L1: Risk and Risk Measurement1 Lecture 1: Risk and Risk Measurement We cover the following topics in this part –Risk –Risk Aversion Absolute risk aversion Relative risk aversion –Risk premium –Certainty equivalent –Increase in risk –Aversion to downside risk –First-degree stochastic dominance

L1: Risk and Risk Measurement2 Risk Risk can be generally defined as “uncertainty”. Sempronius owns goods at home worth a total of 4000 ducats and in addition possesses 8000 decats worth of commodities in foreign countries from where they can only be transported by sea, with ½ chance that the ship will perish. If he puts all the foreign commodity in 1 ship, this wealth, represented by a lottery, is x ~ (4000, ½; 12000, ½) If he put the foreign commodity in 2 ships, assuming the ships follow independent but equally dangerous routes. Sempronius faces a more diversified lottery y ~ (4000, ¼; 8000, ½; 12000, ¼) In either case, Sempronius faces a risk on his wealth. What are the expected values of these two lotteries? In reality, most people prefer the latter. Why?

L1: Risk and Risk Measurement3 Risk Averse Agent and Utility Function There is no linear relationship between wealth and the utility of consuming this wealth Utility function: the relationship between monetary outcome, x, and the degree of satisfaction, u(x). When an agent is risk averse, the relationship is concave.

L1: Risk and Risk Measurement4 Risk Aversion Definition A decision maker with utility function u is risk-averse if and only if u is concave. –What is risk aversion? What is concavity? -- page 8 Risk Premium Arrow-Pratt approximation Holds for small risks

L1: Risk and Risk Measurement5 Deriving the Risk Premium Formula Note: The cost of risk, as measured by risk premium, is approximately proportional to the variance of its payoffs. This is one reason why researchers use a mean-variance decision criterion for modeling behavior under risk. However П=1/2σ 2 *A(w) only holds for small risk (thus we can apply for the 2 nd -order approximation).

L1: Risk and Risk Measurement6 Measuring Risk Aversion The degree of absolute risk aversion For small risks, the risk premium increases with the size of the risk proportionately to the square of the size –Assuming z=k*ε, where E(ε)=0, σ(ε)=σ Accepting a small-mean risk has no effect on the wealth of risk-averse agents ARA is a measure of the degree of concavity of a utility function, i.e., the speed at which marginal utility decreases

L1: Risk and Risk Measurement7 A More Risk Averse Agent Consider two risk averse agents u and v. if v is more risk averse than u, this is equivalent to that A v >A u Conditions leading to more risk Aversion – page 14-15

L1: Risk and Risk Measurement8 Example Two agents’ utility functions are u(w) and v(w).

L1: Risk and Risk Measurement9 CARA and DARA CARA However, ARA typically decreases –Assuming for a square root utility function, what would be the risk premium of an individual having a wealth of dollar 101 versus a guy whose wealth is dollar with a lottery to gain or lose $100 with equal probability? What kind of utility function has a decreasing risk premium?

L1: Risk and Risk Measurement10 Prudence Thus –u’ is a concave transformation of u. Defining risk aversion of (-u’) as –u’’’/u’’ This is known as prudence, P(w) The risk premium associated to any risk z is decreasing in wealth if and only if absolute risk aversion is decreasing or prudence is uniformly larger than absolute risk aversion P(w)≥A(w)

L1: Risk and Risk Measurement11 Relative Risk Aversion Definition Using z for proportion risk, the relation between relative risk premium, П R (z), and absolute risk premium, П A (wz) is This can be used to establish a reasonable range of risk aversion: given that (1) investors have a lottery of a gain or loss of 20% with equal probability and (2) most people is willing to pay between 2% and 8% of their wealth (page 18, EGS). CRRA

L1: Risk and Risk Measurement12 Some Classical Utility Function For more of utility functions commonly used, see HL, pages Also see HL, Page 11 for von Neumann-Morgenstern utility, i.e., utility function having expected value Appropriate expression of expected utility, see HL, page 6 and 7

