ENGINEERING ECONOMICS
Lecture # 3
Cash Flow and Cash Flow Diagram
Rule of 72
Arithmetic Gradient Factor
Geometric Gradient Factor
Total present worth
Examples and Numerical
Cash Flow
Engineering projects generally have economic
consequences that occur over an extended period of
time
Each project is described as cash receipts or
disbursements (expenses) at different points in time
For any practical engineering economy problems, the
cash flows must be:Known with certainty
Estimated
Range of possible realistic values
Generated
simulation
from
assumed
distribution
and
Categories of Cash Flows
The expenses and receipts due to
engineering projects usually fall into one
of the following categories:
First cost: expense to build or to buy and
install
Operations and maintenance (O&M): annual
expense, such as electricity, labor, and minor
repairs
Salvage value: receipt at project termination
for sale or transfer of the equipment (can be a
salvage cost)
Revenues: annual receipts due to sale of
products or services
Overhaul: major capital expenditure that
occurs during the asset’s life
Cash Flow Diagrams
The costs and benefits of engineering
projects over time are summarized on a cash
flow diagram (CFD). Specifically, CFD
illustrates the size, sign, and timing of
individual cash flows, and forms the basis for
engineering economic analysis
A CFD is created by first drawing a
segmented time-based horizontal line,
divided into appropriate time unit. Each time
when there is a cash flow, a vertical arrow is
added  pointing down for costs and up for
revenues or benefits. The cost flows are
drawn to relative scale
An Example of Cash Flow Diagram
A man borrowed $1,000 from a bank at 8%
interest. Two end-of-year payments: at the end of
the first year, he will repay half of the $1000
principal plus the interest that is due. At the end
of the second year, he will repay the remaining
half plus the interest for the second year.
Cash flow for this problem is:
End of year
Cash flow
0
+$1000
1
-$580 (-$500 - $80)
2
-$540 (-$500 - $40)
Cash Flow Diagram
$1,000
1
2
0
$580
$540
Important Aspects of CFD
Extremely valuable analysis tool
First step in solution process
Graphical representation on a time scale
Does not have to be drawn to exact scale
Information in one glance
Cash Flow Diagram
Used to describe any investment opportunity.
Inflow (revenue)
Outflow (costs)
0
P
Make an initial investment (purchase) at
“time 0”
Cash Flow Diagram
The net amount is written on the cash flow diagram
0
P
1
2
T
Receive revenues and pay
expenses over time.
Cash Flow Diagram
0
P
1
2
T
Write as a NET cash flow in
each period.
Cash Flow Diagram
SV
0
P
1
2
T
Receive salvage value at end
of life of project.
Time Value of Money
Generally, money grows (compounds)
into larger future sums and is smaller
(discounted ) in the past
Compound Interest and
Cash Flow Diagrams
Example: P=$1000, i=10%, compounded
annually.
How much accrued after two years?
F = 1210
0
1
2
P = 1000
In general: F = P(1+i)n
Steps to Solve Time Value of
Money Problems
1. Read problem thoroughly
2. Create a time line
3. Put cash flows and arrows on time line
4. Determine if it is a PV or FV problem
5. Determine if solution involves annuity
6. Solve the problem
Rule of 72
Investors most often ask
How long will it take for my investment to
be doubled in the value?
Can
I
have
a
known
or
assumed
compound interest rate in advance?
Rule of 72
The
approximate
time
for
an
investment to be doubled in value
given the compound interest rate is
n = 72 / i
For example if i = 13% then
time = 72 / 13 = 5.54 years
Rule of 72
One can estimate the future required
interest rate for an investment to be
doubled in value over time
i = 72 / n
Assume that we want the investment to be
doubled in 3 years
i = 72 / 3 = 24%
Arithmetic Gradient
It is a cash flow series that either increases
or decreases by constant amount
The cash flow changes by
arithmetic amount each period
the
same
The amount of increase or decrease is the
gradient
If it is predicted that the cost of NOKIA
mobile will increase by Rs 2000 each year, a
gradient series is involved and the amount of
gradient is Rs 2000
G = Constant arithmetic change (+ or -)
Arithmetic Gradient - Formulae
i = annual interest rate
n = interest period
P = present principle amount
A = Equal annual payments
F = Future amount
G = Annual change or gradient
Factors
F / P = Single payment future worth factor
P / F = Single payment present worth factor
F / A = Equal payment series future worth factor
A / F = Equal payment series sinking fund factor
P / A = Equal payment series present worth factor
A / P = Equal payment series capital recovery factor
A / G = Arithmetic gradient series factor
F / G = Arithmetic gradient future worth factor
P / G = Arithmetic gradient present worth factor
Geometric Gradient factor (only definition)
Total Present Worth in Gradient Problems (Pt)
The total present worth of a gradient series must consider the
base and the gradient separately
The base amount is the uniform series amount that begins in
year 1 and extends through year n. It is represented by P1
For an increasing gradient, the gradient amount must be
added to the uniform series amount. It is represented by P2
For a decreasing gradient, the gradient amount must be
subtracted from the uniform series amount. It is represented
by –P2
Pt = P1 + P2
Pt = P1 – P2
F / P = Single payment future
worth factor
n
F = P (1 + i)
F / P = (1 + i)n
P / F = Single payment present
worth factor
1 / (1 + i)n = P/F
F / A = Equal payment series
future worth factor
What will be the future worth of an amount of $
100 deposited at the end of each next five years
and earning 12 % per annum?
A / F = Equal payment series
sinking fund factor
It is desired to accumulate $ 635 by making a
series of five equal annual payments at 12 %
interest annually, what will be the required
amount of each payment?
P / A = Equal payment series
present worth factor
A / P = Equal payment series capital
recovery factor
A car has useful life of 5 years. The
maintenance cost occurs at the end of each
year. The owner wants to set up an account
which earns 12 % annually on an amount of $
3604 to cater for this maintenance cost. What
is the maintenance cost per annum?
Geometric Gradient
It is common for cash flow series such as
operating cost, construction cost and revenues to
increase or decrease by a constant percentage
such as 10 %
This uniform rate of change in %age is called
geometric gradient = g
g = constant rate of change in %age or decimal
form by which amount increase or decreases
from one period to other
Thank You