Table of Contents
Unit 1 – Fundamentals of Engineering Economics and the “Invest and Earn
Problem”
The Basic Concepts of Engineering Economics
The Cash Flow
Tips to Getting a Good Cash Flow
The Problem of Equivalence
The Rate of Return - The Basic Tool for Equivalence
The Time Value of Money
How are Rates of Return Determined?
Calculation Tool Intermission
The Mystery of the Real and Nominal Interest Rate
How do I convert Nominal to Real and vis-versa?
Calculation Tool Intermission
Using the Rate of Return in Money Calculations
Magic Numbers and Moving Money to Equivalent Amounts
Meeting F/P
Calculation Tool Intermission
Using Unit Cancelation to Pick the Correct Magic Number
FE Exam Intermission
Interest Rates and Compounding Periods
Step 1 – Get the period interest rate
Step #2 Get the number of compounding periods for the problem
Impact and use of a shorter compounding period
Bankers Tricks and Yield
Calculation Tool Intermission
Unit 1 – Fundamentals of Engineering Economics and the “Invest and Earn
Problem”
The Basic Concepts of Engineering Economics
This study of Engineering Economics deals with determining which projects or investments are
economically desirable and which are not. As Engineers an important part of anything we design is that
it represent an economically attractive investment for those who will “put up” the money needed to
build it. In a money oriented society the difference between a design that will be built and serve society
and a “paper decoration” will come down to whether the project makes money or achieves a required
goal at the least cost.
This class shows us how we can measure and quantify whether a project investment achieves that goal
of being economically attractive. Fortunately, many of the basic principles of Engineering Economics are
concepts that we have understood since we were small children. Most problems in Engineering
Economics are done in the same way using the same principles. Once those principles and patterns are
grasped the rest of the topic becomes details.
The two most important questions that we have to answer about money are (1)- How much do I get?
And (2)- When do I get it? One scarcely has to explain why we care how much money we get. We all
understand that when it comes to money we want to get as much of it as possible. When do I get it?
Might take a little more thought – although a lot of us can remember being little kids wanting a piece of
candy and the only thing we really understood was that we wanted it “right now”. Consider yourself
wanting to take a spring break vacation with your friends, but then Teacher A offers your $3,000 to do
student work for them over spring break. All of a sudden you start to think that maybe that spring break
vacation is not as important as you thought (if $3,000 doesn’t “do the trick” for you – there is probably
an amount that will – most of us do have our price). A few minutes later Teacher B offers you $3,000 to
do student work for them over spring break. The difference is that Teacher A will pay you $3,000 at the
end of spring break. Teacher B will pay you $3,000 when you graduate. Are both offers equally
attractive to you? You probably are realizing that getting $3,000 right away has a “lot more pull on you”
than $3,000 at some time in the distant future. You are realizing that the value that the money has to
you is not just how much you get, but how soon you get it. That is why in Engineering Economics we
want to know (1)- How Much Do I Get? And (2)- When Do I Get It?
The Cash Flow
This leads us to the first step that we perform in doing Engineering Economics problems – that being to
create a “Cash Flow”. Most real world investment or design programs begin as a story. I do such and
such which costs me X amount of money. My project then does wonderful things that “pay off” and I
make Y amount of money over the time my project is it service. The first thing we do to solve and
Engineering Economics problem is to take that story and write down a list of how much money we make
or spend and when we make it or spend it. This list is called a “Cash Flow”.
Let us illustrate the idea of making a cash flow with an example. The City of Carbondale decides that
rather than raise taxes to pay for needed upgrades and maintenance on its water system that it is better
“privatize” the water system and sell it to private investors. Enter Soak it To You Inc. – a company that
invests in and operates municipal water systems. Soak it to You Inc. pays Carbondale $8,000,000 to buy
the municipal water system. Carbondale sold the system because it was in need of some very costly
maintenance so during the first year of ownership Soak it to You Inc. spends $18,000,000 more than
they make. During the second year they still spend $6,000,000 more than they make. Finally, Soak it to
You Inc. has the water system “back in shape”. At this point they begin making more money than they
spend - $4,000,000 every year. This money making story continues for 10 years. At this point
Carbondale has an election for a new Mayor. Sammy Socialist runs for Mayor promising to buy back the
water system. Sammy points out that Soak it to You Inc. is “ripping the citizens off” and charging them
far more for water than it costs to provide the water and maintain the system. A community owned
water system provides services to its citizens first rather than thinks about dipping into its customers
pockets in order to give money to its share holders (investors). Sammy is elected Mayor and forces Soak
it to You Inc. to sell back the water system for $21,000,000. Suppose we would like to know whether
buying Carbondale’s water system was a good investment for Soak it to You Inc.
Like most real world situations the Carbondale water system investment begins with a story. Our first
step in solving the economics problem is to create the “Cash Flow”. We need to look at the story
problem and pick out how much money moves and when the money moved.
The first money moving event was that initial instant in time when Soak it to You wrote out a check to
Carbondale for $8,000,000 to buy the water system. This is the first point at which money moves. It is
called “time 0” and represents that instant in time when our “business deal” was started.
Time
Money
0
-$8,000,000 ($8,000,000 dollars moved out of Soak it to You’s bank account)
Reading on in our story we find the next movement of money was that during the first year Soak it to
You put another $18,000,000 dollars into fixing up and improving the water system.
Time
0
1
Money
-$8,000,000
-$18,000,000
We note an approximation that we made when we listed the next amount of money to move. Soak it to
You Inc. in fact spent $18,000,000 over a period of one year. They did not just wait till December 31rst
and then spend $18,000,000 on one day. Most of the time when we are planning or evaluating an
investment or project that goes for 40 years we are not going to decide what hour of what day the
secretary is going to buy paper clips. We can estimate approximately how much will be spent or earned
and approximately when, but we usually don’t know, or wouldn’t spend the time and effort to get quite
that detailed. When we build cash flows we frequently take money that will spent over a period of time
and put all that money at one listed time in the cash flow that is “close enough”. In this case this means
that while time 0 is a single instant in time when a check is written, time 1 is the accumulation of
expenses over a period of one year. (I could have made the cash flow out every day or every week or
every month, or as I did every year depending on how much detail is really important to the problem).
Reading on in the story we find that in year 2 Soak it to you spends $6,000,000 and then in years 3, 4, 5,
6, 7, 8, 9, 10, 11, and 12 they earn $4,000,000.
Time
0
1
2
3
4
5
6
7
8
9
10
11
12
Money
-$8,000,000
-$18,000,000
-$6,000,000
$4,000,000 (Note now the money is adding to Soak it to You’s bank account).
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
Our story concludes when Sammy Socialist buy’s back the water system for $21,000,000 the next year.
