Daniel Bernoulli: The Beginnings of Utility Theory and Risk-Adjusted Valuation (1713/1738)

We read earlier how, in the second half of the seventeenth century, Pascal, Fermat, de Witt and Huygens developed and applied the concept of pricing uncertain cashflows by assessing their mathematical expectation. The discounting of expected cashflows to allow for the impact of the time value of money was understood and well demonstrated by the annuity pricing work of de Witt and Halley; in both cases, they explicitly calculated the mathematical expectations of the future cashflows and then discounted them into a present value using a government bond yield. Up until this time, this strand of thinking had not considered whether the valuation should make any allowance for the riskiness of the future cashflows (except insofar as it impacts on the size of the expected cashflows).

In 1713, a seemingly trivial valuation problem was introduced in a letter from Nicolas Bernoulli (who we briefly met earlier as the editor of his Uncle Jacob’s Ars Conjectandi) to another eminent French mathematician Pierre Montmort. This valuation riddle highlighted the possible need for a more refined treatment of risk in the valuation of variable future cashflows. The valuation problem—known as the St Petersburg Paradox—can be described as follows. Suppose we have a fair coin. The coin is tossed and if it lands heads, player A pays player B ?1. If it lands tails, then the coin is tossed again. If it lands heads on this turn, then player A pays player B ?2. If it lands tails, then the coin is tossed again. If it lands heads on this turn, then player A pays player B ?4. And so on, with the amount player A must pay player B when heads lands doubling each time the coin lands tails. An interesting difficulty arises when mathematical expectation is used to determine how much player B should pay player A to play this game: the mathematical expectation of this cashflow is infinite. This was viewed as a paradox because it was thought that no reasonable person would pay more than, say, ?10 or ?20 as player B in the game, even though he would receive a pay-out that had an infinite expectation.

The problem posed by Nicolas was eventually tackled by another Bernoulli— Daniel, another nephew of Jacob Bernoulli. Daniel was born in 1700 and established himself as another Bernoulli mathematical genius, making lasting contributions to mathematics and physics, most notably in fields such as fluid mechanics. In 1738, Daniel published a solution to the paradox in the annals of the University of St Petersburg (from which the paradox derives its name).^{[1]} Bernoulli’s solution proposed the use of a new form of expectation. Instead of pricing a contract using mathematical expectation (i.e. by considering the product of the possible cashflows and their respective probabilities), he proposed using what he termed moral expectation, which he defined as the product of the utility the recipient gained from each possible cashflow and their respective probabilities. This was the first time a quantitative concept of expected utility was considered in the context of valuation of risky cashflows.

Bernoulli went on to show that an intuitive price could be obtained for the St Petersburg game when moral, rather than mathematical, expectation was used. To do this, he supposed that increases in wealth were directly proportional to increases in utility (that is, a logarithmic utility function). He noted that the risk-aversion embedded in this non-linear utility function could rationalise demand for insurance and diversification practises (such as dividing cargo amongst several ships) as well as help to solve the St Petersburg Paradox.

To find the maximum price that someone would pay as player B in the game, Bernoulli argued we should find the price that equates his expected utility after playing the game with the utility he obtains from his wealth without playing the game. If the utility function is linear, then the expected utility from playing the game is infinite for any finite price. The non-linearity of the logarithmic function places less value on the extremely unlikely upside pay?offs of the game. As a result, a finite value for the game is obtained. The risk- aversion embedded in the utility function also implied that the maximum value a player would rationally pay to play the game would be a function of his current level of wealth—the smaller his current wealth, the less he would be prepared to pay to play, even though the expected pay-out from playing is technically infinite.

Interestingly, Nicolas Bernoulli, who originally considered the paradox 25 years earlier, did not accept Daniel’s concept of moral expectation. Nicolas was by this time a professor of law at the University of Basel. He maintained that for a contract to be legally equitable it should be priced such that the buyer and seller have an equal chance of winning or losing. Daniel’s more sophisticated treatment had moved the pricing of uncertain claims from the realm of jurisprudence to that of economics.

His cousin was not the only one to object to Daniel Bernoulli’s conception of risk-adjusted valuation of stochastic cashflows. Jean D’Alembert, another contemporary French mathematician, argued that the introduction of moral expectation was an ad hoc solution to the St Petersburg Paradox. He suggested a simpler way of reducing the value that the mathematical expectation attached to the extremely unlikely but extremely large pay-offs that arose in the game: he argued that beyond a particular probability level, an event is not physically possible, and it should make no contribution to the valuation. But the choice of such a probability level is similarly ad hoc and arbitrary.

One of the implications of Bernoulli’s utility treatment is that the price a player would be willing to pay to play the game increases as his starting wealth goes up relative to the stake of the game. This implication is consistent with modern economic thinking on pricing of risky cashflows. The pay-out from the game is, in economic terms, a diversifiable risk. If the game can be played an increasingly large number of times for increasingly small stakes (relative to starting wealth), the utility function and starting wealth of the player becomes increasingly irrelevant, and the value of the game will tend to its mathematical expectation. So Bernoulli’s solution to the paradox arguably only applies when it is assumed that the game must be played for stakes that are a material portion of the player’s current wealth. This line of argument was pursued by the respected English mathematician Augustus De Morgan in the early nineteenth century. In his discussion of the St Petersburg Paradox he concludes: ‘The results of all which precedes shows us that great risks should not be run, unless for sums so small that the venture can afford to repeat them often enough to secure an average.’^{[2]}

The ultimate modern solution to the paradox, as argued by Samuelson,^{[3]} is that no rational person would accept playing as player A for any finite price, and hence the game cannot be played and no paradox can arise. But this doesn’t really address the original point of the paradox, which was that, if the game was offered, no reasonable person would intuitively be prepared to pay more than ten or twenty times the pay-off from the first coin toss, even though the total expected pay-off is infinite.