Singte Amounts

The distinction between simple and compound interest can best be seen by example. Illustrates the rather dramatic effect that compound interest has on an investment's value over time when compared with the effect of simple interest. From the table it is clear to see why some people have called compound interest the greatest of human inventions. The notion of compound interest is crucial to understanding the mathematics of finance. The term itself merely implies that interest paid (earned) on a loan (an investment) is periodically added to the principal. As a result, interest is earned on interest as well as the initial principal. It is this interest-on-interest, or compounding, effect that accounts for the dramatic difference between simple and compound interest. As we will see, the concept of compound interest can be used to solve a wide variety of problems in finance.

Future (or Compound) Value. To begin with, consider a person who deposits $100 into a savings account. If the interest rate is 8 percent, compounded annually, how much will the $100 be worth at the end of a year? Setting up the problem, we solve for I lie future value (which in this case is also referred to as the compound value) of the account at the end of the year (FV,).

Interestingly, this first-year value is the same number that we would get if simple interest were employed. But this is where the similarity ends.

What if we leave $100 on deposit for two years? The $100 initial deposit will have grown to $108 at the end of the first year at 8 percent compound annual interest. Going to the end of the second year, $108 becomes $116.64, as $8 in interest is earned on the initial $100, and $0.64 is earned on the $8 in interest credited to our account at the end of the first year. In other words, interest is earned on previously earned interest - hence the name compound interest.

At the end of three years, the account would be worth.

In general, FV„, the future (compound) value of a deposit at the end of n periods.

Where we let FVIF;„ (i.e., the future value interest factor at i% for n periods) equal (1 + i)". Showing the future values for our example problem at the end of years 1 to 3 (and beyond), illustrates the concept of interest being earned on interest. A calculator makes Eq. (3.4) very simple to use. In addition, tables have been constructed for values of (1 + ;)" - FVIFU„ ~ for wide ranges of i and n. These tables, called (appropriately) future value interest factor (or terminal value interest factor) tables, arc designed to be used with. Is one example covering various interest rates ranging from 1 to 15 percent. The Interest Rate (i) headings and Period (n) designations on the table are similar to map coordinates. They help us locate the appropriate interest factor. For example, the future value interest factor at 8 percent for nine years (FVIFew) is located at the intersection of the 8% column with the 9-period row and equals 1.999. This 1.999 figure means that $1 invested at 8 percent compound interest for nine years will return roughly $2 - consisting of initial principal plus accumulated interest.