CP

What would you end up with if you left your $100 in the account for five years?

Here is a time line set up to show the amount at the end of each year:

05% 1 2 3 4 5

Initial deposit: 
100 FV 1 ?FV 2 ?FV 3 ?FV 4 ?FV 5 ?

Interest earned: 5.00 5.25 5.51 5.79 6.08

Amount at the end of

each period FVn: 105.00 110.25 115.76 121.55 127.63

Note the following points: (1) You start by depositing $100 in the account—this is

shown as an outflow at t 0. (2) You earn $100(0.05) $5 of interest during the first

year, so the amount at the end of Year 1 (or t 1) is $100 $5 $105. (3) You start

the second year with $105, earn $5.25 on the now larger amount, and end the second

year with $110.25. Your interest during Year 2, $5.25, is higher than the first year’s in-

terest, $5, because you earned $5(0.05) $0.25 interest on the first year’s interest. (4)

This process continues, and because the beginning balance is higher in each succeed-

ing year, the annual interest earned increases. (5) The total interest earned, $27.63, is

reflected in the final balance at t 5, $127.63.

Note that the value at the end of Year 2, $110.25, is equal to

Continuing, the balance at the end of Year 3 is

and

In general, the future value of an initial lump sum at the end of n years can be

found by applying Equation 2-1:

(2-1)

The last term in Equation 2-1 defines the Future Value Interest Factor for i and n,

FVIFi,n,as (1 i)n. This provides a shorthand way to refer to the actual formula in

Equation 2-1.

Equation 2-1 and most other time value of money equations can be solved in three

ways: numerically with a regular calculator, with a financial calculator, or with a com-

puter spreadsheet program.^2 Most work in financial management will be done with a

financial calculator or on a computer, but when learning basic concepts it is best to

also work the problem numerically with a regular calculator.

NUMERICAL SOLUTION

One can use a regular calculator and either multiply (1 i) by itself n 
1 times or else

use the exponential function to raise (1 i) to the nth power. With most calculators, you

FVnPV(1i)nPV(FVIFi,n).

FV 5 $100(1.05)^5 $127.63.

$100(1.05)^3 $115.76,

PV(1i)^3

FV 3 FV 2 (1i)

$100(1.05)^2 $110.25.

PV(1i)^2

PV(1i)(1i)

FV 2 FV 1 (1i)

58 CHAPTER 2 Time Value of Money

(^2) Prior to the widespread use of financial calculators, a fourth method was used. It is called the “tabular ap-
proach,” and it is described in the Chapter 2 Web Extension, available on the textbook’s web site.

56 Time Value of Money