EC 202 Lecture Notes 18: Dealing with Risk

Dealing with Risk

• How people deal with risky situations.
• How risk-averse people reduce risk.
• Investment: risk-taking is a choice; diversification may reduce risk.
• Risk pooling and insurance.
• Risk may be eliminated or reduced by obtaining information.

Mean-Variance Analysis

• Risk-averse individuals dislike risk but might take risks if compensated with a greater expected return.
• Individuals care about the expected return and risk, considering both the expected value and the variance of any lottery.
• Consider $W<em>1, W</em>2, …, W<em>N$ wealth levels with respective probabilities $P</em>1, P<em>2, …, P</em>N$.

Mean-Variance Analysis

• Expected Value: $E[W] = μ = P<em>1W</em>1 + P<em>2W</em>2 + … + P<em>NW</em>N$
• Variance: $Var(W) = σ^2 = P<em>1(W</em>1 – μ)^2 + P<em>2(W</em>2 – μ)^2 + … + P<em>N(W</em>N – μ)^2$
• Variance is the probability-weighted average of the squared deviations between the value of each possible outcome of the lottery and the expected value of the lottery.
• The higher the variance, for any given $μ$, the riskier the lottery.

Quantifying Risk

• Example: Internet Company
  • Probabilities: 0.3, 0.4, 0.3
  • Payoffs: $80, $100, $120
  • Expected Value (EV): $0.3 \times 80 + 0.4 \times 100 + 0.3 \times 120 = 100$

Quantifying Risk

• Example: Public Utility
  • Probabilities: 0.1, 0.8, 0.1
  • Payoffs: $80, $100, $120
  • Expected Value (EV): $0.1 \times 80 + 0.8 \times 100 + 0.1 \times 120 = 100$

Quantifying Risk

• Variance of returns for the internet company: $0.3 \times (80 – 100)^2 + 0.4 \times (100 – 100)^2 + 0.3 \times (120 – 100)^2 = 240$
• Variance of returns for the public utility: $0.1 \times (80 – 100)^2 + 0.8 \times (100 – 100)^2 + 0.1 \times (120 – 100)^2 = 80$
• The internet company stock is riskier than the public utility stock.
• Diversification may help reduce risk in investment.

Diversification

• Investments in a single risky project or asset.
• Simultaneously invest in two risky projects.
• Positively Correlated: Projects succeed or fail together (e.g., stocks in the same industry).
• No Correlation: Outcomes are independent (e.g., gold and tech stocks).
• Negative Correlation: One project succeeds when the other fails and vice versa.

Diversification

• For a risk-averse investor, utility increases with the expected return on the portfolio and decreases with the variance of this return.
• By investing a share of funds in both projects, the investor can reduce the variance associated with a given expected return, as long as the outcomes of the two investment projects are not perfectly positively correlated.
• “Don’t put all your eggs in one basket.”

Diversification: Example

• Project A pays £1.12 for every £1 invested if it rains but only £1.02 if it is sunny.
• Project B pays £1.04 for every £1 invested if it rains and £1.10 if it is sunny.
• The returns of the two projects in this example are negatively correlated.
• Probability of rain = Probability of sunshine = ½.
• Investing £100 in project A gives expected return $\frac{1}{2} \times 112 + \frac{1}{2} \times 102 = £107$.
• Investing £100 in project B gives expected return $\frac{1}{2} \times 104 + \frac{1}{2} \times 110 = £107$. Also project B is less risky than project A.

Diversification: Example

• Investing £Z in project A and £100–Z in project B.
• If it rains, the investor obtains $1.12 \times Z + 1.04 \times (100–Z) = 104 + 0.08Z$.
• If it is sunny, she/he obtains $1.02 \times Z + 1.10 \times (100–Z) = 110 – 0.08Z$.
• The expected return is $\frac{1}{2} \times (104 + 0.08Z) + \frac{1}{2} \times (110 – 0.08Z) = £107$

Diversification: Example

• The variance of the return when one invests £Z in project A and £100–Z in project B is $\frac{1}{2} \times (104 + 0.08Z – 107)^2 + \frac{1}{2} \times (110 – 0.08Z – 107)^2 = 9 – 0.48Z + 0.0064Z^2$
• The variance is lower under a suitable diversification strategy (in particular, for 0 < Z < 75) than under no diversification (i.e. for Z = 0 or Z = 100).
• For Z = 37.5, the variance is zero: no risk at all. The investor obtains £107 whatever the weather.

Diversification: Example

• The numbers for the payoffs were chosen so that the expected return is independent of Z (always £107).
• The investor’s problem was simply to minimise the variance, i.e. the risk. Minimising variance was in this case equivalent to maximising expected utility.
• In more general problems, both the expected return and the variance are likely to differ across the various options. We would then need to make explicit use of the investor’s utility function to find the preferred option, i.e. the one that maximises expected utility.

