Time Value of Money – Present Value of Regular Annuities

In the previous tutorial, we saw how to calculate the future value of an annuity. Here, we will instead find the present value of a regular annuity. There are many examples where you might want to find the present value of an annuity:

• to find the balance of a mortgage or car loan,
• determining the size of an annuity that you might need to provide retirement income
• calculating the value of your pension (if you are lucky enough to have one)
• comparing the cost of an equipment lease to the cost of an outright purchase
• and many others…

A regular annuity is simply an annuity where the first payment is made at the end of the period. The picture below shows an example of a 3-period, $100 regular annuity:

Notice that we can view the annuity as a series of three \$100 lump sums, or we can (and will) treat the cash flows as a package.

Calculating the Present Value of a Regular Annuity

As noted above, according to the principle of value additivity, we can treat an annuity as a series of lump sum cash flows. Well, we have already seen how to calculate the present value of a lump sum. All that we need to do is apply this formula to each of the cash flows individually, and then sum the results:

$$P{V_A} = \sum\limits_{t = 1}^N \frac{C{F_t}}{(1+i)^N}$$

Using the example shown in the timeline (above), and a 9% per period interest rate, we get:

$$P{V_A} = \frac{100}{{\left( {1 + 0.09} \right)}} + \frac{100}{\left( {1 + 0.09} \right)^2} + \frac{100}{\left( {1 + 0.09} \right)^3} = 253.13$$

The formula shown above works fine, but it is tedious if the annuity has more than a few payments. Fortunately, we can derive a closed-form version of that equation, which means that we don’t have to iterate through a series of sums. The closed-form equation is:

$$P{V_A} = Pmt\left[ \frac{1\,-\frac{1}{\left(1\,+\,i\right)^N}}{i}\right]$$

where $Pmt$ is the per period annuity payment amount (\$100 in our example). This formula is much easier to use, no matter how many payments there are. In this case, it gives us:

$$P{V_A} = 100\left[ \frac{1\,-\frac{1}{\left(1\,+\,0.09\right)^3}}{0.09}\right]=253.13$$

which is exactly the same as we got previously. The image below shows a timeline for this process:

Deferred Annuities

Let’s look at another example that combines the concept of the present value of an annuity with the present or future value of a lump sum. This is known as a deferred annuity because the cash flows don’t start until some point after year 1 (in this case, the payments start in year 10):

Imagine that your daughter has announced that she will be attending a local university beginning ten years from today. After some research, you believe that you will need to pay $15,000 per year for each of the four years that she will be attending. If you can earn 6% per year on your investment, how much do you need to have in ten years in order to be able to pay her college tuition?

In this example, $Pmt$ is \$15,000 because you plan to spend that amount each of the four years. The number of periods ($N$) is 4 years, and the interest rate ($i$) is 6% per year. Solving the closed-form equation for the present value, we find that you need to save:

$$P{V_A} = \text{15,000}\left[ \frac{1\,-\frac{1}{\left(1\,+\,0.06\right)^4}}{0.06}\right]=\text{51,976.58}$$

So, we now know that you will need to have \$51,976.58. Note that what we have done is to compress four years of cash flows into one single cash flow located at some future point in time. But when is that “future point in time?”

Recall, from the note above, that the present value from that equation is one period before the first cash flow, and that the first cash flow occurs in year 10 in this problem. Therefore, the PV is as of year 9, not year 10. So, we need to move that PV from year 9 to year 10 in order to answer the question (this is just the FV of a lump sum):

$$P{V_{10}} = \text{51,976.58}\times \left(1+0.06\right)^1 = \text{55,095.18}$$

So, at year 10 you need to have \$55,095.18 in order to be able to pay \$15,000 per year for four years.

Let’s continue the example:

Since you know that you will need to have accumulated $55,095.18 in ten years, the urgent question is how much would you need to invest today (a lump sum) in order to achieve your goal of paying for your daughter's college tuition?

We previously converted the four years of cash flows into a single cash flow at year 10. Now, we need to find the present value of that lump sum cash flow at year 10:

$${PV}_0=\frac{\text{55,095.18}}{\left(1+0.06\right)^{10}} = \text{30,764.86}$$

Let’s take it one step further:

You don't currently have $30,764.86 just laying around, so you need to figure out some other way to accumulate $55,095.18 ten years from now. You decide that making a series of 10 investments might be easier. If you make the first investment one year from now, how much will you need to save each year assuming that you can earn 6% per year on your investments?

We have now changed the problem such that you need to solve for the annuity payment. You know the future value is \$55,095.18, the number of payments is 10, and the interest rate is 6% per year. Using the skills from the future value of a regular annuity tutorial, we can solve this problem.

Recall that the future value of a regular annuity is given by:

$$F{V_A} = Pmt\left[ {\frac{{{{\left( {1 + i} \right)}^N} \,- 1}}{i}} \right]$$

but we need to solve for the $Pmt$ because we already know the future value. Fortunately, a little bit of simple algebra reveals that the annuity $Pmt$ is given by:

$$Pmt = \frac{F{V_A}}{\left[ {\frac{{{{\left( {1 \,+\, i} \right)}^N} \,-\, 1}}{i}} \right]}$$

So, plugging in the known values, we can solve for the amount that you need to invest each year for the next 10 years:

$$Pmt = \frac{55,095.18}{\left[ {\frac{{{{\left( {1 \,+\, 0.06} \right)}^{10}} \,-\, 1}}{i}} \right]}= \text{4,179.96}$$

So, by combining the skills that you learned in these tutorials, you have found that by saving \$4,179.96 each year for the next 10 years, you can afford to pay your daughter’s college tuition of \$15,000 per year for four years.

I hope that you have found this tutorial to be helpful.