Engineering Finance

Project Evaluation
Net Worth

Complex Cash Flows

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.

Moving Money Around
Solution by Decomposition

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)

Uniform series Worth:

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

Future Worth at time 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 N-1 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 open-book exams. You must use the factor tables for closed-book exams. In such cases, the necessary tables will be included with the exam sheet.

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Net Worth Summary


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.

Cash Flow Exercises
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Cash Flow Exercises Solutions

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.

Time Value of Money

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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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