Time
0
1
2
3
4
5
6
Money
-$8,000,000
-$18,000,000
-$6,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
7
8
9
10
11
12
13
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$4,000,000
$21,000,000
We have just taken a story of an investment or project and turned it into a “Cash Flow”. I can also
represent a cash flow graphically.
$21,000,000
$4,000,000 per year
0
1
2
3 …………………………………………………….12
$8,000,000
13
$6,000,000
$18,000,000
Figure 1-1 Graphical Representation of a Cash flow
Both the money list and the money graphic are valid ways of representing a cash flow. A great many
engineering students are “visual learners” and the graphical cash flow may be more meaningful to
people with that type of “learning style”.
Note that a cash flow directly answers the two most important questions in Engineering Economics (1)How Much Do I Get? And (2) – When Do I Get It?
Tips to Getting a Good Cash Flow
The cash flow is the basic starting information you will use to solve an Engineering Economics problem.
Obviously if the cash flow does not represent the problem you are really trying to solve you will likely
get a wrong answer. With costly engineering projects a wrong answer can represent millions of dollars
or worse.
Watch for the following in getting a good cash flow –
(1)- Make sure you do not confuse your pluses and minuses
Whether a number in a cash flow is a positive or a negative number can greatly change the answer. Use
this rule – money coming into my pocket is a positive event – money going out of my pocket is a
negative event. Most of us already think of getting money as positive and having it disappear as
negative. When you get money coming in – give it a positive sign. When you have to give up money and
put it into the project or investment – give it a negative sign.
(2)- Pick the viewpoint of your investor and stay with it
Most engineering projects have lots of people working with them. Some may be contractors or
suppliers, perhaps a project management company. All these people are trading money around. What
may be money going out of one person’s pocket is money coming into another person’s pocket. To
avoid getting numbers with the wrong sign or numbers that don’t even pertain to the problem take the
view point of the investor who is “putting up” the money for the project. Track what happens to your
investor’s wallet (or purse). When your investor takes money out of his/her pocket and puts it into the
project record it with a negative sign. If someone other than the investor takes money out of their
pocket and gives it to someone else – don’t record it as all – it is not an investment your investor made.
If the project causes money to flow back into your investor’s pocket record the money with a positive
sign. If someone other than your investor gets money put into their pocket don’t record it at all – it is
not something your investor is getting as a result of the project.
The Problem of Equivalence
Having developed the Cash Flow for the problem or situation to be evaluated we now have the answers
to the two most important questions (1)- How Much Do I Get? and (2)-When Do I Get It? we are now in
a position to use the information.
When we want to know "How Much Did I Get"? our method of choice is usually to count up or add up
the total. Here the issue of "Equivalence" enters into the problem. Consider the Example of Pedro
Plush and Freddy Flush. Pedro has 23 Pesos and 2 dollars. If we try to add up Pedro's money we might
conclude that Pedro has 25 monies (we probably already sense something is going very wrong). Next
Freddy Flush presents the contents of his pocket - 2 Pesos and 23 dollars. Freddy also has 25
monies. Both Pedro and Freddy must have equal amounts of money (Ok - none of us believe that). The
problem that we have encountered is that you can only add up units of the same value. Where the units
to be added have different value we must first convert the units to a common base. In the case of our
Pesos and Dollars this is accomplished by multiplying either the Dollars or the Pesos by an "Exchange
Rate" so that either the value of our Dollars are expressed in Pesos or the values of our Pesos are
expressed in Dollars. Only then can we add to get a meaningful measure of how much wealth either
Pedro or Freddy poses. The principle then - Units of Different Value can be added only after they are
converted to a common basis.
Now let us return to the story of Soak it To You Inc - the company that bought the water system from
the City of Carbondale. We would like to know whether Soak it To You got a favorable deal by buying
the Carbondale water system. We remember our long held almost instinctive definition - in a good deal
you get more out of the project or investment than you put into it. Pulling out our cash flow we begin to
add. Soak it to you put the following investments into the water system
$8,000,000 to buy it
$16,000,000 in improvements in year 1
$6,000,000 in improvements in year 2
Soak it To You Inc put $32,000,000 into the water system.
Now lets see what they got out of the deal
$4,000,000 per year for 10 years
$21,000,000 to sell the system back to the city
Soak it To You Inc got $61,000,000 out of the deal. Lets see $32,000,000 in and $61,000,000 out sounds pretty good - or does it?
Our minds go back to the story of Pedro Plush and Freddy Flush who both had 25 monies. Then we
remember that first principle Units of Different Value can be added only after they are converted to a
common basis. No - it can't be! - after all these were all dollar bills. Then we remember the story of
the $3,000 to work over spring break. It made a difference to us when we got the money - $3,000 right
now had a lot more buying power with us than $3,000 that we would never see for 5 years! Yipes! could
it be that we just added units of unequal value again! Yes - in as real a way as blindly adding Pesos and
Dollars to get monies adding money from different points in time is adding units of unequal value. A
dollar is equal to a dollar only if I hold both in my "hot little hands" at the same time. The two most
important questions in Engineering Economics are "How Much Do I Get" and "When Do I Get It". The
answers to both questions are needed to establish "equivalence" and let us add up a total. As with the
Pesos and the Dollars there are conversion factors. By multiplying dollars at different points in time to
convert them to a common time we can get our cash flow on a basis where we can add things up.
As an incidental - when this is done to the cash flow for Soak it To You Inc buying the Carbondale water
system it turns out (at a 15% rate or return) that Soak it To You Inc put in $28,189,000 and only got out
$18,593,000. It was not a good deal and just like adding the Pesos and Dollars without converting to
equivalent value first led to a wrong conclusion about the relative wealth of Pedro and Freddy, so to
blindly adding dollars at different points in time can be expected to lead to wrong conclusions.
The Rate of Return - The Basic Tool for Equivalence
So where and how do we get these conversion factors that make money from different points in time
equal in value so we can add it up?
The Time Value of Money
Let us begin with thinking of another situation - Do you, or one of your friends work while you go to
College? You probably know someone who is doing it if you are not already doing it yourself. So why do
you do it? Answers like - "I enjoy having no social life" or "I like having to wait till the last minute to do
my homework, term paper, or study for a test and then having to pull an all nighter just to get a C" or
"There is nothing in life that brings me more joy than cleaning garbage cans and tables" probably are
suspect answers. I would venture to guess that most working College students are willing to put off
gratification and make sacrifices in order to get paid. Taking a lesson from life - one of the most
successful ways of getting people to put off gratification is to pay them for the sacrifice.
