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## Time Value of Money Part X

### The Future Value of Annuities

In this lesson, we’ll expand on what we learned and be able to calculate how much money we’ll have in the future from an annuity. As we’ve been talking about the road trip that I went on with my friend Steve (from Minneapolis to San Diego), we’ll stick on that topic with another example. For our road trip, we had the pleasure of spending multiple hours riding in my Ford Ranger that I purchased from my grandparents when I was 18. Let’s say that starting at age 14, I was able to save $1000 for a car at the end of every year for 4 years total. If I was able to save at 4% in my savings account, how much money would I have at the end of the 4 years? To find this out, we’ll learn how to calculate the future value of annuities in this small business financial management example.

### Saving to the Buy the Ranger with Purple and Blue Pinstripes

Notice that since we’re putting in a set amount of money to the savings account on a routine basis ($1000 every year), we’ve essentially created an annuity. Like when we calculated how much an annuity is worth to us today in the present value of annuities lesson (remember annuities are payments of a set amount of money that occur on a routine basis such as every year), we can use the same method to calculate the future value of annuities. To calculate how much our savings account balance will be worth in 4 years, we can use the equation listed below:

Future Value of Annuities = Value of Each Payment x Future Value Annuity Factor

Like when calculating the present value of the annuity, we can use a table that I created that has future value annuity factors. This table can be found here:

Annuity Factors Future Value Table

For this example, we know that we can save at 4% and the annuity goes for 4 periods (the four years we’re saving for). Using the table above, we can find that the future value annuity factor is 4.2465. Since we know that the value of each payment is $1000, now all we have to do is multiply the $1000 by the future value annuity factor of 4.2465.

Future Value of Annuities = $1000 x 4.2465 = $4246.50

So after depositing $1000 at the end of each of the four years, I would be able to save a total of $4246.50 to buy my Ford Ranger from my grandparents.

### Future Value of Annuities – How Much Do We Need to Save?

Let’s say that I know that my grandparents will sell me the truck for $5000 and that I have a total of five years to save up for it. If I deposit an amount of money at the end of each year for five years and I can save at a 6% interest rate, how much will I have to save each year?

For this example, we can take the equation above and use some algebra to flip it around to solve for the value of each payment:

Future Value of Annuity = Value of Each Payment x Future Value Annuity Factor

Change it to solve for Value of Each Payment:

Value of Each Payment = Future Value of Annuity / Future Value Annuity Factor

Like the last problem, we can use the Annuity Factors Future Value Table to find the annuity factor needed in this equation. Since the interest rate is 6% and the number of periods is 5 (since we’re saving for 5 years), the future value annuity factor is 5.6371. Since we know that we’ll need $5000 in the future, this is the Future Value of Annuity in the equation above. Now we can just fill in these two values to figure out how much we need to save each year:

Value of Each Payment = $5000 / 5.6371 = $886.98

So if we save $886.98 at the end of each year for five years, we’ll have a total of $5000 in our 6% interest savings account.

**Key Takeaway:** We can find the future value of annuities (how much it will be worth in the future) by multiplying the future value annuity factor by the amount of each payment.

In the next lesson, we’ll talk about an annuity where the first payment is paid right away instead of at the end of the year.

Test Your Knowledge: Future Value of Annuities

**To Next Lesson: Annuities Due Formula**

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