Measuring Central Tendency Involving Ratios - An Intuitive Approach
Common methods for estimating the central tendency of dataset values include the median, mean, harmonic mean, weighted harmonic mean, and geometric mean, among others methods. The focus of this article is on distinguishing the mean, harmonic mean, and weighted harmonic mean from each other.
It is well-established that the harmonic mean or weighted harmonic mean is a better indicator of central tendency when the value being measured is comprised of ratios. Why is this? A quick search may provide something along the lines of “the mean overestimates central tendency because it gives disproportionately greater weight to higher multiple values.” But why is that the case, and why is it problematic when estimating the central tendency of multiples?
For some, taking reciprocals of data may remove some of their intuition of what is being measured when applying these methods. Thus, it may help professionals if they initially abandon the mathematical terminology/equations and view these measures as a sum of capital investment or earnings available in the dataset being measured.
Observe the following sample dataset:
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The ratio’s being measured are purchase price (MVIC) and earnings (FCF). The weighted harmonic mean provides the most basic and arguable intuitive method for going about finding the ratio. This is followed closely the harmonic mean. The mean, however, appears to lack any intuitional value whatsoever and is far-removed from how most investors would go about optimizing their investment stratetgy.
Here is the bottom-line:
Weighted Harmonic Mean - An investor would purchase each of the sample companies at their respective purchase prices and the earnings ratios observed are weighted according to the percentage of the corresponding company’s MVIC relative the the entire samples MVIC. In other words, it does not manipulate the data-set at all. Thus applying greater weight to the higher MVIC companies (and their multiples) and lessor weight to the lessor MVIC companies (and their multiples).
Harmonic Mean: An investor would purchase an equal amount of each company listed in the sample, and the earnings ratios observed are thus weighted equally according to the number of companies provided in the sample. Thus, equal weight is applied to all ratios regardless of the cost required to obtain such earnings. This is why the harmonic mean is typically the best method for measuring central tendency involving ratios.
Mean: An investor would purchase an equal amount of EARNINGS of each company listed in the sample. In other words, the investor purchases the pro-rata amount of each sample company in order to obtain exact same number of earnings provided by each company. In other words, harmonic mean = “equal purchasing” and mean = “equal earnings”.
It would be like saying, you would purchase $10,000 of a high P/E company for every $1,000 you spend of a low P/E company merely because you would like for your earnings to be equal among all of your investments.
When viewed this way, it is the well-known “mean” that lacks intuitional appeal and it is the lessor known harmonic mean and weighted harmonic mean that more accurately capture how an investor would go about making these purchases.