If I wait 5 years to get $3,000 for work I perform I'm actually going to consider part of that as pay for
waiting 5 years. The reason $3,000 now is worth more than $3,000 in 5 years is because if I put off my
gratification money for 5 years somebody better be paying me for the sacrifice. $3,000 in 5 years is not
all money for work I perform during spring break. Part of the money is for waiting the 5 years. Thus, I
do not get $3,000 just for working through spring break if I have to wait 5 years. Now all I need is the
ratio for how much of the money I get in 5 years is pay for waiting.
How are Rates of Return Determined?
Since I get paid for waiting for my money does that mean I can name my price? We all have a nasty
feeling we know the answer to that one. Suppose I go to McDonalds and offer to work as a food service
worker. I want $50 per hour, nights, weekends, and holidays off, 40 hours a week, and my shifts in 8
hour blocks. Do you think I will get the job? Of course most of you are speculating that the answer is no
because my rate and terms are not competitive with what they could get someone else to do the job
for. Not surprisingly, the rate that people are paid for waiting for their money is also set by the market.
The market pays people for delays in getting money as a percentage of the money they are waiting
for. By U.S. and international standard this is expressed as a percentage per year. If, for example, the
percentage is 5%, then I will get paid $5 for waiting a year to get $100. ($5 is 5% of $100 dollars). The
percentage rate I am paid for "renting" or waiting for my money is called "the Rate of Return".
The market uses several things in determining the "Rate of Return" on money. We can get an image of
what these things are and why they are considered from another story. It turns out that I am rather
fond of Dairy Queen "Blizzards". Suppose I can buy a Dairy Queen "Blizzard" for $3. Student A asks to
borrow $30 from me for 5 years. I realize that this is equal to the gratification from 10 Dairy Queen
"Blizzards", but being such a nice person I agree. Being such an honest person Student A returns in 5
years and pays me my $30. After suffering "Blizzard Depravation" for so long I rush down to Dairy
Queen to buy a round of "Blizzards" but much to my frustration the price of "Blizzards" is now $3.50
each . I got my money back but it no longer has the power to buy me 10 Dairy Queen "Blizzards". I am
going to be in a bad mood over the situation the rest of the day! The problem I have identified is a
common one - in general the buying power of a unit of money declines with time (think of your own
experience buying gas for your car). This is called "Inflation". What must be done to bring me out of my
"Blizzard" loss bad mood? Well having people pay me back enough to keep up with inflation would be a
nice start - get me $35 at the end of 5 years so I can still buy 10 Dairy Queen "Blizzards". This is one of
the things the market considers in determining a rate of return. The market considers how fast money
is loosing value due to inflation and demands a rate of return high enough to keep pace with inflation.
Well, hearing the story of my lending "Blizzard" money to students my entire class of 100 students
decides one semester they would all like to borrow $35 each for 5 years. Being fresh back from my
"Blizzard" loss experience due to inflation I tell them all I will need $40 back in 5 years to cover
inflation. The students all agree. There is just one little problem - what are the chances that all 100
students will remember their promise and come back faithfully in 5 years and pay me back $40. Ok - I
know you are honest - but what about that Bozo sitting next to you? There is a reason for that old
Chinese proverb about "a bird in hand is worth two in the bush". Even good judges of projects and
investments are wrong part of the time. If I want to make sure I get enough money back to pay for 1000
blizzards, just keeping up with inflation is not going to solve my problem - I need to collect enough extra
to account for the reality that some percentage of my lovely students are going to "stiff" me. If I had
enough statistics to look at (like the market has everything to look at) I could estimate what size "risk
premium" I need to make up for the inevitable failure of some students to return and keep their
promise. Government projects that are backed by the Government's power to "take it out of the hide"
of its citizens with taxes often have low risk premiums around 1 to 4%. Manufacturing which may have
fewer guarantees often has about a 6 to 9% risk premium. Mining where Mother Nature can do all sorts
of things to a well planned project may have a 12% to 15% risk premium. One looks at the risk premium
that similar projects have to determine the risk premium for a new project or investment being
considered.
As the saga of the Diary Queen Blizzard continues I lend 100 people $35 dollars with an agreement that
they will pay me back $50 in 5 years. The arrangement allows for the price of my Diary Queen Blizzard
to go to $4 each and for 20% of my students to fail to pay me back, however, I still have to wait 5 years
for my Blizzard bonanza. We remember the question about "why do you work while you go through
college" and the answer - because someone pays me to sacrifice having control of my own time now by
providing me money for things I need. The fact that I have now lent money with protection against
inflation and the possibility that some people will fail to pay me back still does not cover the fact that I
have put off 1000 Diary Queen Blizzards for 5 years. Why would I do that if I was not being paid for
putting off my gratification? This brings up the 3rd factor in getting a rate of return - you get people to
put off gratification and rent you their money by paying them to do so. It turns out that the market rate
for just waiting to get your money has been about the same from the time someone first learned to spell
Capitalism to the present day. It is about 1 to 2% per year. The market calls this the "Safe Rate of
Return". It is a rate that is paid just for waiting for your money since the issues of inflation and risk have
been dealt with elsewhere.
Most of us have seen or even been involved personally in the 4th thing that is used to determine a rate
of return. Two gas stations across the street from one another. One gas station lowers it price a few
cents. What does the other gas station do? Target advertises a good deal on something? What does
Walmart do? General Motors does advertising for how cool and desirable their cars are. What does
Toyota do? Have you ever been involved in trying to do something to your resume that will make it
stand out or be different from everyone elses? Have you ever spent extra time or money picking out
cloths that will make you stand out? Of course we all know that everyone spends a little something to
try to compete. When it comes to the market picking rates of return that little bit is about 0.1% extra
that does something to compete.
Great - now we know what the market considers in determining a rate of return for an investment or
project - what do I do with these things to get the rate of return the market really will pay for the use of
an investors money? Of course our first impulse when we have 4 components of an interest rate is to
add them all up. That would be wrong. The problem is each of the factors effects not only the original
money but every other factor. If someone fails to come back to pay me for the "Blizzard Money" I lent
them 5 years ago are they just going to forget to pay me back the $35 they borrowed or will they also
forget to pay me the extra $5 for inflation, my "risk premium", may premium for just waiting for my
money, and my competition premium? I bet they will forget to pay me back everything! Will only the
original $35 I lent out be subject to inflation or will my risk premium and safe rate of return money for
waiting also be effected by inflation? If the price of something goes up - can I still get the old price by
paying the cashier with money from an "inflation proof wallet"? Ya - in my dreams maybe! Thus every
factor in a rate of return effects not only the original money, but the money paid to satisfy every other
factor.
