If U is linear, then U[E(Ỹ )] = E[U(Ỹ )], and one is indifferent between lottery and its expectation. One is called risk neutral.

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1 Risk aversion For those preference orderings which (i.e., for those individuals who) satisfy the seven axioms, define risk aversion. Compare a lottery Ỹ = L(a, b, π) (where a, b are fixed monetary outcomes) with receiving E(Ỹ ) = πa + (1 π)b for sure. Whether the lottery or its expectation is preferred, depends on the curvature of U: If U is linear, then U[E(Ỹ )] = E[U(Ỹ )], and one is indifferent between lottery and its expectation. One is called risk neutral. If U is concave, then U[E(Ỹ )] E[U(Ỹ )], and one prefers the expectation. One is called risk averse. If U is convex, then U[E(Ỹ )] E[U(Ỹ )], and one prefers the lottery. One is called risk attracted. The inequalities follow from Jensen s inequality (see Sydsæter, Strøm and Berck, equations , or D&D, p.47). If U is strictly concave or convex, the inequalities are strict, except if Ỹ is constant with probability one. Quite possible that many have U functions which are neither everywhere linear, everywhere concave, nor everywhere convex. Then one does not fall into one of the three categories. 1

2 Assume risk aversion (Does not follow from the seven axioms.) Most common behavior in economic transactions. Explains the existence of insurance markets. But what about money games? Expected net result always negative, so a risk-averse should not participate. Cannot be explained by theories taught in this course. Some of our theories will collapse if someone is risk neutral or risk attracted. Those will take all risk in equlibrium. Does not happen. How measure risk aversion? Natural candidate: U (y). Varies with the argument, e.g., high y may give lower U (y). Is U() twice differentiable? Assume yes. But: The magnitude U (y) is not preserved if c 1 U() + c 0 replaces U(). Use instead: U (y)/u (y) measures absolute risk aversion. U (y)y/u (y) measures relative risk aversion. In general, these also vary with the argument, y. 2

3 Arrow-Pratt measures of risk aversion For small risks: R A (y) U (y)/u (y) meaures how much compensation a person demands for taking the risk. Called the Arrow-Pratt measure of absolute risk aversion. R R (y) U (y) y/u (y) is called the Arrow-Pratt measure of relative risk aversion. Consider the following case (somewhat more general than D&D, sect ): The wealth Y is non-stochastic. A lottery Z has expectation E( Z) = 0. For a person with utility function U() and inital wealth Y, define the risk premium Π associated with the lottery Z by E[U(Y + Z)] = U(Y Π). Will show the relation between Π and absolute risk aversion. 3

4 Risk premium is proportional to risk aversion (The result holds approximately, for small lotteries.) E[U(Y + Z)] = U(Y Π). Take quadratic approximations (second-order Taylor series). LHS: U(Y + z) U(Y ) + zu (Y ) z2 U (Y ) which implies E[U(Y + Z]) U(Y ) E( Z 2 )U (Y ). RHS: U(Y Π) U(Y ) ΠU (Y ) Π2 U (Y ). Assume last term is very small. Let σ 2 z var( Z). Then: which implies 1 2 σ2 zu (Y ) ΠU (Y ) Π U (Y ) U (Y ) 1 2 σ2 z. 4

5 The U function: Forms which are often used Some theoretical results can be derived without specifying form of U. Other results hold for specific classes of U functions. Constant absolute risk aversion (CARA) holds for U(y) e ay, with R A (y) = a. Constant relative risk aversion (CRRA) holds for U(y) 1 1 g y1 g, with R R (y) = g. (Exercise: Verify these two claims. (a, g are constants.) Determine what are the permissible ranges for y, a and g, given that functions should be well defined, increasing, and concave.) Essentially, these are the only functions with CARA and CRRA, respectively, apart from CRRA with R R (y) = 1. (Any constant can be added to the functions, and any positive constant can be multiplied with them.) R R (y) 1 is obtained with U(y) ln(y). Another much used form: U(y) = ay 2 + by + c, quadratic utility. Easy for calculations, U linear. (What are permissible ranges, given that U should be concave? Hint: There is a minus sign in front of a.) Quadratic U has increasing R A (y) (Verify!), perhaps less reasonable. (What happens for this U function when y > b/2a? Is this reasonable?) 5

6 Stochastic dominance Two criteria for making decisions without knowing shape of U(). May be important for delegation, for research, for prediction. Work only in a limited number of comparisons. For other comparisons, these decision criteria are inconclusive. Useful for narrowing down choices by excluding dominated alternatives. First-order stochastic dominance A random variable X A first-order stochastically dominates another random variable X B if every vn-m expected utility maximizer prefers X A to X B. Second-order stochastic dominance A random variable X A second-order stochastically dominates another random variable X B if every risk-averse vn-m expected utility maximizer prefers X A to X B. 6

