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 from assumed distribution and

simulation

Categories of Cash FlowsThe expenses and receipts due to engineering projects usually fall into one of the following categories:

First cost: expense to build or to buy and installOperations and maintenance (O&M): annual expense, such as electricity, labor, and minor repairsSalvage 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 servicesOverhaul: major capital expenditure that occurs during the asset’s life

Cash Flow DiagramsThe 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

0

1 2

$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

PMake an initial investment (purchase) at “time 0”

Cash Flow Diagram

0 1 2 T

Receive revenues and pay expenses over time. P

The net amount is written on the cash flow diagram

Cash Flow Diagram

0 1 2 T

Write as a NET cash flow in each period.

P

Cash Flow Diagram

0 1 2 T

SV

P Receive salvage value at endof life of project.

Time Value of Money

Generally, money grows (compounds)

into larger future sums and is smaller

(discounted ) in the past

Generally, money grows (compounds)

into larger future sums and is smaller

(discounted ) in the past

Compound Interest and Cash Flow Diagrams

0 1 2

P = 1000

F = 1210

In general: F = P(1+i)n

Example: P=$1000, i=10%, compounded annually.

How much accrued after two years?

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

Steps to Solve Time Value of Money Problems

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 GradientIt is a cash flow series that either increases

or decreases by constant amount

The cash flow changes by the same

arithmetic amount each period

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 raten = interest periodP = present principle amountA = Equal annual paymentsF = Future amountG = 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

P / F = Single payment present

worth factor

F = P (1 + i)n

1 / (1 + i)n = P/F

F / P = (1 + i)n

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