It turns out the way to get every premium in the rate of return to affect every other premium is to chain
multiply. We would proceed as follows –
Suppose the rate of inflation is 3%
Suppose the safe rate of return (money just for waiting) is 1.5%
Suppose the risk premium is 7%
And suppose the “motivation premium” to “set us apart” is 0.1%
Before moving on we have to “pick up” one more “trick of the trade”. When we talk or write about
rates of returns or premiums we write or talk about percentages. When we pull out our calculator to do
the math we treat the rate of return or the premium as a decimal fraction.
Thus a 3% rate of inflation will be treated as 0.03 when doing mathematical calculations
A 1.5% safe rate of return will be treated as 0.015 in doing math.
A 7% risk premium will be treated as 0.07 in math
Finally the 0.1% motivation premium becomes 0.001 when doing math.
Now we set up the chain multiplication
(1.03)*(1.015)*(1.07)*(1.001) = (1.rate of return)
Most of us can see the 0.03 for inflation, the 0.015 for safe rate, the 0.07 for risk, and the 0.001 for
motivation in the above equation. That leaves only the question of “Why did we add 1 to everything”?
Lets suppose that I borrow $100 dollars from you and promise to pay you $10 of interest at the end of
the week. Getting 10 dollars of interest for a one week loan probably sounds good to you. (If not – its
bound to sound good to one of your class-mates).
The end of the week comes and I come to you and give you a $10 bill of interest money and thank you
for the loan before walking away. Are you happy? Is something wrong? After all I promised you $10 of
interest and $10 of interest is exactly what you got. I suspect, however, that you are very unhappy
about the fact that I never repaid you your $100. To make the deal work you need your original money
back and interest. We know where the interest factors are in the above equation. Now we ask – what
happens when you multiply any number by 1? You get the number back again – right? So what is that 1
doing in the above equation? That’s right – it guarantees that you get your original money back along
with the interest. Now we have our Rate of Return.
Calculation Tool Intermission
One of the tools we have available for this class is a simple “spreadsheet” written in Microsoft Excel
(part of the Microsoft Office suite). It is called “Class Assistant”. The spreadsheet is capable of
performing most of the common calculations or number crunching operations needed in this class. The
formulas being used can all be read in their respective cells so there is no issue with mysterious “black
boxes” or prohibitions against “decompiling” software.
One of the first functions in this spreadsheet is calculation of an interest rate
Interest Rate Components
Enter Values as Percentages but do not use the % key
Enter on an Annual Rate Basis
Safe Rate
1.5 (usually between 1 and 2)
Inflation
3.5
Risk
9
Motivation
0.1 (usually about 0.1)
Real Interest Rate
Nominal Interest Rate
in % in decimal
10.745635 0.107456
14.62173223 0.146217
Figure 1-2 The Interest Rate Component area of the Class Assistant spreadsheet
This view is a good place to introduce a few “conventions” in Class Assistant. Yellow cells are cells where
you input your values. Red cells with bold letters are where you find “answers”. These cells are also
places where you find formulas. Class Assistant does not have protected or hidden cells. The result is
that nothing (other than your desire to get right answers) stops you from changing formulas. Since you
can change the answers Class Assistant gives you obviously want to “keep your editing fingers off the
red cells” unless you know exactly what you are doing. Bold headings give a summary about the portion
of the spreadsheet below does – in this case compute and interest rate from its component factors.
Blue wording gives you information about what format to use for entering your information in the
yellow cells. Black lettering identifies what a particular cell is.
If we look at the yellow cells we see the four components of an interest rate that we have just discussed.
You simply enter the interest rate components as a percentage. Some of the cells even have tips to the
sides on what usual values would be.
Looking at the red cells we see the interest rate given. One column gives the interest rate as a
percentage (the format of choice for oral and written reporting) and in decimal form (the format of
choice for plugging into a calculation). There is nothing magic about the red cells. A look at the
formulas will show they are doing exactly what was explained above. So why use the spreadsheet? It
can crunch numbers faster and diminish your ability to get wrong answers from typing formulas into
your calculator incorrectly.
The Mystery of the Real and Nominal Interest Rate
We noted in looking at our Class Assistant spreadsheet that the sheet reports both something
called a “nominal” and something called a “real” interest rate. Depending on how snoopy you got with
the spreadsheet you may already know that the “nominal” interest rate corresponds to the formula we
already discussed and that when the spreadsheet computes the “real” interest rate it leaves out the
inflation component.
We measure two kinds of interest rates – real and nominal. A nominal interest rate considers
only how many dollars one is dealing with, ignoring entirely the question of what they will buy. Because
of inflation what a dollar will buy today is very different than what it could have bought in the past.
Some you may have seen nostalgic advertisements for 5 cent fast food hamburgers. What does a fast
food hamburger cost today? In the 1960s the U.S. minimum wage was about $1.25 per hour. What is it
today? Does that mean that a person working for minimum wage today can buy more than a person
working for minimum wage in the 1960’s? If you know your inflation history or have just talked to
enough people struggling to “get by” or maybe been one of those people yourself, you know that the
bigger paycheck of today does not imply more buying power. A nominal interest rate deals with how
many dollars are there. It is indifferent to whether those dollars buy much. By contrast a real interest
rate deals with “constant” dollars. A dollar that continues to buy the same amount. A 1960’s constant
dollar could buy 4 gallons of gasoline. This “constant” or “real” dollar buys 4 gallons of gas today. Now
admittedly, none of us have ever seen a “constant” dollar or a “real” dollar, at least not the kind of
“real” dollar that buys 4 gallons of gasoline, but our theory people tell us such a dollar exists – at least in
our imagination or on paper.
At first talking about “constant” or “real” dollars seems up-surd because they don’t exist – or do
they? Think about what you do as an engineer building a “cash flow” for a project. Suppose you know
that you will need a bulldozer for your project and that it will cost you $100,000 to buy. What number
do you put in your cash flow for the bulldozer? Oh yes -$100,000 (minus because it is money moving
out of your pocket in order to buy the bulldozer). Suppose you know this bulldozer wears out in 7 years
and will need to be replaced. Now what number do you put down? If you are like most engineers
spending your time making wild guesses about the inflation rate at bulldozer factories is not what you
do. There is a good chance you just put down -$100,000. As you build you project cash flow you are
pricing everything in dollars of today’s value. This can even be a great check on numbers because if
things are priced out in dollars that “make sense” in today’s price environment people are much more
likely to detect errors than if all the numbers look foreign to today’s price perspective. Thus, one of the
places that “real dollars” are found is in project planning cash flows.
Now the problem – what happens if I use a “nominal” discount rate on a “real” cash flow?
