The Investor Who Could Only Count to Five
Professor Pierre Pica is a linguist at France’s National Centre for Scientific Research. For 10 years he has been researching a Munduruku tribes. The Munduruku are an indigenous group of about 7,000 to 11,000 people living in the Amazon River basin. The Munduruku are subsistence farmers and hunter-gatherers, living in small villages across a rain forest twice the size of Wales. The focus of Pica’s research has been the Munduruku language: it has no tenses, no plurals and no words for numbers beyond five.
Because of the Munduruku’s natural way of life, that doesn’t involve any commercial trade, they have no need for money, or any major engineering projects, so they never evolved counting skills. In comparison, as Alex Bellos states in his book Alex’s Adventures In Numberland,
it has been argued that the appearance of numbers was triggered by elaborate customs of gift exchange. The Amazon, by contrast, has no such traditions.
When Alex asked Pica how the Munduruku would say ‘six fish’, he replied that ‘It is impossible,’ and that ‘The sentence “I want fish for six people” does not exist.’
Pica is studying the innate mathematical abilities of the Munduruku to ‘discover the nature of our basic numerical intuitions.’ He wants to find out what is universal human intuition, and what is learned through culture. In one of Pica’s experiments he analyses their spatial understanding of numbers. Alex writes,
How did they visualize numbers when spread out on a line?
This seems like a trivial question. Something we almost take for granted in our more engineered world so common it is — roads, parking bays, charts, buildings and object sizes, rulers, and so forth. Pica tested the Munduruku by presenting a screen with a slider with one dot on the left and ten dots on the right end. Each individual was then shown several sets of of dots consisting of a minimum of one and a maximum of ten dots. He or she then had to point at where on the slider they thought the number of dots should be lied. Collating the individual submissions, Pica could see exactly how the Munduruku spaced numbers between one and ten.
An interesting thing happened. The Munduruku reduced the intervals as the number of dots increased, so that the distance between the one dot set and the two dot set was much greater than that between the eight dot set and the nine dot set.
In 2004, in a completely separate research project, Robert Siegler and Julie Booth at Carnegie Mellon University in Pennsylvania conducted a similar experiment with a group of kindergarten pupils, first-graders and second graders. The kindergarten pupils, mapped out the dots in the same way as the Munduruku. As the pupils gained experience with maths, number words and symbols, the intervals became more and more even, until they were completely evenly laid out along the line.
The point is that both the Munduruku and the younger children viewed the dots in logarithmic scale. As we learn more formal mathematics, we think more and more in linear or arithmetic scale, until that becomes the default way in which we view the world.
So the next question is, why do the Munduruku and the young ‘children think that higher numbers are closer together than lower numbers?’ It seems they decide
where numbers lie based on estimating the ratios between amounts. When considering ratios, it is logical that the distance between five and one is much greater than the distance between ten and five. And if you judge amounts using ratios, you will always produce a logarithmic scale.
The Munduruku Investor
Yet, for all the seeming importance of, and interest in precise numbers of the modern world, when it comes to the figures that matter most, the logarithmic scale still rules the financial world. Why is that you ask? For the simple reason that finance is obsessed with the percentage change of, well, just about anything related to money.
Just think about all the various ratios related to company and stock performance: Rate of Return, P/E, Return on Investment, Net Profit Margin, Current Ratio, Quick Ratio, revenue growth rate, income growth rate, Dividend Yield, to name a few.
Consider the Annualized Rate of Return of a financial security. If I told you I made a 15% return it sounds great initially. But if I told you later I had to hold the security ten years to earn it, suddenly the performance doesn’t sound that great anymore. This can be seen in the Annualized Rate of Return calculation:
(((Initial Value + Profit or Loss)/Initial Value)^(1/n))-1
From the table of Rate of Returns over a 10 year period, it’s easy to identify the logarithmic pattern. Obviously with each new year that passes the returns diminish. As you hold the investment longer and longer, your returns actually get less and less. What’s interesting is that the rate of change also gets less and less. As stated earlier when you work with any kind of ratio you will always produce a logarithmic scale. So from year 1 to 2, the change is almost 60%. Compare this to the change from year 9 to 10, that’s just above 10%. So even though the absolute end value of the security stays the same, keeping it longer reduces the true return. Here are also some hints that the passing of time causes a logarithmic change in events.
So what role does logarithmic scale have in the specific financial field of security trading? Well, based on my research on the web, and according to Charles Kirkpatrick and Julie Dahlquist (in their excellent book Technical Analysis), most traders use arithmetic scale. It turns out that logarithmic scale is preferred when
observing long-term price movements. For example, compare
the following two figures.
Both plot the monthly price for Clear Channel Communication (CCU). In January 1995, CCU was trading for under $10 a share; at that time, a $1 increase in price would represent approximately a 10% gain for the investor. By January 2000, CCU’s price had advanced to almost 100$ a share. At that point, a $1 price increase represented only about a 1% gain for an investor owning CCU. On the arithmetic, a $1 price movement is visually the same whether it is a move from $10 to $11 or a move from $99 to $100. This type of scale can be somewhat deceptive; a $1 move is much more significant to an investor if the price of a security is $10 than if the price of the security is $100. The logarithmic scale addresses this issue.
When security prices, on the vertical Y axis of a chart, is in the logarithmic scale each interval represents the same percentage change in price.
The rule of thumb for when to use an arithmetic or logarithmic scale is that when the security’s price range over the period being investigated is greater than 20%, a logarithmic scale is more accurate and useful. As a rule, the truly long-term charts (more than a few years) should always be plotted on logarithmic scales.
For some further reading, see James Hamilton’s short and sweet write-up about logarithmic scale in economics.
What About Log Time?
Thinking about charts, and the logarithmic scale of asset prices, made me wonder about the logarithmic scale of a chart’s x axis — for time. It turns out some very clever scientists have also thought much about this. In turbulence theory, time is considered logarithmic, as William K. George explains in his 2016 research paper, titled “Could time be logarithmic?”
There it is the growth of length scales with time in the decaying homogeneous turbulence which dictates that turbulence evolves in logarithmic time increments. In effect ‘time’ slows down as the turbulence decays.
William goes on to explain that this concept might also apply to the universe, and if they do
what we think we perceive as the universe expanding might simply be the scales growing — as in the turbulence solutions.
Further, he states that he is actually investigating something different, where time itself physically functions logarithmically, not only its scales and they way we perceive it:
The most important difference in a log-time universe from a linear time one
would be that we must modify our definition of mass. And of course velocity
and acceleration as well. This will be seen to have no measurable consequences over the span of our human existence. But it has great consequences for how we interpret the results of applying our physical laws to astronomical observations of events that happened long ago. In particular, when viewed using linear time, mass will appear to us to be missing, even when it is not. The gravitational constant will appear to be time-dependent. And the speed of light will be slowing down. Also the universe might appear to be expanding, even if it is not.
The interesting part is that this seems similar to the long term price series of a security. Relative to the subject domain, it seems that logarithmic scale has a bigger impact the longer the time span gets. For astrophysics this means billions of years (as in since the Big Bang, 13.7 billion years ago), and for trading this probably means something like 5 years or longer.
