Engineering Economic Analysis

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Single Payment Compound Interest Formulas 73

Is the least-costalternative the onethat hasthe lower initial cost andhigher operating costs

or the one with higher initialcost and lower continuing costs? Because of the time value of

money, one cannot add up sums of money at different points in time directly. This means

that alternatives cannotbecompared in actual dollars at different points in time; instead

comparisons must be made in some equivalent comparable sums of money.

It is not sufficientto compare the initial $600 against $850. Inste~d,wemust compute a

value that represents the entire stream of payments. In other words, we want to determine a

sum that is equivalent to Alternative A's cash flow; similarly, we needto compute the

equivalent sum for AlternativeB.By computing equivalent sums at the same point in time

("now"),we willhave values that may be validlycompared. The methods for accomplishing

this willbepresented later in this chapter and Chapter 4..

Thus far we have discussed computingequivalentpresent sums for a cash flow.But the

technique of equivalenceis not limited to a present computation.Instead,wecould compute

the equivalent sum for a cash flow at any point in time. We could compare alternatives in

"Equivalent Year 10" dollars rather than "now" (Year 0) dollars. Further, the equivalence

need not be a single sum; it could be a series of payments or receipts. In Plan 3 of Table 3-1,

the series of equal payments was equivalentto $5000 now. But the equivalencyworks both

ways. Suppose we ask the question,What is the equivalentequal annualpayment continuing

for 5 years,givena presentsumof $5000andinterest at 8%? The answeris $1252.

Single Payment Compound Interest Formulas

Tofacilitate equivalence computations, a series of interest formulas will be derived. To

simplify the presentation, we'll use the following notation:

i =interest rate per interest period.In the equations the interest rate is stated as a

decimal(thatis, 9% interestis 0.09).

n=number of interest periods.

P=a present sum of money.

F=a future sum of money.The future sumFis an amount,ninterest periods from

the present, that is equivalenttoPwith interest ratei.

Suppose a present sum of moneyPis invested for one yearl at interest ratei.At the end

of theyear,weshouldreceivebackourinitialinvestmentP,togetherwithinterestequalto

i P,or a total amountP+i P.FactoringP,the sum at the end of one year isP(1 + i).

Let us assume that, instead of removing our investment at the end of one year,we agree

to let it remain for another year. How much would our investmentbe worth at the end of the

second year? The end-of-first-yearsumP(1+i)will draw interest in the secondyear of.

i P(1+i).This means that, at the end of the second year, the total investment will become

P(1+i)+iP(1+i)

IA more general statement is to specify "one interest period" rather than "one year." Since, however,

it is easier to visualize one year, the derivation will assume that one year is the interest period.