During the Renaissance, the humanist scholar Desiderius Erasmus compiled a list of proverbs whose title in English is Adages. These were largely compiled from ancient Greek and Roman sources. They included a variety of expressions commonly used today, such as “the blind leading the blind,” “kill two birds with one stone,” and “between a rock and a hard place.” One that relates to risk is “trust not all your goods to one ship.”
The study of probability began with a question about gambling. In the seventeenth century, a friend of the mathematician Blaise Pascal asked him to determine how the spoils of a game of chance if it were interrupted before it could be completed. Pascal began a correspondence with the judge and amateur mathematician Pierre de Fermat, and in an exchange of letters, worked out the expected outcome. For several decades, the study of uncertainty centered on averages. The first paper to consider fluctuations around average values was Daniel Bernoulli’s landmark paper in 1738 titled “Exposition of a New Theory on the Measurement of Risk,” which took into consideration the variation around the mean. Bernoulli’s paper is best known for introducing the notions of risk aversion, risk premium, and utility in the study of economics. Bernoulli developed these concepts in order to resolve the St. Petersburg Paradox, a problem that was posed by his cousin. It is less well remembered for considering an example that show that proper risk management also considers the variance around the expected value.
In his paper, Bernoulli shows that more than the mean, or average, outcome should be considered when weighing alternatives. He used the framework of utility theory. However, we will consider it without the unnecessary baggage of the unwieldy concept of utility, a measure of satisfaction from money that is difficult to measure directly. We only need the monetary outcomes to show that more than the expected value should be considered.
As a scholar, Bernoulli may have been familiar with Erasmus’ adage of not putting all your goods on one ship because he came up with a numerical example that illustrates this notion. Bernoulli’s example used a type of coin called a ducat. The ducat was a coin made of precious metals commonly used as a trade coin in Europe until the twentieth century. In Bernoulli’s example, a merchant owns 4,000 ducats worth of goods at home. He possesses another 8,000 ducats of goods in a foreign country. These goods can only be transported by sea, a perilous trip. Based on experience, half of all ships that journey across the sea do not reach their destination because of storms, pirate attacks, and other reasons. If the merchant has all his foreign goods sent home on one ship, there is a 50% chance that he will get his 8,000 ducats worth of goods. There is also a 50% chance that these goods will wind up at the bottom of the ocean, stolen by pirates, or perhaps even in the belly of a large white whale. Thus, after the voyage, there is a 50% chance that his net worth will be 4,000 ducats and a 50% chance that his net worth will be 12,000 ducats.
The expected value is just the weighted average of these two values, which in this case is 4,000*.50 + 12,000*0.50 = 8,000 ducats. (To keep things simple, assume that the cost of shipping the goods is zero.) However, suppose that the merchant decides to diversify his risk. He transports the goods in two ships, with half in each. Three different outcomes are possible. One is that both ships arrive safely, in which case his net worth is 12,000 ducats. Another is that only one ship makes it, in which case his net worth is now 4,000+4,000 = 8,000 ducats. And the third possibility is that neither ship makes it safely across, in which case his net worth is only 4,000 ducats. To determine the expected value of this approach, note that the 50% probability of the goods arriving safely is the same as the result of tossing a fair coin. Looking at the arrangement of two independent coin tosses, there are four possibilities, if you consider the order of the tosses. One is heads, then heads again. The second is heads, then tails. Next is tails, then heads. Lastly, there is tails, then tails again. Each one of these outcomes is equally likely, since the probability of heads or tails on each flip is 50%. Let heads denote the event that a ship arrives safely, and tails denote the event that the ship is lost to at sea. Since these events are all equally likely, and there are four of them, each has a probability of occurrence equal to 25%. That is, the likelihood that both make it is 25%. The chance that neither ship makes it is also 25%. The chance that only one ship makes it is 50%, since there are two equally likely ways this can happen – the first ship makes it, and the second does not, or the first ship does not make it, while the second ship arrives safely. Thus, there is a 25% chance that the merchant’s net worth will be 4,000 ducats, a 25% chance that his net worth will be 12,000 ducats (both ships make it), and a 50% chance that his net worth will be 8,000 ducats (only one ship makes it). The expected value of his wealth is equal to the weighted average of these outcomes:
4,000*.25+12,000*0.25+8,000*0.50 = 1,000+3,000+4,000 = 8,000 ducats.
The expected value of both alternatives is the same, 8,000 ducats. This average, however, does not tell the entire story. In the first case, half the time the merchant has a net worth equal to 4,000 ducats, while in the second case, in three-quarters of the cases he has at least 8,000 ducats in net worth. The second case is preferable. You are better off not putting all your goods in one ship. See the graph below for a comparison.
