Chapter 4:

Equivalence Calculations under

Inflation
Fundamentals of Engineering Economics, 3rd Ed.

By Park, C.S.

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 1

Objectives
 To understand the meaning of inflation
 To learn how to measure inflation
 To understand the differences between actual dollars
and constant dollars
 To be able to transform actual dollars to constant
dollars and vice versa
 To learn how to consider inflation in equivalence
calculations
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 2
Inflation and Economic Analysis

 What is inflation?

 How do we measure inflation?

 How do we incorporate the effect of inflation in

equivalence calculations?

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 3

What is Inflation?
 Inflation is the rate at which the general level of prices
for goods and services is rising, and subsequently,
purchasing power is falling

 It is the loss in the purchasing power of money over

time

 It means that “the same dollar amount buys less of an

item over time”

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 4

What is Inflation?
 Value of Money
• Earning Power How much you currently make at your place
of employment plays a major part in your earning power
• Purchasing power The value of a currency expressed in
terms of the amount of goods or services that one unit of
money can buy

 Purchasing Power
• Decrease in purchasing power (inflation)
• Increase in Purchasing Power (deflation)

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 5

Purchasing Power
 Example:
$100 $100

1990 1990 2008 2012

You could buy 50 Big Macs in You can only buy 23.81 Big
year 1990 Macs in year 2012
210%
$2.00 / unit $4.20 / unit
Price change
due to $100
inflation Loss of purchasing power

The $100 in year 2012 has only $47.62 $47.62

worth of Big Mac purchasing power of 1990

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 6

Deflation
 Example:
$100 $100

2007 2008 2009 2010 2007 2008 2009 2012

You could purchase You can now purchase

63.69 gallons of purified 80 gallons of purified
drinking water in 2008 drink water in 2012

20.38%
$1.57 / gallon $1.25 / gallon
Price change due to
deflation
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 7
Inflation Terminology - I
 Consumer Price Index (CPI)
• CPI is a measure of the average change, over time in the
prices paid by urban consumers for a market basket of
consumer goods and services

• CPI is a statistical measure of change, over time, of the

prices of goods and services in major expenditure groups –
such as food and beverages, housing, apparel,
transportation, entertainment, medical care, personal care
and other goods and services – typically purchased by city
consumers
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 8
 What goods and services does the CPI cover?
 The CPI represents all goods and services purchased for consumption by the
reference population. Bureau of labor and Statistics (BLS) has classified all
expenditure items into more than 200 categories, arranged into eight major
groups. Major groups and examples of categories are:
1. FOOD AND BEVERAGES (breakfast cereal, milk, coffee, chicken, full service
meals, snacks)
2. HOUSING (rent of residence, owners' equivalent rent, fuel oil, bedroom
furniture)
3. APPAREL (men's shirts and sweaters, women's dresses, jewelry)
4. TRANSPORTATION (new vehicles, airline fares, gasoline, motor vehicle
insurance)
5. MEDICAL CARE (prescription drugs and medical supplies, physicians' services,
eyeglasses and eye care, hospital services)
6. RECREATION (televisions, toys, pets and pet products, sports equipment,
admissions)
7. EDUCATION AND COMMUNICATION (college tuition, postage, telephone
services, computer software and accessories)
8. OTHER GOODS AND SERVICES (tobacco and smoking products, haircuts and
other personal services)
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 9
Measuring Inflation
 CPI is a measure of the average change over time in the price
paid by city family for a set of consumer goods and services
 CPI compares the cost of a sample “market basket” of goods
and services in a specific period relative to the cost of the same
“market basket” in an earlier reference period
 This reference period is designated as the base period

Market basket
Base Period (1982-84) 2009
$100 $179.9
CPI for 2009 = 179.9 %
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 10 10
 Measuring Inflation

Original 2011
Measure $669.41
CPI=(669.41/$100)*100
CPI for 2011 = 669.41

2011
$223.47
Revised
CPI=(223.47/$100)*100
Measure CPI for 2011 = 223.47

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 11 11

Inflation Terminology - I
 Producer Price Index (PPI)
• PPI is a statistical measure of wholesale industrial price change,
compiled monthly by the Bureau of labor and Statistics (BLS), to
evaluate wholesale price levels in the economy
• PPI is a family of indices that measure the average change over
time in the selling prices received by domestic producers of
goods and services
• PPI measures price change from the perspective of the seller.
This contrasts with other measures, such as the CPI, that
measure price change from the purchaser's perspective
• CPI is a good measure of the general price increase of
consumer products. However, it is not a good measure of
industrial price increases
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 12
Inflation Terminology - I
 Producer Price Index (PPI)

