Probability, life expectancy, certainty, normal distribution, utility, and inference were the six crucial breakthroughs in insurance, narrates Niall Ferguson in ‘The Ascent of Money: A financial history of the world’ (www.penguin.com). Though insurance in some form or the other existed even during the fourteenth century, it was in a remarkable rush of intellectual innovation, beginning in around 1660, that the theoretical basis for insurance was created, he says.
A key name associated with probability is Blaise Pascal, the French mathematician, who wrote in ‘Ars Cogitandi’ (1662) that ‘fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event,’ attributing the insight to a monk at Port-Royal. Pascal and his friend Pierre de Fermat had been toying with problems of probability for many years, but for the evolution of insurance, this was to be a critical point, Ferguson writes.
Around the same time as Pascal’s book, John Graunt published his ‘Natural and Political Observations… Made upon the Bills of Mortality,’ where he sought to estimate ‘the likelihood of dying from a particular cause on the basis of official London mortality statistics.’ A finer insight came from his fellow member of the Royal Society, Edmund Halley, whose life table, ‘based on 1,238 recorded births and 1,174 recorded deaths, gives the odds of not dying in a given year.’
One of the founding stones of actuarial mathematics, as the author observes, is the statement of the odds of not dying in a given year, thus: ‘It being 100 to 1 that a man of 20 dies not in a year, and but 38 to 1 for a man of 50…’
Certainty, the third advancement, was from Jacob Bernoulli, who proposed in 1705 that ‘under similar conditions, the occurrence (or non-occurrence) of an event in the future will follow the same pattern as was observed in the past.’
Bernoulli’s ‘Law of Large Numbers’ stated that inferences could be drawn with a degree of certainty about, for example, the total contents of a jar filled with two kinds of ball on the basis of a sample, Ferguson explains. “This provides the basis for the concept of statistical significance and modern formulations of probabilities at specified confidence intervals.”
As for normal distribution, the next step forward, Abraham de Moivre, showed that ‘outcomes of any kind of iterated process could be distributed along a curve according to their variance around the mean or standard deviation.’
Sample this snatch from what he wrote in 1733: “Tho’ Chance produces Irregularities, still the Odds will be infinitely great, that in process of Time, those Irregularities will bear no proportion to recurrency of that Order which naturally results from Original Design.”
The fifth progress, utility, is related to the Swiss mathematician Daniel Bernoulli who in 1738 argued that ‘the value of an item must not be based on its price, but rather on the utility that it yields’ and that the ‘utility resulting from any small increase in wealth will be inversely proportional to the quantity of goods previously possessed.’ In other words, as Ferguson puts it, ‘$100 is worth more to someone on the median income than to a hedge fund manager.’
Finally comes ‘inference,’ which attacked the following problem, as outlined in ‘Essay Towards Solving a Problem in the Doctrine of Chances’ of Thomas Bayes (published posthumously in 1764): “Given the number of times in which an unknown event has happened and failed; required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named.”
And, this is how Bayes resolved the problem: ‘The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the chance of the thing expected upon its happening.’ This, as Ferguson finds, anticipated the modern formulation that expected utility is the probability of an event times the payoff received in case of that event.
Great company for finance professionals, during an extended weekend.