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# From risk to opportunity Lecture 12 John Hey and Carmen Pasca.

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From risk to opportunity Lecture 12 John Hey and Carmen Pasca

Lecture 12 Market Implications and Overview The first part of this lecture looks at implications of EU in a market context. We discuss how insurance works. We note that if people are insured then the probabilities of the risks may change (moral hazard). We discuss the problems when the insurer is not sure about the probabilities (adverse selection) We also show how the exchange of risks is mutually beneficial even if everyone is risk-averse – exchange is an opportunity. The final part of this lecture briefly summarises From Risk to Opportunity as a whole.

Lecture 12 Market Implications and Overview: Insurance We consider a simple case of insurance, where there are two possible states of the world, state 1 and state 2, with probabilities p 1 and p 2 (which sum to 1). We assume state 1 is a nasty in that people are worse off and hence that people want to insure against it. An insurance contract specifies a price p per unit of insurance cover C, where pC is paid by the buyer of the insurance to the company if state 1 does not occur and C is paid by the insurance company to the buyer of the insurance if state 1 occurs. Let us suppose that on average the insurance company breaks even so that p 2 pC = p 1 C (average amount paid in = average amount paid out) and hence that p = p 1 /p 2. So the price for fair insurance is p 1 /p 2 the ratio of the probabilities.

Lecture 12 Market Implications and Overview: Optimal purchase Let c 1 and c 2 denote the (EU) individuals consumption in states 1 and 2. Ex ante, he wants to maximise Expected Utility which is given by p 1 u(c 1 ) + p 2 u(c 2 ). On the right we draw indifference curves in a graph with c 1 and c 2 on the two axes. These curves have equation p 1 u(c 1 ) + p 2 u(c 2 ) = constant. We assume that if the nasty state 1 does not happen the individual has consumption 70 but if state 1 does happen his/her consumption is 10. So the start point is point W. In this example we have p 1 =0.4 and p 2 =0.6

Lecture 12 Market Implications and Overview: Optimal purchase Now a crucial property of the (blue) indifference curves is that their slope along the (black) certainty line is –p 1 /p 2......which you will recall is precisely the slope of the fair insurance budget line (which passes through W). Hence it follows that the optimal point to which to move is the point on the certainty line......with fair insurance our risk- averse individual chooses to become completely insured...... and the insurer breaks even. In this example we have p 1 =0.4 and p 2 =0.6

Lecture 12 Market Implications and Overview: Moral Hazard The problem now is that the individual now has no incentive to prevent state 1 happening. Suppose he/she lets it rise to 0.5 (from 0.4). If the insurance company is not aware of this, they will continue to offer the same price, the individual will want to go to the red asterisk on the blue dashed line, when he/she should be operating on the red dashed line. The company makes a loss. This is moral hazard. This is why insurance companies take measures to stop the probabilities changing.

Lecture 12 Market Implications and Overview: Adverse Selection A related problem is when there are different people with different degrees of riskiness. In the picture here the low (high) risk have the blue (red) indifference curves. If the insurance company can distinguish them there is no problem....... they would offer the blue dashed budget line to the low risk and the red to the high risk. All would fully insure and the insurance company would happily break even. But if they could not, all insurers would choose the blue line and the company would lose...

Lecture 12 Market Implications and Overview: A solution to Adverse Selection Now the insurance company can no longer simply offer a price for insurance – it must offer packages...... the blue asterisk and the red asterisk. The low risk accept the blue and the high risk the red...... everyone is happy! (there are other solutions)

Lecture 12 Market Implications and Overview: Exchange of risk Here we consider a simple market in which there are just two risk-averse individuals, A and B, each facing a risk. We show that they can exchange part of the risk and both end up happier than they were before they did the exchange. We keep life simple: two equally likely possible states of the world, 1 and 2. Both individuals are interested in consumption which we will denote by c. Exchange is carried out before it is known which state will occur, and consumption after exchange and after the actual state is known. We assume that both individuals are EU.

Lecture 12 Market Implications and Overview: Each individual We are going to represent the exchange problem inside an Edgeworth Box, with consumption in the two states on the two axes (1 horizontal and 2 vertical). Let c 1 and c 2 denote the ex post consumption in states 1 and 2. Ex ante, each wants a consumption bundle which maximises Expected Utility which is given by ½u(c 1 ) + ½u(c 2 ). We draw indifference curves, first for individual A, in a graph with c 1 and c 2 on the two axes. These curves have equation ½u(c 1 ) + ½u(c 2 ) = constant. I have assumed that A has a utility function u(c)=-exp(-0.01c). (CARA with absolute risk aversion 0.01) Let us suppose that A starts, before exchange, with consumption 75 if state of the world 1 occurs and consumption 50 if state of the world 2 occurs. Later we will assume the same for individual B. You can think of state 2 as a nasty.