Obviously since the discount rate was assuming the amount of money moving around should be
increasing when in fact it is not (the cash flow was written down in non-inflating terms) will cause
predictions of the amount of money being spent or earned to be wrong. This is a hidden but major
problem for businesses. The cost estimators and engineers produce “real” cash flows and the
accounting department says to evaluate it with a “nominal” interest rate and everyone gets the wrong
project value or cost. It is a common way businesses get things wrong and it is a dumb mistake that
people go out of the way to hide. Don’t let this happen to you! Use a “real” non-inflating discount rate
to analyze a “real” cash flow. Use a “nominal” interest rate on a nominal cash flow. Never mix and
match!
How do I convert Nominal to Real and vis-versa?
If you know from the start whether you are trying to get a real or a nominal interest rate and
you are working with the four components of an interest rate it is easy. Take “safe rate”, “risk
premium” and “motivation premium” and chain multiply to get a real interest rate. If you want a
nominal interest rate chain multiply all 4 components of an interest rate. This is easy enough and as was
shown above Class Assistant automatically calculates both real and nominal rates. The trick of course is
that you have to know which one to use on your cash flow – ie you need to know whether the cash flow
has built in an assumption of inflation.
The more common problem is that the engineers and cost estimators produce a real cash flow
and then go to the accounting department to ask what interest rate to use. The accounting department
answers this question by looking at markets (where people are constantly “pricing in” an expectation
that money is losing buying power due to inflation). Accounting then delivers a “nominal” interest rate
and disaster is set up unless you know to ask the second question. The second question is “What
inflation rate did you estimate”? Given that you multiplied the inflation rate by the other three
components to get a “nominal” or inflating interest rate, how do you think you would take the inflation
out? Of course – you divide. The calculation would look like this. Suppose you have a 10% nominal
interest rate and a 3% inflation rate.
((1.1)/(1.03)-1)*100 = 6.8% real interest rate.
I now have the interest rate to use on a real cash flow.
Although not nearly as common a situation one might have a “real” interest rate and then
recognize by systematically rising prices and costs in a cash flow that the cash flow is nominal. In this
case you ask “what inflation rate was assumed in the cash flow”. To adjust for this you just multiply the
inflation rate into your real interest rate. Suppose that you have a 6.8% real interest rate and you come
upon a cash flow that included 3% inflation. I can get a nominal interest rate thus
((1.068)(1.03)-1)*100 = 10% nominal interest rate.
Calculation Tool Intermission
Class Assistant has a spreadsheet area designed for converting interest rates between real and nominal.
Adjust Between Real and Nominal Rates
Enter Interest Rate in Percent but do not use the % key during data entry
Enter on an Annual Rate Basis
Real Interest Rate
Nominal Interest Rate
Rate of Inflation
7
11
3.5
(in %)
10.745 Corresponding nominal int rate
7.2463768 Corresponding real interest rate
Figure 1-3 Adjust Between Real and Nominal Interest Rate area of Class Assistant spreadsheet
The key number is the inflation rate put in the bottom yellow cell in units of percent but without writing
the percent symbol. Above you may place a Real interest rate. If you look at the bold number in the red
box right across from it you see the nominal interest rate where inflation has been multiplied into the
real interest rate. Below this you can put a nominal interest rate and the bold number in the adjacent
red box is the real interest rate with the inflation rate taken out of it. A look at the formulas in the red
cells will show that the spreadsheet is doing exactly the same calculation that was explained above.
Using the Rate of Return in Money Calculations
At this point we know how to get a rate of return – but we do not know how to use that rate of return to
convert money at one point in time to an equivalent amount of money at another point in time. Lets
consider a simple case. We know that to have the equivalent of $500 today you would need more than
that in the future. The rate of return tells us how much more money it would take. Rates of return are
measured as an annual percentage – for example 5% would mean $5 of interest on each $100 each year.
Suppose you put $100 in a bank at 5% interest. In one year you would have $100, plus $5 of interest or
$105.
Now lets suppose you leave the money in the bank for another year. This time you have $105 in the
bank on which to get 5% interest. Setting this up mathematically and remembering that money in
percentages get converted to decimal form before doing math we find
$105 * (1.05) = $110.25
We now note an interesting feature – as time goes on we end up paying interest on interest (5% of our
original $100 is just $5, but we got $5.25 this time because we got interest on last periods interest
money). We say our interest is “compounding” (ie. Collecting interest on interest). We let our story go
on. What if the money is left in the bank another year?
$110.25 * (1.05) = $115.76
What about another year?
$115.76*(1.05) = $121.55
What if the story goes on for 35 years? Oh wait a minute here! – multiplying numbers by 1.05 is not
hard when you have a calculator, but at some point it gets boring and tedious and asking for 35 years of
compounding interest gets us to boring and tedious. There must be a better way.
There must be a god because fortunately there is a better way. Notice what we did to get our 4 year
accumulation of compounding interest
$100 * (1.05)*(1.05)*(1.05)*(1.05) = $121.55
We could also write this out another way.
$100 * (1.05)4 = $121.55
Could the pattern really be that easy? (1+i)n where i is the interest rate expressed in decimal form and n
is the number of times the interest compounds? The answer is yes! Asking for a proof? – Go read
another book – we don’t do proofs here. Well this sets us up to get that 35 year interest compounding
answer real quick.
(1.05)35 = 5.516
$100 * 5.516 = $551.60
One of the things we can note is that we just used an interest based conversion factor to take money
right now and convert it to an equivalent amount of money in 35 years. The conversion factor we just
used has a special name and symbol - F/Pi,n. If we multiply an amount of money we have right now by
the appropriate value of F/P it will be converted to an equivalent amount of money that may not be
available for many years. F/P is a function of two parameters. What is the interest rate i, and how many
times will that interest compound n. The idea that we can convert money at one point in time to an
equivalent amount of money at another point in time has been realized.
Magic Numbers and Moving Money to Equivalent Amounts
Meeting F/P
The idea of having a conversion factor that puts money of different values to a common basis is
common with currencies of different countries but when those conversion factors account for
differences in value depending on when you get the money its kind of magic. In this text “magic
numbers” refers to what many call “discount factors” that convert money at one point in time to an
equivalent amount of money at another time. We have just met the first of those “magic numbers” –
F/P. F/P has the ability to move money from the present time to an equivalent and greater amount of
money in the future. We might say F/P has the slogan – “My name if F over P! Cash moves to the future
with me!” F/P has a formula (1+i)n.
Calculation Tool Intermission
Class Assistant can be used to compute the value of F/P. The spreadsheet has many different areas on
the same sheet that do different calculations.
