micro L 3 risk attitudes

1.3 Decision under Uncertainty: Risk Attitudes

Attitudes Toward Risk Goal: Categorize and compare preferences with regard to uncertainty/risk.

Risk Attitude Toward a Lottery Question: How does individual decide between lottery g and receiving its expected value E[g] (with certainty)?

Definition: Risk Aversion (w.r.t. lottery g) Given a lottery g, we call an individual with vNM-utility function U(g) = E[u(g)] ◮ risk averse w.r.t. g if u(E[g]) ≥ E[u(g)], ◮ risk neutral w.r.t. g if u(E[g]) = E[u(g)], ◮ risk loving w.r.t. g if u(E[g]) ≤ E[u(g)], (and ‘strictly’ in each case if the respective inequality is strict)

 Risk Aversion, Generic Definition: Risk Aversion Call an individual ‘risk averse’ if it is risk averse with respect to any lottery g. (Analogous definition for risk lovingness and risk neutrality.) Can we connect this to properties of the Bernoulli utility function u? Yes, to u ′′, via ‘Jensen’s inequality’. . .

Risk Aversion and u ′′ Thereby: Result: Risk aversion and u ′′ For a vNM-utility maximizer with Bernoulli utility function u(·), we have: ◮ u ′′(x) < 0 for all x: Individual is (strictly) risk averse; ◮ u ′′(x) > 0 for all x: Individual is (strictly) risk loving; ◮ u ′′(x) = 0 for all x: Individual is risk neutral.

Measures of Risk Aversion Next: How measure extent/strength of risk aversion? Or: By how much does an individual prefer E[g] over g? Idea: Utility difference u(E[g]) − E[u(g)]? Not very useful, because that would change with linear transformations of u! Instead: Which amount can I subtract from E[g] before preference flips? → Risk premium P.

Certainty Equivalent Definition: Certainty Equivalent The certainty equivalent of a simple lottery g is an amount CE(g) such that u(CE(g)) = E[u(g)]. So the certainty equivalent of a lottery g tells us the level of a certain payoff for which the decision maker is indifferent between this certain payoff and lottery g. The sign of the difference between certainty equivalent and expected value reflects the decision maker’s (absolute) risk attitude: Falls ◮ CE(g) < E[g]: decision maker is risk averse w.r.t. g, ◮ CE(g) = E[g], decision maker is risk neutral w.r.t. g, ◮ CE(g) > E[g]), decision maker is risk loving w.r.t. g.

Measuring Risk Attitudes ◮ Is risk premium P(g) a useful measure of risk aversion? ◮ We might say: ‘Individual A is more risk averse than individual B’ if (and only if) individual A has a higher risk premium. ◮ Problem: This statement depends on a certain lottery g (because the risk premium does). ◮ Can we find a measure which is independent of g? Which instead directly uses properties of u(·)?

Measuring Risk Attitudes ◮ The more curve the Bernoulli-utility function, the higher the risk premium P of lottery g. ◮ What would be a good measure for curvature? ◮ The curvature of a function is determined by its second derivative (which determines, how strongly its first derivative increases or falls). ◮ Idea: use −u ′′ as measure? ◮ Not suitable! (Why?) the alternative is the arrow pratt measure of absolute risk aversion.

Arrow-Pratt Measure of Absolute Risk Aversion The Arrow-Pratt measure of absolute risk aversion Ra(w) = −u ′′(w)/u ′ (w) ◮ is unaffected by linear transformations to the utility function. ◮ contains all identifying properties of the utility function, i.e. if you know Ra(w) for all w ∈ A, you can deduce the accompanying utility function (unique up to linear transformations). ◮ is a local measure, i.e. it may vary in w! So an individual’s risk aversion might change with the size of payoffs in a lottery, and thereby in the size of the individual’s initial wealth.

Absolute Risk Aversion and Changes in Income ◮ Risk premium P(g) of lottery g is an intuitive measure for the risk attitude of the decision maker (willingness to pay to avoid lottery g). ◮ The risk premium for g might change, though, if the individual’s income changes. ◮ Example: At what price would you be willing to accept a gamble, for which you have an equal probability of winning or loosing 1e?

risky assets

Example 3: “Portfolio Choice” In conclusion: ◮ Demand for risky asset depends on ◮ risk attitude of investor, and ◮ price of income in states H bzw. L (i.e. rH and rL). ◮ First order condition: marginal utility of income in different states is equalized (as always when choosing optimal consumption bundles). Your turn: Can you connect this to the insurance problem above (exercise 2)? What would the example here look like if financial markets were ‘fair’?



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