Lloyd’s of London is the oldest existing commercial insurance company, and it is an institution with an illustrious past. Originally formed in the eighteenth century to help merchants cope with the risks of commerce, it grew in the heyday of the British Empire into a mainstay of imperial trade.

The basic idea of Lloyd’s was simple. In the eighteenth century, shipping goods via sailing vessels was risky: the chance that a ship would sink in a storm or be captured by pirates was fairly high. The merchant who owned the ship and its cargo could easily be ruined financially by such an event. Lloyd’s matched shipowners seeking insurance with wealthy investors who promised to compensate a merchant if his ship was lost. In return, the merchant paid the investor a fee in advance; if his ship didn’t sink, the investor still kept the fee. In effect, the merchant paid a price to relieve himself of risk. By matching people who wanted to purchase insurance with people who wanted to provide it, Lloyd’s performed the functions of a market. The fact that British merchants could use Lloyd’s to reduce their risk made many more people in Britain willing to undertake merchant trade.

Insurance companies have changed quite a lot from the early days of Lloyd’s. They no longer consist of wealthy individuals deciding on insurance deals over port and boiled mutton. But asking why Lloyd’s worked to the mutual benefit of merchants and investors is a good way to understand how the market economy as a whole “trades” and thereby transforms risk.

The insurance industry rests on two principles. The first is that trade in risk, like trade in any good or service, can produce mutual gains. In this case, the gains come when people who are less willing to bear risk transfer it to people who are more willing to bear it. The second is that some risk can be made to disappear through diversification. Let’s consider each principle in turn.

It may seem a bit strange to talk about “trading” risk. After all, risk is a bad thing—and aren’t we supposed to be trading goods and services?

But people often trade away things they don’t like to other people who dislike them less. Suppose you have just bought a house for $100 000, the average price for a house in your community. But you have now learned, to your horror, that the building next door is being turned into a nightclub and bar. You want to sell the house immediately and are willing to accept $95 000 for it. But who will now be willing to buy it? The answer: a person who doesn’t really mind late-night noise. Such a person might be willing to pay up to $100 000. So there is an opportunity here for a mutually beneficial deal—you are willing to sell for as little as $95 000, and the other person is willing to pay as much as $100 000, so any price in between will benefit both of you.

The key point is that the two parties have different sensitivities to noise, which enables those who most dislike noise, in effect, to pay other people to make their lives quieter. Trading risk works exactly the same way: people who want to reduce the risk they face can pay other people who are less sensitive to risk to take some of their risk away.

As we saw in the previous section, individual preferences account for some of the variations in people’s attitudes toward risk, but differences in income and wealth are probably the principal reason behind different risk sensitivities. Lloyd’s made money by matching wealthy investors who were more risk-tolerant with less wealthy and therefore more risk-averse shipowners.

Suppose, staying with our Lloyd’s of London story, that a merchant whose ship went down would lose £1000 (this includes the cost of the ship, all its contents, and the lost revenue) and that there was a 10% chance of such a disaster. The expected loss in this case would be 0.10 × £1000 = £100. But the merchant, whose whole livelihood was at stake, might have been willing to pay £150 to be compensated in the amount of £1000 if the ship sank. Meanwhile, a wealthy investor for whom the loss of £1000 was no big deal would have been willing to take this risk for a return only slightly better than the expected loss—say, £110. Clearly, there is room for a mutually beneficial deal here: the merchant pays something less than £150 and more than £110—say, £130—in return for compensation if the ship goes down. In effect, he has paid a less risk-averse individual to bear the burden of his risk. Everyone has been made better off by this transaction.

The funds that an insurer places at risk when providing insurance are called the insurer’s capital at risk. In our example, the wealthy Lloyd’s investor places capital of £1000 at risk in return for a premium of £130. In general, the amount of capital that potential insurers are willing to place at risk depends, other things equal, on the premium offered. If every ship is worth £1000 and has a 10% chance of going down, nobody would offer insurance for less than a £100 premium, equal to the expected claim. In fact, only an investor who isn’t risk-averse at all—that is, who is risk-neutral—would be willing to offer a policy at that price, because accepting a £100 premium would leave the insurer’s expected income unchanged while increasing his or her risk.

Suppose there is one investor who is risk-neutral; but the next most willing investor is slightly risk-averse and insists on a £105 premium. The next investor, being somewhat more risk-averse, demands a premium of £110, and so on. By varying the premium and asking how many insurers would be willing to provide insurance at that premium, we can trace out a supply curve for insurance, as shown in Figure 20-3. As the premium increases as we move up the supply curve, more risk-averse investors are induced to provide coverage.

Meanwhile, potential buyers will consider their willingness to pay a given premium, defining the demand curve for insurance. In Figure 20-4, the highest premium that any shipowner is willing to pay is £200. Who’s willing to pay this? The most risk-averse shipowner, of course. A slightly less risk-averse shipowner might be willing to pay £190, an even slightly less risk-averse shipowner is willing to pay £180, and so on.

