# Different Types of Averages

What are the differences between the Geometric, Weighted, and Harmonic mean?

Thinking back to your middle school days, you were probably taught what an average or  mean was when describing a data set.  A question you might have seen on a 5th grade level test was to calculate the average daily high temperature during a week given the highest temperature during the day.  To calculate the average, technically called the arithmetic mean, simply add all of the daily high temperatures together and then divide by the number of days you have data for.  Pretty easy, right?

Well, there are actually three more types of means/averages that you were neglected to be taught during your early years in school.  These means, the geometric, weighted, and harmonic, should be applied when determining averages for different financial situations where the arithmetic mean will not accurately represent the data’s average.  Knowing the differences between these four types of means is very important for anyone in the finance field, especially for those studying for any of the CFA exams.

For the sake of this article, and everyone’s time, I will not go over how to calculate the arithmetic mean.  Additionally, the variable N is the number of observations in the entire population, and Xi is the ith observation in a set.

## Geometric Mean

The geometric mean is used to determine average rates of change over a period of time, such as a portfolio return or the return of a stock.   To compute, add 1 to each observation, assuming you are calculating the geometric mean of several rates of return.  Next, multiply all of these together and take the N root of the product (or take the product and raise it to the (1/N) power).  Because you added 1 to each observation, at the end you must subtract 1.  The reason that you must add 1 to each percent in the first place is because if a return for one of the periods is 0 or negative, you will not be able to find the N root or raise it to the (1/N) power.

Example: A portfolio has a return of 8%, 4%, -2%, and 6% in four consecutive years.  Find and compare the geometric mean with the arithmetic mean.

Geometric Mean = ((1.08 * 1.04 * .98 * 1.06) ^ (1/4)) * 100 = 3.93%

Arithmetic Mean = (8% + 4% – 2% + 6%) / 4 = 4%

The difference between the two means is .07%, or 7bps.

## Weighted Mean

Another type of mean that is used often in finance, specifically when analyzing portfolio returns for a given period, is the weighted mean. When calculating the arithmetic mean, all observations are given an equal weight.  However, when determining the weighted mean, you ‘assign’ different weights to different observations.  To calculate the weighted mean, multiply the weight and the return for each observation, and then add them all together.

Example: An analyst wants to find the portfolio return of a \$1MM portfolio that is made up of three different asset classes.  The initial investments into the portfolio were \$500,000 in equities, \$350,000 in bonds, and \$150,000 in international stocks.  What is the portfolio return given equities, bonds, and international stocks had a return of 8%, 3%, and 12%, respectively.

First, to determine the weights of each class, divide the amount of each initial investment by the total initial investment of \$1,000,000.  (The weights end up being 50% for equities, 35% for bonds, and 15% for international stocks.)  Next, add the products of each weight multiplied by the class return.

Weighted Average = ((.5 * .08) + (.35 * .03) + (.15 * .12) * 100) = 6.85%

## Harmonic Mean

Lastly, the harmonic mean is a type of average that only applies to data sets relating to or characterized by musical harmony (…kidding). The harmonic mean is actually a specialized average that is only appropriate when averaging ratios when the ratios are applied to a fixed quantity.  To calculate harmonic mean, divide N by the sum of 1 divided Xi, or each observation.  An example of when the harmonic mean would be used is when you need to determine the average price-per-share an investor paid at different times, a strategy known as cost averaging.

Example: A private wealth adviser wants to find the average price his client paid for an asset.  His client purchased \$2,000 of a security each month for 3 months. The share prices were \$25, \$29, and \$27 respectively.  What is the average price paid for the security?

Harmonic Mean = (3 / (1/25 + 1/29 + 1/27)) = \$26.9

## Summary

Depending on the situation, the arithmetic mean of a data set will not always be your best choice.  Instead, make sure to use one of the three covered in this article.  Disclaimer: this article doesn’t even cover all of the different types of averages.  If you are analyzing a large data set with outliers (values that lies outside of where the majority of the other values are) and need to find the average, you might want to use a truncated or trimmed mean, or even a winsorized mean!  Questions? Leave a comment.