## Mathematics of Future Value

The future value of a present amount can be expressed as:

We will illustrate how this mathematical expression works by using the amounts from the three accounts in Part 1.

**Account #1: Annual Compounding**

A single deposit of $10,000 will earn interest at 8% per year and the
interest will be deposited at the end of one year. Since the interest
is compounded annually, the one-year period can be represented by n = 1
and the corresponding interest rate will be i = 8% per year:

The formula shows that the present value of $10,000 will grow to the FV of $10,800 at the end of one year when interest of 8% is earned and the interest is added to the account only at the end of the year.

**Account #2: Semiannual Compounding**

In Account #2 the $10,000 deposit will earn interest at 8% per year, but
the interest will be deposited at the end of each six-month period for one
year. With semiannual compounding, the life of the investment is stated as
n = 2 six-month periods. The interest rate per six-month period is
i = 4% (8% annually divided by 2 six-month periods).

The present value of $10,000 will grow to a future value of $10,816 (rounded) at the end of two semiannual periods when the 8% annual interest rate is compounded semiannually.

**Account #3: Quarterly Compounding**

In Account #3 the $10,000 deposit will earn interest at 8% per year,
but the interest earned will be deposited at the end of each three-month
period for one year. With quarterly compounding, the life of the investment
is stated as n = 4 quarterly periods. The annual interest rate is restated
to be the quarterly rate of i = 2%
(8% per year divided by 4 three-month periods).

The present value of $10,000 will grow to a future value of $10,824 (rounded) at the end of one year when the 8% annual interest rate is compounded quarterly.

**Future Values for Greater Than One Year**

To be certain that you understand how the number of periods, **n**,
and the interest rate, **i**, must be aligned with the compounding
assumptions, we prepared the following chart. **Note that the chart assumes an
interest rate of 12% per year.**

To be certain you understand the information in the chart, let's assume
that a single amount of $10,000 is deposited on January 1, 2013 and will
remain in the account until December 31, 2017. This will mean a total of five years:
2013, 2014, 2015, 2016, and 2017. If the account will pay interest of 12% per
year **compounded quarterly**, then
**n = 20** quarterly periods (5 years x 4 quarters per year),
and **i = 3% **per
quarter (12% per year divided by 4 quarters per year). The mathematical
expression will be:

Let's try one more example. Assume that a single amount of $10,000
is deposited on January 1, 2013 and will remain in the account until
December 31, 2014 (a total of two years). If the account
will pay interest of 12% per year **compounded monthly**, then **n = 24** months (2 years x 12 months per year),
and **i = 1%** per month (12% per year divided by 12 months per year).
The mathematical expression will be: