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Lecture 8. Extra office hours . Wednesday 2013/6/18, 1 -3pm . Price Theory. Risk and Uncertainty Chapter 18. 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.

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extra office hours
Extra office hours

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

risk and uncertainty chapter 18

Price Theory

Risk and Uncertainty

Chapter 18

introduction
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?
introduction1
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
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
introduction2
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.
risk attitudes and utility functions
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
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 uncertainty1
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 uncertainty2
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 uncertainty3
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 uncertainty4
Preferences Under Uncertainty

12

EU=7

2

$0

$45

$90

Wealth

preferences under uncertainty5
Preferences Under Uncertainty

U($45) > EU  risk-aversion.

12

U($45)

EU=7

2

$0

$45

$90

Wealth

preferences under uncertainty6
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 uncertainty7
Preferences Under Uncertainty

12

EU=7

2

$0

$45

$90

Wealth

preferences under uncertainty8
Preferences Under Uncertainty

U($45) < EU  risk-loving.

12

EU=7

U($45)

2

$0

$45

$90

Wealth

preferences under uncertainty9
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 uncertainty10
Preferences Under Uncertainty

12

EU=7

2

$0

$45

$90

Wealth

preferences under uncertainty11
Preferences Under Uncertainty

U($45) = EU  risk-neutrality.

12

U($45)=EU=7

2

$0

$45

$90

Wealth

preferences under uncertainty12
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 uncertainty13
Preferences Under Uncertainty
  • For risk-neutral individuals, state-contingent consumption plans that give equal expected utility are equally preferred.
preferences under uncertainty14
Preferences Under Uncertainty

Cna

Indifference curvesEU1 < EU2 < EU3

EU3

EU2

EU1

Ca

preferences under uncertainty15
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.
preferences under uncertainty21
Preferences Under Uncertainty

Cna

Indifference curvesEU1 < EU2 < EU3

EU3

EU2

EU1

Ca

choice under uncertainty
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
Measures of risk-aversion
  • Absolute risk aversion
  • Relative risk aversion
implications
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.
implications1
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 vs entrepreneurs
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
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
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
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 insurance1
Market for Insurance
  • Uninsurable risks
    • A risk that cannot be diversified
    • Eg. Group risk
    • Eg. Health risk
    • Eg. Profit or income
futures markets1
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
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
market 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
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
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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