TIME VALUE OF MONEY: Difference in Repayment Plans.

The four plans computed in Table 3-1 are equivalent in nature but different in structure.

Table 3-2 repeats the end-of-year payment schedules fromTable 3-1 and also graphs each plan to show the debt still owed at any point in time. Since $5000 was borrowed at the beginning of the firstyear, all the graphs begin at that point.We see,however, that the four plans result in quite different situations on the amount of money owed at any other point in time. In Plans 1 and 3, the money owed declines as time passes. With Plan 2 the debt remains constant,whilePlan 4 increases the debt until the end of the fifth year.

These graphs showan importantdifferenceamongthe repaymentplans-the areasunder the curvesdiffer greatly. Since the axes areMoney Owed and Time, the area is their product:Money owed x
Time, in years.

In the discussion of the time value of money, we saw that the use of money over a time period was valuable, that people are willing to pay interest to have the use of money for periods of time. When people borrowmoney, they are acquiring the use of money as representedby the area under the curve forMoney owed vs Time-.It follows that, at a given interest rate, the amount of interest to be paid will be proportional to the area under the curve. Since in each case the $5000 loan is repaid, the interest for each plan is the total minus the $5000 principal:

We can use Table3-2 and the data fromTable 3-1, to compute the area under each of the four curves, that is, the area bounded by the abscissa, the ordinate, and the curve itself. We multiply the ordinate (Money owed) by the abscissa (1 year) for each of the fiveyears, then add:

TABLE 3-2  End-of-Year Payment Schedules an their Graphs.

The dollar-years for the four plans would be as follows.

With the area under each curve computed in dollar-years, the ratio of total interest paid to area under the curvemay be obtained:

We see that the ratio of total interest paid to the area under the curve is constant and equal to 8%.Stated anotherway,the total interest paid equals the interest rate times the area under the curve.

Fromour calculations,wemore easily see why there payment plans require the payment of different total sums of money,yet are actually equivalent to each other.The key factor is that the four repaymentplans provide the borrowerwith differentquantitiesof dollar-years.

Since dollar-years times interest rate equals the interest charge, the four plans result in different total interest charges.