Now imagine a market in which there are thousands of shipowners and potential insurers, so that the supply and demand curves for insurance are smooth lines. In this market, as in markets for ordinary goods and services, there will be an equilibrium price and quantity. Figure 20-5 illustrates such a market equilibrium at a premium of £130, with a total quantity of 5000 policies bought and sold, representing a total capital at risk of £5 000 000.

Notice that in this market risk is transferred from the people who most want to get rid of it (the most risk-averse shipowners) to the people least bothered by risk (the least risk-averse investors). So just as markets for goods and services typically produce an efficient allocation of resources, markets for risk also typically lead to an efficient allocation of risk—an allocation of risk in which those who are most willing to bear risk are those who end up bearing it. But as in the case of the markets for goods and services, there is an important qualification to this result: there are well-defined cases in which the market for risk fails to achieve efficiency. These arise from the presence of private information, an important topic that we will cover in the next section.

The trading of risk between individuals who differ in their degree of risk aversion plays an extremely important role in the economy, but it is not the only way that markets can help people cope with risk. Under some circumstances, markets can perform a sort of magic trick: they can make some (though rarely all) of the risk that individuals face simply disappear.

In the early days of Lloyd’s, British merchant ships traversed the world, trading spices and silk from Asia, tobacco and rum from the New World, and textiles and wool from Britain, among many other goods. Each of the many routes that British ships took had its own unique risks—pirates in the Caribbean, gales in the North Atlantic, typhoons in the Indian Ocean.

In the face of all these risks, how were merchants able to survive? One important way was by reducing their risks by not putting all their eggs in one basket: by sending different ships to different destinations, they could reduce the probability that all their ships would be lost. A strategy of investing in such a way as to reduce the probability of severe losses is known as diversification. As we’ll now see, diversification can often make some of the economy’s risk disappear.

Let’s stay with our shipping example. It was all too likely that a pirate might seize a merchant ship in the Caribbean or that a typhoon might sink another ship in the Indian Ocean. But the key point here is that the various threats to shipping didn’t have much to do with each other. So it was considerably less likely that a merchant who had one ship in the Caribbean and another ship in the Indian Ocean in a given year would lose them both, one to a pirate and the other to a typhoon. After all, there was no connection: the actions of cutthroats in the Caribbean had no influence on weather in the Indian Ocean, or vice versa.

Statisticians refer to such events—events that have no connection, so that one is no more likely to happen if the other does than if it does not—as independent events. Many unpredictable events are independent of each other. If you toss a coin twice, the probability that it will come up heads on the second toss is the same whether it came up heads or tails on the first toss. If your house burns down today, it does not affect the probability that my house will burn down the same day (unless we live next door to each other or employ the services of the same incompetent electrician).

There is a simple rule for calculating the probability that two independent events will both happen: multiply the probability that one event would happen on its own by the probability that the other event would happen on its own. If you toss a coin once, the probability that it will come up heads is 0.5; if you toss the coin twice, the probability that it will come up heads both times is 0.5 × 0.5 = 0.25.

But what did it matter to shipowners or Lloyd’s investors that ship losses in the Caribbean and ship losses in the Indian Ocean were independent events? The answer is that by spreading their investments across different parts of the world, shipowners or Lloyd’s investors could make some of the riskiness of the shipping business simply disappear.

Let’s suppose that Joseph Moneypenny, Esq., is wealthy enough to outfit two ships—and let’s ignore for a moment the possibility of insuring his ships. Should Mr. Moneypenny equip two ships for the Caribbean trade and send them off together? Or should he send one ship to Barbados and one to Calcutta?

Assume that both voyages will be equally profitable if successful, yielding £1000 if the voyage is completed (this payoff includes the cost of the ship, its contents, and the revenue). Also assume that there is a 10% chance both that a ship sent to Barbados will run into a pirate and that a ship sent to Calcutta will be sunk by a typhoon. And if two ships travel to the same destination, we will assume that they share the same fate. So if Mr. Moneypenny were to send both his ships to either destination, he would face a probability of 10% of losing all his investment.

But if Mr. Moneypenny were instead to send one ship to Barbados and one ship to Calcutta, the probability that he would lose both of them would be only 0.1 × 0.1 = 0.01, or just 1%. As we will see shortly, his expected payoff would be the same—but the chance of losing it all would be much less. So by engaging in diversification—investing in several different things, where the possible losses are independent events—he could make some of his risk disappear.

Table 20-2 summarizes Mr. Moneypenny’s options and their possible consequences. If he sends both ships to the same destination, he runs a 10% chance of losing them both. If he sends them to different destinations, there are three possible outcomes. Both ships could arrive safely: because there is a 0.9 probability of either one making it, the probability that both will make it is 0.9 × 0.9 = 81%. Both could be lost—but the probability of that happening is only 0.1 × 0.1 = 1%. Finally, there are two ways that only one ship can arrive. The probability that the first ship arrives and the second ship is lost is 0.9 × 0.1 = 9%. The probability that the first ship is lost but the second ship arrives is 0.1 × 0.9 = 9%. So the probability that only one ship makes it is 9% + 9% = 18%.