Class Assistant Spreadsheet
The class assistant is a series of simple formulas and calculations that tend to be useful on homeworks and tests
The color coding of the cells indicates where input is be be received from you and answers provided
Yellow cells require you to enter input
Green cells are just calculations - they are part of the spreadsheet design and should not be changed
Red cells are the answer outputs. The cells usually contain formulas that are part of the spreadsheet design and should not be changed
Interest Rate Components
Adjust Between Real and Nominal Rates
Enter Values as Percentages but do not use the % key
Enter on an Annual Rate Basis
Safe Rate
1.5 (usually between 1 and 2)
Inflation
3.5
Risk
9
Motivation
0.1 (usually about 0.1)
Enter Interest Rate in Percent but do not use the % key during data entry
Enter on an Annual Rate Basis
Real Interest Rate
Nominal Interest Rate
Real Interest Rate
Nominal Interest Rate
Rate of Inflation
7
11
3.5
(in %)
10.745 Corresponding nominal int rate
7.2463768 Corresponding real interest rate
in % in decimal
10.745635 0.107456
14.62173223 0.146217
Period Interest Rate Adjustment
Enter Annual Interest Rate as a Percentage
Do not use the % key during entry
Annual Int Rate
15
12 Enter number of compounding periods/year
(Would be 12 for months 365 for daily)
in % in decimal
Period Int Rate
1.25
0.0125
Period Interest Rate to an Annual Interest Rate
Enter Your Period Interest Rate as a Percentage
Do not use the % key during entry
Period Interest Rate
1.25
12 Enter number of compounding periods/year
Annual Interest Rate
15
Effective Interest Rate 16.07545
Magic # Calculator
Note - It is recommended that you go to the Magic # Calculator Tab for advice on how to use this feature
Enter Annual Interest Rate in %
Do not use the % key during data entry
Annual Int Rate
12
1 Enter the number of compounding periods/year
in % in decimal
Period Int Rate
12
0.12
(Please note - you must enter the number of compounding periods that money
Enter # Compouning Periods to Move Cash (value of n)
33 is to be moved either forward or back for the F/P and P/F numbers to work)
The value should be an interger
(The value you enter here affects F/P and P/F but not P/A, A/P, F/A, A/F)
F/P 42.09153347 (used to move one cash flow element n compounging period into the future)
P/F
0.023757747 (used to move one cash flow element n compounging periods back)
Enter # of payments (or repeating earnings) in the annuity
The value should be an interger
32 (Please note - You must enter the number of payments to use the P/A
A/P, F/A, or A/F values below. This value has no effect on P/F or F/P above)
P/A
8.111594362 (used to convert an annuity to a single sum of money one compounding period before first payment)
A/P
0.123280326 (used to convert a single sum of money into a series of n payments starting one compounding period in the future)
Figure 1-4 General Overview of Class Assistant spreadsheet showing different calculation areas
We have already met the Interest Rate Components area and the Adjust Between Real and Nominal
Rates area. To calculate “Magic Numbers” or discount factors such as F/P we go to the Magic #
Calculator area.
Magic # Calculator
Note - It is recommended that you go to the Magic # Calculator Tab for advice o
Enter Annual Interest Rate in %
Do not use the % key during data entry
Annual Int Rate
12
1 Enter the number of compounding periods/year
in % in decimal
Period Int Rate
12
0.12
Enter # Compouning Periods to Move Cash (value of n)
33
The value should be an interger
F/P 42.09153347 (used to move one cash flow element n compounging period into the future)
Figure 1-5 The F/P calculation area in Class Assistant
The key areas of the spreadsheet for getting F/P are first off the yellow cell where one puts the annual
interest rate. The example above contains the number 12. Note that although you are told to put
numbers in as percent units you are not to enter the % symbol in the yellow box. The yellow box just
below asking for the number of compounding periods in a year is 1 for right now. The next key number
is the number of times that interest is allowed to compound – the value of n. In the example above the
number 33 has been entered. The resulting value of F/P is found in bold in the red box below beside the
entry that says F/P. If one takes a present amount of money and multiplies it by 42.0915, the result will
be converting that money to an equivalent sum in 33 years after growing at an interest rate of 12% per
year.
Using Unit Cancelation to Pick the Correct Magic Number
Of course moving money from the present time to a future time is not always the way we want to move
money before counting. There are 6 basic “magic numbers” like F/P that we can use to move money at
one point in time to another. This of course leads to the question of how do I know which one to use?
Here the unit cancelation trick used often in engineering comes into play. Just like a common number in
the numerator can “cancel” the same number in the denominator, so to a unit in the numerator can
cancel the same unit in the denominator. See how it works with magic numbers.
𝐹𝑢𝑡𝑢𝑟𝑒
*
𝑃𝑟𝑒𝑠𝑒𝑛𝑡
Present
Present cancels Present leaving us with future. F/P cancels present and leaves us with future. It will
have the effect of starting with money in the present and converting it to money in the future.
Watching for unit cancelation will help us to use the correct magic number in the future when we have
more than one available. For right now we only know F/P.
FE Exam Intermission
As part of the path to becoming a Licensed Professional Engineer, engineering students within a year of
graduation from an accredited Engineering School have the chance to take the Fundamentals of
Engineering Exam, which as its name implies tests whether students have mastered the basic skills and
knowledge to become a Professional Engineer. About 8 to 10% of that test covers the subject of
engineering economics. As a result – this text will often stop right in the middle of a concept to explore
ways in which historical and similar questions on the FE exam may test your mastery of a concept.
On the subject of Engineering Economics, one of the most often tested concepts is whether you
understand equivalence – the idea that a certain amount of money at one time is equivalent to a
different amount of money at some other time. Many problems that test this concept have the form
A*B=C, where A is an amount of money found in a story problem, B is one of the magic numbers, and C
is the correct answer to the question.
An example question – “If $500 is put in a bank for 10 years at 4% interest, how much money most
nearly will be in the bank account at the end of 10 years?”
Looking for the pattern - $500 is the amount of money A
The money grows for 10 years at 4% interest (should suggest a magic number)
What amount of money will there be in the future C, the answer.
What magic number takes money in the present and moves it to the future?
𝐹𝑢𝑡𝑢𝑟𝑒
*
𝑃𝑟𝑒𝑠𝑒𝑛𝑡
Present = Future
We want F/P. The interest rate i is 4% and the number of compounding periods n is 10.
Plugging into the formula (1+0.04)10 = 1.4802
Setting up our A ($500) * B (1.4802) = C ($740.12)
The FE exam which is a four answer multiple choice will likely have $740 as one of the answers. We pick
it and are one step closer to passing our FE exam. Our goal is to get to 70% or better. The FE exam is a
pass-fail exam.
Lest one go into early phases of panic about whether you can ever be an engineer, the FE exam does not
expect you to memorize an endless array of formulas. When you take the exam you are given a book of
formulas. The engineering economics portion of the book includes the formula for F/P along with all the
other magic numbers we will eventually learn.
The exam book also includes something called an interest table. This is a quick look up table where we
can find the value of F/P (or our other magic numbers) listed by interest rate and number of
compounding periods.