L1: Risk and Risk Measurement13 Measuring Risks So far, we discuss investors’ attitude on risk when risk is given –I.e., investors have different utility functions, how a give risk affects investors’ wealth Now we move to risk itself – how does a risk change? Definition: A wealth distributions w 1 is preferred to w 2, when Eu(w 1 )≥Eu(w 2 ) –Increasing risk in the sense of Rothschild and Stiglitz (1970) –An increase in downside risk (Menezes, Geiss and Tressler (1980) –First-order stochastic dominance

L1: Risk and Risk Measurement14 Adding Noise w 1 ~(4000, ½; 12000, ½) w 2 ~(4000, ½; ε, ½) [adding price risk; or an additional noise] Then look at the expected utility (page 29) General form: w 1 takes n possible value w 1, w 2,w 3,…, w n. Let p s denotes the probability that w 1 takes the value of w s. If w 2 = w 1 + ε  Eu(w 2 ) ≤ Eu(w 1 ).

L1: Risk and Risk Measurement15 Mean Preserving Spread Transformation Definition: –Assuming all possible final wealth levels are in interval [a,b] and I is a subset of [a, b] –Let f i (w) denote the probability mass of w 2 (i=1, 2) at w. w 2 is a mean-preserving spread (MPS) of w 1 if 1.Ew 2 = Ew 1 2.There exists an interval I such that f 2 (w)≤ f 1 (w) for all w in I Example: the figure in the left-handed panel of page 31 Increasing noise and mean preserving spread (MPS) are equivalent

L1: Risk and Risk Measurement16 Single Crossing Property Mean Preserving spread implies that (integration by parts – page 31) This implies a “single-crossing” property: F 2 must be larger than F 1 to the left of some threshold w and F 2 must be smaller than F 1 to its right. I.e.,

L1: Risk and Risk Measurement17 The Integral Condition

L1: Risk and Risk Measurement18 MPS Conditions Consider two random variable w 1 and w 2 with the same mean, (1)All risk averse agents prefer w 1 to w 2 for all concave function u (2)w 2 is obtained from w 1 by adding zero-mean noise to the possible outcome of w1 (3)w 2 is obtained w 1 by a sequence of mean-preserving spreads (4)S(w)≥0 holds for all w.

L1: Risk and Risk Measurement19 Preference for Diversification Suppose Sempreonius has an initial wealth of w (in term of pounds of spicy). He ships 8000 pounds oversea. Also suppose the probability of a ship being sunk is ½. x takes 0 if the ship sinks and 1 otherwise. If putting spicy in 1 ship, his final wealth is w 2 =w+8000*x If he puts spicy in 2 ships, his final wealth is w 1 =w+8000(x 1 +x 2 )/2 Sempreonius would prefer two ships as long as his utility function is concave Diversification is a risk-reduction device in the sense of Rothschild of Stiglitz (1970).

L1: Risk and Risk Measurement20 Variance and Preference Two risky assets, w 1 and w 2, w 1 is preferred to w 2 iff П2> П1 For a small risk, w 1 is preferred to w 2 iff the variance of w 2 exceeds the variance of w 1 But this does not hold for a large risk. The correct statement is that all risk-averse agents with a quadratic utility function prefer w 1 to w 2 iff the variance of the second is larger than the variance of the first –See page 35.

L1: Risk and Risk Measurement21 Aversion to Downside Risk Definition: agents dislike transferring a zero-mean risk from a richer to a poor state. w 2 ~(4000, ½; ε, ½) w 3 ~(4000+ ε, ½; 12000, ½) Which is more risky

L1: Risk and Risk Measurement22 First-Degree Stochastic Dominance Definition: w2 is dominated by w1 in the sense of the first- degree stochastic dominance order if F2(w)≥F1(w) for all w. No longer mean preserving Three equivalent conditions regarding first-degree stochastic dominance outlined in Proposition 2.5 See page 45, HL

L1: Risk and Risk Measurement23 Second-degree Stochastic Dominance Definition –See page 45, HL (it is exactly same as a MPS transformation) Combining FSD with increase in risk or a mean preserving spread Third-degree stochastic dominance

L1: Risk and Risk Measurement24 Technical Notes Taylor Series Expansion: f(x)= Integration by parts

L1: Risk and Risk Measurement25 Exercises EGS, 1.2; 1.4; 2.2