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Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Table of Contents
1 Choice Under UncertaintyBudget ConstraintPreferences
2 InsuranceChoice FrameworkExpected Utility Theory
3 Diversification & Risk Sharing
4 AIG
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
States of Nature and Contingent Plans
States of Nature:
“fire destroys house” (f) vs. “no fire” (nf)Probability of: fire = πf , no fire = πnf ; πf + πnf = 1Fire causes loss of $L
Contingent Plan:
A state-contingent consumption plan: consumptionlevel/bundle is different in each state (e.g. vacation only if nofire)Contracts may be state-contingent (e.g. insurer pays only ifthere is a fire)
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Endowment Bundle
Deriving a state-contingent budget constraint: where to start?Without insurance:
cnf = mcf = m − L
Graph:
Cnf
Cf
mThe endowment bundle.
m L−
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Budget Constraint
Buy $K of fire insurance at price p.
cnf = m − pKcf = m − L− pK + K = m − L + (1− p)KSolve for K , substitute:
cnf =m − pL
1− p− p
1− pcf
Cnf
Cf
mThe endowment bundle.
ppLm −m L−
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Preferences
We face a risky gamble
U(cf , cnf ) captures attitude towards uncertainty/risk
Risk averse vs. risk neutral
Consider our three favorite examples:
A Perfect SubstitutesB Cobb-DouglasC Perfect ComplementsD Not sureE Don’t have clicker yet
CLICKER VOTE: which reflects extreme risk aversion?
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Optimal Choice (Graph)
Some insurance (+ or -) is preferred
Cnf
Cf
m
optimal affordable plan
ppLm −m L−
Comparative statics:
risk aversion ↑=⇒ K?p ↑=⇒ K ↓L ↑=⇒ K?
What about algebraic solution? But first. . .
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Expected Utility
Example: a lottery
Win $90 or $0 equally likely
U(90) = 12 and U(0) = 2
Expected Utility is
EU = .5 ∗ U(90) + .5 ∗ U(0) = .5 ∗ 12 + .5 ∗ 2 = 7.
Expected Money is
EM = .5 ∗ 90 + .5 ∗ 0 = $45.
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Risk Attitudes
How do we characterize attitude towards risk?Recall: EU = 7 and EM = $45
U(45) > 7 =⇒ risk-averse
U(45) < 7 =⇒ risk-loving
U(45) = 7 =⇒ risk-neutral
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Risk Attitudes
We typically assume diminishing marginal utility (DMU) of wealth.
Wealth$0 $90
2
12
$45
EU=7
So EU < U(EM). . .
this implies risk aversion!
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Risk Attitudes
Example: Risk-loving preferences
Wealth$0 $90
12
2
EU=7
$45
U($45)
EU < U(EM)
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Risk Attitudes
Example: Risk-neutral preferences
Wealth$0 $90
12
2
$45
U($45)=EU=7
EU = U(EM)
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Optimal Choice (Algebra)
Calculating the MRS
EU = πf U(cf ) + πnf U(cnf )
Indifference curve =⇒ constant EU
Differentiate:
dEU = 0 = πf MU(cf )dcf + πnf MU(cnf )dcnf
MRS = dcnf
dcf= − πf MU(cf )
πnf MU(cnf )
Solution satisfies
p
1− p=
πf MU(cf )
πnf MU(cnf ).
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Competitive Insurance
Free entry =⇒ zero expected economic profit
So pK − πf K − (1− πf )0 = (p − πf )K = 0.
=⇒ p = πf
Insurance is fair
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Competitive Insurance
With fair insurance, rational choice satisfies
πf
πnf=
πf
1− πf=
p
1− p=
πf MU(cf )
πnf MU(cnf ).
In other words, MU(cf ) = MU(cnf )
Risk-aversion =⇒ cf = cnf
Full insurance!
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Not-Fair Insurance
Suppose insurers make positive expected economic profit.
pK − πf K − (1− πf )0 = (p − πf )K > 0
Then p > πf and p1−p >
πf1−πf
=⇒ MU(cf ) > MU(cnf )
Risk-averse =⇒ cf < cnf : less than full (not-fair) insurance
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Demand for Insurance: EU Perspective
Certainty Equivalent = dollar amount you would need to havewith certainty to make you indifferent to the gambleU(CE ) = EUEM − CE = willingness to pay for full-insurance (length of redline)
Wealth$0 $90
2
12
$45
EU=7=U(CE)
CE
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Proposed Gamble
I flip a fair coin. Heads: I pay you $120; tails: you pay me $100.Any takers?
CLICKER VOTE:
A Accept
B No thank you!
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Proposed Gamble: II
What if I offered this same gamble at the beginning of every lecture(and you had to tell me today what you would choose each time)?
CLICKER VOTE:
A Accept every time
B Reject every time
C Some combination
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Analysis
Why is the same gamble more attractive when it is repeated?
Each gamble has positive expected value
Each coin toss is independent
Law of Large Numbers: expected money from compoundgamble = N times the EM = a big positive number
Portfolio of gambles is diverse, so very little chance of net loss
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Diversification
Example:
Two firms, A and B. Shares cost $10
With prob = .5, ΠA = 100 and ΠB = 20
With prob = .5, ΠA = 20 and ΠB = 100
You have $100 to invest. How?
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Diversification
Example:
Buy only firm A’s stock?
$100/10 = 10 shares
Earn $1000 w/ prob .5 and $200 w/ prob .5
Expected earning: $500 + $100 = $600
Same for buying only B
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Diversification
Example:
Buy 5 shares of each firm?
Earn $600 for sure
Diversification has maintained expected earnings whilelowering risk
Typically there’s a tradeoff between earnings and risk
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
Recap
What are rational responses to risk?
Buying insurance
A diverse portfolio of contingent consumption goods (assets)
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
AIG: WTF?
How does this help us understand what big insurance/financialcompanies like AIG are supposed to do?
You buy insurance in response to risk
Insurance company gets your premium, but now faces risk ofhaving to pay claim
To the extent that claims are independent, this is ok for thembecause they have a diverse portfolio of risks
Same w/ home lenders: they get your mortgage payments,but lose if you default
To diversify risk, lenders wad mortgages together into bundles,then sell them (in pieces) as relatively safe (diversified)securities
Thus, our risk and insurance courses through the veins of thefinancial system
Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG
AIG: WTF?
So what can and did go wrong?
Diversification works if risks are independent, but not ifcorrelated.
My proposed gamble: imagine if I decided outcome w/ onecoin-toss at the end of the quarter. Taker?
Risk of house burning down: Seattle vs. SoCal
Wildfires, earthquakes, hurricanes can wipe out entirecities/regions at once
Natural disasters are disasters for insurers
Insurers know this: there is an enormous re-insurance industry