# Future Value of a Single Sum of Money

Future value of an single sum of money is the amount that will accumulate at the end of n periods if the a sum of money at time 0 grows at an interest rate i. The future value is the sum of present value and the total interest.

The future value (FV) of a single sum depends on the initial sum of money called present value (PV), interest rate, total time period, nature of interest (simple vs compound) and number of compounding periods per year.

If the present value, the annual percentage interest rate and the time period are the same, a sum of money which grows under the compound interest and has more compounding periods per year will have higher future value than a corresponding sum with grows only at simple interest rate and which has lower number of compounding periods per year.

## Formula

The future value of a single sum of money in case of a simple interest can be computed using the following formula.

Future Value (Simple Interest)
= Present Value × (1 + i × n)

However, compound interest is the most common method of interest accumulation in which case the future value can be calculated using the following formula:

Future Value (Compound Interest)
= Present Value (PV) × (1 + i)n

Where,
i is the periodic interest rate (= annual percentage rate divided by compounding periods per year; and
n are the total number of compounding periods.

(1 + i × n) and (1 + i)n are the future value factors in case of simple interest and compound interest respectively.

## Examples

Example 1: An amount of \$10,000 was invested on Jan 1, 20X1 at annual interest rate of 8%. Calculate the value of the investment on Dec 31, 20X3. Compounding is done on quarterly basis.

Solution

```We have,
Present Value         PV  = \$10,000
Compounding Periods    n  = 3 × 4 = 12
Interest Rate          i  = 8%/4 = 2%
Future Value          FV  = \$10,000 × ( 1 + 2% )^12
= \$10,000 × 1.02^12
≈ \$10,000 × 1.268242
≈ \$12,682.42```

Example 2: An amount of \$25,000 was invested on Jan 1, 20X0 at annual interest rate of 10.8% compounded on quarterly basis. On Jan 1, 20X1 the terms or the agreement were changed such that compounding was to be done twice a month from Jan 1, 20X1. The interest rate remained the same. Calculate the total value of investment on Dec 31, 20X1.

Solution

The problem can be easily solved in two steps:

STEP 1: Jan 1 - Dec 31, 2010

```Present Value         PV1 = \$25,000
Compounding Periods    n  = 4
Interest Rate          i  = 10.8%/4 = 2.7%
Future Value          FV1 = \$25,000 × ( 1 + 2.7% )^4
= \$25,000 × 1.027^4
≈ \$25,000 × 1.112453
≈ \$27,811.33```

STEP 1: Jan 1 - Dec 31, 2011

```Present Value         PV2 = FV1 = \$27,811.33
Compounding Periods    n  = 2 × 12 = 24
Interest Rate          i  = 10.8%/24 = 0.45%
Future Value          FV2 = \$27,811.33 × ( 1 + 0.45% )^24
= \$27,811.33 × 1.0045^24
≈ \$27,811.33 × 1.113778
≈ \$30.975.64```