EC202 Lecture notes 18
EC 202 Lecture Notes (George Symeonidis)
Page 1
Lecture notes 18 from George Symeonidis
Page 2: Dealing with Risk
Key Questions:
How do people deal with risky situations?
How can risk averse individuals reduce risk?
Main Topics:
Investment and risk-taking choices
Diversification as a risk reduction strategy
Discussion of risk pooling and insurance
Elimination/reduction of risk through information
Page 3: Mean-Variance Analysis
Concept of Risk Aversion:
Risk averse individuals dislike risk but may take on risks for greater expected returns.
Mean-Variance Framework:
Individuals care about expected return and the variance of potential outcomes.
Wealth Levels and Probabilities:
Consider a project with wealth results W1, W2,…,WN and respective probabilities P1, P2,…, PN.
Page 4: Expected Value and Variance
Expected Value (0):
Formula: E[W] = μ = P1W1 + P2W2 +…+ PNWN
Variance of Lottery:
Formula: Var(W) = σ² = P1(W1–μ)² + P2(W2–μ)² + …+ PN(WN–μ)²
Interpretation of Variance:
Higher variance indicates a riskier lottery for a fixed expected return (μ).
Page 5: Internet Company
Example Calculation:
Probability EV:
EV = 0.3 × 80 + 0.4 × 100 + 0.3 × 120 = 100
Payoff Range: $80, $100, $120
Page 6: Public Utility
Example Calculation:
Probability EV:
EV = 0.1 × 80 + 0.8 × 100 + 0.1 × 120 = 100
Payoff Range: $80, $100, $120
Page 7: Variance Comparison
Variance Calculation:
Internet Company:
Variance = 0.3 × (80–100)² + 0.4 × (100–100)² + 0.3 × (120–100)² = 240
Public Utility:
Variance = 0.1 × (80–100)² + 0.8 × (100–100)² + 0.1 × (120–100)² = 80
Conclusion: Internet company stock is riskier than public utility stock.
Diversification as a Risk Reduction Strategy:
Discussed in investment context.
Page 8: Diversification Introduction
Investing in Multiple Projects:
Investing in two risky projects can mitigate risk.
Co-Relationship of Projects:
Projects may be positively correlated (same industry) or negatively correlated.
Page 9: Investor Utility and Portfolios
Utility Concepts:
Utility increases with expected return but decreases with variance.
By investing in both projects, variance associated with returns can be reduced.
Adage: "Don't put all your eggs in one basket."
Page 10: Diversification Example
Project Characteristics:
Project A:
Rain: £1.12 per £1
Sunshine: £1.02 per £1
Project B:
Rain: £1.04 per £1
Sunshine: £1.10 per £1
Expected Returns:
Investing £100 in each project gives expected returns of £107 for both.
Page 11: Diversification Impact
Investor Strategies:
Invests Z in Project A and £100-Z in Project B.
Return Calculations:
If rain: Return = 1.12Z + 1.04(100-Z)
If sunny: Return = 1.02Z + 1.10(100-Z)
Expected return remains £107 regardless of Z.
Page 12: Variance with Diversification
Variance Calculation in Investments:
Variance lower under suitable diversification strategy.
Achieving zero variance at optimal Z = 37.5 implies no risk.
Page 13: Expected Utility Maximization
Goal of Minimizing Variance:
Optimal utility maximization equates variance minimization in certain cases.
General considerations:
Expected returns and variance differ across options.
Page 14: Evidence for Diversification
Diversification Benefits:
Common among people and institutions (e.g., pension funds).
Multi-product firms face lower market risks.
Page 15: Understanding Insurance
Inherent Risks:
Insurance is crucial for managing unavoidable risks.
Risk Pooling Concept:
Insurance companies collect premiums and cross-share risks.
Challenges:
Limited insurance options on not largely independent risks (e.g., natural disasters).
Page 16: Risk Pooling Mechanism
Consideration of Multiple Drivers:
N drivers commuting to work, each with an accident probability P.
Individual accidents are independently occurring events.
Page 17: Bet Against Accidents
Individual Wealth Bet Analysis:
Possible outcomes for wealth-based accidents and their probabilities defined.
Page 18: Outcomes of Risk Pooling
Scenario with Two Drivers:
Four outcomes categorized by probabilities regarding accidents.