Diversification

• Diversification reduces risk, significantly so when the outcomes are negatively correlated.
• Evidence for diversification:
  • People, pension funds, etc. typically diversify their savings and investments.
  • Multi-product innovating firms face lower market risk (but not lower technical risk) and are less likely to exit their industry than single-product innovating firms or non-innovating firms.

Insurance

• Risk is often unavoidable rather than a choice.
• Insurance works by pooling risks that are substantially independent.
• Not all people are likely to face a bad state at the same time.
• An insurance company collects “insurance premiums” from many people but has to pay only those who face a bad state at any given time.
• Firms often offer limited insurance or might not offer insurance for risks that are not largely independent, e.g. earthquakes…

Risk Pooling

• Consider N individuals who drive to work every weekday morning.
• The probability that any one driver has an accident is P.
• The probability that a randomly chosen driver A has an accident is independent of whether a randomly chosen driver B has an accident.

Risk Pooling

• The damage involved in an accident is D.
• The “bet” faced by an individual of wealth W is a probability P of ending up with W – D versus a probability 1–P of ending up with W.
• The expected value of this bet is $P \times (W – D) + (1–P) \times W = W – PD$

Risk Pooling

• M drivers can sign an agreement to share equally in paying for the total costs of accidents incurred. Assume M = 2.
• Four possible outcomes:
  1. Both have an accident: probability $P^2$
  2. Only driver A has an accident: prob. $P \times (1–P)$
  3. Only driver B has an accident: prob. $(1–P) \times P$
  4. No accident: probability $(1–P)^2$

Risk Pooling

• Expected Value: W – PD
• The expected value of the bet for any driver is the same under Sharing and No Sharing.

Risk Pooling

• Illustration of Wealth Distribution under No Sharing vs Sharing with P=1/2.

Risk Pooling: Example

• Assume W = 144, D = 44.
• Utility function of any driver: $U(W) = W^{\frac{1}{2}}$.
• Expected utility of a driver without pooling: $EUNP = P \times (W – D)^{\frac{1}{2}} + (1–P) \times W^{\frac{1}{2}} = 12 – 2P$
• Expected utility of a driver with pooling: $EUP = P^2 \times (W – D)^{\frac{1}{2}} + P \times (1–P) \times (W – D/2)^{\frac{1}{2}} + (1–P) \times P \times (W – D/2)^{\frac{1}{2}} + (1–P)^2 \times W^{\frac{1}{2}} \approx 12 – 1.9P – 0.1P^2$
• EUP > EUNP for any $P \in (0,1)$, therefore pooling is preferable, i.e. gives higher expected utility.

Risk Pooling

• Risk pooling increases the expected utility of risk-averse individuals.
• When would risk pooling not work in our example?
  1. If the risks were not largely independent.
  2. If the accident probability was very different across drivers.
• It is usually impractical for people to sign a risk pooling agreement among themselves because of the transaction costs involved.
• An insurance company can organise risk pooling by selling insurance policies separately to each individual.

Insurance

• Consider a risk-averse individual faced with an unavoidable risky situation (e.g. ill-health, driving, unemployment…).
• Insurance available allows the individual to reduce the risk inherent in the situation.
• What would be the individual’s demand for insurance? Under what conditions is the amount of insurance bought efficient?

Demand for Insurance

• A risk-averse individual would never invest in a fair bet.
• A risk-averse individual will always fully insure if insurance is priced in a fair manner.
• An insurance policy is fair if its expected value is zero.
• Fully insure: the individual removes all risk, i.e. chooses a situation where his/her income is independent of the state of nature.

Insurance

• With probability P an individual incurs a loss L.
• Insurance premium r: the price of purchasing one unit of insurance.
• By buying Z units of insurance (“insurance policy”), the individual pays rZ if there is no loss and obtains Z – Zr = (1–r)Z if there is a loss.

Demand for Insurance

• Budget Line Representation

Insurance: Preferences

• Indifference curves (preferences) added to the budget line model the demand for insurance as a standard problem in consumer theory.
• If an individual is risk-averse, his/her indifference curves between $Y<em>1$ and $Y</em>2$ are convex to the origin.
• An insurance policy is fair if its expected value is zero: $P \times (1–r)Z + (1–P) \times (–rZ) = 0 \Leftrightarrow r = P$.
• Equilibrium: When faced with a fair insurance policy, a risk-averse individual chooses the point where the budget constraint meets the certainty line, i.e. a point such that $Y<em>1 = Y</em>2$.

Demand for Fair Insurance: Equilibrium

• The individual fully insures at the point where the budget line (fair odds line) intersects the certainty line.

“Unfair” Insurance

• An insurance firm offering fair insurance, i.e. charging r = P, will make zero expected profit.
• To cover its operating costs, an insurance firm normally needs to charge a premium r > P.
• The budget line will be steeper than the fair odds line.
• The equilibrium must be to the left of the certainty line: the individual does not fully insure against risk.