Lets examine how we can use such a table.
Figure 1-6 A typical Compound Interest Table showing the interest rate for which the table was
computed
The first thing to look for is the interest rate for the table which is usually located in one or both of the
top two corners of the table depending on who published the table. In the example above we see that
the interest rate is ¾%. The table was derived by using the formula for each of the “magic numbers”
using the value of at the top of the table for the interest rate and then calculating the resulting “magic
numbers” for a wide array of numbers of compounding periods. The answers you get from an “interest
table” will thus be the same as you would get using the formula, except that you do not take the time or
potential for error that might result from exhaustively punching everything into a calculator. (The idea
of using a programmable calculator for the FE exam won’t work since the FE exam specifies the types of
calculators that can be used and programmables are on the “do not use list”.
The second thing one looks for on the interest rate is which “magic number” you are looking up. At
present we only know F/P. Look across the top of the table for the “correct” magic number.
Figure 1-7 A typical Compound Interest Table showing where to find the column heading that identify
which “Magic Number” or discount factor is being calculated.
Next we look down the left side of the table at the value of n. Suppose for example that we wanted the
value of F/P at 4% interest and 16 years. We first check in the top corners of the table to make sure we
see 4%. For the view above we definitely see 4%. Then we look across the top of the table for the
column that contains F/P. In the view above it is the first column.
Figure 1-8 Reading down the side of a typical Compound Interest Table and then reading over to get the
value of the desired “Magic Number” or discount factor.
Line up with the value of n. (Remember – n is the number of compounding periods). Follow the pretty
red arrow on the view above down to the number 16. I have 16 compounding periods. Now read over
to the column containing the magic number you are interested in – in this case F/P. We note the value
is 1.873. If I have $1000 today and I put it in an account that pays 4% interest and I leave it there for 16
years the amount of money will become
$1000 * 1.873 = $1,873
This means that when you are doing your FE exam you can get the required “magic numbers” (discount
factors) either by using a formula in your calculator or doing a table look-up. Most of the time a table
look-up will tend to give you a slight time advantage, though individuals differ in the dexterity of their
data entry fingers and the speed of the flipping fingers and eye-balls. One might ask “What about class
assistant”. You are not allowed to use a pre-formula loaded spreadsheet although the FE exam is being
moved from a paper test format to a computer based test.
Interest Rates and Compounding Periods
We have just met interest rates that compound once a year. Indeed it is a government and international
standard that interest rates are reported for a one year period. If someone says “the interest rate is
4.5%” and does not say anything more one can reasonably conclude the interest rate is 4.5% annually
(ok loan sharks and shady characters may not follow the law – but this book is not a shady character).
But does the standard time period for interest reporting have to mean that one year is a mandatory
standard compounding period? No. In fact a great many financial arrangements have a compounding
period that is something other than a year – almost always a shorter compounding period. How do we
deal with compounding periods shorter than one year?
Step 1 – Get the period interest rate
How often do interest rates compound in one year? Common answers for interest compounding are
monthly and daily. There are 12 months in a year and 365 days in a year. Take the yearly interest rate
(which should be reported since it is the legal standard) and divide it by the number of compounding
periods in one year.
Thus 4% interest compounded monthly has a “period interest rate” of
4%/12 = 0.3333%
The same 4% interest rate with daily compounding has a “period interest rate” of
4%/365 = 0.01096%
One could similarly deal with a semi-annual interest rate – semi-annual means twice a year
4%/2 = 2%
A quarterly interest rate would give 4 quarters to a year so the period interest rate is
4%/4 = 1%
Step #2 Get the number of compounding periods for the problem
Just as interest rates are reported on an annual basis, the amount of time that money gathers interest is
also often reported on an annual basis. It would be common to say “the money is in the bank for 5
years” or “the loan is for a period of 2.5 years”. Our first impulse would be to grab this number and use
it for the value of n in the F/P formula, but n is the number of compounding periods. A 5 year loan with
monthly compounding is 60 compounding periods. If the compounding period is daily a 5 year loan is
5*365 = 1825 compounding periods – or for our purists who dot every i and cross every t and know
there has to be a leap year in there 1826.
Impact and use of a shorter compounding period
Lets see how we use a shorter compounding period and what the results are. We previously considered
$100 left in the bank for 35 years at 5%
(1.05)35 = 5.516
$100 * 5.516 = $551.60
Lets switch that off to monthly compounding.
Step #1 – Get the period interest rate
5/12 = 0.4167%
Step #2 – Get the number of compounding periods
35 years * 12 months per year = 420 compounding periods
Plug into the F/P formula
(1.004167)420 = 5.7337 (Did you notice that I took a 0.4167% interest rate and converted it to decimal
form 0.004167 before doing any calculations?)
Now apply F/P to our $100 dollars
$100 * 5.7337 = $573.37
Did you notice what happened when I shortened the compounding period? $100 for 35 years at 5%
interest became $551.60 with annual compounding, but when I shortened the compounding period to 1
month the amount became $573.37. What happened? The more often I add the interest to the account
the more opportunity I have to get interest on interest. Gee that’s exciting! I wonder what happens
when we go to daily compounding?
Step #1 Get the period interest rate
5%/365 = 0.0137%
Step #2 Get the number of compounding periods
35 years * 365 days per year = 12,775 Lets soup up our earnings just a little and throw in those leap
years. Nine leap years are possible (every 4 years) so we will add 9 days.
n= 12,784
Plug into the F/P formula
(1.000137)12784= 5.7610
Apply F/P to our $100
100 * 5.7610 = $576.10 which is up from monthly compounding of $573.37, which is up from yearly
compounding giving $551.60.
Bankers Tricks and Yield
Of course the reality of life is that bankers lend more money and earn more interest than the average
Joe with a savings account. Most of us will spend more time paying loans off than we will collecting
interest. With this in mind it is probably not surprising that bankers have within the law worked the
rules to optimize their interest collection. You saw in the above example above that not all interest
rates reported as 5% earn the same interest. You may have even thought it interesting that I used leap
years in figuring the number of compounding periods, but not when I divided by the number of days in a
year to get the period interest rate. One of the ways I can “soup-up” the interest collected at a given
interest rate is to shorten the compounding period – thus enhancing the amount of interest I collect on
interest and “amping-up” the “yield” I get at a given interest rate. In fact banks actually calculate and
sometimes report both the interest rate and “yield” (effective interest rate). You may have noted that
banks may post the interest rate on their certificates of deposit and then indicate a yield that shows an
even higher interest rate. The yield is the effective rate of interest that results from using a
compounding period shorter than 1 year and getting interest on interest during the year. Lets see how
it works to calculate a yield.
