We Know Zero

I received a set of nontransitive dice in the post this week. Transitive is an interesting word. As we all know in grammar it refers to verbs which do things to something. What I didn’t learn at school was that if they do things to one thing they are called monotransitive, and ditransitive if they have both a direct and indirect object. A verb like to trade is categorised as tritransitive. If a verb does not play with others it is called intransitive, eg an example appropriate to this story, to die. If a verb swings both ways it is called ambitransitive.

In the mathematical world transitive is a description of a relation on a set. For example, if A = B and B = C, then A = C. So = is transitive. Similarly, if A > B  and  B > C  then  A > C.

Or does it? Let’s return to the dice (singular die: cemented in my memory on the occasion a teacher responded to a boy coming into his class and asking to borrow a dice by shouting “die, die, die!” at the startled youngster). Mathematicians do not use the word intransitive, preferring perhaps to avoid the ambiguity of words like flammable and inflammable, but instead use nontransitive. Nontransitive dice have the property that if die A tends to beat die B on average, and die B tends to beat die C on average, then rather counter-intuitively die C tends to beat die A on average. How does this work?

There are many different arrangements of the numbers on the faces of the dice which would achieve this effect. My red die has 4 on all its faces except one, which has a 6. My blue die has half its faces with 2s and the other half with 6s. My green die has 5 on all its faces except one, which is unnumbered (or, in fact, undotted).

If we take the average number we expect to get when throwing each die (the concept of expected value, first introduced by Blaise Pascal of triangle fame, also known as the mean, is the first thing that tends to get calculated in any statistical analysis), then red gives us 4⅓, blue gives us 4 and  green     4 1/6. So we would expect from that to see red beat blue, green beat blue and red beat green.

When we pitch red against blue, if we throw a 2 with the blue die (probability of a ½), then we will always lose to red, since all of its faces are greater than 2. If we throw a 6 with blue, we have a 5/6 chance of beating red (since 5 of its 6 faces are 4s) and a 1/6 chance of drawing. So we have for blue a probability of ½ of losing, a probability of ½ x 5/6 = 5/12 of winning and a probability of ½ x 1/6 = 1/12 of drawing. So, in the long run, red beats blue on average, as we would expect it to.

When we pitch blue against green, blue will always win if we throw a 6 with it, with a probability ½. If we throw a 2, also with a probability ½, we have a 1/6 chance of winning against green (if green’s single blank face comes up) otherwise we will lose to a 5. So we have for blue a probability of losing of ½ x 5/6 = 5/12. And the probability of winning as blue (since no draws are possible this time) of 1 – 5/12 = 7/12. So, in the long run, blue beats green, exactly the opposite of what we would expect just going on the expected values.

Finally, when we pitch red against green, the only time green will beat red is when red has a 4 (with probability 5/6) and green has a 5 (also with probability 5/6). So we have a probability of green beating red of 5/6 x 5/6 = 25/36. And the probability of winning as red (since again no draws are possible as the two dice have no numbers in common) is therefore 1 – 25/36 = 11/36. So, in the long run (when as Keynes once helpfully pointed out, we are all dead) green beats red, again exactly the opposite of what we would expect just going on the expected values.

We only had to mess around a little with the 6 faces of the dice to get this counter-intuitive result. Nearly all financial instruments and products are obviously much more complicated than this, with the probabilities of certain outcomes being largely unknown, and even more so when in combination with each other, and therefore counter-intuitive results turn up almost too frequently to be called counter-intuitive any more. In fact the habit of trying to treat financial markets as if they were games obeying rules as fixed and obvious as those you can play with dice is what Nassim Nicholas Taleb refers to as the Ludic Fallacy.

If we double them up we get another surprise. Red still has the highest expected value (8⅔), followed by green again (8⅓) and then blue (8). But this time each pairing has three possible outcomes. Red and green both beat blue as expected from the expected values, but then green unexpectedly beats red.

This kind of behaviour is called nonlinearity, when adding quantities of things together does not just increase their effects, but instead changes them. Nonlinearity in this case means that blue beats green when we use one die each, but that green beats blue when we use two. Nonlinearity is also the single biggest threat to the financial system.

