Againstthegods

"Against the gods" is a beautiful narrative on the history of risk management. Peter L Bernstein , the author of the book has a done a terrific work of narrating the evolution of risk measurement. The book is divided in to 5 periods where the story of risk is presented.

The five demarcated periods are Uptil 1200 ,1200-1700,1700-1900,1900-1960,Post 1960.

In each of these periods, the author talks about various personalities involved. Let me recap the book in the same manner, listing the main items from each of these time periods.

Firstly, something about the title of the book " against the gods" . It is so named  because the author brings out a pattern in his narrative i.e, through the history of the development of risk, there was one powerful idea that galvanized the development of risk, the idea  that humans are in control of their destiny Vs Gods who control the destiny.

Pre-1200
Numbers Era
Greeks had an immense interest in gambling and hence probability theory was the natural thing for them to take upon and explore. Yet Greeks never took up probability and worked on it. One of the reason that author surmises is the fact that greeks were of the belief that the world was controlled by god and any study to control the universe in whichever manner would only result in a futile exercise.
Pre-1200's was a period which was characterized by folks trying to understand and formulate numbers.

Fibonacci
Fibonacci

Leonardo Pisano (Fibonacci) wrote a called  Liber Aabaci which was the first treatment on the theory and application of various aspects of numbers. This was also a period when the development of 0 made a significant impact in the way numbers were used

Period 1200 – 1700
Outstanding facts Era

Renaissance Gambler:
Pacioli

Lucal Pacioli was the first person to give an exhaustive treatment to accounting in his book summa.He is also the first accountant in the human history.  In his book summa, he gave tables for 60*60 multiplication operations. He was a numbers man and he posed the most famous problem of all times, the problem of balla.
A and B are playing a fair game of balla. They agree to continue until one has won 6 rounds. The game actually stops when A has won 5 and B has won three. How should the stakes be divided

Cardan_3.jpg Cardano

A physician named Cardano was an eternal gambler. He gambled every day of his life and he had seen so many gambling games in his life that he wanted a set of rules for playing the game based on the odds of various outcomes. He wrote a breat book on mathematics Ars Magna ( The great art) which was followed by Liber de Ludo Algae( Book on games of chance). This appears to have been the first serious effort to develop statistical principles of probability. Probability always had 2 meanings one looking in to the future, the other interpreting the past, he former cncerned with our opinions and the latter concerned with what we actually know. However the idea of measuring probability came later which means " How much can we accept of what we know ? . In a sense the book  Liber de Ludo Algae was a primer to risk management. Cardano is credit for an bringing a new terminology like fair dice, circuit, combinations,odds ratio etc. Interestingly , the word "fair dice" came in to being because he had spent years on the gambling table and he could see how various players cheated.  However his book was not accessible for a lot of mathematicians in the renaissance time for various reasons

French Connection :
Pascal_2
Pascal  Chevalier Chevalier Fermat_5Fermat

 There are three important French  personalities who played a significant role in the development of probability theory. First, was Blaise pascal , an outstanding mathematician whose work on cones at a young age of 16 brought great praise for his intellectual faculties. The other was Fermat whose work on the theory of numbers is by far the most comprehensive work done by an individual. He is more popularly known for his last theorem which mathematicians have struggled to solve it for about 350 years.

These two mathematicians were great in their respective fields but it was Chevalier De mere , a noble man, a person with keen interest in gambling and mathematics , posed the old problem of balla. In a series of communications between Pascal and Fermat, Pascal came up with a triangle , popularly referred to as Pascal's triangle to calculate the odds for solving problem of balla. This was the first time  a mathematical tool was used to forecast, in this case, the prize money in the game. Pascal's triangle was a neat way to summarize the events that could happen in a probabilistic sense. For example if there are 5 games are to be played between 2 folks, then the 2 power 5 = 32 corresponds to the 5th row in the triangle from which one can read different types of events that happen.

Remarkable Notions Man:

Graunt
Graunt   Petty
Petty

John Graunt , a merchant and William Petty were folks who used statistical inference techniques. Graunt was a man who was obsessed with verifying common every day notions. With the help of Petty, he developed a method of drawing inferences from a small sample  .Both were extremely interested in the organization of human society rather than the science of nature However they never used the word probability . Estimating the odds of uncertain events had to wait until 1700-1900 , a period appropriately titled "Measurement Unlimited"

Period 1700 – 1900
Measurement Unlimited Era

Meet the Bernoulli family:

Bernoulli family is considered as a family which had a swarm of mathematical descendants whose have made immense contributions to the understanding of uncertainty.

