Risk Analysis in Theory and Practice || The Measurement of Risk

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<ul><li><p>Chapter 2</p><p>The Measurement of Risk</p><p>We define risk as representing any situation where some events are not</p><p>known with certainty. This means that the prospects for risk are prevalent.</p><p>In fact, it is hard to consider any situation where risk does not play a role.</p><p>Risk can relate to weather outcomes (e.g., whether it will rain tomorrow),</p><p>health outcomes (e.g., whether you will catch the flu tomorrow), time</p><p>allocation outcomes (e.g., whether you will get a new job next year), market</p><p>outcomes (e.g.,whether thepriceofwheatwill risenextweek),ormonetaryout-</p><p>comes (e.g., whether you will win the lottery tomorrow). It can also relate to</p><p>events that are relatively rare (e.g., whether an earthquake will occur next</p><p>month in a particular location, or whether a volcano will erupt next year).</p><p>The list of risky events is thus extremely long. First, this creates a significant</p><p>challenge to measure risky events. Indeed, how can we measure what we do</p><p>not know for sure? Second, given that the number of risky events is very</p><p>large, is it realistic to think that risk can be measured? In this chapter, we</p><p>address these questions. We review the progress that has been made evalu-</p><p>ating risk. In particular, we review how probability theory provides a formal</p><p>representation of risk, which greatly contributes to the measurement of risk</p><p>events. We also reflect on the challenges associated with risk assessment.</p><p>Before we proceed, it will be useful to clarify the meaning of two terms:</p><p>risk and uncertainty. Are these two terms equivalent? Or do they mean</p><p>something different? There is no clear consensus. There are at least two</p><p>schools of thought on this issue. One school of thought argues that risk</p><p>and uncertainty are not equivalent. One way to distinguish between the two</p><p>relies on the ability to make probability assessments. Then, risk corresponds</p><p>to events that can be associated with given probabilities; and uncertainty</p><p>5</p><p>Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 5Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 5Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 5Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 5</p></li><li><p>corresponds to events for which probability assessments are not possible.</p><p>This suggests that risky events are easier to evaluate, while uncertain events</p><p>are more difficult to assess. For example, getting tails as the outcome of</p><p>flipping a coin is a risky event (its probability is commonly assessed to be</p><p>0.5), but the occurrence of an earthquake in a particular location is an</p><p>uncertain event. This seems intuitive. However, is it always easy to separate</p><p>risky events from uncertain events? That depends in large part on the</p><p>meaning of a probability. The problem is that there is not a clear consensus</p><p>about the existence and interpretation of a probability. We will briefly</p><p>review this debate. While the debate has generated useful insights on the</p><p>complexity of risk assessment, it has not yet stimulated much empirical</p><p>analysis. As a result, we will not draw a sharp distinction between risk and</p><p>uncertainty. In other words, the reader should know that the terms risk</p><p>and uncertainty are used interchangeably throughout the book. It implicitly</p><p>assumes that individuals can always assess (either objectively or subjectively)</p><p>the relative likelihood of uncertain events, and that such assessment can be</p><p>represented in terms of probabilities.</p><p>DEFINITION</p><p>We define a risky event to be any event that is not known for sure ahead of</p><p>time. This gives some hints about the basic characteristics of risk. First, it</p><p>rules out sure events (e.g., events that already occurred and have been</p><p>observed). Second, it suggests that time is a fundamental characteristic of</p><p>risk. Indeed, allowing for learning, some events that are not known today</p><p>may become known tomorrow (e.g., rainfall in a particular location). This</p><p>stresses the temporal dimension of risk.</p><p>The prevalence of risky events means that there are lots of things that are</p><p>not known at the current time. On one hand, this stresses the importance of</p><p>assessing these risky outcomes in making decisions under uncertainty. On</p><p>the other hand, this raises a serious issue: How do individuals deal with the</p><p>extensive uncertainty found in their environment? Attempting to rationalize</p><p>risky events can come in conflict with the scientific belief, where any event</p><p>can be explained in a causeeffect framework. In this context, one could</p><p>argue that the scientific belief denies the existence of risk. If so, why are there</p><p>risky events?