Diversification of Parameter Uncertainty

Eric R. Ulm

Georgia State University

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.

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.

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?

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%!

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.

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.

yes and NO

• 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”)

Multiple Periods

• Choice of ball color fixed in time

• Urn is fixed for all periods

• Ambiguity aversion is just risk aversion

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%

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!

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.

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

k1 k

2 kn

k k

kkp kp

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

km

kp

ks

ki

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!

n

kk

n

k

kk

i

pi

1

1

n

kk

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1

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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.

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

n

k

kk

n

k

kk wpw

11

n

k

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1

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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.

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!

kk

0 kkij ji

kpk

n

k

kk pw

1

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kkw

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1

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)

kpk

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

n

k

kk

n

k

kk wpw

11

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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

kipm

jpk m km

Worst Math Yet

• It can be shown the that correlation between and given and is:

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mk

km

mmm

kkk

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mmkk

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ppCovi

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11 ),(),(),(

Diversification Step 1

• approaches a distribution with mean and standard deviation

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1111

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

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kkw

1

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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

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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

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

)Pr(

)()|Pr()|Pr(

A

BPRBAAB

Means and Standard DeviationsPeriod 1 Period 2 Prob Premium Claims Cash Flow H H 0.0005 $ 2,500.00 $(100,000.00) $ (97,500.00) H N 0.0195 $ 2,500.00 $ - $ 2,500.00

N H 0.0195 $ 1,989.80 $(100,000.00) $ (98,010.20) N N 0.9605 $ 1,989.80 $ - $ 1,989.80

• Mean = $0, Standard Deviation = $13,999.82

• Standard Deviation is lower than the $14,000 that would occur in the absence of updating!

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!

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

iPO

jPO

ji

0| FPOE j

0Pr| F

j FFPOE

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

iPO

0 ji POEPOE

jiji POPOEPOPOCov ,

0Pr)(| F

ijji FFPOFPOEPOPOE

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

nq

np

iPO

n

jjjj

n

jjjjjji ppqppppqPOE

00

222 )1()1()1(

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:

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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

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%

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?

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).

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.