I’m trying to understand the concept of annualizing economic statistics.
I understand what to do if you have a monthly (n=12) or quarterly (n=4) percent change and you want to annualize it.
( ((percent change / 100 + 1)^n) - 1) x 100
But what if you have a series of n annual percent changes (say inflation rates, i) and you want to annualize the total change over n years?
Here’s my guess –
( ( ((i1+i2+…+in)/n)/100)^(1/n) ) - 1) x 100
Does that make any sense?
Take the geometric mean.
Do we know all the different Ns, and are they different? If so, you’re just solving for x in the equation
i1*i2*i3...iN = x^N
For the sake of clarity of the post, I’m defining i as, say, 1.12, not 12%. I know you know about dividing by 100 and adding 1 and doing the reverse with the answer, so I’m not going to clutter up the equation with it.
Okay, let’s fill in numbers. Say the inflation rates for 2000-2004 are 10.5, 10.7, 5.3, 6.4, and 8.7. What is the annualized rate of inflation for the entire period? Is it just the mean?
(1.1051.1071.0531.0641.087)^(1/5)= 1.083
8.3%
Ah, I missed Ultrafilter’s post. Thanks, Chessic Sense. So the geometric mean takes into account compounding of interest?
Yeah, pretty much.
Let’s define some variables. S is the starting amount. E is the ending amount. The various changes are W, X, Y, and Z, in the decimal form, not percentage. That is, 1.105 not 10.5%.
In reality, the number changes like so:
Swxyz = E
That is, the starting amount gets multiplied by each change. as you go through each iteration. But you want to know what the average is. You want to know A in the following:
SAAAA = E
So by substitution, we get:
SA^4 = SWXY*Z
Cancelling an S and taking the 4th root of both sides yields:
(WXY*Z)^(1/4) = A
If you have a number of variables other than 4, you can generalize it as:
(i1i2i3…iN) ^ (1/N) = A
You messed up when you tried to arithmetically average the percentages instead of geometrically. In other words, you declared that:
AAA*A = W+X+Y+Z
4A = W+X+Y+Z
A = (W+X+Y+Z) / 4
The error is twofold. One: AAA*A is A^4, not 4A, and Two: W+X+Y+Z never shows up in the equation and is thus a meaningless quantity. They’re originally multiplied, not added.
Ah! Thanks for the explanation!