Daniel Bernoulli’s paper, “Exposition of a New Theory on the Measurement of Risk” is, perhaps, the greatest theory of risk ever told. First published in Latin in 1738 in the Papers of the Papers of the Imperial Academy of Sciences in Petersburg, it was only translated into English in 1954, its importance such that this translation was published in Econometrica, one of the great journals of the economics discipline. This article is the first in a series of articles on that great paper, explaining the historical background, his theory and its implications, and finally, how his theory has not only guided some of the greatest investors of all time, notably, the recently late Jim Simons of Renaissance Technologies, and Warren Buffett -although I think that Buffett at least is unaware of this lineage-, but also, how it can help you, dear reader, be a better investor. This is, above all, a theory that assumes that wealth, in the long run, as I have previously said, is destroyed, that investors are in a perpetual duel with catastrophe.
The Problem of Points
Expected value, the probability-weighted average of possible payouts, has been the dominant decision criterion since the seventeenth century. Its genealogy can be traced back to mathematicians’ first attempts to develop a solution to the problem of points. The problem of points occurs when two players, with equal chances of winning future rounds, and who contribute equal amounts to winnings, and agree that the winner of a predetermined number of rounds will win the game, are forced to abandon their game before that predetermined number. Those players are then faced with a dilemma: how to divide the stakes fairly.
The monk Luca Pacioli’s book, Summa de arithmetica, geometrica, proportioni et proportionalità, written in 1494, proposed one of the earliest resolutions to the problem of points. Yet, he, along with other luminaries of the world of Renaissance mathematics, such as Niccolò Tartaglia and Giovan Francesco Peverone could not grasp upon a convincing solution. In my unpublished master’s thesis1, I observed that,
Tartaglia so despaired of finding a solution that he declared that any solution was “judicial rather than mathematical”, and would be so contestable as to lead to litigation.
The Birth of Expected Value Theory
In the late summer of 1654, Antoine Gombaud, known as the Chevalier de Méré, presented the problem to Blaise Pascal. In my thesis I noted that,
Pascal presented his solution to Pierre de Fermat, stating that he had discovered a way to calculate a fair division of stakes, later providing his complete solution in his book, Traité du triangle arithmétique avec quelques autres petits traitez sur la mesme matière. The correspondence between the two mathematicians, in which they developed three solutions to the problem, reveals that they understood as a fundamental principle that a fair division of stakes had to reflect the chance of getting it.
Having seen Pascal’s (1665) solution, Christiaan Huygens’ book, De Ratiociniis in Ludo examined the problem of points and presented Pascal’s expectations-based solution to the world in the world’s first systematic treatise on the mathematical theory of probability. More than a millennia since Aristotle had laid the foundations of a mathematical theory of probability, one had finally emerged. Huygens established the notion of expected value, saying,
“That any one Chance or Expectation to win any thing is worth just such a Sum, as wou’d procure in the same Chance and Expectation at a fair Lay. As for Example, if any one shou’d put 3 Shillings in one Hand, without telling me know which, and 7 in the other, and give me Choice of either of them; I say, it is the same thing as if he shou’d give me 5 Shillings; because with 5 Shillings I can, at a fair Lay, procure the same even Chance or Expectation to win 3 or 7 Shillings.”
In essence, one is faced with two outcomes:
- Outcome 1: Win 3 shillings, with a probability of 50%
- Outcome 2: Win 7 shillings, with a probability of 50%
The expected value, denoted by E[X] is calculated as,
E[X] = (0.5 ·3) + (0.5 · 7) = 5
Thus, the fair value of this gamble is equal to its expected value of 5 shillings, which is another way of saying that, faced with equi-probable payouts of 3 and 7 shillings is the same as receiving 5 shillings with certainty.
Symbolically, Huygens’ gamble can be represented as a discrete random variable X, with a set of possible payouts 𝑥i ∈ {3, 7}, and an associated probability distribution 𝑃(xi) = 0.5 for all 𝑥i ∈ X. The expected value of the game can then be written as:
E[X] = ∑𝑥i𝑃(xi)
The rational gambler, one is led to believe, is the one who seeks to maximise their expected value. Bernouli began his great paper by surveying the emerging consensus, saying:
Ever since mathematicians first began to study the measurement of risk there has been general agreement on the following proposition: Expected values are computed by multiplying each possible gain by the number of ways in which it can occur, and then dividing the sum of these products by the total number of possible cases where, in this theory, the consideration of cases which are all of the same probability is insisted upon. If this rule be accepted, what remains to be done within the framework of this theory amounts to the enumeration of all alterna-tives, their breakdown into equi-probable cases and, finally, their insertion into corresponding classifications.
As Bernoulli realised, expected value theory assumes that risks are identical for all counter-parties, and objective, leaving one to the easy task of simply estimating possible payouts and their associated probabilities and ranking gambles by their expected values.
Again, from my master’s thesis,
Huygens’ belief that the game amounts to being given the expected value is of course obviously both true and untrue. For instance, there is a material difference if one picks the hand with 3 shillings over the hand with just 7 shillings. The truth of Huygens’ statement rests on the dynamics of the game as we repeat it. If we played Huygens’ game 100 times, the mean payout might be 4.8. Per the central limit theorem, as i →∞, the mean payout approaches the expected value. The more we play, the closer we get to the limit of the game, i.e. the expected value. If we play this game 1 million times, the mean payout might be 4.999444.
