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#1




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; 04182010 at 01:07 PM. 
#2




Take the geometric mean.

#3




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 Last edited by Chessic Sense; 04182010 at 01:30 PM. 
#4




Okay, let's fill in numbers. Say the inflation rates for 20002004 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




Quote:
8.3% 
#6




Ah, I missed Ultrafilter's post. Thanks, Chessic Sense. So the geometric mean takes into account compounding of interest?

#7




Quote:
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




Ah! Thanks for the explanation!

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