What Is the Harmonic Mean?

The harmonic mean is a numerical average used in finance to average multiples like the price-to-earnings ratio. It is calculated by dividing the number of observations, or entries in the series, by the reciprocal of each number. Thus, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

Calculation and Formula

🌱To calculate the harmonic mean of 1, 4, and 4, divide the number of observations by the reciprocal of each number, as follows: 

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

The harmonic mea𝓡n has uses in finance and technical analysis of markets. It helps to find multiplicative or divisor relationships between 🅠fractions without worrying about common denominators. Harmonic means are often used when averaging rates such as the average travel speed over several trips.

Using the Weighted Harmonic Mean

The weighted harmonic mean is used in finance to average multiples like the price-to-earnings (P/E) ratio because it gives equal weight to each data point. A weighted 澳洲幸运5官方开奖结果体彩网:arithmetic mean gives greater weight to high data points than low d꧃ata points because P/E ratios aren't price-normalized while the earnings are equalized.

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

$\displaystyle{\frac{\sum^n_{i=1}w_i}{\sum^n_{i=1}\frac{w_i}{x_i}}}$∑i=1n​xi​wi​​∑i=1n​wi​​

Arithmetic Mean and Geometric Mean

Three types of means, the harmonic, arithm𒐪etic, and geometric, are known as the Pythagorean means. The ✃distinctions between the three types of Pythagorean means make them suitable for different uses.

An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers. To find th𒐪e class (arithmetic) average of test scores, simply add up all the test scores of the students, and then divide that sum by the number of students.

The 澳洲幸运5官方开奖结果体彩网: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 澳洲幸运5官方开奖结果体彩网:portfolio. It is technically defined as "the nth root product of n numbers." The geometric mean must be used when working with 澳洲幸运5官方开奖结果体彩网:percentages, which are derived from values, while the standard ♊;ariꦜthmetic mean works with the values themselves.

Example of the Harmonic Mean

One company has a market capitalization of $100 billion and earnings of $4 bil൩lion (P/E of 25), and the other company has a market caꦺpitalization 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: 

$\begin{aligned}&\text{Using the WAM:\ P/E}\ =\ 0.1 \times25+0.9\times250\ =\ 227.5\\\\&\text{Using the WHM:\ P/E}\ =\ \frac{0.1\ +\ 0.9}{\frac{0.1}{25}\ +\ \frac{0.9}{250}}\ \approx\ 131.6\\&\textbf{where:}\\&\text{WAM}=\text{weighted arithmetic mean}\\&\text{P/E}=\text{price-to-earnings ratio}\\&\text{WHM}=\text{weighted harmonic mean}\end{aligned}$​Using the WAM: P/E = 0.1×25+0.9×250 = 227.5Using the WHM: P/E = 250.1​ + 2500.9​0.1 + 0.9​ ≈ 131.6where:WAM=weighted arithmetic meanP/E=price-to-earnings ratioWHM=weighted harmonic mean​

The weighted arithmetic mean significa💯ntly overeܫstimates the mean price-to-earnings ratio.

Advantages and Disadvantages

The harmonic mean incorporates all the entries in the series and remains impossible to compute if any item is disallowed. Using the harmonic mean allows a more significant weighting to be given to smaller values in the series, and it can also be calcula🏅ted for a series that includes negative values. In comparison with the arithmetic mean and geometric mean,ꦑ the harmonic mean generates a straighter curve.

However, there are also a few downsides to using the harmonic mean. It requires using the reciprocals of the numbers in the series, so the calculation of harmonic mean can be complex and time-consuming. It is also not feasible to calculate the harmonic mean if the series contains a zero value. Finally, any ༒extreme values on th✤e high or low end of the series have an intense impact on the results of the harmonic mean.

The Bottom Line

The harmonic mean is calculated by dividing the number of entries in a series by the reciprocal of each number. The harmonic mean stands out from the other types of Pythagorean mean—the arithmetic mean and geometrical mean—by using reciprocals and giving greater weight to smaller values.