Diversification of Parameter Uncertainty

1. Diversification of Parameter Uncertainty Eric R. Ulm Georgia State University

2. Defining an Ambiguous Risk • Case 1: Risk: 2% chance of hurricane next year. • Case 2: Ambiguity: Expert 1 says 1% chance, Expert 2 says 3%. You think there is a 50% chance of each expert being correct.

3. Effect of Ambiguity on Premiums • Hogarth and Kunreuther (1989) surveyed actuaries. They found for risks of this size, the ratio of prices charged for ambiguous and standard risks was in the range 1.50-2.00.

4. Should ambiguity affect premiums? • Yes and no. • One time Ellsberg Game • Game 1: One urn, 50 black, 50 white • Game 2: One urn, X black, 1-X white. X=75 or 25. (Type A or Type B) Payoff of $100,000 for your ball. What do you pick?

5. Probably Not • Game 1, choose white, 50% probability of winning. • Game 2, flip a coin for ball color. • Type A. 50% white choice times 25% white drawn plus 50% black choice times 75% black drawn is 50%! • Type B. 50% white choice times 75% white drawn plus 50% black choice times 25% black drawn is 50%!

6. Hidden Assumptions • Choice of color to bet is voluntary • Choice of color to bet is made after the urn type (A or B) is determined • The game is played only once.

7. Argument Also Valid If: • Urn type (A or B) is a 50-50 choice by the selector. Choice of color need not be voluntary, but still must be made only once. • Game can be played repeatedly if the voluntary choice of white or black can be changed (by a coin flip) every period.

8. yes andNO • Froot and Posner (2002) use a more formal version of this argument to price catastrophe bonds with binary payouts. • They say ambiguity should not affect risks (i.e. ambiguity aversion is “irrational”)

9. Multiple Periods • Choice of ball color fixed in time • Urn is fixed for all periods • Ambiguity aversion is just risk aversion

10. Two periods-Ambiguous • 2 hurricanes .5*.03^2+.5*.01^2=.0005 • 1 hurricane .5*2*.03*.97+.5*2*.01*.99=.0390 • No hurricanes .5*.97^2+.5*.99^2=.9605 • Mean 2%, standard deviation 9.92472%

11. Two Periods - Unambiguous • 2 hurricanes .02^2=.0004 • 1 hurricane 2*.02*.98=.0392 • No hurricanes .98^2=.9604 • Mean 2%, standard deviation 9.89949% • Smaller Sigma!

12. Long Haul • Ambiguous: Mean 2%, Sigma 1% • Unambiguous: Mean 2%, Sigma 0% • No longer insuring hurricanes, but insuring the model! • Effect would disappear if insurers could choose to bet on hurricanes with a coin flip.

13. General Case:Multiple Ambiguous Risks • Probability Parameters: , ,… drawn once • Produce Mean and Standard Dev • From Central Limit Theorem: Mean of the average of a large number of draws from this distribution approaches normal distribution with mean and Standard Dev 0. • is an “ordinary” risk

14. Diversifying the Ambiguity • Type 1 Risks (Hurricanes), Type 2 Risks (Earthquakes), … , Type n Risks • Which “expert” is right is uncorrelated from risk to risk. • Type “k” risks are mean drawn from distribution with mean and standard deviation • Each is an insignificant part of the total

15. More Diversifying Ambiguity • The mean payout per claim is • Central Limit Again. This approaches a normal with mean and standard deviation 0. It is riskless!

16. Complication #1:Correlation of Parameter Means • Hurricanes and Earthquakes are uncorrelated • Correlation Across Models: For example, if Company “High Risk” model for hurricanes is correct, their model for earthquakes is more likely to be right (and vice versa) • Different than bias toward bad results.

17. Markowitz-ish • Mean payout per claim is normal with mean and standard deviation • Risk Averse Insurance Company changes values of weights to maximize utility. • Suspiciously like Markowitz and CAPM

18. Complication #2:Correlation of individual risks • Hurricanes correlated with other hurricanes. • Earthquakes correlated with other earthquakes. • Individual hurricanes and earthquakes are uncorrelated. • Model parameter realizations are uncorrelated.

19. Diversifying Away Ambiguity Again • Correlation if • is a random draw from distribution with mean and standard deviation (not necessarily normal) • Central Limit Theorem approaches mean and standard deviation 0 (if no correlation among different types of risks) • Central Limit Theorem approaches mean and standard deviation 0 (if no correlation among parameter means). No risk!

