The Means

We were at a financial seminar tonight (who isn’t these days?) about the history of the stock market. During the presentation, the presenter was talking about arithmetic means and geometric means. I realized that I didn't really remember the difference, so here is a quick primer on it based on what I learned brushing up on my math.

The arithmetic mean is the sum of a list of numbers divided by the number of items in the list. The geometric mean is the nth root of the product (multiplication) of all the items in a list where n is the number of items in the list. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. It can be confusing to know when to use each type of mean. I’ll use some investment examples here to illustrate all three.

Scenario 1: Suppose you have three friends. Each starts with some sum of money and then each reports back after some period of time of investing (for example, a year) how much money they made. Suppose they report 20%, 10%, and -10%.
Question 1: What is the average rate of return?
Answer 1: In this case you simply use the arithmetic mean to get 6.67%. So far, so good.

Scenario 2: Suppose you start with $100. After year one, you have $120. After year two, you have $132, and after year three you have $118.80. These numbers are equivalent to having a 20% rate of return the first year, a 10% rate of return the second year and a-10% rate of return the third year. So we have similar percentages as in scenario 1.
Question 2: What is the rate of return such that for each year if you applied that rate you would end up with $118.80?
Answer 2: In this case you don’t use the arithmetic mean, you use the geometric mean to get (1.20*1.10*0.9) ^ (1/3) = 0.0591 or 5.91%. To check this we can simply calculate $100*1.0591*1.0591*1.0591 = $118.80.
If you thought the arithmetic mean would work simply run through the calculation ($100*1.0667*1.0667*1.0667 = $121.36) to find it is not correct. If you thought you could just take the $118.80 - $100 to get $18.80 and divide that by 3 (number of years) you get 6.27% which if you check the calculation gives you $120.01. Again, not correct. Both of these approaches are not correct because the compounding nature (multiplicative) of the calculation means that an arithmetic mean is not appropriate.

Scenario 3: Suppose you save $1,200 by saving $100 every month. Then, you save another $1,200 by saving $120 every month followed, finally by saving yet another $1,200 by saving $200 every month. At the end you would have $3,600 and it would have taken you 12+10+6=28 months to save.
Question 3: What is the average savings rate per month?
Answer 3: Your first response is to take $3,600 and divide by 28 which is $128.57. In this case it is easy to calculate, but if you had lots of saving periods, say over 20 years, you might want a short cut to calculate this. This is where you the harmonic mean comes into play. The harmonic mean would be 3/ ((1/$100) + (1/$120) + (1/$200)) = $128.57. The harmonic mean is used is situations when you dealing with rates.