The Greek Gifts?

Commentary by Dr. James McCann, Chief Scientist, Plotinus Asset Management

The first week at university is disorientating for most new arrivals. One of the surprises that mathematics students find in the mass of paperwork is a one-page listing of the Greek alphabet along with a phonetic guide on how to pronounce these strange characters. This alphabet appeared in the 8th Century BCE when the Greeks adapted the Phoenician (from where we get phonetic) script and added vowels. Vowels were previously considered as unnecessary filler in written text.

Mathematics students are not being trained in classics so that they can quote Aeschylus or Aristotle. Rather they are being introduced to a symbolic notation that pervades mathematics around the world. The ubiquity of the symbols is not intended to make life complicated, although it often offers a veneer of sophistication to otherwise simple concepts, it simply helps distinguish symbols from letters, in particular the Roman alphabet. This symbolic notation then became a convention in certain topics, so it is not necessary to look up a table every time a letter appears. For example in statistics the symbol μ (pronounced ‘mew’ like a cat) represents the mean or average value, while in Physics it is used for the magnetic permeability of a material, as well as the label of the muon, a particle that is heavier-than-an-electron but lighter-than-a-proton.

For equity investors, two Greek letters have special meaning: alpha, α, and beta, β. The reason is quite simple. Suppose you have an investment which produces a series of monthly yields, with values s1, s2… and so on. It is natural to benchmark your investment against another. Suppose that the benchmark gives yields, over the same period, with corresponding values r1, r2… etc. Then we can relate the yields by a linear regression (trend line): si = α + βri + εi. And this is how one defines alpha and beta. Note that the roman letters r and s are variables (taken from the data) whereas α and β are parameters imposed externally by the model (the linear regression we have chosen). Note that I am being inconsistent, the symbol ε (epsilon) is a Greek letter but is actually a variable. Clearly epsilon did not get the memo that the dress code was for a toga party. This “error term” is of vital importance, because if it is not small, then the model is a poor approximation.

If only the dividend to our understanding of the human condition was as powerful as our α is to our bank account.

Suppose your investment is in shares of Netflix and your benchmark is the S&P 500, then over the last three years you will have 36 data points to fit to the trend line, giving you α=2.41% (per month) and β=+0.833 with ε around 10%. The value of β tells you that your investment is strongly correlated to the market, which can be good or bad depending how the market did. If the market went up by 10% you would get (on average) 8.3 % in concert with this rise, and vice versa if it fell. The value of ε is so big that the model is quite poor, but at least you can surmise that this is a good investment given that α, your steady baseline performance, is very good. You can think of β as your exposure to the volatility of the market. A purely passive tracker fund, which replicates the index, would have α=0 and β=1, for example. A slight negative α might indicate the amount of investment charges taken by the fund manager.

Active investors looking to outperform the index generally look for “big alpha” or “smart beta” strategies. Smart beta is a hybrid approach to stock picking, in which the aim is to diversify, but outperform the market, that is a β greater than 1. The investment is weighted among a group of stocks for which the correlations are assumed known. This is usually more volatile that the passive investment but less risky than choosing just a random handful of stocks.

The meaning of “big alpha” is obvious. It is an approach that aims to decouple your investment returns from the market (so it could also be called small beta). The aim of this method is to reduce the volatility of the market yet outperform the passive funds. This is the approach that Plotinus favors.

Reducing volatility can be achieved by the judicious use of hedging using options, for example. This brings up a whole different set of “greeks,” for example the Δ (delta) which expresses the appropriate ratio of stock to options in order to reduce uncertainty. Somewhere along the line we ran out of greek letters so invented new ones such as vega and vomma!

And it all started with the simple ideas of Euclid’s γεωμετρία (geometria). Geometry was highly-prized by the Greek philosophers. According to some, the entrance to the Academy founded by Plato in Athens had engraved above the entrance “Let no one ignorant of geometry enter.” The geometry of Euclid was founded on logic and proof, a practice that is still being applied to modern philosophy. If only the dividend to our understanding of the human condition was as powerful as our α is to our bank account.

March 2021

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Image shows the Alexander Sarcophagus. Credit: Muharrem Zengin at at 123RF.

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