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Comparison of Choices in Risky Scenarios: Finance Quiz

Analyze preferences in risk-taking through a series of financial scenarios. Understand expected value, standard deviation, and utility functions in decision-making.

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Comparison of Choices in Risky Scenarios: Finance Quiz

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  1. Finance 30210: Managerial Economics Risk, Uncertainty, and Information

  2. Rationality Quiz: Which would you prefer • Question #1: You are offered the following choice • $1M in Cash • A lottery ticket with a 10% chance of winning $5M, an 89% chance of winning $1M and a 1% chance of winning nothing • Question #2: You are offered the following choice • A lottery ticket with an 11% chance of winning $1M • A lottery ticket with a 10% chance of winning $5M • Question #3: You are offered the following choice • $1M in Cash • A lottery ticket with a 10/11 chance of winning $5M

  3. When dealing with, uncertain events, we need a way to characterize the level of “risk” that you face. Expected Value refers to the “most likely” outcome (i.e. the average) Probability of Event i Payout of Event i Note: if all the probabilities are equal, then the expected value is the average.

  4. Question #1: You are offered the following choice • $1M in Cash • A lottery ticket with a 10% chance of winning $5M, an 89% chance of winning $1M and a 1% chance of winning nothing

  5. When dealing with, uncertain events, we need a way to characterize the level of “risk” that you face. Standard Deviation measures the “spread” around the mean – this is what we mean by risk. Probability of Event i Squared difference between each event and the expected value Note: Standard Deviation is the (square root of) the expected value of squared differences from the mean.

  6. Question #1: You are offered the following choice • $1M in Cash • A lottery ticket with a 10% chance of winning $5M, an 89% chance of winning $1M and a 1% chance of winning nothing

  7. Risk versus Return We can calculate the expected payout and the standard deviation for each choice.

  8. Preferences towards risk Suppose that you have a utility function defined as follows: (Linear in Income) Utility • Suppose that this individual were to choose between: • $100 with certainty • A 50% chance of earning $200 Income

  9. Suppose that this individual were to choose between: • $100 with certainty • A 50% chance of earning $200 Utility Choice A gives this individual 200 units of utility with certainty 400 200 Choice B gives this individual a 50% chance at 400 units of utility E(Utility) = 200 0 $0 $100 $200 Income We would describe this individual as “Risk Neutral”

  10. Preferences towards risk Suppose that you have a utility function defined as follows: (Convex in Income) U • Suppose that this individual were to choose between: • $100 with certainty • A 50% chance of earning $200 I

  11. Suppose that this individual were to choose between: • $100 with certainty • A 50% chance of earning $200 U Choice A gives this individual 10,000 units of utility with certainty 40,000 20,000 Choice B gives this individual a 50% chance at 40,000 units of utility E(Utility) = 20,000 10,000 0 $0 $100 $200 I We would describe this individual as “Risk Loving”

  12. Preferences towards risk Suppose that you have a utility function defined as follows: (Concave in Income) U • Suppose that this individual were to choose between: • $100 with certainty • A 50% chance of earning $200 I

  13. Suppose that this individual were to choose between: • $100 with certainty • A 50% chance of earning $200 U Choice A gives this individual 10 units of utility with certainty 14 10 7 Choice B gives this individual a 50% chance at 14 units of utility E(Utility) = 7 0 $0 $100 $200 I We would describe this individual as “Risk Adverse”

  14. Preferences towards risk • Suppose that this individual were to choose between: • $100 with certainty (Choice A) • A 50% chance of earning $200 (Choice B)

  15. Back to our Quiz… How would a rational, risk neutral individual answer this quiz?

  16. Question #2: You are offered the following choice • A lottery ticket with an 11% chance of winning $1M • A lottery ticket with a 10% chance of winning $5M • Question #3: You are offered the following choice • $1M in Cash • A lottery ticket with a 10/11 chance of winning $5M If you look closely, you will see that both of the choices in Question #2 are 11% of the values in Question #3 (Question #2 gives you an 11% chance of obtaining the choices in question #3) Your answer to Question #2 = Your answer to Question #3

  17. Question #1: You are offered the following choice • $1M in Cash • A lottery ticket with a 10% chance of winning $5M, an 89% chance of winning $1M and a 1% chance of winning nothing • Question #3: You are offered the following choice • $1M in Cash • A lottery ticket with a 10/11 chance of winning $5M Lets write these a bit differently Question #1: A: 11% Chance of a win ($1M) (Consolation Prize of $1M) B: 11% Chance of a 10/11 Chance of a win ($5M) (Consolation prize of $1M) Question #3: A: 100% Chance of a win ($1M) B: 10/11 Chance of a win ($5M)

  18. Question #1: A: 11% Chance of a win ($1M) (Consolation Prize of $1M) B: 11% Chance of a 10/11 chance of a win ($5M) (Consolation prize of $1M) Question #3: A: 100% Chance of a win ($1M) B: 10/11 Chance of a win ($5M) If you look closely, you will see that both of the choices in Question #1 are 11% of the values in Question #3 – with the addition of a $1M consolation prize in the event of a loss Your answer to Question #1 = Your answer to Question #3

