Risk and Uncertainty

Let us begin by making sure we understand what risk and uncertainty mean. (Here we will use the terms more or less interchangeably, although people sometimes reserve the term uncertainty for cases where it is hard to quantify the risks that we face.) Probably the simplest example of risk is familiar to us all: the toss of a coin. Imagine flipping a coin five times. Each time, the outcome will be either a head or a tail. Table 5.8 "Coin-Flipping Experiment" shows an example of such an experiment. In this experiment, the outcome was three heads and two tails. For each flip of the coin, there was uncertainty about the outcome. We did not know ahead of time whether there would be heads or tails. The outcome reported in Table 5.8 "Coin-Flipping Experiment" is only one example. If you were to carry out this experiment right now, you would almost certainly end up with a different outcome.

Coin tosses are special because the flips of the coin are independent of each other (that is, the history of previous tosses has no effect on the current toss of a coin). In Table 5.8 "Coin-Flipping Experiment", the coin was not more likely to come up tails on the third toss because the previous tosses were both heads. Even if you have 100 heads in a row, this does not affect the outcome of the 101st toss of the coin. If you think that the coin is “fair,” meaning that heads and tails are equally likely, then the 101st toss is still just as likely to be heads as tails. By contrast, the likelihood that it will be raining an hour from now is not independent of whether or not it is raining at this moment.

Financial Risk and Expected Value

Some of the risks that we confront are nonfinancial. An example of nonfinancial uncertainty is the risk that you might break your ankle playing basketball or the possibility that your favorite sporting team will win a big game and make you happy. Here, we will focus on financial uncertainty, by which we mean situations where there is money at stake. In other words, we are thinking about risks where you can measure the implications in monetary terms. An obvious example is the money you could win or lose from buying a lottery ticket or playing poker. Another is the money you would have to pay for repairs or medical expenses following a car accident. Another is the gains or losses from buying stocks, government bonds, or other financial assets. Another is the income you would lose if you were laid off from your job.

When we evaluate risky situations, we must have a way of describing the kinds of gambles that we confront. In general, we do this by listing all the possible outcomes together with the likelihood of each outcome. For example, Table 5.9 "Outcomes and Probabilities from a Coin Toss" lists the outcomes and the probabilityThe percentage chance that an outcome will occur. (that is, the likelihood of each outcome) for the experiment of tossing a coin one time.

Think about rolling a normal six-sided die one time and describing outcomes and probabilities.

• We must make sure that we include every outcome. We cannot list as possible outcomes “less than or equal to 2” and “greater than or equal to 4.” Such a list ignores the possibility of rolling a 3.
• We cannot list as possible outcomes “less than or equal to 4” and “greater than or equal to 3.” These categories overlap because a roll of a 3 or a 4 would show up in both categories.
• The outcome “less than or equal to 6” has a probability of 1 because it is certain.
• The outcome “9” has a probability of 0.
• Provided we have a complete list of outcomes (for example “less than or equal to 4” and “greater than or equal to 5”), the probabilities of all the outcomes will always sum to 1. (In this case, the probability of the first outcome is 2/3, and the probability of the second outcome is 1/3.)

Now suppose you are playing a gambling game based on a toss of a coin. If the coin comes up heads, you win $1. If it comes up tails, you win $0. When we look at a situation such as this, we are often interested in how much you would get, on average, if you played the game many times. In this example, it is easy to guess the answer. On average, you would expect to win half the time, so half the time you get $1, and half the time you get nothing. We say that the expected valueThe measure of how much you would expect to win (or lose) on average, if the situation were to be replayed a large number of times. of each flip of the coin is 50 cents.

Table 5.10 "Outcomes and Probabilities from Investment in Internet Venture" gives another example of expected value. Suppose a friend is planning on establishing a small Internet business and asks you to invest $1,000. He tells you (and you believe him) that there is a 50 percent chance that the business will fail, so you will lose your money. There is a 40 percent chance that the business will just break even, so you will get your $1,000 back but nothing more. And there is a 10 percent chance that the business will be very successful, so you will earn $16,000.

In this case, the expected value of the investment is given by the following:

expected value = (0.5 × $0) + (0.4 × $1,000) + (0.1 × $16,000) = $2,000.

Thus for your investment of $1,000, you could expect to get $2,000 back on average. This seems like a good investment. It is important to remember, though, what “on average” means. You will never actually get $2,000. You will receive either $16,000, $1,000, or nothing. Even though this is a good investment on average, you might still decide that you don’t want any part of it. Yes, you might get the big net gain of $15,000. But there’s also a 50 percent chance that you will be out $1,000. The gamble might seem too risky for you.

Coin tosses are special because it is relatively easy to determine the probability of a head or a tail. This is not the case for all the types of uncertainty you might face. In some cases, financial instruments—such as the mortgage-backed securities that played a big role in the financial crisis of 2007–2009—are so complex that investors find it difficult to assess the probabilities of various outcomes.

