In this article, we’re going to explore one of the most important concepts and formulas in Finance – the Present Value of an Annuity. Let’s get into it.

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## What is Present Value of an Annuity?

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

- the Present Value (PV), and
- an Annuity

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 an Annuity.

### 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 an Annuity?

An annuity can be described as a constant stream of cash flows for a defined period of time.

In other words, it’s anything that gives you the *same* *amount of cash* (equal payments) at the same pre-defined intervals (periodic payment). For example, each of the following can be seen as annuities:

- rent payments of $1,000 every quarter for 20 quarters
- monthly payment for car finance/loan of £300 for 48 months
- pension payments of €9,600 every year for 25 years

Although the examples are quite distinct – being rent, loan repayments, and pension payments – they all involve paying or receiving the same cash flow at the same pre-defined intervals.

These types of cash flows are sometimes dubbed/called an annuity stream.

### The Intuition Behind the Present Value of an Annuity

Let’s think about the last example; the one with pension payments.

Suppose a pension plan offers to pay you €9,600 every year for 25 years from when you retire.

In total, this means the pension plan will pay you €240,000

Naturally, the pension plan won’t give you that money for free. They’ll ask you to make contributions towards the pension.

So how much would *you* be willing to pay in pension contributions in order to receive €9,600 every year for 25 years from when you retire?

You might think that you’re willing to pay up to €240,000 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.

Thus, if you pay €240,000 today to receive 25 payments of €9,600 each year, you’d be significantly overpaying. Frankly, it would be a rip-off.

So, what is a fair price to pay?

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That depends on how much those pension payments are worth right here, right now.

In other words, it depends on the *present value* of those pension payments.

And since the pension payments are an annuity, we can say that it depends on the *present value of an Annuity*.

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

## Present Value of an Annuity Formula

The Present Value of an Annuity formula is:

Where:

- is the future cash flow
- represents the discount rate
- denotes the timeframe of the annuity

Strictly, this relates to an ordinary annuity (as opposed to a deferred annuity).

We’re only going to be focusing on the ordinary annuity since that’s the one that’s more common.

And once you get your head around the ordinary annuity, it’s much easier to understand the deferred annuity.

But 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

When dealing with the Future Value, it’s common to denote this as “interest rate” instead of “discount rate”.

That’s because when we’re calculating the Future Value, we’re *compounding cash flows *into the future.

But when we’re calculating the Present Value, we’re *discounting future cash flows* back to the present.

Okay, now that the jargon’s pretty clear, let’s focus our attention on the present value of an annuity formula again.

The equation above is not the only way to write it out. We can also express the present value of an annuity formula like this:

Writing it out this way is closer to the idea that the PV of an Annuity is equal to the *difference* between the Present Value of 2 Perpetuities.

**Related: How to Calculate Present Value of a Perpetuity**

Both forms are identical, so that…

Now that you can see both versions in one line, can you figure out *why* the two are identical?

If you have, well done! It is of course because we can just expand the brackets in the first version to get to the second version.

If you couldn’t figure it out, or if the equations are starting to freak you out, then relax. Calm down. Breathe.

It’s a lot easier than it looks.

And if you do want to:

- get past your fear of equations,
- understand how equations work from scratch,
- be able to prove the PV of an Annuity formula from scratch

Then check out our course on Financial Math Primer for Absolute Beginners where you’ll achieve all of that and a whole lot more.

Okay, we’re going to assume you’re more or less alright now, so let’s think about when to use Present Value of Annuity formula.

## When to Use Present Value of Annuity Formula

Hopefully, it’s already clear that you should only use the Present Value of Annuity formula when you’re *dealing with an annuity*.

But how do you know if you’re really dealing with an annuity?

It’s not sufficient for the cash flows to be equal alone. That’s just one of the three conditions required.

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

- cash flows remain constant
- the discount rate remains unchanged, and
- the time period is finite (i.e., you’re not dealing with a
*perpetuity*)

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 an Annuity formula.

Indeed, we use these 3 conditions in our PV of an Annuity Calculator.

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

## Present Value of Annuity Example

Assume you’re now 20 years of age and that you’re considering investing in a 40-year fund that is promising to pay you $10,000 every year until you turn 60 of age. If the appropriate discount rate is 18%, up to how much should you be willing to pay to buy this fund today? Please round off your answer to 2 decimal places.

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.

You should be willing to pay up to $55,481.52 to buy the fund today.

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

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

Recall that we calculate the Present Value of an Annuity like this…

In this example, the is equal to $10,000 because that’s what the fund promises to pay you each year.

is equal to 18% = 0.18 because remember that is the discount rate (aka cost of capital).

And finally, is equal to 40 because that’s the timeframe of the fund. It’s promising to pay you $10,000 every year for 40 years.

With the 3 variables established, it’s now just a simple case of plugging in our numbers into the Present Value of Annuity formula, so we have…

And that’s it!

Feel free to verify this result using our Present Value of Annuity Calculator.

Now, although we’ve solved this particular question using the formula/equation, there is another way. In fact, the other approach is arguably a lot easier.

We can solve this by using what’s called a Present Value of an Annuity Factor.

