Engineering Economic Analysis

Equivalence 71

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

Dollar-Years

Plan 1 Plan 2 Plan 3 Plan 4

(Money owed in Year 1)(1 year)

(Money owed in Year 2)(1 year)

(Money owed in Year 3)(1 year)

(Money owed in Year 4)(1 year)

(Money owed in Year 5)(1 year)

Total dollar-years

$ 5,000

4,000

3,000

2,000

1,000

$15,000

$ 5,000

5,000

5,000

5,000

5,000

$25,000

$ 5,000

4,148

3,227

2,233

1,159

$15,767

$ 5,000

5,400

5,832

6,299

6,803

$29,334

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

area under the curve may be obtained:

We see that the ratio of total interest paid to the area under the curve is constant and equal
to 8%. Stated another way,the total interest paid equals the interest ratetimesthe area under
the curve.
From our calculations,wemore easily see why the repaymentplans require the payment
of different total sums of money,yet are actually equivalentto each other. The key factor is
that the four repaymentplans provide the borrower with differentquantities of dollar-years.
Since dollar-years times interest rate equals the interest charge, the four plans result in
different to~ interest charges.

Equivalence Is Dependent on Interest Rate

In the example of Plans 1-4, all calculations were made at an 8% interest rate. At this
interest rate, it has been shown that all four plans are equivalentto a present sum of $5000.
But what would happen if we were to change the problem by changing the interest rate?
If the interest rate were increased to 9%, we know that the required interest payment
for each plan would increase,and the calculated repayment schedules (Table3-1, column f)

could no longer repay the $5000 debt with the higher interest. Instead, each plan would
repay a sumlessthan the principal of $5000, because more money would have to be used
to repay the higher interest rate. By some calculations (to be explained later in this chapter -
and in Chapter 4), the equivalentpresent sum that each plan will repay at 9% interest is:

Plan

1

2

3

4

Repay a Present Sum of

$4877

4806

4870

4775

Total Interest Area Under Curve Ratio of Total Interest Paid

Plan Paid (dollar-years) to Area Under Curve

(^1) $1200 15,000 0.08
(^22000) 25,000 0.08
(^31260) 15,767 0.08
(^42347) 29,334 0.08