“Unfair” Insurance: Equilibrium

• Graphical illustration of equilibrium with unfair insurance (r > P), showing the individual does not fully insure.

Demand for Insurance

• A risk-averse individual will always fully insure if insurance is priced in a fair manner, i.e. if r = P.
• If insurance firms need to cover operating costs and therefore charge a premium r > P, the individual will not fully insure.
• When a risk-averse individual does not buy full insurance from a risk-neutral firm, there is a form of inefficiency: a loss in total social surplus.

Value of Information

• A final way to deal with risk is to eliminate or reduce uncertainty by obtaining information.
• An economic agent may find it worthwhile to incur a cost in order to acquire information.
• How much would the agent be willing to pay for information?

Value of Information: An Example

• Jack wants to invest in a house.
• Home town: Net return is £100,000 for sure.
• France: Inspecting costs C. Renovation may be needed (£50,000) with probability ½.
• Net return on the house in France is £120,000 if renovation is not needed, and £120,000 – £50,000 = £70,000 if renovation is needed.

Decision Tree

• What should Jack do? Buy in his home town, buy in France, or fly to France and then decide? To analyse this problem (and other similar problems under uncertainty), we can use a decision tree.
• A decision tree is a diagram that describes the options available to a decision-maker as well as the risky events that can occur at each point in time.
• Elements of the decision tree:
  • Decision Nodes: when a decision is made
  • Chance Nodes: when a risky event occurs
  • Probabilities
  • Payoffs

Decision Tree: No Information

• Diagram representing Jack's decision without obtaining information.

Expected Utility Maximisation

• Jack is risk-neutral and has utility function U = Y, where Y is the net return from the investment.
• If Jack does not fly to France, he compares the utility from buying in his home town, which is 100k, to the expected utility from buying in France, which is: $\frac{1}{2} \times 120k + \frac{1}{2} \times 70k = 95k$.
• If Jack does not fly to France, then to maximise his expected utility he buys in his home town. He gets utility of 100k.

Jack - No Information

• Diagram illustrating payoffs without additional information.

Value of Information: Example

• Would Jack care to find out whether the house in France needs renovation before making his investment decision?
• Suppose Jack could have this information at zero cost. Then if the house in France needed renovation, Jack would prefer to buy in his home town and get utility of 100k rather than buy in France and get 70k.
• But if the house in France did not need renovation, Jack would prefer to buy the house in France and get utility of 120k rather than buy home and get 100k.
• Information matters for Jack because it would affect his decision.

Decision Tree: Costless Information

• Decision tree illustrating the scenario with costless information.

Value of Information: Example

• Jack’s expected utility if he could have costless information would be: $\frac{1}{2} \times 100k + \frac{1}{2} \times 120k = 110k$.
• This is higher than his utility if uninformed (100k), so Jack would definitely choose to obtain information if this were costless, or not too costly.

Decision Tree: Costless Information

• Decision tree illustrating Jack's decision process with costless information.

Value of Information: Example

• Information can be obtained at cost C, as already mentioned. The higher the cost C, the lower Jack’s expected utility from obtaining information before making his decision.

Decision Tree: Costly Information

• Decision tree showing costs deducted from each payoff based on information obtained.

Value of Information: Example

• What is the maximum amount that Jack will be willing to pay to obtain information about the state of the house in France?
• We need to compare:
  1. his expected utility if he has no information (in which case he buys in his home town), i.e. 100k, to
  2. his expected utility if he obtains information before deciding, which is 110k – C.
• For any C such that 110k – C > 100k \Leftrightarrow C < 10k, Jack prefers to pay for information.
• The value of information for Jack is 10,000.

Value of Information: Example

• The value of information is the maximum amount an investor is willing to pay to obtain information.
• It is the difference between the expected utility with costless information (110k in our example) and the expected utility without information (100k in our example).

Asymmetric Information

• Choice under uncertainty by one person or by many persons all of whom had the same probability of winning or losing.
• What if the probability of an accident differs across individuals and insurance companies cannot separate high risk from low risk individuals?
• Asymmetric information: a party in a transaction or a player in a game has more information than another.
• Two types of problems: hidden characteristics and hidden actions.

Hidden Actions

• Actions taken by one agent that the other agent cannot observe.
• Examples:
  • a manager cannot observe the exact amount of effort an employee exerts on a job
  • the shareholders or the creditors of a firm cannot fully observe the activities of the manager and the firm’s key personnel
  • an insurance firm cannot observe how healthily you eat or how carefully you drive.

Hidden Characteristics

• Relevant characteristics of one agent known to herself but not to the other agent.
• Examples:
  • you have better information about your health or driving ability than an insurance company
  • you know better your skills than a prospective employer
  • a firm applying for a loan to invest on a project knows more about the project than the bank that considers offering the loan.