Various Cost Components that Affect the Retail Gasoline Price

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 13
Inflation Terminology - I
 Producer Price Index (PPI)
Year (Base New CPI Old CPI Gasoline Automobile

Period) (1982–84) (1967) (1982) Steel (1982) (1982)

2002 178.9 535.8 89.0 114.1 134.9

2003 183.8 550.5 100.1 121.5 135.1

2004 188.0 563.2 126.1 162.4 136.5

2005 194.6 582.9 162.5 171.1 135.1

2006 201.5 600.9 217.7 186.6 130.6

2007 206.7 619.1 232.7 193.4 136.6

2008 214.8 643.5 294.3 207.5 135.2

2009 213.2 638.8 177.3 156.5 134.9

2010 218.0 653.1 244.8 195.8 138.2

2011 223.5 669.4 303.6 216.0 140.9

Selected Price Indexes between 2002 and 2011

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 14
Inflation Terminology - I
 Average Inflation Rate (𝑓)
• It a single rate that accounts for the effect of unstable yearly
inflation rates over a period of several years

 Example:
• Fact: Base Price = $100 (year 0)
Inflation rate (year 1) = 4%
Inflation rate (year 2) = 8%
Average inflation rate over 2 years?

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 15

Inflation Terminology - I
Solution:
• Step 1: Find the actual inflated price at the end of year 2
$100 (1 + 0.04) (1 + 0.08) = $112.32

• Step 2: Find the average inflation rate by solving the

following equivalence equation
$112.32
$100 (1+ f)2 = $112.32
f = 5.98%
0 1

$100
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 16
 Average Inflation Rate (𝑓)
 Example: Yearly & average inflation rates Year Cost
• What are the annual inflation rates and 0 $504,000
the average inflation rate over 3 years? 1 538,000
Solution: 2 577,000
Annual inflation rates: 3 629,500
Inflation rate during year 1 (f1):
$504,000 (1+ f1) = $538,000 f1 = 0.0683 = 6.83%
Inflation rate during year 2 (f2):
$538,000 (1+ f2) = $577,000 f2 = 0.0717 = 7.17 %
Inflation rate during year 3 (f3):
$577,000 (1+ f3) = $629,500 f3 = 0.091= 9.10%
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 17
 Average Inflation Rate (𝑓)
 Example: Yearly & average inflation rates Year Cost
• What are the annual inflation rates and 0 $504,000
the average inflation rate over 3 years? 1 538,000
Solution: 2 577,000
Average inflation rate over 3 years: 3 629,500

$504,000 (𝟏+𝒇)𝟑= $629,500 or (1+ f1)(1+ f2)(1+ f3) = (𝟏+𝒇)𝟑

$629,500 1/ 3
f ( )  1  0.0769  7.69%
$504,000

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 18

CPIs for 1963 and 2004

91.7 100 561.23

2004
1963 1967

Average inflation rate = 4.52%

561.23  91.70(1  f ) 41
f  41 6.1203  1
 4.5176%

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 19

Average Inflation Rate (𝑓)
 Example: Calculating Average Inflation Rate
F = P (1+ f )N $22,218 = $15,518 (1+ f )6
f = –1 = 0.0616 f = 6.16%
Item 2006 Price 2000 Price Average Inflation
(CPI) Base Period: 1982 - 84 = 100 F P Rate (%)
Consumer price index (CPI) $200.43 $171.20 2.66
Postage 0.39 0.33 2.82
Homeowners Insurance 617.00 500.00 3.57
Private college tuition and fees 22,218 15,518 6.16
Gasoline 2.56 1.56 8.61
Haircut 15.00 10.50 6.12
Car (Toyota Camry) 22,900 21,000 1.45
Natural gas (MBTU) 7.08 3.17 14.33
Baseball tickets 171.19 132.44 4.37
Health care (per year)
ISE 307 - Term 192 2,351.00
Dr. Yasser Almoghathawi, KFUPM 1,656.00 6.0120
Inflation Terminology - I
 General Inflation Rate (𝑓)ҧ
• It is an average inflation rate calculated based on the CPI for
all items in the market basket

