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 multi-period 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.
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 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 end-of-the-period 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 end-of-the-period 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(t-1). 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 (N-1)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 0-10 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.

 Simple Time Value

Click the tools icon to open the calculator. This is a Flash document.

 Factor Calculator

Click the Quicktime symbol to see the presentation.

 Equivalence Factors

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.

 Factor 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.

 Factor Tables
 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.

 Limit Values for Factors
 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.

 Effective Interest

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.

 The Economics of Loans
 Summary

 Equivalence 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.

 Simple Time Value
 Toolbox

Engineering Finance
by Paul A. Jensen