# Return on Investment (ROI)

ROI 1
Return on Investment (ROI)
Prepared by Sarah Major
What is ROI?
Return on investment (ROI) is a measure that investigates the amount of
this calculation to compare different scenarios for investments to see
which would produce the greatest profit and benefit for the company.
However, this calculation can also be used to analyze the best scenario for
other forms of investment, such as if someone wishes to purchase a car,
buy a computer, pay for college, etc.
Simple ROI Formula
The simplest form of the formula for ROI involves only two values: the cost
of the investment and the gain from the investment. The formula is as
follows:
��� (%) = ���� ���� ���������� − ���� �� ����������
���� �� ����������
×100
The ratio is multiplied by 100, making it a percent. This way, a person is
able to see what percentage of their investment has been gained back after
a period of time. Some, however, prefer to leave it in decimal form, or ratio
form.
Simple ROI Problems
Here are a few examples to get the hang of calculating ROI.

1. Gains = \$535,000 and cost = 400,000. What is ROI?
��� = 535,000 − 400,000
400,000 = 135,000
400,000 = 0.34 × 100 = 34%
2. Gains = \$3,640 and cost = \$1,880. What is ROI?
��� = 3,640 − 1,880
1,880 = 1,760
1,880 = 0.94 × 100 = 94%
ROI 2
Simple ROI Over Time
3. You buy a car for \$26,450. Because you now have reliable
transportation, you are able to obtain a job and earn \$10,860 in your
first year. Calculate your return on investment for that year.
��� = 10,860 − 26,450
26,450 = −15,590
26,450 = −0.59 × 100 = 59%
4. A business purchases a new form of information system technology
for \$500,000. Because of this purchase, the company begins earning
\$50,000 a year. Find the ROI for the first year.
��� = 50,000 − 500,000
500,000 = −450,000
500,000 = −0.9 × 100 = −90%
What does a negative ROI mean? Let’s take a step back and think about a
different question: what would it mean if we had a zero ROI? This only
occurs when the numerator of our formula is zero, and this can only
happen if our gains were the same as our costs, meaning we broke even.
Therefore, if ROI is negative, the costs must be greater than the gains, or we
have yet to achieve an amount of gain great enough to cover the cost of the
investment. Once ROI is positive, that means we have earned more than the
cost we put into the investment. When it’s positive, we have actually
returned a profit!
Investors calculate ROI over time to see how the value changes or when a
positive ROI will occur. This gives them a better timeframe of how long it
will take them to get an adequate return on their purchase.
5. Take the business investment in Problem 4 and calculate their ROI
for the first four years.
���! = 50,000 − 500,000
500,000 = −450,000
500,000 = −0.9 × 100 = 90%
���! = 100,000 − 500,000
500,000 = −400,000
500,000 = −0.8 × 100 = −80%
���! = 150,000 − 500,000
500,000 = −350,000
500,000 = −0.7 × 100 = −70%
���! = 200,000 − 500,000
500,000 = −300,000
500,000 = −0.6 × 100 = −60%
ROI 3
6. Take Problem 3. Calculate the ROI for each consecutive year until
you obtain a positive return on their investment.
���! = 10,860 − 26,450
26,450 = −15,590
26,450 = −0.59 × 100 = −59%
���! = 21,720 − 26,450
26,450 = −4,730
26,450 = −0.18 × 100 = −18%
���! = 32,580 − 26,450
26,450 = 6,130
26,450 = 0.23 × 100 = 23%
Analyzing Different Scenarios
To find the best investment, investors must analyze ROI calculations for
different scenarios to see which produces the higher number, or higher
return. This is so they know which purchase to make before they actually
invest in a product. There are two different ways of predicting which
investment in a series of scenarios will give the best return. The first
method is to see which will give a positive return in the shortest amount of
time.
7. You crashed the car you bought in Problem 3, so you need to
purchase a new one. You’ve narrowed your choice down to two cars.
Car A is \$19,345, and since gas would cost you about the same as for
your old car, you would still be pocketing about \$10,860 a year. Car B
is \$27,120 but is extremely good on gas mileage, so after paying for
gas, you would be pocketing about \$13,430 a year. Which car should
you buy based on which one will give you a positive ROI faster?
���!! = 10,860 − 19,346
19,346 = −8,486
19,346 = −0.44 × 100 = −44%
���!! = 21,720 − 19,346
19,346 = 2,374
19,346 = 0.12 × 100 = 12%
���!! = 13,430 − 27,120
27,120 = −13,690
27,120 = −0.50 × 100 = −50%
���!! = 26,860 − 27,120
27,120 = −260
27,120 = −0.01 × 100 = −1%
���!! = 40,290 − 27,120
27,120 = 13,170
27,120 = 0.49 × 100 = 49%
Car A will give you a positive ROI faster and is thus is the best investment.
