VIEWS: 39 PAGES: 30 POSTED ON: 6/4/2012 Public Domain
Micro 1015 Risk and Uncertainty Introducing risk and uncertainty • Many economic decisions have to be taken without being certain of the consequences • When we buy certain goods, we cannot tell for sure in advance what their quality is, so we won’t know how much we will appreciate them • When we decide to train for a particular job, we do not know for sure what the future wage levels in that job are going to be • When we buy some type of capital asset we cannot tell for sure what return it will earn, or what it’s future price is going to be • so in these cases, how do consumers decide? decisions under uncertainty • Even when there is uncertainty of the above types, consumers have some idea of the possible outcomes that can occur • It is also reasonable to assume that, as in the case when there is no uncertainty, consumers’ preferences over outcomes can be represented by a utility function • So if we represent an outcome by a vector x of relevant variables, we know the consumer’s utility u(x). The problem, however, is that we are not sure that the outcome will be x; it may also be y, say, where y is a vector with different elements. We also know u(y) – so here x and y may stand for one of the goods we buy being of high or low quality, the wages of our future job being high or low, the return on the asset we buy being high or low … decisions under uncertainty • Next, it is reasonable to believe that we also have some idea about the likelihood of the different outcomes. To capture this we make the strong assumption that in the case where the outcome may be x or y, the consumer is willing to assign a probability p that the outcome will be x • Note that the probability that the outcome will be x must be less than or equal to 1. Moreover, if the outcomes can only be x or y, then the probability that it will be y is just (1-p) • Now suppose the outcome if the consumer does not take some action will be z with certainty. If he does take the action, the outcome will be either x (with probability p) or y decisions under uncertainty • So if he does not take the action, his utility will be u(z), while if he takes the action his utility will be either u(x) or u(y) • The expected utility hypothesis states that in this kind of situation with uncertainty but specified outcomes and probabilities, the consumer will choose the action that maximizes his expected utility • Expected utility in this case is defined as the weighted average of the utilities of each possible outcome, with the weights being the probabilities of each outcome • Thus in the above example, the consumer will choose the action that may result in either x or y if pu(x)+(1-p)u(y)>u(z) expected utility • Note that this way of making the decision doesn’t guarantee that the consumer will make the right decision. For example, suppose the inequality holds and the consumer chooses to take the action, but that u(y)<u(z). Then if the consumer is unlucky and outcome y occurs, he will be worse off than if he hadn’t taken the action • However, on average a consumer who consistently follows the expected utility rule will be better off than if he does not • So the theory of decision under uncertainty is similar to the theory of decisions under certainty expect that now we maximize expected utility, not just “actual” utility restrictions on the utility function • In the no-uncertainty case, we found that a given preference ordering could be represented by many different utility functions – for example, if u(x) could represent the preference ordering, it could also be represented by f(u(x)) as long as f was an increasing function • in the uncertainty case, a given preference ordering can also be represented by many different utility functions • however, if u(x) represents the consumer’s preference ordering, then any other utility function representing the same preference ordering (say, v(x)) must be of the form v(x) = A + Bu(x), B greater than or equal to zero A, B constants restriction on the utility function • As an example, suppose that u(x) represents the utility function of the consumer in the previous example, and suppose that u(x)=16, u(y)=49, and u(z)=31.36. Suppose also that p=0.5. Since 0.5(16+49)=32.5>31.36, we conclude that the consumer should take the action • Now suppose we instead represent the utility function by the square root of u(x). The square root certainly is an increasing function. But the square root of 31.36 is 5.6, which is larger than 0.5(4+7), so with this utility function we conclude that the consumer should not take the action • So the square root of u(x) does not represent the same preference ordering under uncertainty as u(x) does restrictions on the utility function • To see that a transformation of the form v(x)=A+Bu(x) will properly represent the preference ordering, suppose we represent all possible outcomes under decision 1 as vectors x(i) which occur with probabilities p(i), while the outcomes under decision 2 are vectors y(j) with probabilities p(j). Then the inequality p [ A Bu( x )] i i i j p j [ A Bu( y j )] will hold for any A and any B>0 if it holds for A=0, B=1. • In the next few slides, we consider a consumer whose utility depends on one variable only, namely his income Y, but whose income is uncertain (may take on any of many possible values) Risk • Notice that in this case, if Y can take on many values Y(i) and we know the probabilities p(i), we can compute the mean, variance, and standard deviation of Y. • The for a given mean value of of Y, the variance (or equivalently, the standard deviation) can be thought of as a measure of the risk that the consumer faces because the outcome of the income variable is uncertain • The next question is: if we think of consumers who face different probability distributions of income, with different means and variances, which