# Deriving the Formula for Future Value with an Annuity

In order to derive the formula for future value with an annuity, we need to first understand the formula for finding the future value without an annuity, as well as the formula for geometric series.

An annuity is a reoccurring amount, such as a payment or reoccurring investment. Imagine if you decided to save \$2,000 per year for the next 30 years. If you are obtaining interest on your invested money, each annuity deposit of \$2,000 will receive one year less interest than the previous deposit. By the end of the 30 years, your last deposit of \$2,000 will receive no interest, while the first deposit of \$2,000 may have quadrupled depending on the interest rate.

How much money would you have total? In order to manually figure out the total amount, you would have to solve for the future value for each annuity, based on the number of years it was invested, and add them all together (which would be very time-consuming).

However, if we know the annuity is a fixed amount (e.g. \$2,000) then we can derive a formula for the future value of our investment with this fixed annuity. To begin, let's discuss the basic formula for determining future value.

## Deriving the Formula for Future Value

In order to determine the future value of an investment, we assume a yearly percentage increase and multiply the original amount by that percentage each year. If we invested \$2,000 one time, and we wanted to know what the future value of that investment will be in 5 years (assuming that it will grow by 5% each year), we would have:

\$2,000 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05

We multiply by 1.05 because we want 100% of the value plus the additional 5%, which as decimals is 1.00 + 0.05 = 1.05. Whenever we repeatedly multiply the same number, we use powers to represent the amount of times this multiplication occurs. Therefore, this formula can be simplified into \$2,000(1.05)^5:

Or we can also rewrite it generically as FV = PV(1 + r)^n:

In this future value formula, FV represents future value, PV represents present value (the \$2,000), r represents the percentage rate as a decimal (5% = 0.05, so 1 + 0.05 = 1.05), and n represents the number of periods (the 5).

Note: We will use r to represent interest rate instead of i, in order to avoid confusion, because i represents imaginary numbers in algebra. The n number of periods could be years, or if interest is accrued semi-annually, monthly, or daily then n would represent the number of those periods.

Now we can begin to derive the formula for future value with a fixed annuity.

## Deriving the Formula for Future Value with an Annuity

Once we add a fixed annuity to the formula, we have to modify our equation. Remember that the equation without an annuity was simply PV(1 + r)^n:

However, now we have to do this formula for each individual annuity and add them together. Imagine if we invest \$2,000 each year for 5 years. The first \$2,000 was invested at the end of the year and gains interest for 4 years FV = PV(1 + r)^4:

The second \$2,000 was invested at the end of the second year and gains interest for 3 years FV = PV(1 + r)^3:

Continuing this for each annuity, we have to add them all together to get the total future value

FV = PV(1 + r)^4 + PV(1 + r)^3 + PV(1 + r)^2 + PV(1 + r)^1 + PV(1 + r)^0:

Notice that when investing the \$2,000 for 5 years, it actually only accrued interest for 4 years, because usually annuity payments are considered to be invested at the end of the year (the final formula changes slightly if it is invested at the beginning of the year). Therefore, the first power can be generalized to (n – 1) since 5 – 1 = 4. The second annuity payment can be generalized to (n – 2) since 5 – 2 = 3, and so on.

If we generalize the equation for n periods, then it produces this formula

FV = PV(1 + r)^n-1 + PV(1 + r)^n-2 + PV(1 + r)^n-3 … + PV(1 + r)^0:

This type of formula is a geometric series, which is any series where each unit is transformed by multiplication or division. In this situation, each unit of (1 + r) is being transformed by a different power, which is multiplication. The generic formula for a geometric series is s = x(1 - y^n) / (1 – y):

S represents "geometric series" and y represents a constant. At the end of this article, we will replace s with FV, and we will replace x with PMT since this number represents the annuity payment.

The only difference between our equation for future value and this generic simplified geometric equation is that we need to replace the y with the constant that is present in the annuity formula.