 Specific Inflation Rate (𝑓𝑗 )

• It is an average inflation rate calculated based on a price
index (other than CPI) specific to segment j of the economy
such as labor, material, housing, or gasoline

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 21

General Inflation Rate (𝑓)ҧ
 This average inflation rate is calculated on the basis of CPI for
all items in the market basket. The market interest rate is
expected to respond to this general inflation rate. In terms of
CPI, we define the general inflation rate as:
_
CPIn  CPI0 (1  f )n ,
1/ n
_  CPIn 
f   1
CPI
 0
_
where f  The genreal inflation rate,
CPIn  The consumer price index at the end period n,
CPI0  The consumer price index for the base period.
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 22
General Inflation Rate vs. Specific Inflation Rate
Specific inflation rates

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM

General inflation rate 23
 Average Inflation Rate
 Example 4.2: Developing Specific Inflation rate for Baseball Tickets
• What are the annual inflation rates and
the average inflation rate over 6 years?
Solution:
Annual inflation rates:
Inflation rate between 2005 and 2006 (f1):
($287.84 - $276.24) / $276.24 = 4.20%
Inflation rate between 2006 and 2007 (f2):
($313.83 - $287.84) / $287.84 = 9.03 %
Inflation rate between 2007 and 2008 (f3):
($320.71 - $313.83) / $313.83 = 2.19%
Average inflation rate over 6 years:
1/6
 $339.01 
f    1  0.0347  3.47%
ISE 307 - Term 192
 $276.24 Dr. Yasser Almoghathawi, KFUPM 24
Actual vs. Constant Dollars
 Due to inflation, the purchasing power of the dollar
changes over time
 To compare dollar values of different purchasing power
from one period to another, they need to be converted
to dollar values of common purchasing power –
conversion from actual to constant dollars or from
constant to actual dollars
 To introduce the effect of inflation into our economic
analysis, we need to define two inflation – related terms
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 25
Inflation Terminology - II
 Actual (current) Dollars (𝐴𝑛)
• Estimates of future cash flows for year n that take into
account any anticipated changes in amount caused by
inflationary or deflationary effects

 Constant (real) Dollars (𝐴′𝑛 )

• Estimates of future cash flows for year n in constant
purchasing power, independent of the passage of time (or
base period)

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 26

 Conversion from Constant to Actual Dollars

𝐴𝑛 = 𝐴′𝑛 (1 + 𝑓)ҧ 𝑛 ↔ 𝐴′𝑛 (𝐹/𝑃, 𝑓,ҧ 𝑛)

n3
$1,000 $1,260
_
f  8%

0 3 3
Constant Actual
Base period 3
Dollars $1,000 (1 + 0.08) Dollars
= $1,260

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 27

Conversion from Constant to Actual Dollars
Average inflation rate = 5%

Net Cash Flow in Conversion Cash Flow in

Period
Constant $ Factor Actual $
0 -$250,000 (1+0.05)0 -$250,000

1 100,000 (1+0.05)1 105,000

2 110,000 (1+0.05)2 121,275

3 120,000 (1+0.05)3 138,915

4 130,000 (1+0.05)4 158,016

5 120,000 (1+0.05)5 153,154

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 28

Conversion from Constant to Actual Dollars

 Example 4.3: What would $30,000 earned In 1995 be equal

to in 2011? The CPIs for the two years are 152.4 and 223.47,
respectively.
Solution:
𝑛
CPI2011 = CPI1995 1 + 𝑓ҧ
16
233.47 = 152.4 1 + 𝑓 ҧ
1
𝑓 ҧ = 233.47/152.4 16 − 1 = 2.4211%
16
Equivalent actual dollars = 30000 1 + 𝒇ത = $43,990.16