ones will they prefer? • The answer depends on the nature of their utility functions Risk aversion • In the picture on the next slide, we measure income on the horizontal axis and utility on the vertical axis. A consumer with a utility function like solid curve A displays risk aversion, whereas a consumer with a utility function like dashed curve B is a risk lover • For simplicity, we assume that income can take on only two values, Y(1) and Y(2), and that the probability of each is ½. As the curves have been drawn, two consumers with utility functions depicted by A and B get the same expected utility in this situation • However, consider now a consumer who is offered the choice between this distribution on the one hand, and the mean income with certainty, on the other hand Utility B A EA E EB Y(1) mean Y Y(2) Y Risk aversion and risk premium • In this diagram, the expected utility of both consumer A and consumer B is E when income is either Y(1) or Y(2) with probability 0.5. • The utility the consumer would get from receiving mean Y with certainty is different, however. Consumer A would receive utility EA (at the level where the vertical through mean Y intersects curve A) while consumer B would receive utility EB • Thus consumer A is risk averse: he gets a higher level of (expected) utility from receiving the mean level of income with certainty than he gets from the lottery with Y(1) and Y(2) Risk aversion and risk premium • Consumer B, on the other hand, is a risk lover. He receives less (expected) utility from getting mean income with certainty than he gets from taking the gamble – note: actual utility and expected utility is the same for the case where the consumer gets a given level of utility with certainty • Note also the thick horizontal line segment to the right of E. It measures the risk premium for (risk averse) consumer A • It can be interpreted as the amount of money that person A would be willing to give up in order to eliminate the risk when he will either get Y(1) or Y(2). That is, his expected utility if he receives (mean Y – risk premium) for certain is the same as his expected utility in the risky situation Measuring risk aversion • Can we measure the degree of a person’s risk aversion? One possible answer is by looking at the second derivative of the utility function relating utility to income • The first derivative is the slope of utility function (which is arbitrary, since units of utility are arbitrary). The second derivative tells us how fast the slope changes (the “curvature” of the utility function) • The Arrow-Pratt measure of (absolute) risk aversion, denoted by r, is given by: d 2u (Y ) u ' ' (Y ) r 2 dY u ' (Y ) Risk aversion • Note that this measure is positive for a person who is risk-averse (that is, whose utility function displays diminishing marginal utility of income). It is zero for a risk-neutral person, and negative for a risk lover (a person such as person B in the earlier example • It can be proved that the Arrow-Pratt measure of risk aversion is related to the size of a person’s risk premium. Specifically, the larger is r, the larger is the risk premium associated with any given level of risk • An example of a utility function with a constant level of absolute risk aversion is u (Y ) e rY , r 0 Risk aversion • One can verify that the absolute risk aversion for this utility function is r. • One can also define a measure of Relative Risk Aversion (RAV): u ' ' (Y )Y RAV rY u ' (Y ) • An example of a utility function that displays constant relative risk aversion (treating utility as a function of consumption), is C 1 u (C ) 1 Risk aversion • This utility function is frequently used in macroeconomics and other models. It has a relative coefficient of risk aversion given by σ, and draws attention to the fact that the coefficient of risk aversion is the same as the elasticity of the marginal utility of consumption or income • Note also that if the consumer behaves in accordance with expected utility theory, the parameter σ of the utility function can, in principle, be estimated on the basis of observable data (by looking at how much the consumer is willing to pay in order to reduce risk) Risk and insurance • If most people are risk averse, they must be paid a risk premium in order to take on risk. But conversely, this also means that they are willing to pay in order to reduce risk. • Specifically, in the earlier example, suppose that the difference between Y(2) and Y(1) is interpreted as a random loss that happens to half the population – that is, if there is no loss, a person’s annual income is Y(2). If there is a loss, it is Y(1). In the example, half the population suffers this loss in a given year – This is consistent with the idea that in this population, mean income is Y(mean), but in a given year the actual income is either Y(2) or Y(1) Risk and insurance • In this situation, there should be a market for risk pooling (insurance) • Specifically, suppose insurers could offer plans under which everyone in the plan would pay a premium ahead of time. The insurance plan would then promise that for those who were unlucky and suffered a loss (that is, received an income of Y(1)), the plan would pay out a lump-sum amount X. Those who did not suffer a loss (that is, received an income of Y(2) would not get a payout). • Suppose we consider a very generous plan: it would actually pay the whole difference [Y(2)-Y(1)] to those who otherwise would have an income of Y(1)! Risk and insurance, cont’d • What would be the expected payout from such an insurance plan for each insured person? Answ: simply 0.5(Y(2)-Y(1)). Since half the population would receive an income of Y(1), the plan would have to pay out this amount to half the population • How large would the cost of the plan have to be? Answ: more than 0.5(Y(2)-Y(1)) per insured person, since this is the average