1 The economics of insurance demand and portfolio choice Lecture 1 Christian Gollier

2 General introduction Risks are everywhere. Managing them efficiently is an important aspect of modern society. There is no field of economics without some risk analysis. Insurance economics is an excellent basis for expansion.

3 General introduction Two parts: –Part 1: Risk management in the classical framework: Standard comparative statics of risk transfers; Optimal dynamic risk management; Pension; Equilibrium risk transfers with heterogeneous beliefs; –Part 2: Risk management with richer psychological characters: Ambiguity aversion; Conformism and envy; Aversion to regret; Anxiety.

4 The economics of insurance demand and portfolio choice Lecture 1 Christian Gollier

5 Introduction to Lecture 1 Background material for the next 7 lectures. A quick overview of the first half of my MIT book (2001). Analysis of insurance demand and portfolio choice. Two static models: –The complete insurance model; –The coinsurance model. Comparative static analysis. Risk pricing. Prerequisite: some knowledge of the EU model.

6 Evaluate your own degree of risk aversion Suppose that your wealth is currently equal to 100. There is a fifty-fifty chance of gaining or losing  % of this wealth. How much are you ready to pay to eliminate this risk?

7 Evaluate your own degree of risk aversion Utility function: Certainty equivalent  :

8 Complete insurance markets

9 Description of the model The uncertainty is described by the set of possible states of nature, and their corresponding probabilities. Insurance markets offer flexible contracts. Arrow-Debreu framework. Two branches of the theory: the optimal insurance and the theory of finance.

10 The model One period. {1,...,S}= set of possible states of the world. p(s)=probability of state s.  (s)=initial wealth in state s. c(s)=consumption in state s.  (s)=price of state s, per unit of probability.

11 Two interpretations Interpretation for individual risks: Optimal insurance. Interpretation for macroeconomic risk: Optimal asset portfolio.

12 The decision problem Select (c(.)) such that it

13 A simple property FOC: u’(c(s))=  (s). c(s) is smaller when  (s) is larger. If  (s)=  (s’), then c(s)=c(s’). Full insurance is optimal with actuarially fair prices. c(s)=C(  (s)) with u’(C(  ))= .

14 The optimal exposure to risk C’(  ) measures the exposure to risk (locally). C’(  )=-T(C(  ))/  <0. If u 1 is more risk-averse than u 2, then C 1 single-crosses C 2 from below. C1C1 

15 A complete Risk Pricing Model

16 The simplest equilibrium model Lucas’ tree economy. Agents are identical; utility function u. They consume fruits at the end of the single period. They are each endowed with a tree. Each tree will produce a random number  of fruits. The individual risks are perfectly correlated.

17 The equilibrium Equilibrium condition: c(s)=  (s) for all s. It implies that the equilibrium prices are:  (s)=  u’(  s)). Risk aversion: Consumption is relatively more expensive in poorer states. Insurability of catastrophic risk? Pricing kernel: the core of all asset pricing models. We take

18 The equity premium Price of one share of the entire economy ("equity"): The equity premium is equal to If CRRA + LogNormal distribution:

19 The equity premium puzzle USA : Equity premium= 0.06 . We need to have a RRA larger than 40 to explain the existing prices. Invest all your wealth in stocks! $1 invested –at 1% over 40 years = $1.48 –at 7% over 40 years = $14.97

20 Markets for coinsurance

21 Chapter 4: The standard portfolio problem The simplest model of decision under risk.

22 The model An agent who lives for a single period; Initial sure wealth w 0 ; One risk free asset with a zero real return; One risky asset with real return X; Investment in the risky asset:  dollars.

23 Other interpretations Demand for insurance: –initial wealth z subject to a random loss L. –Transfer a share b of the loss to an insurance against a premium bP. –Final wealth: Capacity choice under an uncertain profit margin. Technological risks.

24 FOC and SOC  V(  )

25 A special case Suppose that u is exponential and X is N(  2 ). In that case, the Arrow-Pratt approximation is exact: Optimal solution:

26 The impact of more risk aversion More risk aversion less risk-taking? u 2 more concave than u 1       V1()V1() 11

27 A useful tool Consider two real-valued functions f1 and f2. Under which conditions on these functions is it always true that

28 Searching for the best lottery Search for the r.v. that is the most likely to violate the property.

29 A useful tool Consider two real-valued functions f 1 and f 2. Under which conditions on these functions is it always true that Theorem: This is true if and only if there exists a scalar m such that for all x.

30 A useful tool Consider two real-valued functions f 1 and f 2. Under which conditions on these functions is it always true that Theorem: This is true if and only if there exists a scalar m such that for all x. Suppose that f 1 (0)=f 2 (0)=0. Then, the only possible m is m=f' 1 (0)/f' 2 (0).

31 A useful tool Consider two real-valued functions f 1 and f 2. Under which conditions on these functions is it always true that Theorem: This is true if and only if there exists a scalar m such that for all x. Suppose that f 1 (0)=f 2 (0)=0. Then, the only possible m is m=f' 1 (0)/f' 2 (0). A necessary condition is

32 The impact of risk aversion on the optimal risk exposure Conclusion: More risk-averse agents purchase less stocks. more insurance.

33 The impact of more risk Notions of stochastic dominance orders: – Such changes in risk reduce the optimal risk exposure if and only if Examine the shape of f(x)=xu'(w 0 +x).

34 The impact of an FSD-deterioration in risk Examine the slope of f(x)=xu'(w 0 +x). Theorem: A FSD-deterioration in the equity return reduces the demand for equity if relative risk aversion is less than unity.

35 The impact of a Rothschild-Stiglitz increase in risk Examine the concavity of f(x)=xu'(w 0 +x). Theorem: A Rothschild-Stiglitz increase in risk of the equity return reduces the demand for equity if relative prudence is positive and less than 2.

36 Central dominance Theorem: Conditions 1 and 2 are equivalent: Example: MLR: f 2 (t)/f 1 (t) is decreasing in t. Corollary: A MLR-deterioration in equity returns reduces the demand for equity.

37 Conclusion Two choice models under risk. An increase in risk aversion reduces the optimal risk exposure. But the observed decisions/prices suggest unrealistically large risk aversion. The impact of a change in risk on the optimal risk exposure is problematic...