Each equivalence factor changes one simple cash
flow pattern to another. When a simple pattern does not conform with the convention used by the equivalence factors
it is often necessary to use combinations of factors to find
the equivalent present worth, future worth or annual worth.

For a cash flow that is not
one of the simple patterns it may be necessary to
use more than one equivalence factor to find its
present worth or annual worth.


The Flash demonstration illustrates some
simple variations that are often useful for the analysis of complex
cash flows.

We find the equivalent value of a complex cash flow by decomposing
it into several simple patterns.

The present
worth of a complex cash flow is the sum of the
present worth values of its components.


Complex Cash Flow 
Component Cash Flows 

P(1) = 30 
P(2) = 10(P/A, i, 10) 
P(3) = 5(P/G, i, 7)(P/F, i,
3)
Note that the gradient begins at 4
with 0 cash flow. The P/G factor finds the value
at 3, and the P/F factor brings that value to 0.


Present Worth of combination:

P = P(1) + P(2) + P(3) = 30 + 10(P/A, i, 10)
+ 5(P/G, i, 7)(P/F, i, 3) 

A = P(A/P, i, 10) 

F = P(F/P, i, 10) = A(F/A, i, 10) 

The easiest way
to solve most problems is to first find the present worth.
If the uniform series or future worth equivalent is required,
it can be computed from the present worth. 

There are many different formulas that can derive the equivalent
values. Unless otherwise indicated the following rule is used
to judge alternative expressions.

For this course, expressions
for the time value of money are judged by the number
of terms in the expression. The fewer the better.


When using a computational aid such as Excel, it is often convenient to add the present
worth values for all the individual cash flows, as shown in the formula for P below. This approach
is not highly valued when the goal is to minimize the number
of terms because it has N1 terms, but it is sufficient
when automatic calculation is available.

There are several ways to evaluate the NPW and
NAW. Traditional hand computations use factor tables, which are available in the Toolbox. Two Flash calculators
that simplify computations, the cash flow calculator and the
project calculator, are also provided. Click the icon below to open a Toolbox. Click
an icon in the box to open the desired tool. Unless otherwise
stated, you can use the calculators for homework and openbook
exams. You must use the factor tables for closedbook exams.
In such cases, the necessary tables will be included with the exam sheet.

The first linked document below is a pdf file
with exercises that ask you to express cash flows as equivalent amounts
using equivalence factors. The second link opens a scorecard.
The third link is the answer key. Solve the questions and use
the scorecard and the answer key to see how you did.
The ability to build an expression for the present
worth of a complex cash flow is important throughout this course,
so you should get a lot of practice. Click the icon to view
several more examples. Work out the answers before looking at the
solutions.