Lecture 12 Market Implications and Overview: As starting position As starting position (before trade) He or she starts at point E and the blue lines are his or her indifference curves. If he or she can move by trade above the curve passing through E he or she would become happier. (On the vertical axis is As consumption if state 2 occurs.)

Lecture 12 Market Implications and Overview: Bs starting position Let us assume that B faces the same risk and also starts at point E. However let us assume that he/she is more risk-averse, with utility function u(c)=-exp(-0.03c). and the red lines are his or her indifference curves. They are more convex. (On the vertical axis is As consumption if state 2 occurs.)

Lecture 12 Market Implications and Overview: B upside down Let us now construct an Edgeworth Box to show the issues of trade between A and B. We first draw B upside down.

Lecture 12 Market Implications and Overview: The Edgeworth Box Now we put B on top of A, thus forming the Box Note that we measure A (B)s consumption from the bottom left (top right) corner. Each starts with 75 if state 1 happens and 50 if state 2 happens. Collectively society has 150 if state 1 happens and 100 if state 2 happens – the dimensions of the box.

Lecture 12 Market Implications and Overview: Competitive Trading? The start point is at point E in the middle of the box. The competitive equilibrium is at point C – the intersection of their price-offer curves. What happens? The less risk- averse takes on more risk and the more risk-averse takes on less risk. They are both happy and have taken the opportunity of trade.

Lecture 12 Overview in retrospect 2.An historical perspective 3.Different approaches to risk 4.Philosophical approaches to risk 5.Epistemological approaches to risk 6.Political approaches to risk 7.Sociological approaches to risk emphasis on collectiveness 8.Psychological approaches to risk emphasis on processes 9.Economic approaches to risk emphasis on axioms of rationality 10.Expected Utility theory how elegant and compelling are the axioms 11.Implications of Expected Utility elegance and power 12.Market Implications and Overview shedding light on real life problems/solutions

Lecture 2 Risk an historical perspective: Conclusion Over the decades the concepts of risk (and associate concepts) have evolved and changed. There is beginning to be a consensus that there are risks that can be quantified and are in some way objective and uncertainties which may be unmeasurable or only partially measurable. The distinction is not always clear; neither is the distinction between objective and subjective risks. We shall see what social sciences make of all this!

Lecture 3 Different approaches to risk: Conclusion Different disciplines have different approaches to modelling and assessing risk. Economists focus on individuals and like to have axioms of rationality. Psychologists also focus on individuals but are more concerned about processes. Sociologists are concerned more with cultural aspects of society and the interaction between individuals. Their analysis is at a more aggregative level – that of society rather than the individual.

Lecture 4 Philosophical Approaches to risk: Conclusion Early philosophers did not try to quantify risk but merely to classify situations of certainty, risk, uncertainty and doubt. Later philosophers tried to distinguish between objective and subjective risks. Rawls, rather than try and quantify risk, specified how people should react to uncertain situations. In contrast, Sen argued that we should take into account differences in risk attitude.

Lecture 5 Epistemological approaches to risk: Conclusions Keynes saw risk as characterised by objective probabilities based on logic and reason. Savage constructs a structure in which probabilities, both in conditions of risk and of uncertainty, are subjective (and therefore may have no connection with reality – whatever that is). Allais added to subjectivism the limits of human cognitive ability and hence the need for behaviourism.

Lecture 6 Political and social approaches to risk: Conclusions Politics has a dramatic effect on the way that risk impacts on society and the world. Politicians can reduce risk by various schemes of social insurance and by manipulating the population. They can also increase it. Politicians can make a huge difference on the risks faced by companies. Changes in society (either induced by political actions or by technological change) are changing the way that risk impacts on members of society. In present modern society it is less true that institutions affect the impact of risk on its members. In a paradoxical way it is individual members of society who can change the way that risk impacts on them and their fellow members. Risks are different now than they were before, partly as a result of political, but also social, changes. Risk is becoming globalised.

Lecture 7 MidTerm Exam preparation The examination is not a test of memory (though you obviously have to be aware of key concepts). It is a test of argument. You will not, on the second and third questions, be marked on whether your answers are correct or not......but on the logic of your arguments, and, on question 3, on your intellectual creativity. Enjoy!

