## Monday, July 24, 2017

### The Case for the Harmonic Mean P/E Calculation

The most recent "analysis" seemingly spreading like wildfire across the perma-bear community was performed by Horizon Kinetics in their most recent quarterly commentary. Their claim is that the price-to-earnings of the Nasdaq (or any index really) is much higher than reported because we are being fed a manipulated harmonic mean rather than arithmetic mean for the price to earnings ratio (don't worry, I'll explain the difference). While the piece also claims excluding non-earners from the calculation is wrong (something I also don't agree with), I'll ignore that portion for now* as it is more nuanced, a separate argument in their piece, and because their specific argument for the arithmetic mean is so clearly wrong.

CASE STUDY #1

Let's start with a case study Horizon Kinetics provides outlining how they believe the P/E for an equal weighted three stock portfolio (with an investment of \$1 million to each) should be calculated.

One business earns \$100,000 per year, so it has a price‐to‐earnings ratio of 10x; the second earns \$50,000, for a P/E ratio of 20, and the third earns only \$20,000 and so has a P/E of 50. This last one is probably situated on a high‐ growth street corner. Averaging the three P/E ratios of 10, 20 and 50 means that the average P/E of the 3‐ company portfolio is 26.7x. So far, so good.
Not a good start...

The 3-company portfolio clearly does not have a P/E of 26.7x when you take a step back and think about what you as an investor own in aggregate. The companies in the case study earn \$100,000 (10% yield on \$1 million) + \$50,000 (5% yield on \$1 million) + \$20,000 (2% yield on \$1 million) = \$170,000, which is a 5.7% yield on \$3 million total investment. A \$3 million total investment divided by \$170,000 of earnings = (1/ 5.7% yield) = a P/E of 17.65x, which is 66% LOWER than their calculation.

The easy way to view the correct harmonic mean calculation is to think about what you own in terms of earnings yield (getting to an average earnings yield and then backing into the P/E is the harmonic mean calculation). In this example:
• Company A: 10x P/E = 10% earnings yield (1/10)
• Company B: 20x P/E = 5% earnings yield (1/20)
• Company C: 50x P/E = 2% earnings yield (1/50)
(10% + 5% + 2%) = average yield of 5.67%. 1/5.67% = the correct 17.65x aggregate P/E.

Visualizing this makes it clearer. The left-hand chart shows the earnings yield for each company, while the right hand chart shows the contribution from each company in total (the earnings of each company divided by the whole \$3 million investment - then stacked). We'll revisit the right hand chart to show how an extreme multiple can overly influence the arithmetic mean of the P/Es when we review their second case study.

CASE STUDY #2

Horizon Kinetic's next case study is worse because the error in the result is so obvious as it includes a company with an extreme high P/E ratio.
Observe the following hypothetical equal‐weighted 4‐stock portfolio consisting of a range of low, somewhat high and egregiously high‐valuations, ranging from 10x to 300x. A simple average results in a portfolio P/E of 90x.
• Company A: 10x P/E or 10% yield
• Company B: 20x P/E or 5% yield
• Company C: 30x P/E or 3.3% yield
• Company D: 300x P/E or 0.33% yield
An average P/E of (10x + 20x + 30x + 300x) / 4 = 90x implies an earnings yield of just over 1% (1/90). Compare this to the average earnings yield of 10% + 5% + 3.3% + 0.33% = 4.67% average, which gets you to a correct aggregate portfolio P/E of 21.4x (1 / 4.67%).

Visualizing this case study again shows their error more clearly. On the right hand side we can see that the earnings contribution of a 25% weight to the first three stocks alone yields more than 4.5% (10% x 25% + 5% x 25% + 3.3% + 25% = 4.575%), so by their rationale the earnings of company D contributes -3% to the overall portfolio (i.e. something akin to company D losing \$140,000 on their \$1,000,000 investment instead of having small, but positive earnings).

And of course, their ridiculous conclusion.
That completes the strange journey of transforming a fairly understandable, if alarming, P/E of 90x into the more comforting Harmonic Mean P/E ratio of only 21.5x.
And the even more bearish takeaway of an investment in the Nasdaq 100.
No active manager would be permitted to manage a concentrated, high‐P/E portfolio for an institutional client.

* you are paying a price to own the lack of historical earnings (which is a case for including these companies), but the fact is these non-earners have often been the fastest growing companies in the Nasdaq, thus including their negative historical earnings ignores their future potential (a case for excluding these companies from the valuations calculation)