Over 2 million + professionals use CFI to learn accounting, financial analysis, modeling and more. Unlock the essentials of corporate finance with our free resources and get an exclusive sneak peek at the first module of each course.
Start Free
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.
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
Related Readings
Thank you for reading CFI’s guide to the Harmonic Mean. To keep learning and advancing your career, the additional CFI resources below will be useful:
Take your learning and productivity to the next level with our Premium Templates.
Upgrading to a paid membership gives you access to our extensive collection of plug-and-play Templates designed to power your performance—as well as CFI's full course catalog and accredited Certification Programs.
Gain unlimited access to more than 250 productivity Templates, CFI's full course catalog and accredited Certification Programs, hundreds of resources, expert reviews and support, the chance to work with real-world finance and research tools, and more.