ROI 4
The second method of predicting which scenario will give you the best
return is by seeing which investment will give you the highest ROI after a
predetermined amount of time.
8. Take the car decision you are trying to make in Problem 7. Which will
be the best buy if you look at the ROI after three years?
���!! = 10,860 − 19,346
19,346 = −8,486
19,346 = −0.44 × 100 = −44%
���!! = 21,720 − 19,346
19,346 = 2,374
19,346 = 0.12 × 100 = 12%
���!! = 32,580 − 19,346
19,346 = 0.68 × 100 = 68%
���!! = 13,430 − 27,120
27,120 = −13,690
27,120 = −0.50 × 100 = −50%
���!! = 26,860 − 27,120
27,120 = −260
27,120 = −0.01 × 100 = −1%
���!! = 40,290 − 27,120
27,120 = 13,170
27,120 = 0.49 × 100 = 49%
Car A will give you the highest ROI after three years and thus is the best
investment.
What would it look like if we graphed ROI over time as points on a
coordinate plane? In the space below, graph the two scenarios on a
coordinate plane. Let the x-axis be time and the y-axis by ROI. Connect the
points for each scenario to see what type of growth is produced. 80
70
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0.5 1 1.5 2 2.5 3 3.5 4
ROI 5
As the graphs of the points show, the growth of ROI in these scenarios is
linear. What if we wanted to find the ROI after ten years? We can
approximate the equation of the line so that we don’t have to keep
calculating the different ROI calculations over time. With the equation, we
can simply plug in the unit of time, and the output will produce the ROI at
that point in time. So, first, approximate the equation of the line using the
points you calculated from the best of the two scenarios. Then, find the ROI
after ten years.
����� = � = �! − �!
�! − �!
= 68 + 44
3 − 1 = 112
2 = 56
� = �� + �
⇒ 68 = 56 3 + �
⇒ � = −100
� = 56� − 100
⇒ � = 56 10 − 100
⇒ � = 460%
Factors Affecting Cost and Gains from Investment
It is easy to calculate ROI when the cost and gains are constant but this is
rarely the actual case. Different factors may affect the cost and gains over
time. For instance what if a loan had to be taken out to pay for the
investment? The investor would have to pay interest on the amount owed.
However as the money is paid back the amount of interest would decrease
over time because it is calculated by how much money is owed. Another
factor may include growth of revenue over time. If a business grows over
time their revenue will increase causing their gains to increase over time.
This may affect the way the growth of ROI looks over time.
ROI 6
9. Take the same car decision from Problems 7 and 8. Use the scenario
from Car A except you receive a 20% raise each year from your job.
Calculate the ROI over the first five years and then graph the results
on a coordinate plane.
���!! = 10,860 − 19,346
19,346 = −8,486
19,346 = −0.44 × 100 = −44%
���!! = 10,860 + 10,860 + 10,860 . 2 − 19,346
19,346
= 10,860 + 13,032 − 19,346
19,346 = 4,546
19,346 = 0.23 × 100 = 23%
���!! = 23,892 + 13,032 + 13,032 . 2 − 19,346
19,346
= 23,892 + 15,638.4 − 19,346
19,346 = 20,184.4
19,346 = 1.04 × 100 = 104%
���!! = 39,530.4 + 15,638.4 + 15,638.4 . 2 − 19,346
19,346
= 39,530.4 + 18,766.08 − 19,346
19,346 = 38,950.48
19,346 = 2.01 × 100
= 201%
���!! = 58,296.48 + 18,766.08 + 18,766.08 . 2 − 19,346
19,346
= 58,296.48 + 22,520,02 − 19,346
19,346 = 61,470.5
19,346 = 3.18 × 100
= 318%
600
500
400
300
200
100
–100
–1 1 2 3 4 5
ROI 7
As you can see, the points look to produce exponential growth. Let’s try
another example with other factors involved.
10. Use the information from Problem 4. What if instead of just earning
\$50,000 a year, the company earned \$50,000 more dollars each
year than the year before? Calculate the ROI for the first five years
of business. Then, graph the points.
���! = 50,000 − 500,000
500,000 = −450,000
500,000 = −0.9 × 100 = −90%
���! = 150,000 − 500,000
500,000 = −350,000
500,000 = −0.7 × 100 = −70%
���! = 300,000 − 500,000
500,000 = −200,000
500,000 = −0.4 × 100 = −40%
���! = 500,000 − 500,000
500,000 = 0
500,000 = 0 × 100 = 0%
���! = 750,000 − 500,000
500,000 = 250,000
500,000 = 0.5 × 100 = 50%
100
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ROI 8
Once again, we can see that the graph of the points looks like an
exponential function. Typically, this is the type of growth we see when
dealing with ROI since it is usually used in business calculations. Over time,
businesses grow as they become more productive, successful, and popular.
