Lecture 21 Prospects & Risk Attitude

Sairam

Introduction
- EUT dominated analysis of DM under risk
  - normative model of rational choice
  - descriptive model of economic behavior
- So, assume that all reasonable people obey these axioms
Economic BUU
- Prospect Theory questioned EUT axioms, offering more realistic modelling of BUU
Prospects
- DM under risk = choice between prospects / gambles
- A Prospect: $(x_{1}, p_{1};\dots;x_{n},p_{n})$ is a contract that yields $x_{i}$ with probability $p_{i}\ni \sum_{p_{i}} = 1$
- Without null outcomes: $(x,p)$ to denote prospect $(x,p; 0,1-p)$... like Bernoulli trial
- Riskless Prospect: $x$ with certainty is $(x)$ ...objective or standard probabilities
- Application of EUT to choices between prospects are based on:
  1. Expectation principle
  2. Asset integration
  3. Risk aversion
Expectation principle
- $X$ is a random variable¹
- Realized values: $X_{1}, X_{2} \dots X_{m}$ to which probabilities $P_{1}, P_{2},\dots P_{m}$ are assigned
- $E(X) = \sum_{i=1}^m P_{i}X_{i}$ is the expected value
Expected Utility
- Situation: Dice roll
  - $3 if $\{ 1,2 \}$ comes, OR
  - $12 if $\{ 3,4,5,6 \}$ comes.
  - $E(X) = 9$.
- Alternative: getting $9 for sure! Prospect: $(9,1)$
- Most people choose $9... Fact: Many people try to avoid risk in economic choices in real lives. Risk aversion principle
- EUT explains: she will choose a lottery that gives her highest expected value of utility rather than highest expected value of money prizes.
  - Meaning, though $3\times 4 = 12$
  - $4u(3) \neq u(12)$ no can guarantee
- Expectation Principle considers Expected values of utilities, $u(x_{i})$ associated with outcomes$U(x_{1}, p_{1};\dots;x_{n},p_{n}) = p_{1}u(x_{1}) + \dots p_{n}u(x_{n})$
Asset Integration
- Lottery vs Surety
  - Initial endowment: $e$
  - Additional gains from P&L from stock price movements: $x$
  - Expected utility = $E(u(e+x)) = \sum_{i=1}^{n} P_{i}u(e+x_{i})$
  - Suppose $u(x) = \ln(x)$
  - $e = 10$. Prospect = $(3, \dfrac{1}{3};12, \dfrac{2}{3})$
  - Expected utility from lottery = $\dfrac{1}{3} \times \ln(13) + \dfrac{2}{3}\times \ln(22) \cong 2.91$
  - Expected utility from $(9)$ = $\ln(9) \cong 2.94$.
  - So she chooses the sure $9.
- A prospect is acceptable if $U(w + x_{1}, p_{1};\dots) > u(w)$
- Domain of the utility function is its final state (including one's asset position) i.e. not just profit/loss but also $e$.
Preferences for risk
- Depending on the amount of money she already has, she will take a call.
- If she has about $8.45 then she would be indifferent between receiving the bet and receiving a certain amount of money.
- Here, certainty equivalent, $y$ is $8.45... i.e. $u(e+y) = E(u(e+x))$
- Risk premium = $E[x] - y$ = $0.55 in this case.
Attitudes Towards Risk
- $E[x] - y \gt 0:$ risk averse (tries to avoid risk by being happy with a smaller $y$)
  - Most people are risk averse, and their risk premium for lottery is positive
  - If he prefers $(x)$ to any risky prospect with $E[\cdot] =x$
  - EUT: risk aversion $\implies$ utility function is concave, $u'' \lt 0$
- $E[x] - y \lt 0:$ risk loving
- $E[x] - y = 0:$ risk neutral
- Situation:
  - Car worth $50k.
  - Money in Bank $20k
  - $P(\text{Accident or Car-theft}) = 0.05$
  - $P(\text{Police Intervention}) = 0.20$

Choice	Cost	Prospect	Comments	Expected Payoff
$A:$Full Coverage	7k	$(63)$	$20+50 - 7$	63
$B:$Third-party	4k	$(66,0.95; 16, 0.05)$	$20 + 50 - 4$ if no accident, else $20-4$	63.5
$C:$No Insurance	0k	$(70,0.75; 0, 0.20; 20, 0.05 )$		63.5

Next consider three alternative utility functions: (smallest utilities computed for each case) - Linear: $u(x) = x = 63.5$ $\implies$ $B\sim C$ - Concave: $u(x) = 10\sqrt{ x } = 79.37$ $\implies$Full coverage (risk averse) - Convex: $u(x) = \dfrac{x^2}{50} = 83.90$ $\implies$ No insurance (risk loving, since for no insurance: $E[x] - y = 63.5 - 63 = 0.5$) #doubt What should be taken as the certainty equivalent here?

Real valued function with domain of an outcome or probability space ↩

Choice	Cost	Prospect	Comments	Expected Payoff
\(A:\)Full Coverage	7k	\((63)\)	\(20+50 - 7\)	63
\(B:\)Third-party	4k	\((66,0.95; 16, 0.05)\)	\(20 + 50 - 4\) if no accident, else \(20-4\)	63.5
\(C:\)No Insurance	0k	\((70,0.75; 0, 0.20; 20, 0.05 )\)		63.5