In this example I will use the bankers favorite for “milking” interest out of people – the credit card. An
average credit card has an interest rate of about 14% - but what is the effective rate or yield if the bank
follows the customary practice of daily compounding?
Step #1 – Get the period interest rate
14%/365 = 0.0384% Daily
Step #2 – Use the F/P formula to get the effective rate of interest
(1+0.000384)365 = 1.1502
Step #3 – break out the interest
(1.1502 -1)*100 = 15.02%
So why is 15.02% the “Effective Annual Interest Rate”? We remember that to calculate the interest for
one compounding period we just multiply the amount of money by the interest rate and the interest
charged pops out the other end.
We also remember that if you multiply an amount of money by F/P it will move the money to an
equivalent amount at a future time period. So how much interest gets charged in one year if we have
daily compounding. By law and international convention all interest rates must be reported as a plain
annual rate. So how much interest do we get from $100 at a 14% interest rate. Using our standard
formula of money times interest rate equals interest we would go
$100 * 0.14 = $14
This would be the right amount if the interest compounded once per year, but what if it doesn’t? What
if it is daily compounding? We know we can use F/P to move money forward in time. To move money
one year at 14% interest with daily compounding we calculate
$100* 1.1502 = $115.02
How much interest got added to the account in one year?
$115.02 - $100 = $15.02
But that would be like multiplying by 15.02% interest
$100 * 0.1502 = $15.02
We of course did not really have a 15.02% interest rate, but because we were getting interest on
interest during the year due to daily compounding we effectively accumulated as much interest in one
years time as if we had 15.02% plain annual interest. Thus we call it the yield or “effective annual
interest rate”.
As can be seen, by using daily compounding banks commonly get a full percentage point more effective
interest than what they are legally required to disclose in their annual interest rate. As the interest rate
on the credit card becomes higher, the premium the bank gets by going to daily interest becomes
greater.
This leaves one more question – if daily compounding is better than monthly and monthly is better than
yearly for bank for banks trying to extract maximum interest from their victi - woops I mean customers
then why not compound every second? There are in fact formulas that apply to continuous
compounding. Lets note what happens to effective interest rate or yield as the compounding period
shortens.
Quarterly Compounding
14%/4 = 3.5%
(1.035)4 = 1.1475 which implies 14.75% interest
Monthly Compounding
14%/12 = 1.167%
(1.01167)12 = 1.1493 which implies 14.93% interest
Daily Compounding 15.02%
As can be seen progressive shortening of the compounding period is producing an exponentially rising
amount of work and an exponentially decreasing amount of earnings so bankers can’t gain much more
revenue by compounding every second. Instead they come up with an even better “rule”. It is called
“Average Daily Balance”. Transactions that come in at any point in the day are treated for interest
purposes as if the money earned interest all day. Thus bankers do quite well with daily compounding
and about the only place one really sees continuous compounding is in textbooks somewhere. (You
won’t see it in this text book).
Calculation Tool Intermission
Class Assistant can be used for getting period and effective interest rates (Yield).
Period Interest Rate Adjustment
Enter Annual Interest Rate as a Percentage
Do not use the % key during entry
Annual Int Rate
15
12 Enter number of compounding periods/year
(Would be 12 for months 365 for daily)
in % in decimal
Period Int Rate
1.25
0.0125
Period Interest Rate to an Annual Interest Rate
Enter Your Period Interest Rate as a Percentage
Do not use the % key during entry
Period Interest Rate
1.25
12 Enter number of compounding periods/year
Annual Interest Rate
15
Effective Interest Rate 16.07545
Figure 1-9 Portion of Class Assistant Used to Obtain the Yield or Effective Annual Interest Rate
Period Interest Rate Adjustment
Enter Annual Interest Rate as a Percentage
Do not use the % key during entry
Annual Int Rate
15
12 Enter number of compounding periods/year
(Would be 12 for months 365 for daily)
in % in decimal
Period Int Rate
1.25
0.0125
Figure 1-10 Calculation of the Period Interest Rate with Class Assistant
The period interest adjustment uses a yellow cell inputs the normal yearly interest rate and the number
of compounding periods per year. Examination of the red cells below shows the expected calculation of
annual interest rate divided by number of compounding periods in a year. Beside it is the period
interest rate expressed as a decimal form. Obviously it contains the interest rate as a percent divided by
100.
The effective interest rate can be obtained in the area right beside
Period Interest Rate to an Annual Interest Rate
Enter Your Period Interest Rate as a Percentage
Do not use the % key during entry
Period Interest Rate
1.25
12 Enter number of compounding periods/year
Annual Interest Rate
15
Effective Interest Rate 16.07545
Figure 1-11 Calculation of the Yield or “Effective Annual Interest Rate” using Class Assistant
Just take the annual interest rate and number of compounding periods per year and get the period
interest rate in the Period Interest rate adjustment area. Then take that period interest rate and plug it
in for the period interest rate in the yellow cells of the Period Interest Rate to Annual interest rate area.
Enter the number of compounding periods per year and the red cells below will report both the official
reported annual interest rate and the effective interest rate or yield.
There is one more way to get a yield out of Class Assistant, but one has to know that effective annual
interest rates are calculated using the P/F formula. Go to the Magic Number calculation area.
Magic # Calculator
Note - It is recommended that you go to the Magic # Calculator Tab for advice o
Enter Annual Interest Rate in %
Do not use the % key during data entry
Annual Int Rate
14
365 Enter the number of compounding periods/year
in % in decimal
Period Int Rate 0.038356164 0.000384
(Please note - you must enter the nu
Enter # Compouning Periods to Move Cash (value of n)
365 is to be moved either forward or back
The value should be an interger
(The value you enter here affects F/P
F/P 1.150242923 (used to move one cash flow element n compounging period into the future)
Figure 1-12 Using F/P in the Magic # Calculator to get Yields from Annual Interest Rates
Enter the annual interest rate in the yellow cell labeled annual interest rate. Below enter the number of
compounding periods in a year. The magic number calculator can introduce us to a new cell color –
green. A green cell like a red cell contains a formula and a financial calculation. Like the red cells one
should not change the formula unless one knows exactly what they are doing. Green cells contain
numbers that come from the necessary steps to get to the actual answer located in a red cell in bold
lettering. In this case the green cells contain the period interest rate, which we know is a necessary step
in getting to the effective annual interest rate. To finish the effective interest rate calculation we enter
the number of compounding periods a second time for the value of n and then look below at the value
of F/P. Drop one from the F/P value, and multiply the rest by 100 and you have the effective annual
interest rate. Most engineers can drop the one and read the interest rate in decimal form in their minds
without ever doing a calculation. It will be noted the set-up in the view above is the calculation we just
did above for a credit card with a 14% reported annual interest rate and daily compounding.