Engineering Economic Analysis

86 MORE INTERESTFORMULAS

Chapter 3 presented the fundamentalcomponents of engineeringeconomic analysis,includ-

ing formulas to compute equivalent single sums of money at different points in time. Most

problems we will encounter are much more complex. Thus this chapter develops formulas

for payments that are a uniform series or are increasing on an arithmetic or geometric gra-

dient. Later in the chapter; noilliriaIand effective interest are discussed. Finally, equations

are derived for situations where interest is continuously compounded.

UNIFORM SERIES COMPOUND INTEREST FORMULAS

Many times we will find uniform series of receipts or disbursements. Automobile loans,

house payments, and many other loans are based on a uniform payment series. It will

often be convenient to use tables based on a uniform series of receipts or disbursements.

The seriesAis defined as follows:

A=An end-of-periodl cash receipt or disbursementin a uniform series, continuing

fornperiods, the entire series equivalent toPorFat interest ratei

The horizontal line in Figure 4-1 is a representation of time with four interest periods

illustrated. Uniform paymentsAhave been placed at the end of each interest period, and

there are as manyA'sas there are interest periodsn.(Both these conditions are specifiedin

the definition ofA.)Figure 4-1 uses January 1 and December 31, but other I-year periods

could be used.

In Chapter 3's section on single payment formulas, we saw that a sumPat one point

in time would increase to a sumFinnperiods, according to the equation

F=P(1+it

We will use this relationship in our uniform series derivation.

FIGURE 4-1 The general relationship betweenA

andF.

Year

A A A A

t t t t

0-1-2-3-4

I I I I

II II II II

I I I I

I I I I

I I I I

I I I I

I I I I

I I I I

I I I I

=~ ~~ 0 ~ ~~ 0 ~ ~~ 0 ~

v v v....

~ ~ ~

n=4

F

1In textbooks on economic analysis, it is customary to defineAas an end-of-period event rather than a.

beginning-of-period or, possibly, middle-of-period event. The derivations that follow are based on this

end-of-period assumption. One could, of course, derive other equations based on beginning-of-period

or mid period assumptions.