Geometric Mean

[zeroEthix]

I've been using the CAGR/IRR formula to figure return data...I understand the difference between arithmetic mean and geometric mean: simply (a+b)/2 and (a*b)^.5 respectively. What I can't figure out why the following example uses 1.60 and 1.20 for the second and third year returns. Using CAGR it gives me a return of 28.966% ((ending/beginning)^1/n) - 1...Little help please

For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20. The relevant quantity is the geometric mean of these three numbers.

The question about finding the average rate of return can be rephrased as: "by what constant factor would your investment need to be multiplied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60 the next, and 1.20 the third?" The answer is the geometric mean . If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%, and 20% would give you).

[Baron Von Raschke]

My guess is it is either a typo or the example was originally supposed to prove that returns with the same arithmetic mean would not yield the same geometric mean. The author must have got distracted or something.

If it was supposed to be the latter, a more educational example for investment purposes is to compare the aritmetic and geometric means for yearly returns of something like 100%, 100%, -100%. People often read about much higher arithmetic means of more volatile investments like small cap stocks and are mislead to believe that they also translate into much higher geometric means.

[atomic]

Originally Posted by zeroEthix

I've been using the CAGR/IRR formula to figure return data...I understand the difference between arithmetic mean and geometric mean: simply (a+b)/2 and (a*b)^.5 respectively. What I can't figure out why the following example uses 1.60 and 1.20 for the second and third year returns. Using CAGR it gives me a return of 28.966% ((ending/beginning)^1/n) - 1...Little help please

For example, suppose you have an investment which earns 10% the first year, 50% the second year, and 30% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20. The relevant quantity is the geometric mean of these three numbers.

The question about finding the average rate of return can be rephrased as: "by what constant factor would your investment need to be multiplied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60 the next, and 1.20 the third?" The answer is the geometric mean . If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%, and 20% would give you).

I compute , just as you mentioned.

[4sigma]

1.283 is the correct answer if the annual returns are 10%, 60%, and 20%. It looks like someone changed their mind mid writing the problem to use 10/50/30 instead of 10/60/20.