Engineering Economic Analysis

Equivalence 69

interest rate. Then we can judge the relative attractiveness of the two alternatives,not from
their cash flows, but from comparable equivalentvalues. Since Plan 1, like Plan 2, repays a
presentsum of $5000 with interest at 8%, the plans are equivalentto $5000now;therefore,
the alternatives are equally attractive. This cannot be deduced from the given cash flows
alone. It is necessary to learn this by deieimining the equivalentvalues for each alternative
at some point in time, which in this case is "the present."

Differencein RepaymentPlans

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 from Table 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 first year, 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, while Plan 4 increases the debt until the end of the fifth year.These graphs

show an important differenceamong the repaymentplans-the areas under the curvesdiffer

greatly. Since the axes areMoney OwedandTime,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 borrow money, they are acquiring the use of money as

represented by the area under the curve for Money 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

minusthe $5000 principal:

Plan Total Interest Paid

1 $1200

2 2000

3 1260

4 2347

We can use Table 3-2 and the data from Table 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 five years,

thenadd:

Shaded area=(Money owed in Year 1)(1 year)

• (Moneyowedin Year2)(1year)

+...

• (Moneyowedin Year5)(1year)

or

Shaded area [(Money owed)(Time)] =Dollar-Years