</p><p>Three main factors contribute to the existence and prevalence of risky</p><p>events. First, risk exists because of our inability to control and/or measure</p><p>precisely some causal factors of events. A good example (commonly used in</p><p>teaching probability) is the outcome of flipping a coin. Ask a physicist or an</p><p>engineer if there is anything that is not understood in the process of flipping</p><p>Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 6Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 6Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 6Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 6</p><p>6 Risk Analysis in Theory and Practice</p></li><li><p>a coin. The answer is no. The laws of physics that govern the path followed</p><p>by the coin are well understood. So, why is the outcome not known ahead of</p><p>time? The answer is that a coin is never flipped exactly the same way twice.</p><p>As a result, as long as the coin trajectory is long enough, it is hard to predict</p><p>how it will land. What creates the uncertainty here is the fact that the initial</p><p>conditions for the coin trajectory are not precisely controlled. It is this lack</p><p>of control that makes the coin-flipping outcome appear as a risky event. A</p><p>second example is the pseudo-random number generator commonly found</p><p>nowadays in calculators. It generates numbers that are difficult to predict.</p><p>But how can a calculator create uncertainty? It cannot. All it does is go</p><p>through a deterministic process. But this process has a special characteristic:</p><p>It is a chaotic process that is sensitive to initial conditions. It means that some</p><p>small change in initial conditions generates diverging paths and different</p><p>long-term trajectories. Here, the initial conditions are given by the fraction</p><p>of a second at which you push the random number generator button on the</p><p>calculator. Each time you push the button, you likely pick a different seed</p><p>and start the chaotic process at a different point, thus generating a different</p><p>outcome. In this case, it is our inability to control precisely our use of a</p><p>calculator that makes the outcome appear as a risky event. A final example is</p><p>the weather. Again, the weather is difficult to predict because it is the</p><p>outcome of a chaotic process. This holds even if the laws of thermodynamics</p><p>generating weather patterns are well understood. Indeed, in a chaotic pro-</p><p>cess, any imprecise assessment of the initial conditions is sufficient to imply</p><p>long-term unpredictability. It is our inability to measure all current weather</p><p>conditions everywhere that generates some uncertainty about tomorrows</p><p>weather.</p><p>Second, risk exists because of our limited ability to process information. A</p><p>good example is the outcome of playing a chess game. A chess game involves</p><p>well-defined rules and given initial conditions. As such, there is no uncer-</p><p>tainty about the game. And there are only three possible outcomes: A given</p><p>player can win, lose, or draw. So why is the outcome of a chess game</p><p>uncertain? Because there is no known playing strategy that can guarantee</p><p>a win. Even the largest computer cannot find such a strategy. Interestingly,</p><p>even large computers using sophisticated programs have a difficult time</p><p>winning against the best chess players in the world. This indicates that the</p><p>human brain has an amazing power at processing information compared to</p><p>computers. But it is the brains limited power that prevents anyone from</p><p>devising a strategy that would guarantee a win. It is precisely the reason why</p><p>playing chess is interesting: One cannot be sure which player is going to win</p><p>ahead of time. This is a good example to the extent that chess is a simple</p><p>game with restricted moves and few outcomes. In that sense, playing chess</p><p>is less complex than most human decision-making. This stresses the</p><p>Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 7Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 7Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 7Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 7</p><p>The Measurement of Risk 7</p></li><li><p>importance of information processing in the choice of decision rules. The</p><p>analysis of decision rules under some limited ability to process information</p><p>has been called bounded rationality. As just noted, the outcome of a chess</p><p>game is uncertain precisely because the players have a limited ability to</p><p>process information about the payoff of all available strategies (otherwise,</p><p>the outcome of the game would be known with the identification of the first</p><p>mover). Once we realize that no one is able to process all the information</p><p>available about our human environment, it becomes clear that risky events</p><p>are very common.