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
additional profits produced due to a certain investment. Businesses use
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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    10
    –10
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    –30
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    –50
    –60
    –70
    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
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    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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    1 2 3 4 5
    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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    1 2 3 4 5
    ROI 12
    Project
    Come up with your own situation involving ROI. It can be something that
    pertains to your life (buying a care, paying for college, buying a computer,
    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
    on your poster, present a problem about your scenario that the class will
    solve, whether it be just values that you plug into your function or a word
    problem involving your scenarios. We will present these projects to the
    class.

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