legal entity.

And unless the company goes bankrupt, there’s no particular reason why it can’t last forever (assuming it can sustain itself financially of course).

Now, how do we value companies? Well, there are different ways to value stocks, but the fundamental idea is that the value of a stock/company is based on the present value of its future expectations.

Since companies can last forever, those future expectations can relate to an infinite timeframe.

And we, as investors, want to obtain an estimate for how much all of those cash flows are worth to us today, right here, right now.

As a simple example, imagine a company expects to pay $8 of dividends per share every year forever. For simplicity, assume it pays all of its profits as dividends, and that the profits are approximately equal to its cash flows (meaning there are no accruals accounting distortions).

How much would you be willing to pay for 1 share of this company?

Perhaps you’re thinking, well, infinity is a massive number. But *I* am not going to live forever. The maximum I’ll live is up to 100, say.

Assuming you’re 25 years of age, that’d give you up to 75 years of dividends from the company.

Over those 75 years then, the company would pay you $8 75 = $600 (for every share you own).

You might think that you’re willing to pay up to $600 but this would not be financially wise.

Remember we talked about the Time Value of Money? Well, we know from that that money loses value over time.

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Thus, if you pay $600 today to receive payments of $8 each year until you die, you’d be significantly overpaying. Frankly, it would be a rip-off.

And moreover, you could pass on your shares to your children and grandchildren, meaning it’s useful to think about the timeframe for this stock beyond your life. Remember, we said the stock would pay $8 *forever*.

So, what is a fair price to pay?

That depends on how much those infinite payments of $8 are worth right here, right now.

In other words, it depends on the *present value* of those future cash flows.

And since the cash flows are a perpetuity (in that they occur *perpetually*), we can say that it depends on the *Present Value of a Perpetuity.*

Okay, now that you have an idea of the intuition behind the PV of a Perpetuity, let’s take a look at the formula for the PV of a Perpetuity.

## Present Value of a Perpetuity Formula

The Present Value of a Perpetuity formula is:

Where:

- is the future cash flow
- represents the discount rate

Before we proceed any further, let’s just get some jargon out of the way.

is sometimes also written as , which denotes Free Cash Flow. is the cash flow that’s *freely distributable* to debt as well as equity investors.

, the discount rate, is also known as:

- cost of capital
- opportunity cost of capital
- cost of equity (if it’s an
*equity-only cash flow*) - hurdle rate

Okay, now that the jargon’s pretty clear, let’s focus our attention on the Present Value of a Perpetuity formula again, and learn when to use it.

## When to Use Present Value of a Perpetuity Formula

Hopefully, it’s already clear that you should only use the Present Value of a Perpetuity formula when you’re *dealing with a perpetuity*.

But how do you know if you’re really dealing with a perpetuity?

It’s not sufficient for it to be paid indefinitely, although that is of course the most important condition. There are 2 other conditions, however.

To figure out when to use Present Value of a Perpetuity formula, you want to look out for 3 conditions. They are:

- cash flows remain constant (i.e., identical cash flows throughout time)
- the discount rate remains unchanged, and
- the time period is infinite (i.e., you’re dealing with a
*perpetual*timeframe)

Mathematically, we can express these three conditions as follows:

If and only if *all three* conditions hold, we can go ahead and use the PV of a Perpetuity formula.

Okay, now that you know when to use Present Value of a Perpetuity formula, let’s go ahead and apply it in an example and learn how to calculate Present Value of a Perpetuity.

## Present Value of a Perpetuity Example

The Republic of Utopia is promising investors equal payments of $5,000 every year forever in exchange for an upfront investment of $60,000 today. Suppose the appropriate discount rate is 8%. What is the fair price of the upfront investment cost today?

Can you try solving it on your own? Go on! Give it a go!

We’re going to assume you did that, so let’s go ahead and solve it together now.

The fair price of the investment is $62,500 (based on an annual interest rate / discount rate of 8% and annual perpetual cash flows of $5,000)

Why? Because that’s what the Present Value of the future cash flows is equal to. That’s the fair price of the investment as of today.

How do we calculate that? It’s simple! We just use the Present Value of a Perpetuity formula.

Recall that we calculate the Present Value of a Perpetuity like this…

In this example, the is equal to $5,000 because that’s the annual equal payments the Government of Utopia promises to pay investors forever.

is equal to 8% = 0.08 because remember that is the discount rate (aka cost of capital).

With the 2 variables established, it’s now just a simple case of plugging in our numbers into the Present Value of a Perpetuity formula, so we have…

And that’s it!

### Interpreting the result

Now, in this example, if you buy the investment opportunity for $60,000 (the price charged by the Government of Utopia), you’re essentially paying the $60,000 for something that’s worth $62,500.

Put differently, buying this particular investment would give you a positive Net Present Value (NPV) of $2,500. If you’d like to learn more about the Net Present Value (and other investment appraisal or capital budgeting techniques), do check out our Investment Appraisal Mastery course.

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Alright, but that’s it for this particular article.

Hopefully, you now know what the Present Value of a Perpetuity is, and how to calculate it.

And with that, you now also know a little bit about how to value stocks, since the PV of a Perpetuity is the fundamental basis of the Dividend Discount Model (DDM)!

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