Harmonic Mean

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What is a harmonic mean?

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

The harmonic mean of 1.4 and 4 is:


3(11 + 14 + 14) = 31.5 = 2 frac {3} { left ( frac {1} {1} + frac {1} {4} + frac {1} {4} right)} = frac {3 } {1,5} = 2

(11The + 41The + 41The)3The = 1.53The = 2The

[Important: the reciprocal of a number n is simply 1 / n.]

The basics of a harmonic mean

The harmonic mean allows to find multiplicative or divisive relationships between fractions without worrying about common denominators. Harmonic means are often used to average things like fares (for example, average travel speed given the duration of multiple trips).

The weighted harmonic mean is used in finance to average multiples like the price-benefit ratio because it gives equal weight to each data point. Using a weighted arithmetic mean to average these ratios would give more weight to high data points than to weak data points, since price-benefit ratios are not normalized to prices while benefits 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 corresponding weights w1, w2, w3 is given as:


ΣI=1notwIΣI=1notwIXI displaystyle { frac { sum ^ n_ {i = 1} w_i} { sum ^ n_ {i = 1} frac {w_i} {x_i}}}


Key points to remember

  • The harmonic mean is the inverse of the arithmetic mean of the inverses.
  • Harmonic means are used in finance to average data like price multiples.
  • Harmonic means can also be used by market technicians to identify patterns such as Fibonacci sequences.

Harmonic mean versus arithmetic mean and geometric mean

Other ways to calculate the means include the simple arithmetic mean and the geometric mean. An arithmetic mean is the sum of a series of numbers divided by the number of that series of numbers. If you were asked to find the (arithmetic) average of the test scores, you would simply add up all the student scores and then divide that sum by the number of students. For example, if five students took an exam and their results were 60%, 70%, 80%, 90% and 100%, the average for the arithmetic class 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 not “The geometric mean should 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 harmonic mean

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


WAM usage: P / E = 0.1×25+0.9×250 = 22seven.5WHM usage: P / E = 0.1 + 0.90.125 + 0.9250 131.6or:WAM=weighted arithmetic meanP / E=price / earnings ratio begin {aligned} & text {Use of WAM: P / E} = 0.1 times25 + 0.9 times250 = 227.5 \\ & text {Use of 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 / profit ratio} \ & text {WHM} = text {weighted harmonic mean} fin { aligned}

TheWAM usage: P / E = 0.1×25+0.9×250 = 22seven.5WHM usage: P / E = 250.1The + 2500.9The0.1 + 0.9The 131.6or:WAM=weighted arithmetic meanP / E=price / earnings ratioTheThe

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

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