### TIME VALUE OF MONEY: Repaying a Debt.

*In Plan 1*, $1000will be paid at the end of each year plus the interest due at the end of the year for the use ofmoneyto that point. Thus, at the end of the firstyear, we will havehad the use of $5000. The interest owed is 8% x $5000 =$400. The end-of-year payment is, therefore, $1000 principal plus $400 interest, for a total payment of $1400.At the end of the second year, another$1000 principal plus interestwill be repaid on themoney owed during the year.This time the amountowedhas declinedfrom$5000 to $4000becauseof the $1000 principal payment at the end of the first year.The interest payment is 8% x $4000=$320, making the end-of-year payment a total of $1320. As indicated in Table 3-1, the series of payments' continues each year until the loan is fully repaid at the end of the fifth year.

*Plan 2*is another way to repay $5000 in 5 years with interest at 8%. This time the end-of-year payment is limited to the interest due, with no principal payment. Instead, the $5000 owed is repaid in a lump sum at the end of the fifth year.The end-of-year payment in each of the first four years of Plan 2 is 8% x $5000 = $400.

*Plan 3*calls for five equal end-of-yearpayments of $1252 each. At this point, we have not shown how the figure of $1252 was computed (see later: Example 4-3). However, it isclear that there is some equal end-of-year amount that would repay the loan. By following the computations in Table 3-1, we see that this series of five payments of $1252 repays a $5000 debt in 5 years with interest at 8%.

**TABLE 3-1**Four Plans for Repayment of $5000 in 5 Yearswith Interest at 8%

*Plan 4*is still another method of repaying the $5000 debt. In this plan, no payment is made until the end of the fifth year when the loan is completely repaid.Note what happens at the end of the first year: the interest due for the first year-8% x $5000=$400-is not paid; instead, it is added to the debt. At the second year, then, the debt has increased to $5400. The second year interest is thus 8%x $5400=$432.This amount, again unpaid, is added to the debt, increasing it further to $5832.At the end of the fifth year, the total sum due has grown to $7347 and is paid at that time (see Example 3-4).

Note that when the $400 interest was not paid at the end of the first year, it was added to the debt and, in the second ye~, tilerewas interest chargedon this unpaid interest. That is, the $400 of unpaid interest resulted in 8% x $400= $32 of additional interest charge in the second year.That $32, together with 8% x $5000=$400interest on the $5000 original debt,brought the total interes tchargeat the end of the second year to$432.Charging interest on unpaid interest is called compound interest. We will deal extensively with compound interest calculations later in this chapter.

With Table 3-1 we have illustrated four different ways of accomplishing the same task, that is, to repay a debt of $5000 in 5 years with interest at 8%.Having described the alternatives, we will now use them to present the important concept of equivalence.

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