How to calculate yearly average inflation rate having only percent changes?

Question

I'm trying to calculate yearly average inflation rate across a period of years displayed in my screenshot having only percent changes for each year. I'm unsure how that should be done and have tried 3 different ways using a spreadsheet application. In my first attempt (first row) I simply sum up the percent changes for each year and divide by the number of years. In the second attempt (second row in the screenshot) I thought of having a fixed capital (the 1 under the 2004 column) and think of each yearly inflation rate as if it were yearly capital growth. In the third attempt (third row) I again start with an initial capital (1 under the 2004 column) and apply yearly inflation rate on the initial capital as if it were yearly loss of buying power on that capital. As you can see I get three different values. I believe the third calculation is the correct one but I'm not completely sure about that.

The Sum column values are in the three cases obtained as follows

• in the first case/row, as already said, it is the sum of all the yearly percent rate changes,
• in the second case/row it is the second row value under the 2023 column - 1 and
• in the third case/row as 1 - the third row value under the 2023 column .

Answer 1

Let's say you have 3 years with 2%, 5% and 7%

1. Multiply "1 + percentage" for all years. 1.02*1.05*1.07 = 1.14597 (or 14.6% over three years)
2. Take the cube root (1.14597)^(1/3) = 1.0465
3. Subtract 1 and turn into percent -> 4.65% is the average annual inflation rate.

Answer 2

If you already have the annual inflation rate, what is it you're trying to calculate? Do you mean you're trying to calculate the average rate for some period of time? Like year 1 is 1.0%, year 2 is 2.4%, year 3 is 1.8%, what's the average for all 3 years?

That's a question of definitions. What do you mean by "average"?

The simplest definition would be to find the average of the numbers. Just add the numbers up and divide by the number of values. Like if you had 1.0, 2.4, and 2.0, the sum is 5.4, divided by 3 gives 1.8. (This is the "arithmetic mean".)

Perhaps you are thinking, If the inflation rate had been the same every year, what would it have had to be to give the same final set of prices? In that case, convert each inflation rate to a "growth factor" by adding 1. Multiply all the numbers together. Then find the nth root, where n is the number of values. For example, to use 1.0, 2.4, and 2.0 again, multiply 1.010, 1.024, and 1.020. That gives 1.055. As we have 3 number, then take the 3rd root of 1.055, which comes to a shade under 1.018, or 1.8%. In this case very close to the simple average, but I had only 3 numbers and all fairly small. (Called a "geometric mean".)

There isn't really one "right answer" until you carefully define what you mean by "average".

Answer 3

Percent changes convert to multipliers by dividing by 100 and adding one --that is, 5% increase is a multiplier of 1. 05, and 5% decrease us a multiplier if 0.95.

To get cumulative change from yearly change, multiply the multipliers together then convert back to percentage if desired. This, two years of 5% increase and one of 6%, in any order, is 1.05 * 1.05 * 1.06, yielding 1.16865, or 16.9% total increase.

Note that a 5% increase and a 5% decrease, in either order, do not cancel out as you might expect if trying to average these; 1.95 * 0.95 comes out to 0.9975, or a 0.25% loss.

(Forgot to say:)

From that, we can see that a steady gain of XX% per year for Y years results in a total gain of

[ (1+XX)^Y - 1] %

where ^ is the exponentiation operator. (In some programming languages ** is used instead )

And that shows us how to get the equivalent average/constant yearly gain for those Y years: Take the Yth root:

[ (1+XX)^(1/Y) - 1] %

The same formula can be used to get quarterly or monthly average yield, just use the appropriate Y for the number of time periods you want to divide the results over.

Answer 4

I applied what @Hilmar suggested here and then verified that indeed the obtained average inflation rate applied across the same period of years yielded the same exact overall result and the two correspond perfectly as you can see from the screenshot.

The only doubt I have now is if inflation should be added or subtracted. I'm for subtracting it because a positive inflation reduces the buying power of your capital, it doesn't increase it as we are implying when we sum it.