# Harmonic Mean

A type of average that is calculated by dividing the number of values in the data series by the sum of reciprocals of each value in the series

## What is Harmonic Mean?

Harmonic mean is a type of average that is calculated by dividing the number of values in a data series by the sum of the reciprocals (1/x_i) of each value in the data series. A harmonic mean is one of the three Pythagorean means (the other two are arithmetic mean and geometric mean). The harmonic mean always shows the lowest value among the Pythagorean means. The harmonic mean is often used to calculate the average of the ratios or rates. It is the most appropriate measure for ratios and rates because it equalizes the weights of each data point. For instance, the arithmetic mean places a high weight on large data points, while the geometric mean gives a lower weight to the smaller data points.

In finance, the harmonic mean is used to determine the average for financial multiples such as the price-to-earnings (P/E) ratio. The financial multiples should not be averaged using the arithmetic mean because it is biased toward larger values. One of the most common problems in finance that uses the harmonic mean is the calculation of the ratio of a portfolio that consists of several securities.

### Formula for Harmonic Mean

The general formula for calculating a harmonic mean is:

Harmonic mean = n / (∑1/x_i)

Where:

• n – the number of the values in a dataset
• x_i – the point in a dataset

The weighted harmonic mean can be calculated using the following formula:

Weighted Harmonic Mean = (∑w_i ) / (∑w_i/x_i)

Where:

• w_i – the weight of the data point
• x_i – the point in a dataset

### Example of Harmonic Mean

You are a stock analyst in an investment bank. Your manager asked you to determine the P/E ratio of the index of the stocks of Company A and Company B. Company A reports a market capitalization of \$1 billion and earnings of \$20 million, while Company B reports a market capitalization of \$20 billion and earnings of \$5 billion. The index consists of 40% of Company A and 60% of Company B.

Firstly, we need to find the P/E ratios of each company. Remember that the P/E ratio is essentially the market capitalization divided by the earnings.

P/E (Company A) = (\$1 billion) / (\$20 million) = 50
P/E (Company B) = (\$20 billion) / (\$5 billion) = 4

We must use the weighted harmonic mean to calculate the P/E ratio of the index. Using the formula for the weighted harmonic mean, the P/E ratio of the index can be found in the following way:

P/E (Index) = (0.4+0.6) / (0.4/50 + 0.6/4) = 6.33

Note that if we calculate the P/E ratio of the index using the weighted arithmetic mean, it would be significantly overstated:

P/E (Index) = 0.4×50 + 0.6×4 = 22.4