amount that the plan would have to pay out, per insured person – in reality, it would have to charge a bit more than that, in order to also earn enough revenue to pay the costs of plan administration, profits on invested funds, etc. Actuarially fair insurance • To start with, let’s simplify and suppose that it costs nothing to run the insurance plans, so that its costs actually equal 0.5(Y(2)-Y(1)), and that this is the premium it charges – a hypothetical insurance plan for which the premium is equal to the expected benefit payouts (with no administration costs, etc) is called an actuarially fair insurance plan • Note also that implicitly, we are making use here of the law of large numbers: even though we don’t know how much we have to pay out as benefits for an individual insured person, we know that on average, we have to pay out 0.5(Y(2)-Y(1)) for each person – the law of large numbers says that if we have a large number of insured persons, the average payment will be very close to that amount Actuarially fair insurance • But for a person who is insured by such a plan, what is his expected utility? • Answer: it is the same as the expected utility of a person who receives the mean income for certain – if the person receives an income of Y(2), his net income (given that he has paid the premium of 0.5(Y(2)-Y(1)) ahead of time) is just Y(2)- 0.5(Y(2)-Y(1))=0.5(Y(1)+Y(2)), the mean income – if the person receives an income of Y(1), he receives a payment of (Y(2)-Y(1)). However, he has already paid the premium 0.5(Y(2)-Y(1)). Taking this into account, his net income will also be 0.5(Y(1)+Y(2)), the mean income • So his expected utility is EA in the diagram: the same as a person who gets the mean income, with no risk! Risk, return, and portfolio choices • A large part of the literature regarding choice under risk deals with the problem of how consumers should allocate their private savings (“wealth”, or “assets”) • A whole range of assets are available in the market. They may differ in terms of their expected (mean) yield, but also in their risk (variability of yield) • So buying a given asset is like buying a lottery, but to some extent one can choose the odds • Intuitively, if most people in the market are risk averse (so that, in order to buy a more risky asset, they must be compensated by a higher expected return), one would expect that riskier assets (with higher variance, or standard deviation) would have a higher yield on average Diversification and risk pooling • However, tendency for riskier assets to have higher yields can, to a considerable extent, be offset by the technique of diversification – diversification consists in pooling a large number of assets with uncorrelated risks; because of the law of large numbers, a portfolio of such assets will have a much lower risk than any individual asset would have • Note also that risk pooling can be privately profitable: if risky assets earn returns that are much higher than less risky ones, financial professionals can earn a profit by buying such assets and pooling them, earning a high rate of return with low risk (after pooling) • They can then resell the pooled assets to risk-averse investors Risk pooling and diversification • Nevertheless, risk pooling is not costless, so it will remain true that, in equilibrium, assets with higher risk will earn a higher average return. Moreover, “market risk” cannot be eliminated through risk pooling for a given asset class. Thus, for example, funds invested in the stock market (even in pooled “mutual funds”) will tend to carry higher risk than funds kept in savings accounts • Therefore, an investor with a certain amount of funds to invest faces the problem of what portion of his assets to invest in safe (low-risk) assets with a low expected return, and what portion in higher-risk assets with higher expected return • The text gives an example, pp 179-183 in my copy Asset allocation example • Suppose the consumer has access to two assets, one that has an expected return Rf with zero risk (variance), and one that has an expected return of Rm, with an actual return of rm, and standard deviation σm. Suppose also that the investor has some degree of risk aversion, so that her expected utility can be written U=U(Rp,σp), where the subscript p refers to her entire portfolio – “portfolio”: weighted sum of all assets • Let b be the fraction of her portfolio that she allocates to the risky asset (with subscript m). We thus have Rp=(1-b)Rf+bRm • We also have σp=b σm Asset allocation, concluded • These two equations can be used to create a curve that gives us the consumer’s tradeoff between risk and return – the consumer can trade return for risk be changing b – the text refers to this as a “budget line” • Also note that one can think of U(Rp, σp) as a utility function, and construct a map of indifference curves, each one corresponding to a given level of utility • Note, though, that these indifference curves will slope upwards, not downwards (why? because risk is not a “good”, it’s a bad • So what is the consumer’s optimum choice? Answ: where she reaches the highest possible utility; at that point an indifference curve must be tangent to the budget line (as usual) Rp Risk-return tradeoff Rm Rp* Rf σp* σm σp The “price of risk” • The test refers to the ratio (Rm-Rf)/ σm as “the price of risk”. It tells the consumer how much extra return she can get by assuming more risk • Note that the curve showing the risk-return tradeoff in the diagram has been extended beyond the point where σp=σm; that is beyond the point where b = 1. Does that make sense? – yes in principle; the concept of “leveraging” • Note that the equilibrium price of risk in financial markets depends not only the degree of consumer risk aversion, but also on the efficiency of the financial system in eliminating risk by diversification and risk pooling