</p><p>Third, even if the human brain can obtain and process a large amount of</p><p>information, this does not mean that such information will be used. Indeed,</p><p>obtaining and processing information is typically costly. The cost of infor-</p><p>mation can take many forms. It can involve a monetary cost (e.g., purchasing</p><p>a newspaper or paying for consulting services) as well as nonmonetary cost</p><p>(e.g., the opportunity cost of time spent learning). Given that human learn-</p><p>ing is time consuming and that time is a scarce resource, it becomes relevant</p><p>to decide what each individual should learn. Given bounded rationality, no</p><p>one can be expected to know a lot about everything. This suggests a strong</p><p>incentive for individuals to specialize in areas where they can develop special</p><p>expertise (e.g., plumber specializing in plumbing, medical doctors specializ-</p><p>ing in medical care, etc.). The social benefits of specialization can be quite</p><p>significant and generate large improvements in productivity (e.g., the case</p><p>of the industrial revolution). If information is costly, this suggests that</p><p>obtaining and processing information is not always worth it. Intuitively,</p><p>information should be obtained only if its benefits are greater than its</p><p>cost. Otherwise, it may make sense not to collect and/or process informa-</p><p>tion. These are the issues addressed in Chapter 10 on the economics of</p><p>information. But if some information is not being used because of its</p><p>cost, this also means that there is greater uncertainty about our environ-</p><p>ment. In other words, costly information contributes to the prevalence of</p><p>risky events.</p><p>So there are many reasons why there is imperfect information about many</p><p>events. Whatever the reasons, all risky events have a unique characteristic:</p><p>They are not known for sure ahead of time. This means that there is always</p><p>more than one possibility that can occur. This common feature has been</p><p>captured by a unified theory that has attempted to put some structure on</p><p>risky events. This is the theory of probability. The scientific community has</p><p>advanced probability theory as a formal structure that can describe and</p><p>represent risky events. A review of probability theory is presented in Appen-</p><p>dix A. Given the prevalence of risk, probability theory has been widely</p><p>adopted and used. We will make extensive use of it throughout this book.</p><p>We will also briefly reflect about some of its limitations.</p><p>Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 8Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 8Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 8Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 8</p><p>8 Risk Analysis in Theory and Practice</p></li><li><p>Note that it is possible that one person knows something that is unknown</p><p>to another person. This suggests that imperfect knowledge is typically indi-</p><p>vidual specific (as you might suspect, this has created a large debate about</p><p>the exact interpretation of probabilities). It is also possible for individuals</p><p>to learn over time. This means that imperfect knowledge is situation and</p><p>time specific. As a result, we define imperfect knowledge as any</p><p>situation where, at a given time, an individual does not have perfect infor-</p><p>mation about the occurrences in his/her physical and socioeconomic envir-</p><p>onment.</p><p>In the context of probabilities, any event A has a probability Pr(A), such</p><p>that 0Pr(A) 1. This includes as a special case sure events, wherePr(A) 1. Since risky events and sure events are defined to be mutuallyexclusive, it follows that risky events are characterized by Pr(A) &lt; 1. Acommon example is the outcome of flipping a coin. Even if this is the</p><p>outcome of a deterministic process (as discussed previously), it behaves as</p><p>if it were a risky event. All it takes for a risky event is that its outcome is</p><p>not known for sure ahead of time. As discussed above, a particular event</p><p>may or may not be risky depending on the ability to measure it, the ability to</p><p>control it, the ability to obtain and process information, and the cost of</p><p>information.</p><p>In general, in a particular situation, denote the set of all possible outcomes</p><p>byS. The setS is called the sample space. Particular elementsA1, A2, A3, . . . ,of the set S represent particular events. The statement Ai Aj reads Ai is asubset ofAj andmeans that all elementary events that are inAi are also inAj.</p><p>The set (Ai [ Aj) represents the union ofAi andAj, that is the set of elementaryevents in S that occur either in Ai or in Aj. The set (Ai \ Aj) represents theintersection ofAi andAj , that is the set of elementary events in S that occur in</p><p>both Ai and Aj . Two events Ai and Aj are said to be disjoint if they have no</p><p>point in common, that is if (Ai \ Aj) 1, where 1 denotes the empty set).Then, for a given sample space S, a probability distribution Pr is a function</p><p>satisfying the following properties:</p><p>1. Pr(Ai) 0 for all events Ai in S.2. Pr(S) 1:3. If A1, A2, A3, . . ., are disjoint events, then</p><p>Pr(A1 [ A2 [ A3 . . . ) P</p><p>i Pr(Ai):</p><p>In the case where risky events are measured by real numbers, this gener-</p><p>ates random variables. A random variable X is a function that takes a specific</p><p>real value X(s) at each point s in the sample space S. Then, the distribution</p><p>function of the random variable X is the function F satisfying</p><p>F (t) Pr(X t). Thus, the distribution function measures the probabilitythat X will be less than or equal to t. See Appendix A for more details.</p><p>Chavas / Risk Analysis in Theory and Practice Final 21.4.2004 10:49am page 9Chavas / Risk Analysis in Theory and Practice Fi...</p></li></ul>

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