Mid-Term Examination 1.Suppose you have a committee of some firm formed of members who are all risk-loving. The committee must come to an agreed decision as to which of two favourable (in the sense that all possible outcomes are positive sums of money) gambles they should jointly accept. One of the two gambles is more risky than the other, though they have the same mean (expected value). Once they have decided, all members of the committee independently get to play out whichever gamble the committee has chosen, and individually receive the outcome (which is always some positive sum of money). If they do not reach agreement, no-one gets anything. Assume that all committee members prefer to have a positive sum of money rather than nothing. Argue that the committee, irrespective of what decision rule based on individual expressed preferences they use (for example, majority voting, with everyone voting for what they say is their preferred gamble), will chose the most risky gamble. Argue that there is no scope for strategic voting (for example, when someone votes for his or her least preferred option in the hope of getting something better). (Easy) 2.Over the centuries scientists from all disciplines have debated whether probabilities are objective or subjective. Give examples of each kind of probability, unless you think that one type does not exist – in which case you should argue why it does not exist. (Moderately difficult) 3.How do you verify whether some stated probability is true or not? (Almost impossible)

Comments on answers to Mid-Term Examination In question 1 very few (if any) discuss the decision rule adopted by the committee. We are surprised by the convoluted nature of the answers. It really is a simple question. One or two mentioned that it was important to know whether members were aware of the preferences of the other members – in case they get nothing by disagreeing, but this depends upon the decision rule. Very few of the candidates try to define probability and hence give a framework in which to answer questions 2 and 3. In question 2 there were very few attempts to discuss the consistency of subjective probabilities. We infer them from behaviour, but do we get the same inferences in different decision problems? That is a crucial point. It is not that subjective probabilities are the same as everyone elses or the same as objective probabilities but whether the individual uses them consistently. There is also a lot of confusion between risk attitude and risk perception; in SEU these are completely independent. There was almost no discussion of whether a true stated subjective probability needs to be objectively true. There are many students who were confused about the description of risk and peoples reaction to it. In question 3 many discuss the impossibility of testing by repeating an infinite number of times but very few have talked about statistical testing with fewer observations. There is very little discussion of what randomness means. Many candidates do not understand that axioms are assumptions. Some candidates have a very odd (implicit) concept of probability – seeming to think of it as certainty. (Black swans – life on Mars.)

Lecture 8 Psychological approaches to risk: Conclusion The standard economics model (SEU) appears from some experimental evidence to have difficulties in explaining some behaviour. Psychologists (particularly K&T and Busemeyer) have proposed new theories to rectify these apparent deficiencies. PT, for example, includes reference point effects, isolation and weighting functions, while DFT includes specifically randomness. These theories obviously explain better...... but do they predict better?

Lecture 9 Economic approaches to risk: Conclusion (Subjective) Expected Utility is the workhorse of economics. It is built on reasonable axioms of rationality. It is normatively sound. It avoids problems with dynamic inconsistency However, many experimental findings shed doubt on its universal applicability. Perhaps these findings are special cases and are driven by noise/error? Perhaps SEU plus noise is not bad as a description … … while SEU is good as a prescription?

Lecture 10 Expected Utility Theory: Conclusion (Subjective) Expected Utility Theory is based on very plausible and reasonable axioms. It leads to a very elegant solution for problems under risk. It leads to an even more elegant solution for problems under ambiguity/uncertainty. In dynamic problems it avoids the difficulty of dynamically inconsistent preferences.

Lecture 11 Implications of EUT: Conclusions The great joy of EUT is its elegance and tractability. It is easy to find your (EU) utility function. It is concave (linear, convex) where you are risk- averse (-neutral, -loving). The degree of risk-aversion can be measured by the degree of concavity of the utility function (using either an absolute or a relative measure). CARA and CRRA are to useful special cases… … which lead to insightful results.

Final Examination Preparation As before there will be 3 questions, one easy, one moderately difficult and one almost impossible. They all carry the same weight – so spend the same amount of time on each. There will be an emphasise on the second part of the course, from Lecture 8 onwards. If you are not sure what the question means, ask the invigilator (who has been instructed to contact us). If you do not want to do this, start any question with a statement of what you understand by it. Start any question with a clear statement of any key words (for example, from the Mid-Term Examination – probability). Use, particularly in the final question, the material from the entire course. All disciplines have useful things to say about risk. Dont waffle. Do avoid irrelevancies and things which are simply wrong. We are looking for ideas and excitement, particularly in question 3.

Lecture 12 Goodbye and thank you for your attendance.

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