Page 19: Expected Value in Risk Pooling
Expected Value Calculation:
Expected value remains W – PD for both_shared and no shared scenarios.
Page 20: Wealth and Sharing Outcomes
Diagram Illustrating Wealth Situations:
Comparison between sharing and no sharing of accident costs.
Page 21: Utility Function Analysis
Utility Function Example:
Expected Utility Dynamics in shared vs non-shared accidents.
Conclusion:
Pooling raises expected utility for all drivers.
Page 22: Limitations of Risk Pooling
Conditions for Effective Pooling:
Risks must largely be independent for effective pooling.
High variance in accident probabilities across drivers hinders pooling.
Practical Agreements:
Insurance simplifies risk pooling by aggregating individuals.
Page 23: Addressing Unavoidable Risks
Risk Averse Individuals:
Examine demands for insurance against unavoidable risks like health or unemployment.
Page 24: Full Insurance Demand
Risk Adverse Responses:
Individuals might fully insure if insurance priced fair.
Definition of fair insurance: Expected value equals zero.
Page 25: Buying Insurance Units
Loss Scenarios:
Individual loss mechanism illustrated through insurance premium payments.
Page 26: Budget Line Impact
Movement on Budget Line:
Analysis of how purchasing insurance impacts wealth across states of occurrence.
Page 27: Preferences in Insurance Demand
Demand Modeling:
Indifference curves represent risk-averse preferences.
Fair Insurance Conditions:
Correlate premium rate to probabilities to maintain zero expected value.
Page 28: Equilibrium in Insurance Demand
Overview of Equilibrium Conditions:
Risk-averse individuals meet budget constraints resulting in equilibrium at fair pricing.
Page 29: “Unfair” Insurance Implications
Insurer Profitability:
Higher premiums than probabilities lead to under-coverage.
Page 30: Equilibrium Analysis with Unfair Insurance
Shift in Equilibrium:
The budget line moves leftward; individuals cannot fully insure against risk.
Page 31: Summary on Insurance Demand
Conclusions on Risk Averse Individuals:
Full insurance demanded under fair conditions; inefficiencies occur under high premium environments.
Page 32: Reducing Risk Through Information
Additional Strategies:
Individuals may seek information to lessen uncertainty around risks.
Page 33: Example of Value of Information
Jack's Housing Investment Scenario:
Home town investment yields certain returns; French home's risk varies with renovation costs.
Page 34: Decision Trees Usage
Decision Analysis Technique:
Describes available options and risks for decision-making.
Page 35: No Information Decision Tree
Decision Tree Example without Information:
Jack's options compared between home town and French investments.
Page 36: Expected Utility without Information
Jack's Calculated Outcomes:
Risk-neutral assessment of net returns indicates home town yields higher utility.
Page 37: Without Information Decisions
Summary of Outcomes:
Jack's risk comparison leading to a preference for local investment.
Page 38: Impact of Information on Decisions
Decision Importance:
Costless information improves Jack's decision-making outcome significantly.
Page 39: Costless Information Decision Tree
Analysis when Information is Free:
Options compared more favorably with gained knowledge of house condition.
Page 40: Information Value Calculation
Discrepancy in Expected Utilities:
Jack's preference to obtain information when its value exceeds cost.
Page 41: Costless Information Diagrams
Reinforcement of better decisions through no-cost information acquisition.
Page 42: Cost Implications on Information Value
Variations on Utility with Increasing Costs for Information:
Balance of expense vs. benefit in acquiring valuable data.
Page 43: Costly Information Decision Tree
Example of costly information depicts diminishing returns on expected utility.
Page 44: Maximum Willingness to Pay for Information
Comparative Analysis of Utilities:
Deriving maximum cost useful to inform decision paths.
Page 45: Summary of Information Value
Overall conclusion on information:
It represents the maximum payment an agent is prepared to make to clarify uncertain outcomes.
Page 46: Asymmetric Information Context
Definition:
Situations where one party possesses more information than another, leading to risk uncertainties.
Page 47: Hidden Actions in Information Asymmetry
Examples of Agent Actions:
Factors influencing perceived effort and risk in markets with hidden parameters.
Page 48: Hidden Characteristics Affecting Decision-making
Individual Knowledge Disparities:
Relevant information undetected by other parties impacting transaction terms.