Is it possible to approximate an equation for a function that goes through
the points just like we did with the linear functions so we don’t have to
make so many calculations to get the value we want?
The answer is yes, but we must review what we know about exponential
functions to be able to find this process. First of all, we know that these
functions appear in the form of � = ��!, where � is a fixed constant that is
where the function crosses the y-axis and � is the base. Having a positive
exponent will produce exponential growth, and having a negative
exponent will produce exponential decay (exponential decrease). We also
know that the base for the exponential component has to be positive, but
we are also dealing with negative values. However, we know that a vertical
shift results if we add or subtract a value from our function. We can use
this information to approximate a function for the previous problem.
First of all, we need to find the constant so that we can form an equation.
This can be done by finding where the function will cross the x-axis. This
point occurs where � = 0 on the graph. Because � is a function of time in
this example, and we know that when � = 0, no amount of time has passed,
and therefore, no gains have been made yet. Accordingly, our ROI
calculation is:
��� = 0 − 19,346
19,346 = −19,346
19,346 = −1 ×100 = −100%
So, the point in question is (0, −100). But wait, remember about the
negative values? Let’s shift all of the points up 200 so that we don’t have to
deal with those nasty negatives. Therefore, our graph looks like this:
250
200
150
100
50
1 2 3 4 5
ROI 9
Now, we just need to add 200 to all of our � values. This makes our initial
point (0,100).
We can then plug these initial values in to find the constant �.
� = ��!
⇒ 100 = ��!
⇒ � = 100
Now, we have the formula � = 100�!. We can take the last point, (5,50),
which we changed to (5,250), and plug in for the corresponding values to
find the other constant, �.
� = 100�!
⇒ 250 = 100�!
⇒ 2.50 = �!
⇒ � = 2.50 !
⇒ � ≈ 1.20
Now, we have the full equation � = 100(1.39)!. But wait! Don’t forget that
we shifted the points up 200. We can shift them down by subtracting back
the 200, so our actual equation is:
� = 100(1.20)! − 200
Let’s graph this function along with our points to see how close of an
approximation it is.
60
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–2 2 4
f(x) = 100·(1.2)
x – 200
ROI 10
So, close, but not exact. It goes through the initial and last point because
those are the points we used to find the values for our function. But why
couldn’t we find a better model using our method?
Why didn’t our approximation of the exponential function for our data
points work? Let’s look at the � values of our points: -90, -70, -40, 0, and
11. Is there a relationship between these points that might help us find a
better approximation? Let’s find the differences between these values:
−90 − (−100) = −90 + 100 = 10
−70 − −90 = −70 + 90 = 20
−40 − −70 = −40 + 70 = 30
0 − −40 = 0 + 40 = 40
50 − 0 = 50
There is definitely a pattern between the differences of these values. The
differences increase by 10 as the points progress. However, this does not
seem to be growing exponentially but rather arithmetically. Therefore, this
is not considered exponential growth. When growth LOOKS exponential
but is NOT actually exponential, it usually means that a power function can
be used to represent the points.
Power Functions
Power functions appear in the form:
� � = ��!
where � is a scaling factor and � is the power that controls the growth or
decay (in this case, our growth).
Notice that when we plug in 0 for �, we get:
� � − � 0 ! = 0
Therefore, power functions always pass through the point (0,0). This
means that once again, we’ll need to move all of our points up so that the
initial point passes through the origin. In light of this, let’s add 100 to all of
our points so that we produce the graph:
ROI 11
Now, how do we find an equation for a power function using our points?
Let’s plug in one of our points and see if we can find one of these values we
need. Let’s use (1,10).
� � = ��!
⇒ 10 = �(1)!
⇒ � = 10
Let’s use another point, (5,150), to find �.
� � = 10�!
⇒ 150 = 10(5)!
⇒ 15 = 5!
⇒ ln 15 = ln (5!)
⇒ ln 15 = � ln (5)
⇒ � = ln (15)
ln (5)
⇒ � ≈ 1.68
Therefore, our equation is:
� � = 10�!.!”
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ROI 12
Project
Come up with your own situation involving ROI. It can be something that
etc.), a business example, or anything that involves putting money into
some kind of investment and calculating its return. Think of all the
possible factors that will affect calculations for ROI. These may include
salary, bonuses, loans, interest rates, costs, insurance payments, etc.
Produce at least three different scenarios and calculate the first five points
for each. Your unit of time doesn’t necessarily have to be years. It may be
more practical to use months, decades, etc. Find the best investment out of
your scenarios and the parameters for it being the best. Next, find the type
of function that is the best fit for your data points. Make sure you try AT
LEAST linear, exponential, and power functions for your growth if not
some other types of functions to find which one best fits your data. Then,
produce a graph of a this function on a piece of poster board so that the
class can see how your growth looks over time. Make sure the equation for
the actual function is visible somewhere on your poster. Also, somewhere