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.