  19. Did you pass (Are you rational)? Possibility #1: You are Risk Loving Risk lovers prefer situations with more risk. Therefore, a risk loving person would always choose B Possibility #2: You are Risk Neutral Risk neutral people ignore risk and only look at expected payouts. Therefore, a risk neutral person would always choose B Possibility #3: You are Risk Averse Risk averse people try to avoid risk. Note that in each case, choice B offers a higher expected payout, but higher risk. Therefore, we can’t say which choice a risk averse person would make – we can only say that they will either always choose A or always choose B

  20. What are the expected returns from the Lottery? * Grote, Kent and Victor Matheson, ‘In Search of a Fair Bet in the Lottery”

  21. Who Plays the Lottery? * Clotfelter, Charles, et al , “Report to the National Gambling Impact Study Commission”

  22. Who Plays the Lottery? * Clotfelter, Charles, et al , “Report to the National Gambling Impact Study Commission”

  23. Lottery Data and Risk Aversion The data on Lottery Participation Suggests that at low levels of income, utility is convex (low income individuals are risk loving), but becomes concave at higher levels of income. Utility In other words, those who play the lottery are precisely those who shouldn’t!! Income Risk Loving Risk Averse

  24. Risk Aversion and the Value of insurance Suppose that the probability of being involved in a traffic accident is 1%. Further, the average damage from an accident is $400,000. How much would you be willing to pay for insurance? (For Simplicity, assume that you earn $400,000 per year U 632 • You are choosing between: • Income – Premium (with certainty) • A 1% chance of earning $0, and a 99% chance of earning $400K 0 $0 (Income – Premium) $400K I

  25. Risk Aversion and the Value of insurance Expected Utility without insurance What income level generates 625 units of happiness? U 632 625 You would pay $9,000 for this policy 0 $0 $391K $400K I

  26. Can we make a deal? We have already determined that you would pay up to $9,000 for this policy Is it worthwhile for the insurance company to offer you this policy? Expected Payout The insurance company should be willing to sell this policy for any price above $4,000 (ignoring other costs)

  27. Insurance Markets Rely on Risk Aversion to make mutually beneficial agreements This is why there are mandatory insurance laws!

  28. Suppose that there are two types of drivers (safe and unsafe). Safe drivers have a 1% chance of an accident ($400,000) cost while unsafe drivers have a 2% chance ($400,000) cost. Safe (Cost = $4,000) Unsafe (Cost = $8,000) If the insurance agent can tell them apart, he charges each an amount (at least) equal to their expected cost. Would both policies be sold?

  29. U 632 625 The safe driver would pay $9,000 for this policy 0 $0 $391K $400K I U 632 619 The unsafe driver would pay $16,000 for this policy 0 $0 $384K $400K I

  30. If the insurance agent can tell them apart, the each is charged a price according to their risk Safe Unsafe Value = $9,000 Value = $16,000 Cost = $4,000 Cost = $8,000 What If the insurance agent can’t tell them apart?

  31. Problems With Asymmetric Information Adverse selection refers to situations where, prior to a deal being made, one party lacks information about the other that would be useful (The insurance agent can’t tell good drivers from bad drivers) Suppose that the agent knows there are an equal number of good drivers and bad drivers

  32. With a $6,000 premium to both groups, the safe driver is penalized while the unsafe driver benefits Safe Unsafe Value = $9,000 Value = $16,000 Cost = $4,000 Cost = $8,000 What would happen if the unsafe driver had a 4% chance of getting in a wreck?

  33. As before, we can calculate the expected cost to the insurance company of the unsafe driver If, again, the agent knows there are an equal number of good drivers and bad drivers Value = $9,000 With a $10,000 premium, the safe drivers get priced out of the market!!

  34. Adverse selection refers to situations where, prior to a deal being made, one party lacks information about the other that would be useful How can we deal with adverse selection? • Signaling involves using visible data to classify individuals • Sports cars cost more to insure than sedans of equal value • Smokers pay more for life/health insurance • Banks use credit scoring to assess credit risk • Regulation attempts to eliminate the risk that adverse selection creates • “Lemon Laws” protect used car buyers • FDIC protects bank depositors • Mandatory car insurance keeps insurance prices from exploding

  35. If the safe and unsafe drivers can be identified by the insurance company, they will be charged a rate according to their risk. However, what if the safe driver chooses to become an unsafe driver (after all, he’s insured!) Safe (Cost = $4,000) Moral Hazard refers to situations where, after a deal is made, one party lack information about the behavior of the other party Unsafe (Cost = $8,000)

  36. Moral Hazard refers to situations where, after a deal is made, one party lack information about the behavior of the other party How can we deal with moral hazard? • Optimal Contracting involves the structuring of deals to align individual incentives • Car insurance policies have a deductible • Banks add restrictive covenants to bank loans • Collaterals • Monitoring attempts to directly observe the other party • Regulatory agencies monitor banks • Some employers use timecards

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