## 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, 2019 and will remain in the account until December 31, 2023. This will mean a total of five years: 2019, 2020, 2021, 2022, and 2023. 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, 2019 and will remain in the account until December 31, 2020 (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: