Equivalence
Factors
The subjects of Engineering Economics and Business
Finance have traditionally used equivalence factors to
express a complicated cash flow as a single equivalent number.

Use the equivalence factors to
change from one cash flow pattern to an equivalent
pattern.




Goals 



Use equivalence factors to express
a multiperiod cash flow as a single value equivalent.

Answer problems that involve
a single equivalency factor.

Use the factor calculator.

Use equivalence factors for the special cases of
an infinite time horizon and zero interest rate.

Compute effective interest
rates.




Text 

Sections 3.2.1 and 3.2.2 support
this page.


3.2 Compound
Interest Formulas





Cash Flow 


An important concept throughout this course
is the cash flow. The cash flow shows a collection
of receipts and expenditures over time, and may
be presented as a table or as a graph. Usually receipts are
positive numbers with arrows pointing up on the graph, while disbursements
are negative numbers with arrows pointing down. There will
be cases when costs are the most important. Then we define
disbursements as positive with arrows pointing up.
Time is measured on the horizontal axis. For discrete
compounding, time is measured in compounding periods. Although
the compounding period is often one year, it may be some
fraction of a year (one month, one quarter, one half).
The numbers on the axis indicate compounding periods. In
general the beginning and end points of the time range
are arbitrary. The length of the range is N, and we
assume that the start and end points are integer multiples of the
compounding period. It is most common to have the range
start at 0 and end at N. In the example on the left, N is
10. 
Usually, we restrict cash flows to be located at integer time
values. In practical problems, operating costs and revenues occur
continuously, but for analysis, we collect all cash flows during
a period and show them at the end of the period. This is called
the endoftheperiod assumption. For example, the cash
flows for the first period are shown at time 1. Major receipts
or disbursements are shown where they occur, but only at integer
values of time. The example has an investment at time 0.
Dollar values are measured by the vertical axis. These values may
be scaled to accommodate large amounts.
On this page we consider simple patterns of cash
flow. In the next lesson we show how more complicated cash flows like the one above can
be represented by a collection of simple patterns. 

Patterns of Cash Flow 

We recognize four patterns of cash flow in terms
of their location in time: present worth,
future worth, uniform series and arithmetic gradient series.
P: Present Worth
This is the present worth (or present value). Without
a subscript it is a single amount at time 0, where time
0 usually represents the present.


F: Future Worth
This is the future worth of a cash flow.
Without a subscript, it is a single value at the end of
the interval.


A:
Uniform Series
This is a series of equal cash flows over
a given interval. The individual cash flows occur at
the end of each period in the interval. This is the endoftheperiod assumption.
The series name is based on the first letter of annuity.
An annuity is a series of annual payments, but we allow
the series to be defined for any compounding period. 

G:
Gradient
This is the step in values for a series
of arithmetically increasing cash flows. The value of
the cash flow at time t is G(t1).
For time 1 the cash flow is 0, for time 2, the cash flow
is G, for time 3, the cash flow is 2G,
and so on until for time N the cash flow is
(N1)G. It is important to note that
the first term of a gradient series is always zero. 


We use subscripts to refer
to a single cash flow at time t. When working with
an interval, such as 010 for these examples, the value of t need
not be in the interval. For example might
refer to a single value at time 1. 


Equivalence Factors 

For most of our applications, it will be necessary to move
the components of a cash flow so that the result is a single
value or a uniform series over a given interval. This is done
using equivalence
factors. Each factor changes one cash flow pattern to an
equivalent pattern.
To understand how the factors are used, a set of questions is given in the window called Simple
Time Value. To perform the time value of money calculations, we can use the Factor
Calculator, which resides in the Toolbox in the Computation Directory, or the factor tables discussed below.
To begin, click the Q icon to open the questions.
Click the tools icon to open the calculator. This is
a Flash document.
Click the Quicktime symbol
to see the presentation.
Factors move money around. The factors used in this course and
their purposes are listed below.
Factor 
Purpose 
(F/P, i, N) 
Moves a single payment to N periods
later in time 
(P/F, i, N) 
Moves a single payment to N periods
earlier in time 
(A/F,i, N) 
Takes a single payment and spreads into
a uniform series over N earlier periods. The last
payment in the series occurs at the same time as F. 
(F/A, i, N) 
Takes a uniform series and moves it to
a single value at the time of the last payment in the series. 
(A/P, i, N) 
Takes a single payment and spreads it
into a uniform series over N later periods. The
first payment in the series occurs one period later than P. 
(P/A, i, N) 
Takes a uniform series and moves it to
a single payment one period earlier than the first payment
of the series. 
(P/G, i, N) 
Takes an arithmetic gradient series and
moves it to a single payment two periods earlier than the
first nonzero payment of the series. 
(A/G, i, N) 
Takes an arithmetic gradient series and
converts it to a uniform series. The two series cover the
same interval, but the first payment of the gradient series
is 0. 
Factors depend on interest and time. Click the icon for a summary
of the factors and their formulas.
Traditional textbooks evaluate equivalence factors with Factor
Tables. These give the
values of the factors for selected interest rates and selected
periods. In the class we require the tables for hand calculations
during closed book examinations, so learn to use them.
Instructions for using the tables are in the Computation Directory of this website. Click the icon for a list of interest rates.
Clicking a rate will open a PDF document with values for the
factors.


Special Cases 

There are two important special cases  when the
interest is zero and when the interval goes to infinity. When i = 0, the value of money at one time is the same as it is at
any other. It is like "putting your money in your mattress."
Some of the factors cannot be computed with the formulas, but
can be evaluated with L'Hopital's rule. N going to infinity
is an interesting case because many practical problems can be
approximated with an unbounded interval. Click the icon to open
a window with the limit values.


Effective Interest 

Effective interest rates are relevant when there
is more than one compounding period in a year. There are two
reasons to learn about this. First, many personal loans use a
compounding period other than a year, generally a month. Every
time a payment is due, interest is computed and paid. Second,
in our economic analysis of engineering projects, it may sometimes
be convenient to place payments at times that are fractions of
a year. Since the equivalence factors are valid only for integer
values of time, it will be convenient to define intervals of
time smaller than one year.

When there is more than one compounding
period per year, the factors used for compounding
and discounting must use the interest rate per compounding
period and time is measured in number of compounding
periods.


Click the Quicktime symbol to see the presentation.
Click the icon to open a window with the effective interest
rate formulas.


Effective
Interest Formulas





Loans 

For a loan, a person borrows money from someone else. The principal,
interest rate, and the number of payments determine how much
each payment will be. Click the icon for a summary of the equations
for loans.


Summary 



Problems 

Click the icon to view the simple time value
of money problems with answers. Use the factor calculator or the tables to find numerical
answers. Both tools are reached through the Toolbox.

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