Anyone for darts instead?

I think if I were to ask you what you thought the best way to manage risk was, there would be a significant risk that you would give me a very boring answer. I imagine it would involve complicated mathematical valuation systems, stochastic models and spreadsheets, lots of spreadsheets, risk indicators, traffic light arrangements, risk registers. If you work for an insurance company, particularly on the actuarial side, it would be very quantified, with calculations of the reserves required to meet “1 in 200 year” risks featuring heavily. Recently even operational risk is increasingly being approached from a more quantifiable angle, with Big Data being collected across many users to pool and estimate risk probabilities.

Now you can argue about these approaches, and particularly about the Value at Risk (VaR) tool which has brought this 1 in 200 probability over the next year into nearly every risk calculation carried out in the financial sector, and the Gaussian copula which allows you to take advantage of a correlation matrix to take credit for the “fact” that combinations of very bad things happening are vanishingly rare (the “Gaussian” referring to the normal distribution that makes events more than three standard deviations or “sigma” away from the average vanishingly rare), rather than actually quite likely once the market environment gets bleak enough. The losses at Fortis and AIG 2008 were over 5 and 16 sigma above their averages respectively.

The news last week that the US Attorney for the Southern District of New York had charged two JP Morgan traders with fraud in connection with the recent $6.2 billion “London whale” trading losses reminded me that VaR as it is currently used was largely cooked up at JP Morgan in the early 90s. VaR is now inescapable in the financial industry, having now effectively been baked into both the Basel 2 regulatory structure for banks and Solvency 2 for insurers.

The common approaches to so-called “quantifiable” risk may have their critics, but at least they are being widely discussed (the famous debate from 1997 between Phillippe Jorion and Nassim Nicholas Taleb just one such discussion). However, one of the other big problems with risk management is that we rarely get off the above “boring” topics, and people who don’t get the maths often think therefore that risk management is difficult to understand. In my view we should be talking much more about what companies are famous for (because this is also where their vulnerability lies) and the small number of key people they totally rely on (not all of whom they may even be aware of).

If you asked most financial firms what they were famous for, I imagine that having a good reputation as a company that can be trusted with your money would score pretty highly.

A recent survey of the impact of the loss of reputation amongst financial services companies on Wall Street revealed that 44% of them lost 5% or more in business in the past 12 months due to ongoing reputation and customer satisfaction issues. Losses based on total sales of these companies are estimated at hundreds of millions of dollars. There was an average loss of 9% of business among all companies surveyed.

And the key people we totally rely on? Well, just looking at the top five rogue traders (before the London Whale), we have:

1. SocGen losing 4.9 billion Euros in 2008 when Jerome Kerviel was found guilty of breach of trust, forgery and unauthorized use of the bank’s computers in their Paris office with respect to European Stock Index futures.
2. Sumitomo Corp losing $2.6 billion in 1996 when Yasuo Hamanaka made unauthorised trades while controlling 5% of the world’s copper market from Tokyo.
3. UBS losing $2.3 billion in 2011 when Kweku Adoboli was found guilty of abusing his position as an equity trade support analyst in London with unauthorised futures trading.
4. Barings Bank losing $1.3 billion in 1995 when Nick Leeson made unauthorised speculative trades (specifically in Nikkei Index futures) as a derivatives broker also in London.
5. Resona Holdings losing $1.1 billion in 1995 when Toshihide Iguchi made 30,000 unauthorized trades over a period of 11 years beginning in 1984 in US Treasury bonds in Osaka and New York.

None of these traders will, of course, have done anything for the reputations of their respective organisations either.

These are risks that can’t be managed by just throwing money at them or constructing complicated mathematical models. Managing them effectively requires intimate knowledge of your customers and what is most important in your relationship with them, who your key people are (not necessarily the most senior, Jerome Kerviel was only a junior trader at his bank) and what they are up to on a daily basis, ie what has always been understood as good business management.

And that doesn’t involve any boring mathematics at all.