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 29

Conversion from Actual to Constant Dollars

𝐴′𝑛 = 𝐴𝑛 (1 + 𝑓)ҧ −𝑛 ↔ 𝐴𝑛 (𝑃/𝐹,𝑓,ҧ 𝑛)

n3
$1,000 $1,260
_
f  8%

0 3 3
Constant -3 Actual
Dollars $1,260 (1 + 0.08)
Dollars
Base period = $1,000

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 30

Conversion from Actual to Constant Dollars
Average inflation rate = 5%

Loss in
End of Cash Flow in Conversion Cash Flow in
Purchasing
period Actual $ at f = 5% Constant $
Power

0 $20,000 (1+0.05)0 $20,000 0%

1 20,000 (1+0.05)-1 19,048 4.76

2 20,000 (1+0.05)-2 18,141 9.30

3 20,000 (1+0.05)-3 17,277 13.62

4 20,000 (1+0.05)-4 16,454 17.73

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 31

 Conversion from Actual to Constant Dollars
 Example:
The following table lists the winners and their prize monies in actual dollars
from the U. S. Open Golf Championship from 2002 to 2006. Convert the prize
monies into equivalent dollars of 2006. In doing so,
The prize money Consumer Inflation Equivalent Prize
Year Winner
(in actual dollars) price index rate money in 2006 dollars
2002 Tiger Woods $1,000,000 179.80
2003 Jim Furyk $1,080,000 183.80
2004 Retief Goosen $1,125,000 188.00
2005 Micheal Campbell $1,170,000 194.60
2006 Geoff Ogilvy $1,225,000 200.43
a. Determine the growth rate of the prize money in actual dollars over the 4-year period
b. Find the equivalent prize money for each winner, stated in terms of year 2006 dollars
c. Determine the growth rate of the prize money in constant dollars over the 4-year
period
d. If the current trend continues, what would the expected prize money be in actual
dollars for the winner in 2007?
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 32
Conversion from Actual to Constant Dollars
 Example:
The prize money Consumer Inflation Equivalent Prize
Year Winner
(in actual dollars) price index rate money in 2006 dollars
2002 Tiger Woods $1,000,000 179.80
2003 Jim Furyk $1,080,000 183.80
2004 Retief Goosen $1,125,000 188.00
2005 Micheal Campbell $1,170,000 194.60
2006 Geoff Ogilvy $1,225,000 200.43
Solution:

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 33

Conversion from Actual to Constant Dollars
 Example:
The prize money Consumer Inflation Equivalent Prize
Year Winner
(in actual dollars) price index rate money in 2006 dollars
2002 Tiger Woods $1,000,000 179.80
2003 Jim Furyk $1,080,000 183.80
2004 Retief Goosen $1,125,000 188.00
2005 Micheal Campbell $1,170,000 194.60
2006 Geoff Ogilvy $1,225,000 200.43
Solution:

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 34

Conversion from Actual to Constant Dollars
 Example:
The prize money Consumer Inflation Equivalent Prize
Year Winner
(in actual dollars) price index rate money in 2006 dollars
2002 Tiger Woods $1,000,000 179.80 $1,114,779
2003 Jim Furyk $1,080,000 183.80 2.22% $1,177,813
2004 Retief Goosen $1,125,000 188.00 2.29% $1,199,422
2005 Micheal Campbell $1,170,000 194.60 3.51% $1,205,100
2006 Geoff Ogilvy $1,225,000 200.43 3.00% $1,225,000
Solution:

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 35

Conversion from Actual to Constant Dollars
 Example:
The prize money Consumer Inflation Equivalent Prize
Year Winner
(in actual dollars) price index rate money in 2006 dollars
2002 Tiger Woods $1,000,000 179.80 $1,114,779
2003 Jim Furyk $1,080,000 183.80 2.22% $1,177,813
2004 Retief Goosen $1,125,000 188.00 2.29% $1,199,422
2005 Micheal Campbell $1,170,000 194.60 3.51% $1,205,100
2006 Geoff Ogilvy $1,225,000 200.43 3.00% $1,225,000
Solution:

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 36

Conversion from Actual to Constant Dollars
 Example 4.5:

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 37

Conversion from Actual to Constant Dollars
 Example 4.5: calculate the equivalent college cost in constant dollar (2000)
for each year

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 38

 Conversion from Actual to Constant Dollars
 Example 4.5: calculate the equivalent college cost in constant dollar (2000)
for each year

