Micro 6: Risk

Question 1

What is a lottery?

Answer 1

A lottery is a probability distribution over a fixed set of outcomes.

Question 2

What is state-contingent income space?

Answer 2

State-contingent income space takes probabilities of events as fixed, and plots lotteries according to the values of outcomes. Usually, this is 2D space, and a point (x,y) denotes a lottery where the payoff is x if some event does not occur, and y if it does.

Question 3

What is the Marschack-Machina triangle?

Answer 3

Taking payoff values as fixed, the triangle visualises variations in probabilities of three events.

Question 4

What does the St Petersburg paradox illustrate about expected value?

Answer 4

The St Petersburg paradox involves a game where a fair coin is repeatedly tossed. If it lands head on the first throw, you gain £1. If it lands tails, then heads, you win £2. If it is tails twice, then heads, you win £4. If it is tails three times, then heads, you win £8. This continues.

The expected value of this game is infinite. However, it is clearly not irrational to not want to give away your life savings in order to play this game. Instead, we should model money as having decreasing marginal utility.

Question 5

What is a ‘certainty equivalent’ of a lottery?

Answer 5

The amount of money you could be transferred for certain that you value the same as the lottery. For risk-averse agents, the certainty equivalent is below the expected value of the lottery.

Question 6

What is a risk premium?

Answer 6

The risk premium is the difference between the expected value of the lottery and its certainty equivalent.

This is the additional amount of money you’d have to pay the agent (for certain) in order for them to want to take the lottery, rather than its expected value.

Question 7

What is the Independence Axiom?

Answer 7

The independence axiom states that the decision maker does not worry about counterfactuals; her preferences over lotteries are only concerned with what actually happens, not ‘what might have been’.

Question 8

What is the Expected Utility Theorem?

Answer 8

The expected utility theorem states that if an agent’s preferences over lotteries are complete, transitive, continuous and satisfy the Independence Axiom, they can be represented as a utility maximiser.

Question 9

What are Allais paradoxes?

Answer 9

Allais paradoxes are experiments that produce violations of the EU hypothesis.

Question 10

What transformations are allowable to EU preferences?

Answer 10

Positive affine transformations: that is, any u2(x) = a + bu(x) for positive a,b are allowable. Nonlinear transformations are impermissible.

Question 11

What, precisely, is a risk-averse agent?

Answer 11

An agent is risk averse if she prefers the expected value of a lottery to the lottery itself.

Question 12

What is the Arrow-Pratt measure of risk aversion? Why does it measure risk aversion?

Answer 12

The Arrow-Pratt measure of risk aversion is -u’‘(x)/u’(x). This is a measure of concavity. High positive numbers reflect highly risk-averse agents.

Question 13

What is constant absolute risk aversion?

Answer 13

An agent with constant absolute risk aversion will have A(x)=c. A utility function with this property is -e^(-ax).

Question 14

What is constant relative risk aversion?

Answer 14

An agent with constant relative risk aversion will have xA(x)=r; that is, their risk aversion is in inverse proportion to their wealth. u(x) = lnx has this property.

Question 15

What is First-Order Stochastic Dominance? How can you determine if one lottery first-order stochastically dominates another?

Answer 15

L is FOSD better than L’ if any expected utility maximiser prefers L to L’.

If the CDF of L never ‘overtakes’ L’, L >_FOSD L’.

Question 16

What is Second-Order Stochastic Dominance? How can you determine if one lottery second-order stochastically dominates another?

Answer 16

If L is SOSD better to L’, any strictly risk-averse EU maximiser strictly prefers L to L’, whereas a risk neutral agent is indifferent.

A sufficient condition for second-order stochastic dominance is that EV(L) = EV(L’) and that the CDF of L crosses that of L’ once from below. Smaller weights are assigned to extreme final wealth.

Question 17

What is a mean-preserving spread?

Answer 17

A is a mean preserving spread of B iff A and B have equal means, but portions of the PDF of A are spread out to form B.

Question 18

Suppose A is a mean-preserving spread of B. Does either of these lotteries first- or second-order stochastically dominate the other?

Answer 18

If A is a mean-preserving spread of B, then B second-order stochastically dominates A.

Question 19

What is risk pooling? How does it work?

Answer 19

Under risk pooling, individuals face independent risks and insure each other in order to end up with less exposure. This is a SOSD improvement for each agent.

Question 20

What is risk sharing? How does it work?

Answer 20

Under risk sharing, individuals buy fractional shares in a risky gamble, splitting both gains and losses.

Question 21

Why does expected utility theory imply that the best portfolio of investments is diversified?

Answer 21

If investments are negatively correlated, we have risk pooling; your different gambles are in a sense ‘insuring’ each other. We have shown that this is desirable for risk-averse agents. Therefore, we should have diversified portfolios of investments.