How to Calculate an Annual Percentage Growth Rate

Suppose that between 1989 and 2001 the price of a box of cigars rises from \$60 to \$150. What is the annual percentage rate of increase in the price of cigars (the annual inflation rate)?

1. Your first thought might be the following. (a) Take the difference between the final price of the box cigars (\$150) and the initial price (\$60), which would be \$90. (b) Divide by the initial price (\$60) to get the total percentage increase. (c) Divide by the number of years, [12 = 2001-1989]. And (d) multiply by 100% to get the annual percentage rate of increase. The result would be 12.50 percent per year.

(((\$150-\$60)/\$60)/12)*100% = 12.50%

But this is not completely satisfactory. One problem is that this method doesn't give the same answer going up as going down. Suppose that after having risen to \$150, the price then falls over the next 12 years back to \$60. Then we get the following result.

(((\$60-\$150)/\$150)/12)*100% = -5.00%

So the price rose at 12.5 percent per year for 12 years, and then fell at 5.0.percent per year for 12 years, but we ended back at the same place!

2. A much better way to go is to use the average of the initial and final values as the base. In the example the average is 105 [(60+150)/2 = 110]. So the annual percentage increase is

[(((\$150-\$60)/110)/12)*100] = 6.82% per year.

Now at least, the rate going up, +6.82%, is equal in absolute value to the rate going down, -6.82%.

3. For most calculations, using the average value for the base gives a result similar to the method preferred by economists, and the one, therefore, that you should use. (a) Take the natural logarithm of final value. (b) Subtract the natural logarithm of the initial value. (c) Divide by the number of intervening years. And (d) multiply by 100% to put the answer in percent per year. For the example, the result would be

{(natural logarithm (150) - natural logarithm (60))/ 12}*100% =

{(5.0106-4.0943)/12}*100 = 7.64% per year.

Many hand held calculators have the natural logarithm operator under LN. This is also the symbol for taking the natural logarithm in Excel. This calculation is similar to the one you would make if you were thinking about an investment. You placed \$60 in a bank account and 12 years later your account was worth \$150. At what rate was your money growing? What was the rate of interest you received?

4. Sometimes you have to watch out for unusual endpoints. If the price of a box of cigars was unusually high or low in 1989, or unusually high or low in 2001, the natural logarithm method described above might give a misleading answer about the typical rate of growth over the period. To fix this up you could use an average of the prices near the starting point or the ending point. For example, you could use the average of the prices in 1988, 1989, and 1990 for the initial price and the average of the prices in 2000, 2001, and 2002 for the final price. Another way to adjust for this problem would be to regress the natural logarithm of the price on time. But you do not need to do either of these refinements for your class exercise.