(𝐹/𝑃, 𝑓, 𝑛)
−𝑛
= 1+𝑓
1
= 𝑛
1+𝑓

(1+0.0225)

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 39

Conversion from Actual to Constant Dollars
 Example 4.5:

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 40

Equivalence Calculation Under Inflation
 Types of Interest Rate
• Market Interest rate (i)
• Inflation-free interest rate (i')

 Types of Cash Flow

• In Constant Dollars
• In Actual Dollars

 Types of Analysis Method

• Constant Dollar Analysis
• Actual Dollar Analysis
• Deflation Method
• Adjusted-discount method
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 41
Inflation Terminology - III
 Inflation-free Interest Rate (𝑖 ′ )
• An estimate of the true earning power of money when the
inflation effects have been removed

• This rate is known as real interest rate, and it can be

computed if the market interest rate and the inflation rate
are known.

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 42

Inflation Terminology - III
 Market Interest Rate (𝑖)
• known as the nominal interest rate, which takes into account
the combined effects of the earning value of capital (earning
power) and any anticipated inflation or deflation (purchasing
power).

• Most firms use a market interest rate in evaluating their

investment projects.

• It is known as inflation-adjusted required rate of return

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 43

Inflation and Cash Flow Analysis

 Constant Dollar analysis (𝐴′𝑛 ) (inflation-free interest rate 𝑖’)

• All cash flow elements are given in constant dollars
• Compute the equivalent present worth of constant dollars
(𝐴′𝑛 ) in year n
• In the absence of inflationary effect, we use 𝑖′ to account
for the earning power of money

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 44

Inflation and Cash Flow Analysis

 Actual Dollar analysis (𝐴𝑛 ) (market interest rate 𝑖)

• All the cash flow elements are estimated in actual dollars
• To find the equivalent present worth of this actual dollar
amount (𝐴𝑛 ) in year n
• In case only the inflation-free interest rate 𝑖′ is given, we
use two steps to convert actual dollars into equivalent
present worth dollars

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 45

Inflation and Cash Flow Analysis

 Constant Dollar analysis

Interest Rate to Estimate Cash Choose Analysis
Use Flow Method

• Inflation-free • In Constant $ • Constant $

interest rate analysis

 Actual Dollar analysis

Interest Rate to Estimate Cash Choose Analysis
Use Flow Method

• Market • In Actual $ • Actual $

interest rate analysis

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 46

Actual Dollars (An) Analysis
 Method 1: Deflation Method
• Convert actual dollars into equivalent constant dollars by
discounting with the general inflation rate, a step that
removes the inflationary effect
• Use 𝑖′ to find the equivalent present worth

 Method 2: Adjusted-discount Method

• Combine two steps into one step, which performs
deflation and discounting in one step
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 47
 Actual Dollars (An) Analysis
 Example: Equivalence Calculation when cash flows are in
actual dollars
Applied instrumentation, a small manufacturer of custom electronics to make
investment to produce sensors and control systems that have been requested
by a fruit drying company. The work would be done under a contract that
would terminate in five years. The project is expected to generate the above
cash flows in actual dollars: Net Cash Flows in
n
Actual Dollars
a. What are the equivalent constant dollars
0 -$75,000
if the general inflation rate is 5% per year
1 32,000
a. Compute the present worth of these cash 2 35,700
flows in constant dollars at i' = 10% 3 32,800
4 29,000
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 5 58,000 48
 Actual Dollars (An) Analysis
 Example: Equivalence Calculation when cash flows are in
actual dollars
Solution: Deflation Method
Step 1: Convert Actual dollars to Constant dollars
Cash Flows in Multiplied by Cash Flows in
n
Actual Dollars Deflation Factor 5% Constant Dollars
0 -$75,000 1 -$75,000
1 32,000 (1+0.05)-1 30,476
2 35,700 (1+0.05)-2 32,381
3 32,800 (1+0.05)-3 28,334
4 29,000 (1+0.05)-4 23,858
5 58,000 (1+0.05)-5 45,445

