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#1
Old 04-18-2010, 01:06 PM
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Annualized percent change

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?

Last edited by Acsenray; 04-18-2010 at 01:07 PM.
#2
Old 04-18-2010, 01:22 PM
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Take the geometric mean.
#3
Old 04-18-2010, 01:30 PM
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Do we know all the different Ns, and are they different? If so, you're just solving for x in the equation

Code:
 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.

Last edited by Chessic Sense; 04-18-2010 at 01:30 PM.
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Old 04-18-2010, 01:50 PM
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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?
#5
Old 04-18-2010, 01:53 PM
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Quote:
Originally Posted by acsenray View Post
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?
(1.105*1.107*1.053*1.064*1.087)^(1/5)= 1.083

8.3%
#6
Old 04-18-2010, 01:58 PM
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Ah, I missed Ultrafilter's post. Thanks, Chessic Sense. So the geometric mean takes into account compounding of interest?
#7
Old 04-18-2010, 02:35 PM
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Quote:
Originally Posted by acsenray View Post
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:

S*w*x*y*z = 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:

S*A*A*A*A = E

So by substitution, we get:

S*A^4 = S*W*X*Y*Z

Cancelling an S and taking the 4th root of both sides yields:

(W*X*Y*Z)^(1/4) = A

If you have a number of variables other than 4, you can generalize it as:

(i1*i2*i3...iN) ^ (1/N) = A



You messed up when you tried to arithmetically average the percentages instead of geometrically. In other words, you declared that:

A*A*A*A = W+X+Y+Z
4A = W+X+Y+Z
A = (W+X+Y+Z) / 4

The error is twofold. One: A*A*A*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.
#8
Old 04-18-2010, 02:50 PM
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Ah! Thanks for the explanation!
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