Maths can be a funny beast. It is mathematically possible, for example, to prove that all four sides of a square are minus two centimetres in length and then of course there is the solution to the so-called Monty Hall problem that you may have come across while reading The Curious Incident of the Dog in the Night-Time and which is pretty hard to swallow.
But surely we are on safe ground with averages? Good old averages on which, seeing as value investment revolves around prices reverting to the mean, pretty much our entire philosophy is based – they at least are secure from the less intuitive aspects of mathematics, yes? The average of, say, a price/earnings (P/E) ratio of 10x and a P/E ratio of 50x is a P/E ratio of 30x, right?
Wrong. Or so at least argues Gerard Minack of Minack Advisors in a piece distinguishing between an arithmetic mean and a ‘harmonic’ mean. Normally such a prospect would see a finger moving swiftly to the ‘delete’ button but we read on a little and then a little more and it was fascinating – one of those moments that has you questioning everything you know, or thought you knew, about something.
While an arithmetic mean adds up numbers and divides the result by the number of numbers, a harmonised mean is the inverse of the average of the inverses. “In an equity context,” says Minack, “the arithmetic mean is the average of the P/E ratios, while the harmonic mean is the P/E implied by the average earnings yield – as the earnings yield is the inverse of the P/E.”
When you think about it, then, since so many aspects of finance are thought of in terms of yield – in addition to earnings yields, two other obvious examples are bond yields and equity yields – using harmonic means makes a lot of sense. But how do they work in practice and how do they reach a different answer from arithmetic means?
There is no place left to go here but an example so let’s return to those two P/E ratios above. Say your portfolio contains two stocks, equally weighted, with one on a P/E ratio of 10x and the other on a P/E ratio of 50x – why should it be that, when asked what the average P/E ratio is, we should henceforth fearlessly answer not 30x but 162/3x?
Since the harmonic mean takes the inverse of the average of the inverses, 10x becomes 1/10 so 10%, and 50x becomes 1/50 so 2%. The average of 10 and 2 is of course 6, which inversed once again gives you a P/E ratio of 162/3x. Well, OK – the maths makes sense so far but why should we believe that is the ‘correct’ answer when every brain cell we possess is telling us it is still 30x?
The way Minack goes about illustrating which is the correct ‘correct’ answer is through a familiar real-world example of driving from home to work and back again. If you drive to work at 50mph and then drive home at 10mph, what was your average speed? The arithmetic mean says it is 30mph but the harmonic mean – as with our P/E ratios example – tells us it is 162/3mph.
“In this example it is straightforward to determine which of these two averages is correct,” Minack assures us. “If it is 50 miles between home and work, then it took one hour to drive to work and five hours to drive home. This is a total of six hours to drive 100 miles – an average speed of 162/3mph. The arithmetic mean is wrong; the harmonic mean is correct.”
Returning to our two-P/E example, say you invested £50 in both stocks. The stock on a P/E of 10x will generate £5 of earnings per share (EPS), while the stock on a P/E ratio of 50x will generates £1 of EPS. That leads to a total EPS figure of £6 generated from an initial outlay of £100, which again proves – at 100 divided by 6 – that the average P/E ratio must be 162/3x. Score one more for the harmonic mean.
Abracadabra – no smoke, no mirrors and all we have hidden up our sleeve is a calculator for the trickier inverse calculations. The important investment point here is that higher numbers can end up having a hugely distorting influence on arithmetic means. Thus, as Minack says, if the average is used as ‘fair value’, equities will appear more overvalued using a harmonic mean than an arithmetic mean.
We can all think of good examples from the past when P/E ratios seemed extraordinarily high – the tech boom of the late 1990s being the almost obligatory reference point here – but just think how much more overvalued some of the leading players in these episodes would have looked if everyone had been using harmonic rather than arithmetic means.
This epiphany does of course raise the possibility of a whole world of secondary analysis and, since having had our eyes opened here on The Value Perspective, we have been busy reweighting all our Graham & Dodd P/E ratio sector analysis to take account of harmonic means. That though is for another day as we suspect that too much more on this subject now would be, well, just plain mean.