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 49

Actual Dollars (An) Analysis
 Example: Equivalence Calculation when cash flows are in
actual dollars
Solution: Deflation Method
Step 2: Convert Constant dollars to Equivalent Present Worth
Cash Flows in Multiplied by Discounting Equivalent
n
Constant Dollars Factor i' = 10% Present Worth
0 -$75,000 1 -$75,000
1 30,476 (1+0.10)-1 27,706
2 32,381 (1+0.10)-2 26,761
3 28,334 (1+0.10)-3 21,288
4 23,858 (1+0.10)-4 16,295
5 45,445 (1+0.10)-5 28,218
$45,268
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 50
 Actual Dollars (An) Analysis
 Example: Equivalence Calculation when cash flows are in
actual dollars
Solution: Deflation Method
Converting actual dollars to constant dollars and then to equivalent present worth

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 51

Adjusted-discount Method

 Perform Deflation and Discounting in One Step

An
Pn 
An (1  i ) n
Step 1
(1  f ) n An

An
Pn  (1  i ) n  (1  f ) n (1  i ') 
n

(1  i '-) n Step 2  
(1  i )  (1  f )(1  i ')
An  1  i ' f  i ' f

(1  f ) n (1  i ') n
i  i ' f  i ' f
An
 n  
(1  f ) n (1  i ')  i  i  f  i f
 
_ _
i  f  i(1  f )
If inflation is 0, 𝑖 and 𝑖′ are equal…


For continuous compounding i f

i  
𝑖 = 𝑖′ + 𝑓 1 f
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 52
Actual Dollars (An) Analysis
 Example: Equivalence Calculation when cash flows are in
actual dollars i  i'  f  i' f
 0.10  0.05  ( 0.10 )(0.05)
Solution: Adjusted - Discounted Method
 15.5%

Cash Flows in Equivalent

n Multiplied By (15.5%)
Actual Dollars Present Worth
0 -$75,000 1 -$75,000
1 32,000 (1+0.155)-1 27,706
2 35,700 (1+0.155)-2 26,761
3 32,800 (1+0.155)-3 21,288
4 29,000 (1+0.155)-4 16,296
5 58,000 (1+0.155)-5 28,217
$45,268
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 53
 Actual Dollars (An) Analysis
 Example: Equivalence Calculation when cash flows are in
actual dollars
Solution: Adjusted - Discounted Method
Converting actual dollars to present worth dollars by applying the market interest rate

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 54

 Actual Dollars (An) Analysis
 Example 4.6:
Financial facts:
• Annual fuel savings in constant $ = $244,560
• Market interest rate = 12%
• Fuel escalation (inflation) rate = 5% per year
• General inflation rate = 3% per year
• Project period = 15 years
Solution:
Inflation-free interest rate:
0.12  i ' 0.03  0.03i '
i '  8.74%

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 55

 Actual Dollars (An) Analysis
 Example 4.6:
Solution:
Real growth rate (g):
1+0.05
g= - 1 = 1.94%
1+0.03

First year cash flow (A1):

A1= $244,560 (1+0.0194) = $249,304

Present equivalent fuel savings (P):

P = $249,304(P/A, 1.94%, 8.74%, 15)
= $2,275,096
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 56
 Actual Dollars (An) Analysis
 Example 4.6:
Solution:
Market interest rate (i): 12%
𝑖 = 𝑖’ + 𝑓 ҧ + 𝑖’𝑓 ҧ
= 0.0874 + 0.03 + 0.0874(0.03) = 12%

First year cash flow (A1) in actual dollars:

A1= $244,560 (1+0.05) = $256,788

Present equivalent fuel savings (P):

P = $256,788(P/A1, 5%, 12%, 15)
= $2,275,096

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 57

Mixed Dollar Analysis
 Situation: Some cash flow elements are expressed in
constant dollars and other elements in actual dollars
 What to do: Convert all cash flow elements into the same
dollar units (either in actual dollars or constant dollars)
 Analysis:
• If all cash flow elements are converted into actual dollars, use
the market interest rate (i) to find the equivalent value
• If all cash flow elements are converted into constant dollars,
use the inflation-free interest rate (i') to find the equivalent
value

