## Future Value and Compounding

Suppose an individual were to make a loan of $1. At the end of the first year, the borrower would owe the lender the principal amount of $1 plus the interest on the loan at the interest rate of r. For the specific case where the interest rate is, say, 9 percent, the borrower owes the lender

At the end of the year, though, the lender has two choices. She can either take the $1.09— or, more generally, (1 + r)—out of the capital market, or she can leave it in and lend it again for a second year. The process of leaving the money in the capital market and lending it for another year is called compounding.

Suppose that the lender decides to compound her loan for another year. She does this by taking the proceeds from her first one-year loan, $1.09, and lending this amount for the next year. At the end of next year, then, the borrower will owe her

$1 X (1 + r) X (1 + r) = $1 X (1 + r)2 = 1 + 2r + r2 $1 X (1.09) X (1.09) = $1 X (1.09)2 = $1 + $0.18 + 0.0081 = $1.1881

This is the total she will receive two years from now by compounding the loan.

In other words, the capital market enables the investor, by providing a ready opportunity for lending, to transform $1 today into $1.1881 at the end of two years. At the end of three years, the cash will be $1 X (1.09)3 = $1.2950.

The most important point to notice is that the total amount that the lender receives is not just the $1 that she lent out plus two years' worth of interest on $1:

The lender also gets back an amount r2, which is the interest in the second year on the interest that was earned in the first year. The term, 2 X r, represents simple interest over the two years, and the term, r2, is referred to as the interest on interest. In our example this latter amount is exactly r2 = ($0.09)2 = $0.0081

When cash is invested at compound interest, each interest payment is reinvested. With simple interest, the interest is not reinvested. Benjamin Franklin's statement, "Money makes money and the money that money makes makes more money," is a colorful way of explaining compound interest. The difference between compound interest and simple interest is illustrated

Ross-Westerfield-Jaffe: I II. Value and Capital I 4. Net Present Value I I © The McGraw-Hill

Corporate Finance, Sixth Budgeting Companies, 2002

Edition

Chapter 4 Net Present Value

■ Figure 4.4 Simple and Compound Interest

1 year 2 years 3 years The dark-shaded area indicates the difference between compound and simple interest. The difference is substantial over a period of many years or decades.

in Figure 4.4. In this example the difference does not amount to much because the loan is for $1. If the loan were for $1 million, the lender would receive $1,188,100 in two years' time. Of this amount, $8,100 is interest on interest. The lesson is that those small numbers beyond the decimal point can add up to big dollar amounts when the transactions are for big amounts. In addition, the longer-lasting the loan, the more important interest on interest becomes. The general formula for an investment over many periods can be written as

Future Value of an Investment:

where C0 is the cash to be invested at date 0, r is the interest rate, and T is the number of periods over which the cash is invested.

## Sell Your Annuity

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