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

In an uncertain world, risk is unavoidable, a fact that underpins the entire insurance industry. We are often making judgements based on probability, whether we know it or not. In mathematical terms, the probability of an event lies between 0 (the impossible event) and 1 (the certain event). Why can’t probability be greater than one? Well that would require us to have an event that is more probable that a certain event: but more of that later. Beyond these simple ideas, life gets very complicated.

Suppose you buy a new phone costing $1,000 and the shop offers insurance for $10 per month, just in case something happens. One could guess at the probability of such an event. In effect, the price you are offered is an implied probability that you will lose or break your phone in the next month with probability P = 10/1000 = 0.01. If you buy this insurance you are, in effect, estimating that the probability you lose your phone in the next month is 1% or more. We say that the cost of insurance is a ‘fair price’ if the implied probability (in this case 0.01) is equal to the true probability. Of course, no one knows the true probability, although the insurance company might be able to estimate this based on previous claims. In practice, the insurer offers a flat rate rather than trying to model the uncertainty. The shop acts as a broker, after selling you the insurance for $10, buys the same insurance for $8 from an insurer. This is a simple form of arbitrage trading in which the shop buys and sells, at different prices, on the same market to ensure that the shop makes a risk-free profit, with certainty.

Like any market, the customer aims for the lowest price available. Conversely, the insurance company must avoid quoting prices at values so low that they risk going bankrupt through claims (payouts) exceeding the premiums (income). Insurance prices are dictated more by the market than statistical data. Even if we had a large amount of data, it would be very difficult to determine what is uncertain and what is random. There is a myth that a cornucopia of data leads to information and wisdom about uncertainty. However, probability can be an expression of uncertainty as well as randomness, and one can express this in financial markets.

There is a myth that a cornucopia of data leads to information and wisdom about uncertainty.

Probability as uncertainty is a radically different way of thinking. It is called the Bayesian approach. Thomas Bayes was an 18th century Presbyterian minister and philosopher, who combined logic with theology as a university student. In fact, his ideas were given prominence by the Marquis de Laplace, one of the intellectual giants of that century. Though, it’s not clear whether Bayes would have approved of Laplace’s remark to Napoleon on presenting his theory of the solar system, that God was not necessary as a hypothesis.

The Bayesian approach is a process of discovery: estimates of probability are not pure guesswork, but rather conditional probabilities based on information. A prior belief (or probability) is modified by new information to a posterior probability. Unearthing the connection between known information and uncertainties of events is at the heart of the Bayesian approach to data analysis.

The upside to this Bayesian method is that one can apply probability to any number of scientific problems. The downside is that these probabilities (based on incomplete knowledge) may be entirely subjective, speculative and, at worst, completely wrong. There is another ’fly in the ointment’ with this approach. If, for example, in your opinion, the probability of your bus arriving in the next five minutes is 10%, in order to be consistent then you must logically conclude that the probability that it does not arrives in the next five minutes is 90%. The requirement that probability estimates (whatever their veracity) should be self-consistent is called coherence in this context. The more probability estimates you make, the more effort is required to tidy up the consequences of these guesses. In financial markets, of many assets and instruments, correlations between these prices will often occur.

Any imbalances, or incoherence, in these prices can be exploited for risk-free trading (arbitrage). Trading strategy based on this technique is called statistical arbitrage and is a favorite tool of hedge funds. That is, one is trading in probability, not in terms of prices as a game of skill. A quotation often attributed to Napoleon comes to mind, when offered the services of an expensive fund manager: “Is he skillful, or is he lucky?” ■

November 2020

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