regular annuity formula

Time Value of Money:
Future Value of Regular Annuities

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An annuity is a series of equal cash flows, equally distributed over time. Examples of annuities abound: Mortgage payments, car loan payments, leases, rent payments, insurance payouts, and so on. If you are paying or receiving the same amount of money every month (or week, or year, or whatever time frame), then you have an annuity.

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

Picture of 3-year $100 regular annuity time line

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 Future 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 future 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:

\[F{V_A} = \sum\limits_{t = 1}^N {C{F_t}{{\left( {1 + i} \right)}^{N - t}}} \]

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

\[F{V_A} = 100{\left( {1 + 0.09} \right)^2} + 100\left( {1 + 0.09} \right) + 100 = 327.81\]

Note that the future value of a regular annuity is, by definition, in the same period as the last cash flow. So, the first cash flow must be taken two periods forward, the second cash flow must be moved one period ahead, and the last cash flow is already there so it doesn't move at all. Therefore, if the interest rate is 9%, then the future value of this annuity is $327.81 at the end of period 3.

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:

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

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:

\[F{V_A} = 100\left[ {\frac{{{{\left( {1 + 0.09} \right)}^3} - 1}}{{0.09}}} \right] = 327.81\]

which is exactly the same as we got previously. Let's look at another example.

Imagine that you are planning for retirement. You expect to retire in 35 years, and you think that you can afford to save $500 per month. Furthermore, you believe that you can reasonably expect to earn about 8% per year without taking too much risk. How much will you have accumulated at the time that you retire?

In this example, Pmt is $500 because you plan to save that amount each and every month. The number of periods (N) is 420 months, and the interest rate (i) is 0.667% per month. (Remember, as I have said many times on these pages, all of the variables must be on a per-period basis. In this case, we are making monthly investments, so both N and i must be converted to monthly values.) Solving the closed-form equation, we find that you will have:

\[F{V_A} = 500\left[ {\frac{{{{\left( {1 + 0.00667} \right)}^{420}} - 1}}{{0.00667}}} \right] = 1,146,941.24\]

Whew! Can you imagine solving this problem using the open-form version of the equation? In any case, we just found that investing $500 per month at 8% per year will result in a nestegg of $1,146,941 after 35 years. Not too bad, especially when you consider that $500 per month is pretty much equal to a car payment. (Note: if you solve the above equation and get $1,148,067.778, that is due to rounding the interest rate. Instead of using 0.00667, as shown, you should calculate it as 0.08/12 and use that result.)

Please continue on to the next page to learn how to calculate the present value of a regular annuity.

Category: Annuity

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