In our formula, the constant is (1 + r). We can derive this constant, also called the common ratio, by dividing each term by the following term. This division leaves (1 + r) left over. For example if you divide PV(1 + r)^n-1 by PV(1 + r)^n-2, then the PV and all but one (1 + r) are canceled out. Using real numbers, if we invest for 5 years then PV(1 + r)^4 divided by PV(1 + r)^3 would simplify to (1 + r).

PV * (1 + r) * (1 + r) * (1 + r) * (1 + r)
PV * (1 + r) * (1 + r) * (1 + r)

Now, to derive the formula for the future value with an annuity, we must insert our constant into the y position of the generic geometric equation, which gives us s = x[1 – (1 + r)^n / 1 – (1 + r)]:

We can immediately simplify the denominator by distributing the negative sign, which transforms the bottom from (1 – (1 + r)) into (1 – 1 – r). Then, since 1 – 1 = 0, we are left with – r in the denominator, giving us s = x(1 – (1 + r)^n) / (– r):

Next, we will want to get rid of the negative sign in the denominator, which we can accomplish by multiplying the top and bottom by (– 1). Remember that multiplying anything over itself is the same as multiplying by 1. (– 1)/(– 1) simplifies to 1. This results in the formula becoming s = x(– 1 + (1 + r)^n) / (r):

Finally, we can rearrange the numerator and replace s and x for FV and PMT in order to reveal the familiar formula seen in textbooks FV = PMT[(1 + r)^n – 1 / r]:

## Deriving the Formula for Future Value with an Annuity Due

Normally an annuity is considered to be paid or invested at the end of the year. However, when an annuity payment is invested at the beginning of the year it is called an annuity due. This changes our formula by multiplying everything by (1 + r) to give us the formula FV = PMT{[(1 + r)^n – 1] / r} * (1 + r)

Multiplying our formula by (1 + r) basically adds one year of interest earned to each annuity payment, because all of the payments are multiplied collectively by (1 + r). Remember that \$2,000(1.05)^4 was \$2,000 * 1.05 * 1.05 * 1.05 * 1.05, so by multiplying all these units collectively by the additional (1 + r), it essentially transforms (1.05)^4 into (1.05)^5 while simultaneously doing the same for (1.05)^3, (1.05)^2, and so on.

## Write for Us

Get more great stuff like this delivered to your inbox

## YOU MAY ALSO LIKE

How to Survive Your First Year in Business [Infographic] Here's the first thing you should know about your first year as an entrepreneur - it's tough. There will be financial and psychological...

2. ### 52+ Common Interview Questions and Answers

Most job interviews consist of only 5 to 20 questions, almost all of which are variations of these 52 common interview questions and answers. The following guide provides suggestions on how best...

3. ### The Millennial Job Dilemma: Why Does Everyone Want Experience?

Many job-seeking Millennials find themselves asking the same set of questions: Why Does Everyone Want Experience? How am I supposed to get experience if no one gives me a chance? Often Millennials...

4. ### 5 Expensive Common Mistakes Made by Millennial Entrepreneurs

As a millennial entrepreneur, handling the funds of your business can be quite tricky. Our friends over at FindMyWorkspace.com discuss the expensive common mistakes made by millennial entrepreneurs...

Want to exchange links for your business-related website? The following is a list of ways to get backlinks, as well as a list of websites willing to participate in link exchanges. Submitting your...

6. ### Forex: How Much Money Do You Need to Trade the Daily Timeframe

The answer might surprise you how much money you need to trade the daily timeframe in forex. If your forex broker uses standard lots, then in order to trade one micro-lot of 1,000 you only need...

7. ### How to Trade Forex with \$100: Turn \$100 to \$1 Million in 3 Years Realistically

Not only am I going to show you how to trade forex with \$100, but I am also going to show you how to turn \$100 to \$1 Million in 3 years realistically in forex. Sound impossible? Let me prove it...

8. ### Forex Trend Trading is How to Win in Forex

I’m going to prove to you that forex trend trading is how to win in forex. There are some who claim that it is foolish to buy something that is rising in value, but I think they are foolish for...