Lecture 1: Risk and Risk Measurement

1. 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 Measurement

2. 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 Measurement

3. 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 Measurement

4. 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 Measurement

5. 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=1/2σ2*A(w) only holds for small risk (thus we can apply for the 2nd-order approximation). L1: Risk and Risk Measurement

6. 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 Measurement

7. 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 Av>Au • Conditions leading to more risk Aversion – page 14-15 L1: Risk and Risk Measurement

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

9. 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 100000 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 Measurement

10. 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 Measurement

11. Relative Risk Aversion • Definition • Using z for proportion risk, the relation between relative risk premium, ПR(z), and absolute risk premium, ПA(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 Measurement

12. Some Classical Utility Function For more of utility functions commonly used, see HL, pages 25-28 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 Measurement

13. 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 w1 is preferred to w2, when Eu(w1)≥Eu(w2) • 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 Measurement

14. Adding Noise • w1~(4000, ½; 12000, ½) • w2~(4000, ½; 12000+ε, ½) [adding price risk; or an additional noise] • Then look at the expected utility (page 29) • General form: w1 takes n possible value w1,w2,w3,…,wn. Let psdenotes the probability that w1 takes the value of ws. If w2 = w1+ ε Eu(w2) ≤ Eu(w1). L1: Risk and Risk Measurement

15. 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 fi(w) denote the probability mass of w2 (i=1, 2) at w. w2 is a mean-preserving spread (MPS) of w1 if • Ew2= Ew1 • There exists an interval I such that f2(w)≤ f1(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 Measurement

16. Single Crossing Property • Mean Preserving spread implies that (integration by parts – page 31) • This implies a “single-crossing” property: F2 must be larger than F1 to the left of some threshold w and F2 must be smaller than F1 to its right. I.e., L1: Risk and Risk Measurement

17. The Integral Condition L1: Risk and Risk Measurement

18. MPS Conditions Consider two random variable w1 and w2 with the same mean, • All risk averse agents prefer w1 to w2 for all concave function u • w2 is obtained from w1 by adding zero-mean noise to the possible outcome of w1 • w2 is obtained w1 by a sequence of mean-preserving spreads • S(w)≥0 holds for all w. L1: Risk and Risk Measurement

19. 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 w2=w+8000*x • If he puts spicy in 2 ships, his final wealth is w1=w+8000(x1+x2)/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 Measurement

20. Variance and Preference • Two risky assets, w1 and w2, w1 is preferred to w2 iff П2> П1 • For a small risk, w1 is preferred to w2 iff the variance of w2 exceeds the variance of w1 • But this does not hold for a large risk. The correct statement is that all risk-averse agents with a quadratic utility function prefer w1 to w2 iff the variance of the second is larger than the variance of the first • See page 35. L1: Risk and Risk Measurement

21. Aversion to Downside Risk • Definition: agents dislike transferring a zero-mean risk from a richer to a poor state. • w2~(4000, ½; 12000+ε, ½) • w3~(4000+ ε, ½; 12000, ½) • Which is more risky L1: Risk and Risk Measurement

22. 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 Measurement

23. 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 Measurement

24. Technical Notes • Taylor Series Expansion: f(x)= • Integration by parts L1: Risk and Risk Measurement

25. Exercises • EGS, 1.2; 1.4; 2.2 L1: Risk and Risk Measurement