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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? . Solution: The percentage change can be calculated using the all-purpose percentage change formula: . ((EndingValue-BeginningValue)/BeginningValue) * 100 . In formal typesetting, the equation is: . . The values you need are given in the problem statement: Ending Value = 6.60 Beginning Value = 1.50 . Note that to determine the percentage change you do not need: Number of Years = 30 . Using a calculator, you will find: . ((6.6-1.5)/1.5) * 100 = 340 . That means there was a 340% change. . We can check this by multiplying: 3.4*1.5 = 5.1. Adding the $1.50 and $5.10 results in $6.60. . So, there was a 340% increase in hourly wages from $1.50 to $6.60. . That sounds pretty good, until you consider that we did not consider the number of years. . 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. . 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. . 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. . The compound annual growth rate is commonly known as CAGR. CAGR (pronounced "cagger", which rhymes with Jagger) is an important financial ratio. CAGR equals . (EndingValue/BeginningValue)^((1/NumberOfYears)) -1 . In formal typesetting the equation is: . . The values you need are given in the problem: Ending Value = 6.60 Beginning Value = 1.50 Number of Years = 30 . Plug these values into the equation and you have your answer. Again, you can use a calculator. . ((6.6 / 1.5)^(1 / 30)) - 1 = 0.0506266735 . So, CAGR = .0506266735 . That is the compound average growth rate. However, that is not the average percentage change. . You need to remember that to talk about percentages you need to multiply the calculated rate of change by 100. . So, the compound average percentage change is about 5%. . Notice that this result is vastly different from what you might have expected if you thought in terms of: 340%/30 = 11.33%. . 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%. . 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. This lesson has been accessed 33730 times. |
