Harmonic Mean

What Is a Harmonic Mean?

The harmonic mean is a type of numerical average. It is calculated by dividing the number of observations by the reciprocal of each number in the series. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

The harmonic mean of 1, 4, and 4 is: 

3 ( 1 1   +   1 4   +   1 4 )   =   3 1 . 5   =   2 \frac{3}{\left(\frac{1}{1}\ +\ \frac{1}{4}\ +\ \frac{1}{4}\right)}\ =\ \frac{3}{1.5}\ =\ 2 (11 + 41 + 41)3 = 1.53 = 2

The reciprocal of a number n is simply 1 / n.

The Basics of a Harmonic Mean

The harmonic mean helps to find multiplicative or divisor relationships between fractions without worrying about common denominators. Harmonic means are often used in averaging things like rates (e.g., the average travel speed given a duration of several trips).

The weighted harmonic mean is used in finance to average multiples like the price-earnings ratio because it gives equal weight to each data point. Using a weighted arithmetic mean to average these ratios would give greater weight to high data points than low data points because price-earnings ratios aren't price-normalized while the earnings are equalized.

The harmonic mean is the weighted harmonic mean, where the weights are equal to 1. The weighted harmonic mean of x1, x2, x3 with the corresponding weights w1, w2, w3 is given as:

∑ i = 1 n w i ∑ i = 1 n w i x i \displaystyle{\frac{\sum^n_{i=1}w_i}{\sum^n_{i=1}\frac{w_i}{x_i}}} ∑i=1nxiwi∑i=1nwi

Harmonic Mean Versus Arithmetic Mean and Geometric Mean

Other ways to calculate averages include the simple arithmetic mean and the geometric mean. An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers. If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students. For example, if five students took an exam and their scores were 60%, 70%, 80%, 90%, and 100%, the arithmetic class average would be 80%.

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as "the nth root product of n numbers." The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

The harmonic mean is best used for fractions such as rates or multiples.

Example of the Harmonic Mean

As an example, take two firms. One has a market capitalization of $100 billion and earnings of $4 billion (P/E of 25) and one with a market capitalization of $1 billion and earnings of $4 million (P/E of 250). In an index made of the two stocks, with 10% invested in the first and 90% invested in the second, the P/E ratio of the index is: 

Using the WAM: P/E   =   0 . 1 × 2 5 + 0 . 9 × 2 5 0   =   2 2 7 . 5 Using the WHM: P/E   =   0 . 1   +   0 . 9 0 . 1 2 5   +   0 . 9 2 5 0   ≈   1 3 1 . 6 where: WAM = weighted arithmetic mean P/E = price-to-earnings ratio \begin{aligned}&\text{Using the WAM:\ P/E}\ =\ 0.1 \times25+0.9\times250\ =\ 227.5\\\\&\text{Using the WHM:\ P/E}\ =\ \frac{0.1\ +\ 0.9}{\frac{0.1}{25}\ +\ \frac{0.9}{250}}\ \approx\ 131.6\\&\textbf{where:}\\&\text{WAM}=\text{weighted arithmetic mean}\\&\text{P/E}=\text{price-to-earnings ratio}\\&\text{WHM}=\text{weighted harmonic mean}\end{aligned} Using the WAM: P/E = 0.1×25+0.9×250 = 227.5Using the WHM: P/E = 250.1 + 2500.90.1 + 0.9 ≈ 131.6where:WAM=weighted arithmetic meanP/E=price-to-earnings ratio

As can be seen, the weighted arithmetic mean significantly overestimates the mean price-earnings ratio.