Family_tree

Bernoulli_Daniel_3
Daniel Bernoulli

Daniel Bernoulli is credited to have brought in the element of risk taker in to the whole game of risk. He hypothesized that the importance of wealth for an individual is inversely proportional to the amount of wealth accumulated. From the world of simple dice, roulette wheels, the inclusion of the player brought in a whole new dimension to the risk management development. Utility as a concept had a tremendous influence on the way risk management principles were developed in the later years.Petersburg paradox is a classic example of utility concept explained by Bernoulli,

 

Bernoulli_Jacob
Jacob I Bernoulli

Bernoulli was interested in a-posteriori probabilities , i.e computing probabilities of something after fact.His example of a glass jar having 3000 white pebbles and 2000 black is often quoted in literature. The problem goes something like this :
A pebble is drawn, its color is noted and then put it back . How many pebbles need to be drawn so that we  can reasonably certain of the true ration, result should be with in 2% of the true ratio. The answer turns out to be 25,550
Experimenting on the same lines he formulated the Law of Large numbers which says that :Average of large number of throws will be more likely than the average of a small number of throws to differ from the true average by less than some stated amount..

DeMoivre_gr_12
De moivre

Demoivre then picked the concept and formulated the normal curve distribution.The third person who belonged to the same era and contributed to the formulation of a posterior probabilities is Bayes. Though none of his work got published when he was alive, his work had a great influence later. Possibly, the most important contribution of bayes was the precise problem formulation :

Bayes
Bayes

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 ?
Then came gauss who formulated the single most important theorem in statistics, the central limit theorem

 Gauss_1803
Gauss

Gauss was conducting research on geodesic measurements, i.e, the distance between any two points on the earth and the direct distance between the 2 points. As earth has a curvature, the 2 metrics are going to be different and they are going to be different in different places. But what Guass  found an amazing pattern. The average value of errors between actual and observed , even though they were different in various places, the average of average errors followed De moivres bell curve. Thus central limit theorem deals with the average of averages.
In simple words, this theorem says, if you pick a large sample, take its average, do it multiple times, and plot the averages of all the sample, the frequency distribution is a normal distribution..This is an amazing pattern because the actual distribution of random variables can be anything, but sample averages tend to be normal.

MW_Galton
Francis Galton

Galton was an amateur scientist with a keen interest in heredity but with no interest in business or economics.He was a measurement freak and he studied extensively numbers. He treatise on heredity is supposed to have evoked great praise from Charles Darwin. His contribution to statistics is "Regression to mean" . In the long term, the high and low values of a variable stabilize to an average value. He also hypothesized that influences on a variable themselves had to be normally distributed for the dependent variable to be normally distributed. This is nothing but the popular theorem that sum of N normal random variables is another normal random variable.

Period 1900 – 1960
Clouds of Vagueness and the demand for Precision

The essence of risk management lies in maximizing the areas where we have some control over the outcome while minimizing the areas where we have absolutely no control over the outcome and the linkage between effect and cause is hidden from us.Two people from this era wanted to attribute causality to the every day events. One was Laplace and other was Henri Poincare. However both agreed to the fact that not always, there is complete information to attribute the causality.

    

Laplace_6
Laplace Poincare
Poincare

Hence the development of reject or non-reject hypothesis came in to being . It was thought one can never be certain about any thing. only one can reject or not reject a hypothesis with some confidence level. Thus statistical inference and hypothesis testing concepts flourished .

Soon, the winds in the development of risk management started to change after the world war. The so called happy state that was in the imagination of most people was shattered by the destruction all around. More information was only adding to the uncertainty around.Francis Galton died in 1911 and Henri Poincare died the followin year, Their passing marked the end of the grand age of measurement. Subsequently ideas of Keynes becoming wide spread where he hailed uncertainity and critiqued all the classical ways of dealing with uncertainty using law of large numbers .

Markowitz, Harry M
Markowitz

At the same time, a graduate from chicago , Markowitz, applied mathematics to portfolio selection and came up with a model for selecting stocks.  Inspite of many assumptions , this was widely adopted by the street. For quite some time, the notion that investors are rational was in vogue, when a professor from chicago advocated the behavioral aspect of investing, which opened up a new branch of economics called behavioral economics/ finance. Thus the stage was set for understanding the degrees of belief and exploring uncertainty

Post 1960
Degrees of Disbelief & Exploring uncertainty

The last part of book focuses on prospect theory, derivatives such as futures and options to tame uncertainty.

I have tried to give a pretty elaborate summary of the book. However there are a lot of aspects which you can cherish if you go through the details behind the chronology of events. .

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