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 58

Mixed Dollar Analysis
 Example 4.7:
• Parents wish to establish a college fund at a bank for their 5 year-old
child
• The college fund will earn 8% interest compounded quarterly
• Assume that the child enters college at 18, the parents estimate an
amount of $30,000 per year in terms of today's dollars will be
required to support the child education for 4 years
• College expenses are estimated to increase at a rate of 6%
• College expenses are paid at the beginning of each year
• The first deposit will be made at the end of the first quarter and will
continue until the child reaches age 17
• Determine the equal quarterly deposits the parents must make until
they send their child to college
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 59
Mixed Dollar Analysis
 Example 4.7:
Solution:
Approach:
• Convert any cash flow elements in constant dollars into actual dollars
• Then use the market interest rate to find the equivalent present value

College expenses College expenses in actual

Age
in today’s dollars dollars

18 (Freshman) $30,000 $30,000(F/P,6%,13) = $63,988

19 (Sophomore) 30,000 $30,000(F/P,6%,14) = $67,827
20 (Junior) 30,000 $30,000(F/P,6%,15) = $71,897
21 (senior) 30,000 $30,000(F/P,6%,16) = $76,211

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 60

Mixed Dollar Analysis
 Example 4.7:
Solution:
Required Quarterly Contributions to College Funds

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 61

Problem
Suppose you borrow $20,000 at an 8% annul interest over 5
years to finance your college expenses. If the general
inflation rate is 4% compounded annually, find the constant
dollar value of the 5th payment
Solution:
First, find the monthly payments in actual dollars:
A  $20,000(A / P,8%,5)  $5,009.13
Compute the inflation-free interest rate:
0.08  0.04
i'   3.85%
1.04
Convert the actual dollars into constant dollars
A '5  $5,009.13(P / F ,4%,5)  $4,117.14
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 62
Problem
Suppose you borrow $20,000 at an 8% annul interest over 5
years to finance your college expenses. If the general
inflation rate is 4% compounded annually, find the constant
dollar value of the 5th payment
Solution: End of Payment in Payment in Equivalent Equal-Payment
Year (n) Actual $ Constant $ Series in Constant $
0
1 $5,009.13 $4,816.47 $4,473.63
2 $5,009.13 $4,631.22 $4,473.63
3 $5,009.13 $4,453.09 $4,473.63
4 $5,009.13 $4,281.82 $4,473.63
5 $5,009.13 $4,117.14 $4,473.63

P  $4,816.47(P / F ,3.85%,1)   $4,117.14(P / F ,3.85%,5)  $20,000

A  $20,000(A / P ,3.85%,5)  $4,473.63
P  $4,473.63(P / A,3.85%,5)  $20,000
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 63
 Key Points
 The Consumer Price Index (CPI) is a statistical measure of
change, over time, of the prices of goods and services in major
expenditure groups—such as food, housing, apparel,
transportation, and medical care — typically purchased by
urban consumers

 Inflation is the term used to describe a decline in purchasing

power evidenced in an economic environment of rising prices

 Deflation is the opposite of inflation: It is an increase in

purchasing power evidenced by falling prices.
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 64
Key Points
 The general inflation rate is an average inflation rate based on
the CPI. An annual general inflation rate ( f ) can be calculated
using the following equation:

CPI n  CPI n 1
fn
CPI n1

 Specific, individual commodities do not always reflect the

general inflation rate in their price changes. We can calculate
an average inflation rate for a specific commodity (j) if we have
an index (i.e., a record of historical costs) for that commodity
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 65
Key Points
 Project cash flows may be stated in one of two forms
• Actual dollars (An): Dollars that reflect the inflation or deflation rate
• Constant dollars (A’n): Year 0 (or base year) dollars

 Interest rates for project evaluation may be stated in one of

two forms:
• Market interest rate (i): A rate which combines the effects of interest
and inflation; used with actual dollar analysis
• Inflation-free interest rate (i’): A rate from which the effects of
inflation have been removed; this rate is used with constant dollar
analysis
ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 66
Key Points
To calculate the present worth of actual dollars, we can use a two-
step or a one-step process:

 Deflation method (two steps):

1. Convert actual dollars by deflating with the general inflation rate ( f )
2. Calculate the PW of constant dollars by discounting at i’

 Adjusted-discount method (one step):

1. Compute the market interest rate
2. Use the market interest rate directly to find the present value

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 67

Reading Assignment
 Read Chapter 4 from the textbook

ISE 307 - Term 192 Dr. Yasser Almoghathawi, KFUPM 68