Expected Value Theory Assumes that Wealth is Additive
The trouble with Huygens’ framing is that it creates a no-loss scenario. Regardless of the payout size, in this gamble, it is impossible to lose for one will win either 3 shillings or 7 shillings, with equal probability, and the fair value of this gamble is 5 shillings. When one introduces the possibility of loss into a gamble, the pitfalls of expected value are flung open.
Let w0 be one’s initial wealth with a value of $5,000 and Xi be a random variable representing the multiplier applied to wealth at the ith gamble:
- Outcome 1: 1.4 with a probability of 50%
- Outcome 2: 0.6 with a probability of 50%
The expected value of this gamble is, of course,
E[X] = $5,000*1.4*0.5+$5,000*0.6*0.5 = $5,000
This gamble leaves the value of one’s wealth untouched. It is the sort of gamble one would take purely for intellectual amusement but not to make any money. Although one’s wealth may decline one round, and increase another, over time, one’s payout will, according to our understanding of expected value theory, converge upon one’s initial wealth. However, there is a flaw in this thinking. Expected value is a one-round view of a gamble, it is as if an infinite ensemble of people played one round of a game and their average result was calculated. It is a theory of snapshots. Wealth, however, is not static, it is dynamic, unfolding through time, the outcome of one round in a gamble becoming the starting point of the next. To use a very ugly but very technically correct framing, wealth compounds multiplicatively, not additively, breaking the back of classical expected value theory in the context of long-term investment decisions. What one needs is not an ensemble average but a time average, for while in ergodic processes ensemble averages are equal to time averages, investors act through non-ergodic processes, where the ensemble average is distinct from the time average.
The Prospect of Ruin Stalks Every Investor
When one invests through time, each gain or loss changes the initial or prior wealth from which future gains or losses are calculated. Consequently, the arithmetic mean, which is essentially what the expected value is, does not capture the relevant quantity. What concerns investors, whether they understand the theoretical aspects or not, is the geometric mean, or, more precisely, the expected logarithmic growth rate. In our example, while the expected multiplier per round is 1 (leaving wealth ostensibly unchanged), the expected value of the logarithm of the multiplier is negative. That is,
E[logX] = 0.5⋅log(1.4) + 0.5⋅log(0.6) < 0
So, even when a gamble is “fair”, which is to say that there are equal chances of winning and losing, an investor is likely to lose over time. Wealth tends towards its own destruction. The arithmetic mean of payouts is irrelevant when the process that governs outcomes is multiplicative. The trap of expected value is that it hides this dynamic.
In a letter to Fermat in 1656, two years after the fecund correspondence on the problem of points, Pascal gave a formulation of what is now known as the gambler’s ruin problem. A ruin problem is one in which the outcome of a gamble has some chance of being an unrecoverable loss. Pierre de Carcavi summarised Pascal’s view in a letter that year to Huygens, saying,
Let two men play with three dice, the first player scoring a point whenever 11 is thrown, and the second whenever 14 is thrown. But instead of the points accumulating in the ordinary way, let a point be added to a player’s score only if his opponent’s score is nil, but otherwise let it be subtracted from his opponent’s score. It is as if opposing points form pairs, and annihilate each other, so that the trailing player always has zero points. The winner is the first to reach twelve points; what are the relative chances of each player winning?
Huygens went on to give the classic formulation of the problem in De ratiociniis in ludo aleae in the following way:
Problem (2-1) Each player starts with 12 points, and a successful roll of the three dice for a player (getting an 11 for the first player or a 14 for the second) adds one to that player’s score and subtracts one from the other player’s score; the loser of the game is the first to reach zero points. What is the probability of victory for each player?
Two players facing off for a fixed pot, exchanging points until one player is ruined. This is the world that investors inhabit. In many ways, the fundamental insights here were likely not novel even in the time of Pascal, Fermat, Huygens and Bernoulli. Although the pre-Pascalian era did not give birth to a mathematical theory of probability, traders estimated a price for risk and aleatory contracts were written in which the prospect of ruin was clearly foreseen. In my masters thesis, I cited James Franklin’s fabulous book, The Science of Conjecture, when explaining that,
Under Roman law, the risk of a shipwreck was assumed by the state. Risk was reified and regulations defined under which risk could move from one party to another. Maritime loans were allowed to have higher interest rates than permitted for other loan products, because of the heightened uncertainty of maritime trade, with the Digest saying, ‘the price is for the peril’.
Even today, although the typical analyst, portfolio manager or investor is unaware of the theory and its implications, economic and financial literature and education understand the superiority of compound returns over expected returns, and the ruin problem is a building block in actuarial math. However, vast areas of influential thought are bereft of any notion of ergodic and non-ergodic processes, such as behavioral economics. Daniel Kahenman and Amos Tsversky’s Prospect Theory is, as I will show later, built upon a misunderstanding of wealth dynamics. What follows in this series is an attempt to provide investors with a more holistic vision of risk and uncertainty and how one navigates them.
- Noko, Joseph. ‘The Nature of Risk’. Mémoire (French Master’s’ Thesis), Université d’Angers, 2022. ↩︎