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					   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

				
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