You might think that diversification is a strategy available only to those with a lot of money to begin with. Can Mr. Moneypenny diversify if he is able to afford only one ship? There are ways for even small investors to diversify. Even if Mr. Moneypenny is only wealthy enough to equip one ship, he can enter a partnership with another merchant. They can jointly outfit two ships, agreeing to share the profits equally, and then send those ships to different destinations. That way each faces less risk than if he equips one ship alone.

In the modern economy, diversification is made much easier for investors by the fact that they can easily buy shares in many companies by using the stock market. The owner of a share in a company is the owner of part of that company—typically a very small part, one-millionth or less. An individual who put all of his or her wealth in shares of a single company would lose all of that wealth if the company went bankrupt. But most investors hold shares in many companies, which makes the chance of losing all their investment very small.

In fact, Lloyd’s of London wasn’t just a way to trade risks; it was also a way for investors to diversify. To see how this worked, let’s introduce Lady Penelope Smedley-Smythe, a wealthy aristocrat, who decides to increase her income by placing £1000 of her capital at risk via Lloyd’s. She could use that capital to insure just one ship. But more typically she would enter a “syndicate,” a group of investors, who would jointly insure a number of ships going to different destinations, agreeing to share the cost if any one of those ships went down. Because it would be much less likely for all the ships insured by the syndicate to sink than for any one of them to go down, Lady Penelope would be at much less risk of losing her entire capital.

In some cases, an investor can make risk almost entirely disappear by taking a small share of the risk in many independent events. This strategy is known as pooling. Consider the case of a supplementary health insurance company, which has millions of policyholders, with thousands of them requiring expensive treatment each year. The insurance company can’t know whether any given individual will, say, require heart medication. But heart problems for two different individuals are pretty much independent events. And when there are many possible independent events, it is possible, using statistical analysis, to predict with great accuracy how many events of a given type will happen. For example, if you toss a coin 1000 times, it will come up heads about 500 times—and it is very unlikely to be more than a percent or two off that figure.

So a company offering fire insurance can predict very accurately how many of its clients’ homes will burn down in a given year; a company offering supplementary health insurance can predict very accurately how many of its clients will need heart medication in a given year; a life insurance company can predict how many of its clients will … Well, you get the idea.

When an insurance company is able to take advantage of the predictability that comes from aggregating a large number of independent events, it is said to engage in pooling of risks. And this pooling often means that even though insurance companies protect people from risk, the owners of the insurance companies may not themselves face much risk.

Lloyd’s of London wasn’t just a way for wealthy individuals to get paid for taking on some of the risks of less wealthy merchants. It was also a vehicle for pooling some of those risks. The effect of that pooling was to shift the supply curve in Figure 20-5 rightward: to make investors willing to accept more risk, at a lower price, than would otherwise have been possible.

Diversification can reduce risk. In some cases it can eliminate it. But these cases are not typical, because there are important limits to diversification. We can see the most important reason for these limits by returning to Lloyd’s one more time.

During the period when Lloyd’s was creating its legend, there was one important hazard facing British shipping other than pirates or storms: war. Between 1690 and 1815, Britain fought a series of wars, mainly with France (which, among other things, led to the British takeover of New France during the Seven Years’ War). Each time, France would sponsor “privateers”—basically pirates with official backing—to raid British ships and thus indirectly damage Britain’s war effort.

Whenever war broke out between Britain and France, losses of British merchant ships would suddenly increase. Unfortunately, merchants could not protect themselves against this eventuality by sending ships to different ports: the privateers would prey on British ships anywhere in the world. So the loss of a ship to French privateers in the Caribbean and the loss of another ship to French privateers in the Indian Ocean would not be independent events. It would be quite likely that they would happen in the same year.

When an event is more likely to occur if some other event occurs, these two events are said to be positively correlated. And like the risk of having a ship seized by French privateers, many financial risks are, alas, positively correlated.

Here are some of the positively correlated financial risks that investors in the modern world face:

When events are positively correlated, the risks they pose cannot be diversified away. An investor can protect herself from the risk that any one company will do badly by investing in many companies; she cannot use the same technique to protect against an economic slump in which all companies do badly.

An insurance company can protect itself against the risk of losses from local flooding by insuring houses in many different places; but a global weather pattern that produces floods in many places will defeat this strategy. Not surprisingly, and as we explained in the opening story, insurers pulled back from writing as many weather-related policies when it became clear that extreme weather patterns had become worse. They could no longer be confident that profits from policies written in good weather areas would be sufficient to compensate for losses incurred on policies in flood-, hurricane-, and tornado-prone areas.

So institutions like insurance companies and stock markets usually cannot make risk go away completely. There is usually an irreducible core of risk that cannot be diversified. Markets for risk, however, do accomplish two things: First, they enable the economy to eliminate the risk that can be diversified. Second, they allocate the risk that remains to the people most willing to bear it.