7 First-order stochastic dominance, FSD (Let the cumulative distribution functions be F A (x) Pr( X A x) and F B (x) Pr( X B x).) Possible to show that X A X B by all is equivalent to the following, which is one possible definition of first-order s.d.: F A (w) F B (w) for all w, and F A (w i ) < F B (w i ) for some w i. For any level of wealth w, the probability that X A ends up below that level is less than the probability that X B ends up below it. 7

8 Second-order stochastic dominance, SSD Possible to show that X A X B by all risk averters is equivalent to the following, which is one possible definition of second-order s.d.: and w i F A(w)dw w i F B(w)dw for all w i, F A (w i ) F B (w i ) for some w i. One distribution is more dispersed ( more uncertain ) than the other. If we restrict attention to variables X A and X B with the same expected value, Theorem 3.4 in D&D states that SSD is equivalent to: XA can be written as X B + z, where the difference z is some random noise. 8

9 Risk aversion and simple portfolio problem (Chapter 4 in Danthine and Donaldson.) Sections are covered in Varian, Microeconomic Analysis, chapter 13. Will not have time to discuss here. For simple portfolio problem (one risky, one risk free asset), main result is: Absolute sum invested in risky asset is independent of wealth for CARA, increasing for DARA, decreasing for IARA Fraction of wealth invested in risky asset is independent of wealth for CRRA, increasing for DRRA, decreasing for IRRA Risk aversion and saving (Sect. 4.6, D&D.) How does saving depend on riskiness of return? Rate of return is r, (gross) return is R 1 + r. Consider choice of saving, s, when probability distribution of R is taken as given: max s R + E[U(Y 0 s) + δu(s R)] where Y 0 is a given wealth, δ is (time) discount factor for utility. 9

10 Risk aversion and saving, contd. Savings decision well known topic in microec. without risk Typical questions: Depedence of s on Y 0 and on E( R) Focus here: How does saving depend on riskiness of R? Consider mean-preserving spread: Keep E( R) fixed Assuming risk aversion, answer is not obvious: R more risky means saving is less attractive, save less R more risky means probability of low R higher, willing to give up more of today s consumption to avoid low consumption levels next period, save more Need to look carefully at first-order condition U (Y 0 s) = δe[u (s R) R] What happens to right-hand side as R becomes more risky? Cannot conclude in general, but for some conditions on U (Jensen s inequality:) Depends on concavity of g(r) = U (sr)r If, e.g., g is concave: May compare risk with no risk: E[g( R)] < g[e( R)] But also some risk with more risk, cf. Theorem 4.7 in D&D. 10

11 Risk aversion and saving, contd. max s R + E[U(Y 0 s) + δu(s R)] (assuming, all the time here, U > 0 and risk aversion, U < 0) When R B = R A + ε, E( R B ) = E( R A ), will show: If R R(Y ) 0 and R R (Y ) > 1, then s A < s B. If R R(Y ) 0 and R R (Y ) < 1, then s A > s B. First condition on each line concerns IRRA vs. DRRA, but both contain CRRA. Second condition on each line concerns magnitude of R R (also called RRA): Higher risk aversion implies save more when risk is high. Lower risk aversion (than R R = 1) implies save less when risk is high. But none of these claims hold generally; need the respective conditions on sign of R R. Interpretation: When risk aversion is high, it is very important to avoid the bad outcomes in the future, thus more is saved when the risk is increased. Reminder: This does not mean that a highly risk averse person puts more money into any asset the more risky the asset is. In this model, the portfolio choice is assumed away. If there had been a risk free asset as well, the more risk averse would save in that asset instead. 11

12 Risk aversion and saving, contd. Proof for the first case, R R(Y ) 0 and R R (Y ) > 1: Use g (R) = U (sr)sr + U (sr) and g (R) = U (sr)s 2 R + 2U (sr)s. For g to be convex, need U (sr)sr + 2U (sr) > 0. To prove that this holds, use R R(Y ) = [ U (Y )Y U (Y )]U (Y ) [ U (Y )Y ]U (Y ) [U (Y )] 2, which implies that R R(Y ) has the same sign as U (Y )Y U (Y ) [ U (Y )Y ]U (Y )/U (Y ) = U (Y )Y U (Y )[1 + R R (Y )]. When R R(Y ) < 0, and R R (Y ) > 1, this means that 0 < U (Y )Y + U (Y )[1 + R R (Y )] < U (Y )Y + U (Y ) 2. (This expression has a typo in D&D, p. 70: U instead of U.) Since this holds for all Y, in particular for Y = sr, we find and g is thus convex. U (sr)sr + 2U (sr) > 0, 12