But, if you’re just starting out, we recommend working with the formula exclusively, so you really understand how it works. And once you get comfortable with using the formula, feel free to use the Present Value of an Annuity Factor to calculate things faster.

Alright, let’s now explore what this ‘factor’ is.

## Present Value of an Annuity Factor

A “factor”, in a nutshell, is just a number we tend to multiply another number by.

In this specific case, the Present Value of an Annuity Factor is the number we multiply the cash flow by, in order to calculate the Present Value of an Annuity.

We promise it’s easier than it sounds.

Let’s first take another look at the Present Value of Annuity formula:

Since the cash flow is constant, we can actually rewrite the formula like this…

Or, to make things a tad bit clearer, we can rewrite it like this…

Now, remember we said in this specific case, the Present Value of an Annuity Factor is the *number we multiply the cash flow by*, in order to calculate the Present Value of an Annuity?

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In the equation above, what are we multiplying the cash flow by? Of course, we’re multiplying it by the stuff inside the square brackets.

And that right there *is* the Present Value of an Annuity Factor!

Let’s call that (Present Value of Annuity Factor). We can then say…

And now that we’ve defined formally, we can substitute it into the original Present Value of Annuity formula so that…

Thus, you can either calculate the Present Value of an Annuity using the “full formula” or traditional formula, or you can use the Annuity Factor approach. To verify this, let’s calculate the Present Value of an Annuity for the example question we saw earlier in this article.

Remember that in the example, the was equal to $10,000 because that’s what the fund promised to pay you each year.

was equal to 18% = 0.18

And finally, was equal to 40 because that was the timeframe of the fund. It promised to pay you $10,000 every year for 40 years.

Let’s now calculate the Present Value of an Annuity Factor first. Remember, we said that’s equal to…

Plugging in our numbers for and we have…

Thus, our Present Value of Annuity Factor is equal to 5.548151883

Now it’s just a simple case of multiplying the cash flow of $10,000 with this factor! Because remember…

Plug in the numbers and solve to get…

And that’s it! You now know how to calculate Present Value of an Annuity using the formula and the annuity discount factor.

Now, in fairness, although the approach with the annuity discount factor is a tad bit easier, arguably, it still does take time.

Fortunately, there is an *even quicker* way to calculate the Present Value of an Annuity.

You’d still use the Annuity Factor, but instead of calculating it yourself manually, you can use what’s called a Present Value of an Annuity Table.

## Present Value of an Annuity Table

Firstly, let’s get some jargon out of the way. The Present Value of an Annuity Table is also known as:

- Annuity Discount Factor Table
- Present Value Annuity Factor Table
- Present Value Annuity Discount Factor Table

Now, what is this table? It’s literally just a list of annuity discount factors for different values of and

Remember, and are the only two variables in the Annuity Factor:

Thus, if you change either , , or both, you’ll end up with a different value for .

So people decided to compile a variety of annuity factor values for different discount rates and timeframes into a single table. And that’s called the Present Value of an Annuity Table.

These tables are easily “googlable”, but we’ve provided our own versions below. The first one here relates is a Present Value Discount Factor Table for *single cash flows* (NOT annuities).

This is sometimes also called a Present Value Factor Table.

You can use the table below to calculate Present Value for single cash flows.

The second table (see below) relates to annuities.

In other words, this is the Present Value of an Annuity Table:

Sure, the table looks complicated, but as with most things in Finance, it’s a lot easier than it looks!

Let’s go ahead and see how to use this table with an example.

### How to Use Present Value of Annuity Table

Imagine you want to buy a Tesla car for $50,000. You can either pay upfront or take on car finance (borrow money). Suppose you can get a loan wherein you pay $12,000 a year for 5 years (including interest and repayments). Imagine that the appropriate interest rate is 8%.

Should you buy the car by paying upfront, or should you take on the loan?

At first glance, it seems like paying upfront is the better bet – even a no-brainer perhaps? After all, you’re paying $50,000 if you pay upfront, but $60,000 if you take on the loan!

But is that really true?

Let’s find out, by calculating the Present Value of the loan repayments.

Notice that the loan repayments are in fact an annuity. All 3 conditions of the annuity hold, too:

- cash flows remain constant (at $12,000 per year)
- the discount rate remains unchanged (at 8%), and
- the time period is finite ( is 5 years)

We can therefore use the Present Value of an Annuity formula to estimate the Present Value of this cash flow stream.

Alternatively, of course, we can use the Annuity Discount Factor from the Discount Factor Table above.

Notice that the value in the cell that relates to the 8% column (under *Interest rates (r))* and the 5 periods row (*Periods (n)) *is equal to 3.993

We know that the cash flow is equal to $12,000 so we can estimate the Present Value of this annuity as…

Plugging in our numbers, we have…

Thus, in this example, if you buy the Tesla car via the loan, you’re essentially paying the equivalent of $47,916 in today’s terms. If you pay upfront, however, you would pay $50,000 in today’s terms.

Put differently, buying the Tesla via a loan, in this example, would be a positive NPV decision. If you’d like to learn more about the Net Present Value (and other investment appraisal or capital budgeting techniques), do check out our course on Investment Appraisal Mastery.

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Okay, that’s it for this particular article. We hope you found it useful!

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