# Choice Under Uncertainty Insurance Diversification & Risk Sharing AIG. Uncertainty

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1 Uncertainty

2 Table of Contents 1 Choice Under Uncertainty Budget Constraint Preferences 2 Insurance Choice Framework Expected Utility Theory 3 Diversification & Risk Sharing 4 AIG

3 States of Nature and Contingent Plans States of Nature: fire destroys house (f) vs. no fire (nf) Probability of: fire = π f, no fire = π nf ; π f + π nf = 1 Fire causes loss of \$L Contingent Plan: A state-contingent consumption plan: consumption level/bundle is different in each state (e.g. vacation only if no fire) Contracts may be state-contingent (e.g. insurer pays only if there is a fire)

4 Endowment Bundle Deriving a state-contingent budget constraint: where to start? Without insurance: c nf = m c f = m L Graph: C nf m The endowment bundle. m L C f

5 Budget Constraint Buy \$K of fire insurance at price p. c nf = m pk c f = m L pk + K = m L + (1 p)k Solve for K, substitute: c nf = m pl 1 p p 1 p c f C nf m The endowment bundle. m L m pl p C f

6 Preferences We face a risky gamble U(c f, c nf ) captures attitude towards uncertainty/risk Risk averse vs. risk neutral Consider our three favorite examples: A Perfect Substitutes B Cobb-Douglas C Perfect Complements D Not sure E Don t have clicker yet CLICKER VOTE: which reflects extreme risk aversion?

7 Optimal Choice (Graph) Some insurance (+ or -) is preferred C nf m optimal affordable plan m L m pl p C f Comparative statics: risk aversion = K? p = K L = K? What about algebraic solution? But first...

8 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) = = 7. Expected Money is EM = = \$45.

9 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

10 Risk Attitudes We typically assume diminishing marginal utility (DMU) of wealth. 12 EU=7 2 \$0 \$45 \$90 Wealth So EU < U(EM)... this implies risk aversion!

11 Risk Attitudes Example: Risk-loving preferences 12 EU=7 U(\$45) 2 \$0 \$45 \$90 Wealth EU < U(EM)

12 Risk Attitudes Example: Risk-neutral preferences 12 U(\$45)= EU=7 2 \$0 \$45 \$90 Wealth EU = U(EM)

13 Optimal Choice (Algebra) Calculating the MRS EU = π f U(c f ) + π nf U(c nf ) Indifference curve = constant EU Differentiate: deu = 0 = π f MU(c f )dc f + π nf MU(c nf )dc nf MRS = dc nf dc f = π f MU(c f ) π nf MU(c nf ) Solution satisfies p 1 p = π f MU(c f ) π nf MU(c nf ).

14 Competitive Insurance Free entry = zero expected economic profit So pk π f K (1 π f )0 = (p π f )K = 0. = p = π f Insurance is fair

15 Competitive Insurance With fair insurance, rational choice satisfies π f π nf = π f 1 π f = p 1 p = π f MU(c f ) π nf MU(c nf ). In other words, MU(c f ) = MU(c nf ) Risk-aversion = c f = c n f Full insurance!

16 Not-Fair Insurance Suppose insurers make positive expected economic profit. pk π f K (1 π f )0 = (p π f )K > 0 Then p > π f and p 1 p > = MU(c f ) > MU(c nf ) π f 1 π f Risk-averse = c f < c nf : less than full (not-fair) insurance

17 Demand for Insurance: EU Perspective Certainty Equivalent = dollar amount you would need to have with certainty to make you indifferent to the gamble U(CE) = EU EM CE = willingness to pay for full-insurance (length of red line) 12 EU=7 =U(CE) 2 \$0 CE \$45 \$90 Wealth

18 Proposed Gamble I flip a fair coin. Heads: I pay you \$120; tails: you pay me \$100. Any takers? A Accept B No thank you! CLICKER VOTE:

19 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)? A Accept every time B Reject every time C Some combination CLICKER VOTE:

20 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 compound gamble = N times the EM = a big positive number Portfolio of gambles is diverse, so very little chance of net loss

21 Diversification Example: Two firms, A and B. Shares cost \$10 With prob =.5, Π A = 100 and Π B = 20 With prob =.5, Π A = 20 and Π B = 100 You have \$100 to invest. How?

22 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

23 Diversification Example: Buy 5 shares of each firm? Earn \$600 for sure Diversification has maintained expected earnings while lowering risk Typically there s a tradeoff between earnings and risk

24 Recap What are rational responses to risk? Buying insurance A diverse portfolio of contingent consumption goods (assets)

25 AIG: WTF? How does this help us understand what big insurance/financial companies like AIG are supposed to do? You buy insurance in response to risk Insurance company gets your premium, but now faces risk of having to pay claim To the extent that claims are independent, this is ok for them because 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 the financial system

26 AIG: WTF? So what can and did go wrong? Diversification works if risks are independent, but not if correlated. My proposed gamble: imagine if I decided outcome w/ one coin-toss at the end of the quarter. Taker? Risk of house burning down: Seattle vs. SoCal Wildfires, earthquakes, hurricanes can wipe out entire cities/regions at once Natural disasters are disasters for insurers Insurers know this: there is an enormous re-insurance industry

27 AIG: WTF? So what can and did go wrong? Lenders/financiers were not prepared for the collapse of the housing bubble Housing crisis = financial crisis = credit crisis = baaad recession

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