We often do a bad job of estimating probabilities. One reason for this is because we are unduly influenced by things that we can easily bring to mind. Psychologists call this the “availability heuristic.” For example, we tend to overestimate certain causes of death, such as car accidents, tornadoes, and homicides, and underestimate others, such as diabetes, stroke, and asthma.See Paul Slovic, Baruch Fischoff, and Sarah Lichtenstein, “Facts versus Fears: Understanding Perceived Risk,” in Judgment under Uncertainty: Heuristics and Biases, ed. Daniel Kahneman, Paul Slovic, and Amos Tversky (Cambridge, MA: Cambridge University Press, 1982), 463–89. We also often do a poor job at using probabilities; in particular, we often put too much emphasis on small probabilities. For example, consider two drugs that are equally effective in treating a disease, but suppose the older drug has a 1 in 10 million chance of having a certain side effect and the newer drug has a 1 in a 100 million chance of having the same side effect. Consumers might view the new drug as much more appealing, even though the side effect was already highly improbable with the older drug.

Diversification of Risk

In many cases, we would like to find some way of getting rid of—at least to some degree—the risks that we face. One way we eliminate risk is through insurance. Sometimes we purchase insurance on the market. Sometimes our employer provides us with insurance. Sometimes the government provides us with insurance. In the following subsections, we look at many different kinds of insurance, including property insurance, unemployment insurance, and deposit insurance.We do not discuss health insurance here. In Chapter 16 "A Healthy Economy", we discuss the provision of health care and the problems of health insurance in detail.

First, though, we need to understand how and why insurance works. Suppose you have a bicycle worth $1,000, and (for some reason) you cannot purchase insurance. You think that, in any given year, there is about a 1 percent chance that your bike will have to be replaced (because it is either stolen or written off in an accident).

Now, in expected value terms, this may not look too bad. Your expected loss from an accident is $0.01 × $1,000 = $10. So on average, you can expect to lose $10 a year. But the problem is that, if you are unlucky, you are stuck with a very big expense. Most of us dislike this kind of risk.

You are complaining about this to a friend, and she sympathizes, saying that she faces exactly (and we mean exactly) the same problem. She also has a bike worth $1,000 and thinks there is a 1 percent chance each year that she will need to replace it. And that’s when you have the brilliant idea. You can make an agreement that, if either one of you has to replace your bikes, you will share the costs. So if you have to replace your bike, she will pay $500 of your costs, and if she has to replace her bike, then you will pay $500 of her costs. It is (almost exactly) twice as likely that you will have to pay something, but if you do, you will only have to pay half as much. With this scheme, your expected loss is unchanged. But you and your friend prefer this scheme because it is less risky; it is much less likely that you will have to make the big $1,000 payout.

We are implicitly assuming here that your chances of having to replace your bike are independent of the chance that she will have to replace her bike. (If you are likely to crash into her, or both of your bikes are stolen, then it is a different story.) There is also still a chance that you will both experience the unlucky 1-in-100 chance, in which case you would both still have to pay $1,000. But the likelihood of this happening is now tiny. (To be precise, the probability of both of you having an accident in the same year is 1 in 10,000 [that is, 0.0001]). This is because the probability that two independent events occur equals the probability of one multiplied by the probability of the other.)

But why stop here? If you can find two more friends with the same problem, then you can make it almost certain that you will have to pay out no more than $250. It is true that you would be even more likely to have to make a payment because you will have to pay if you or one of your friends has to replace his or her bike. But because the payment is now being shared four ways, you will have to pay only 25 percent of the expenses. This is an example of diversificationThe insight that underlies insurance in which people can share their risks., which is the insight that underlies insurance: people share their risks, so it is less likely that any single individual will face a large loss.

Diversification and insurance don’t prevent bad stuff from happening. We live in a world where bicycles are stolen; where houses are destroyed by floods, fires, or storms; where people have accidents or become ill; and so on. There is not a lot we can do about the fact that bad things happen. But we can make the consequences of these bad things easier to deal with. Insurance is a means of sharing—diversifying—these risks.

Continuing with our bicycle insurance example, suppose you could find thousands of friends who would agree to be part of this arrangement. As more and more people join the scheme, it becomes increasingly likely that you have to make a payment each year, but the amount you would have to pay becomes smaller and smaller. With a very large number of people, you would end up very close to a situation where you pay out $10 with certainty each year. Of course, organizing thousands of your friends into such a scheme would present all sorts of practical problems. This is where insurance companies come in.

Insurance companies charge you a premium (an annual payment). In return, they promise to pay you an indemnityIn an insurance contract, a value equal to the full amount of the loss minus the deductible. in the event you suffer a loss. The indemnity is usually not the full amount of the loss. The part of the loss that is not covered is called the deductibleIn an insurance contract, the value not covered in the event of a loss.. In our example, there is no deductible, and the indemnity is $1,000. An insurance company would charge you a premium equal to the expected loss of $10 plus a little extra. The extra payment is how the insurance company makes money. You and everyone else are willing to pay this extra amount in return for the removal of risk.

The idea of diversification can also be applied to investment.We discuss this in Chapter 10 "Making and Losing Money on Wall Street". Think back to our example of your friend with the Internet venture. You might not want to invest $1,000 in his scheme because it seems too risky. But if you had 100 friends with 100 similar (but independent) schemes, you might be willing to invest $10 in each. Again, you would be diversifying your risk.

Different Kinds of Insurance

There are many different kinds of insurance available to you. We briefly discuss some of the most important.