20. Combine Complications 1 and 2 • is a random draw from distribution with mean and standard deviation (not necessarily normal) • Central Limit Theorem approaches mean and standard deviation 0 (if no correlation among different types of risks)

21. Markowitz-ish Again • Mean payout per claim is normal with mean and standard deviation • Risk Averse Insurance Company changes values of weights to maximize utility. • Suspiciously like Markowitz and CAPM

22. Complication 3 (and 1 and 2) • 1: Hurricanes correlated with other hurricanes and earthquakes correlated with other earthquakes. • 2: Model Parameters are correlated • 3: Individual hurricanes and earthquakes are correlated, modeled by correlation between and given and is

23. Worst Math Yet • It can be shown the that correlation between and given and is:

24. Diversification Step 1 • approaches a distribution with mean and standard deviation

25. Diversification Step 2 • approaches a distribution with mean and standard deviation • Combining these facts, is distributed with mean and standard deviation • Still Markowitz and CAPM-ish

26. Uncertainty in Asset Risks • Volatility cannot be diversified away independently of the parameter uncertainty. • Mathematics is identical. is distributed with mean and standard deviation • Investors see a larger risk ex ante, but ex post risk is only the measured sigmas, not s’s. • Could explain equity premium puzzle

27. Learning and Multiple Period Risks • Full Bayesian Updating of Premium • Model A, 1% hurricanes, 50% true • Model B, 3% hurricanes, 50% true • $100,000 home is insured, $2,000 is charged • 2% chance of $98,000 loss, 98% chance of $2,000 gain • Mean $0, standard deviation $14,000

28. Period 2 • Premium from Bayes Theorem: • Hurricane occurs: 75% Model B is correct, Premium = $2,500 • Hurricane doesn’t occur: 50.51% Model A is correct, Premium = $1,989.80

29. Means and Standard Deviations • Mean = $0, Standard Deviation = $13,999.82 • Standard Deviation is lower than the $14,000 that would occur in the absence of updating!

30. Combine at 0% discounting • Unambiguous: Sigma = $19,798.99. Losses are uncorrelated • Ambiguous, but no Bayesian updating of premiums: Sigma = $19,849.93. Loss correlation is 0.0051 • Ambiguous with Bayesian updating of premiums: Sigma = $19,798.86. Loss correlation is 0, second period sigma is reduced! • Implies LOWER risk premium for ambiguous multiperiod risks!

31. Does this always work? • Proof that correlation = 0 if premiums are fully Bayesian updated. • Find the correlation between and for • At time j, the Filtration F representing the time sequence of existence or lack of hurricanes through time j-1 is completely known. The Bayesian premium given F is set such that: and the expected value of the total payout is

32. Proof continued • Since we have • is a function only of F! Therefore: because the first term in the sum is zero! • Because the covariance is 0, so is the correlation

33. Proof that variance at time i is less than that at time 1. • is the probability of n hurricanes in periods 1 through i-1 • is the Bayesian updated probability of a hurricane in period i. This is also the Bayesian updated premium given n previous hurricanes. • Variance of equals

34. Proof Continued • Where is the expected value of the Bayesian updated premium, which is also the probability of a hurricane in period 1. • Now, as this is the variance of the Bayesian premium in period i. So:

35. More Pronounced with Profit/Risk Loading • 15% Premium Loading makes mean value $300 in both periods. Standard Deviations are unchanged. Period 1 to Period 2 correlation drops below zero to -0.0008 • 100,000 scenarios for 100 years at 15% loading and 5% discount factor. • Ambiguous with updating: Mean $6,054.97, Sigma $43,119.06 • Unambiguous: Mean $6,055.11, Sigma $43,119.06

36. Bayesian Updating Limited by Regulators • Worst case: NO premium increases allowed, full Bayesian decreases. Determine loading “L” to make NPV = 0 at 5%. • Charge $2,000(1+L) in first period, and either $1,989.80(1+L) or $2,000(1+L) in the second. • L solves for 0.0025 or ¼%

37. Simulation of No Increases and Full Bayesian Decreases • 100,000 scenarios, 100 years. EPV of premiums and claims at 5% are equal if the loading L equals 7.16%. • Surprisingly small. Ambiguity and rate regulation appear unable to account for the high premiums charged by actuaries for ambiguous risks. • Further Research: What if hurricane probabilities drift randomly with time?

38. Conclusions • When should ambiguity not be priced? • 1) One time game. • 2) Multiple Period games where choice of the “side” of the contract to take is voluntary and chosen randomly. • 3) Different types of ambiguous risks can be insured, with no correlation among the parameters of the distributions of different risks. • 4) Multiple Period games with Bayesian updating (if anything, the price should be decreased slightly).

39. More Conclusions • When is ambiguity aversion just risk aversion? • 1) Multiple Period games with only one type of ambiguous risk with “side” taken being involuntary, and no updating of premiums (or updating limited by regulators) • 2)Multiple types of ambiguous risks with correlation among the parameter realizations. • 3) Could explain some part of the “equity premium puzzle” • 4) Unlikely to explain large premiums suggested by actuaries for ambiguous risks.