Lesson Percentage Change vs. Compound Average Growth Rate

Source code of 'Percentage Change vs. Compound Average Growth Rate'

This Lesson (Percentage Change vs. Compound Average Growth Rate) was created by by oberobic(2304)

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Problem: Over a period of 30 years, the minimum wage increased from $1.50 per hour to $6.60 per hour. What was the percentage change? What was the annual rate of change?
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Solution: The percentage change can be calculated using the all-purpose percentage change formula:
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((EndingValue-BeginningValue)/BeginningValue) * 100
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In formal typesetting, the equation is:
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{{{ PctChange = ((EndingValue-BeginningValue)/BeginningValue) * 100 }}}
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The values you need are given in the problem statement:
Ending Value = 6.60
Beginning Value = 1.50
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Note that to determine the percentage change you do not need:
Number of Years = 30
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Using a calculator, you will find:
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((6.6-1.5)/1.5) * 100 = 340
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That means there was a 340% change.
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We can check this by multiplying: 3.4*1.5 = 5.1. Adding the $1.50 and $5.10 results in $6.60.
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So, there was a 340% increase in hourly wages from $1.50 to $6.60.
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That sounds pretty good, until you consider that we did not consider the number of years. 
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Of course, to find the percentage change in absolute terms, you don't care how long it has taken. If you need a tree to grow from 2 ft. to 40 ft tall, you have to wait for the tree to do it. The percentage change is calculated the same way: (40-2)/2 * 100 = 1900%. If it takes 50 years, so be it. If you plant two trees, it will not happen in half the time. You just end up with two trees.
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However, you might wonder if a 340% increase is a "good investment" given the 30 years involved. In that case, you would, in fact, be asking what the "compound annual growth rate" is for the hypothetical hourly wages. 
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The word "compound" is critical because it mean that the annual increase affects the most recent value, not the beginning value. For example, if wages had grown to $4.00 per hour at some point, then a 10% increase would be 10% of $4.00 = 40 cents, not 10% of $1.50 = 15 cents.
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The compound annual growth rate is commonly known as CAGR. CAGR (pronounced "cagger", which rhymes with Jagger) is an important financial ratio. CAGR equals
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(EndingValue/BeginningValue)^((1/NumberOfYears)) -1 
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In formal typesetting the equation is:
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{{{ CAGR = (EndingValue/BeginningValue)^((1/NumberOfYears)) -1 }}}
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The values you need are given in the problem:
Ending Value = 6.60
Beginning Value = 1.50
Number of Years = 30
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Plug these values into the equation and you have your answer. Again, you can use a calculator.
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((6.6 / 1.5)^(1 / 30)) - 1 = 0.0506266735
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So, CAGR = .0506266735
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That is the compound average growth rate. However, that is not the average percentage change.
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You need to remember that to talk about percentages you need to multiply the calculated rate of change by 100. 
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So, the compound average percentage change is about 5%. 
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Notice that this result is vastly different from what you might have expected if you thought in terms of: 340%/30 = 11.33%. 
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Dividing the total change by the number of years tends to be misleading. The worker earning minimum wage experienced only a 5% increase per year, not 11%. The investor who saw his investment grow by 340% must understand that his CAGR was only 5%.
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Answer: Over a period of 30 years, wages rose from $1.50/hr to $6.60/hr. In absolute terms, that is a 340% increase. From a compound annual rate growth rate perspective, the change was 5% per year.