Lecture 10: Expected Utility Theory and Risk Attitudes

Question 1

What is the problem of using expected value to evaluate risky alternatives?

Answer 1

• Expected Value fails to capture risk attitudes
• Many people are risk averse, they are willing to give up a part of the expected value to decrease uncertainty
• Better concept: Take probability weighted average of utilities instead 1

Question 2

How can risk attitudes be analysed?

Answer 2

1. Shape of the utility function
2. Certainty Equivalent
3. Risk premium

Question 3

How can we rank risky alternatives?

Answer 3

Compare the Expect utility (EU) or the Certainty Equivalent (CE) but not the Expected Value (EV).

Question 4

Utility function for risk-averse people

Answer 4

• concave

- u(EV) > EV

Question 5

Utility function for risk-neutral people

Answer 5

• linear

- u(EV) = EV

Question 6

Utility function for risk-prone people

Answer 6

• convex

- u(EV) < EV

Question 7

Certainty Equivalent

Answer 7

CE (a) = Certain outcome that has the same expected utility as lottery a.

Question 8

Risk Premium

Answer 8

RP (a) = Amount of expected value that the DM would give up to avoid the risk of the lottery (= 0 for a risk neutral person).

-> RP = EV - CE

Question 9

Connection between EV, CE and RP

Answer 9

• The distance between EV and CE decreases the closer we are to one of the possible outcomes

Question 10

How can we check how is more risk averse?

Answer 10

Use CE or RP:

• Lower CE = more risk averse
• Higher RP = more risk averse

Don’t use EU!

Question 11

On what does the risk attitude depend?

Answer 11

The risk attitude is reflected by the strength and kind of curvature of the utility function.

Question 12

How can we precisely measure the attitudes towards risk?

Answer 12

1. Absolute risk aversion (Arrow/Pratt)

2. Relative risk aversion

Question 13

Absolute risk aversion

Answer 13

Curvature can be measured by the absolute risk aversion coefficient.

Arrow/Pratt: r(x) = - u''(x) / u'(x)
> 0 = risk averse
= 0 = risk neutral
< 0 = risk prone
-> if r(x) > r(y); x is more risk averse

Question 14

Relative risk aversion

Answer 14

Risk attitude is measured in relation to the consequence of an alternative.

r*(x) = x * r(x)

Question 15

Constant absolut risk aversion (CARA)

Answer 15

• r(x) is constant for all x
• e.g. exponential utility function
• corresponds to increasing relative risk aversion
• Risk tolerance: R = 1/r(x) -> the larger R the more the individual is able to tolerate risk (constant here)
• For increasing wealth (invested amount) RP stays constant and CE increases by the same amount

Question 16

Constant relative risk aversion (CRRA)

Answer 16

• r*(x) = x * r(x) is constant for all x
• corresponds to decreasing absolute risk aversion
• Risk tolerance: R = 1/r(x) = x
• For increasing wealth RP decreases

Question 17

Portfolio Choice: How much should be invested under CARA and CRRA?

Answer 17

CARA: Amount invested is independent of the initial wealth level -> invest the same amount if you get wealthier

CRRA: Percentage invested is independent on the initial wealth level -> invest same percentage with higher wealth level

Question 18

What to do if utility functions are unknown?

Answer 18

Compare payoff distributions instead of expected utilities.

Question 19

Absolute dominance

Answer 19

Alternative a dominates b absolutely, if the worst outcome of a ist still better than the best outcome of b.

Implies statewise dominance!

Question 20

Statewise dominance

Answer 20

Alternative a dominates b statewide, if alternative a has a better outcome than alternative in each state.

Question 21

Stochastic dominance

Answer 21

Alternative a dominates b stochastically, if the risk profile of alternative a is always equal and at least for one consequence above the risk profile of b.

Question 22

How can we rank the dominance states?

Answer 22

Absolut > Statewise > Stochastic