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Lecture 8

Extra office hours

Wednesday 2013/6/18, 1-3pm.

Introduction

- States of the world can be considered as different goods
- States can be traded. How?
- As there is trade, there is a price. What’s the price?

Introduction

- Key elements: a complete set of states, and a probability distribution across these states.
- Expected value: the average value of all the states, weighted by the corresponding probabilities.
- Graphically, we can use iso-expected value lines.

Opportunities

- Budget line and prices
- Fair odds
- Reflect true probabilities of various states of world
- Expected value of betting same as expected value of not betting

- Individual offered fair odds
- Budget line coincides with expected value line

Introduction

- How to measure the risk of a stochastic variable (or a risk asset, or a gamble…)?
- You need a variation index, such as variance or standard deviation.
- Note that these measures are different from risk-attitude measures.

Attitudes toward risk

Risk attitudes and utility functions

- Define utility U as a function of wealth (W)
- If U is concave (U’’<0), the individual is considered risk-averse
- If linear (U’’=0) , then risk-neutral
- If convex (U’’>0), risk-loving

Preferences Under Uncertainty

- Think of a lottery.
- Win $90 with probability 1/2 and win $0 with probability 1/2.
- U($90) = 12, U($0) = 2.
- Expected utility is

Preferences Under Uncertainty

- Think of a lottery.
- Win $90 with probability 1/2 and win $0 with probability 1/2.
- U($90) = 12, U($0) = 2.
- Expected utility is

Preferences Under Uncertainty

- Think of a lottery.
- Win $90 with probability 1/2 and win $0 with probability 1/2.
- Expected money value of the lottery is

Preferences Under Uncertainty

- EU = 7 and EM = $45.
- U($45) > 7 $45 for sure is preferred to the lottery risk-aversion.
- U($45) < 7 the lottery is preferred to $45 for sure risk-loving.
- U($45) = 7 the lottery is preferred equally to $45 for sure risk-neutrality.

Preferences Under Uncertainty

U($45) > EU risk-aversion.

12

MU declines as wealth

rises.

U($45)

EU=7

2

$0

$45

$90

Wealth

Preferences Under Uncertainty

U($45) < EU risk-loving.

12

MU rises as wealth

rises.

EU=7

U($45)

2

$0

$45

$90

Wealth

Preferences Under Uncertainty

U($45) = EU risk-neutrality.

12

MU constant as wealth

rises.

U($45)=EU=7

2

$0

$45

$90

Wealth

Preferences Under Uncertainty

- For risk-neutral individuals, state-contingent consumption plans that give equal expected utility are equally preferred.

Preferences Under Uncertainty

- What is the MRS of an indifference curve?
- Get consumption c1 with prob. 1 andc2 with prob. 2 (1 + 2 = 1).
- EU = 1U(c1) + 2U(c2).
- For constant EU, dEU = 0.

Choice Under Uncertainty

- Q: How is a rational choice made under uncertainty?
- A: Choose the most preferred affordable state-contingent consumption plan.

Measures of risk-aversion

- Absolute risk aversion
- Relative risk aversion

Implications

- Considering forming a portfolio with one risky asset and one risk-free asset.
- If the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the number of dollarsof the risky asset held in the portfolio if absolute risk aversion is decreasing (or constant, or increasing).
- Thus economists avoid using utility functions, such as the quadratic, which exhibit increasing absolute risk aversion, because they have an unrealistic behavioral implication.

Implications

- Similarly, if the person experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the fraction of the portfolio held in the risky asset if relative risk aversion is decreasing (or constant, or increasing).

Individuals vsentrepreneurs

- In general, individuals are more risk-averse than businesses
- Possibly because businesses know better about risk diversification.
- Budget constraint is less a concern for businesses.

Risk and Society

- Societies’ desire for risk neutrality in some instances
- Individual entrepreneurial endeavors promote risk aversion
- Underinvest in risky projects
- Provides corporations with a good buffer

Market for Insurance

- Transfer of risk from one party to another – transferring risks from RA agents to RN or RL ones increases efficiency.
- Imperfect information
- Moral hazard
- Adverse selection

An example of moral hazard

- The prevailing health insurance programs in Taiwan provide compensations to women who have a hysterectomy before age 45.
- This design leads to a typical phenomenon of moral hazard, creating a spike in the number of hysterectomies before insurants reaching age 45.

Market for Insurance

- Uninsurable risks
- A risk that cannot be diversified
- Eg. Group risk
- Eg. Health risk
- Eg. Profit or income

Futures markets

Futures Markets

- Futures contract
- Deliver specified good at specified future date at specified price

- Futures market
- Market for futures contracts

- Spot market
- Market for goods for immediate delivery

- Spot price
- Price in spot market

Speculation

- Speculator
- Attempts to earn profits in futures market
- Predicts future changes in supply or demand

- Speculation and welfare
- Guess future correctly
- Earn profit
- Increase social welfare

- Guess future correctly

Markets for risky assets

Market for Risky Assets

- Returns
- Valued assets not for uses in consumption but for potential increase to owners’ wealth

- Expected returns
- Expected present value of those returns

- Standard deviation
- Measure of risk
- Spread across possible outcomes

- Investors

Portfolios

- Combination of risky assets
- Standard deviation of portfolio at most equal to average standard deviation of individual stocks
- Expected return to portfolio exactly equal to average expected returns of individual stocks

- Efficient set and portfolios

Investor’s Choice

- Capital asset pricing model
- Assumes investor cares only about expected return and risk
- Measures risk by standard deviation

- Risk-free asset
- Market line and portfolios

- Rational investor
- Combines risk-free asset with market portfolio in some proportions in hold portfolio

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