The economics of insurance demand and portfolio choice

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 a% 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 p:

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. • w(s)=initial wealth in state s. • c(s)=consumption in state s. • p(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))=xp(s). • c(s) is smaller whenp(s) is larger. • If p(s)=p(s’), then c(s)=c(s’). • Full insurance is optimal with actuarially fair prices. • c(s)=C(p(s)) with u’(C(p))=xp.

14. The optimal exposure to risk • C’(p ) measures the exposure to risk (locally). • C’(p)=-T(C(p))/p <0. • If u1 is more risk-averse than u2, then C1 single-crosses C2 from below. C1 p

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 w of fruits. • The individual risks are perfectly correlated.

17. The equilibrium • Equilibrium condition: c(s)=w(s) for all s. • It implies that the equilibrium prices are: p(s)=xu’(w(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 1963-1992: • Equity premium= 0.06 g. • 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 w0; • One risk free asset with a zero real return; • One risky asset with real return X; • Investment in the risky asset: a 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(a) a

25. A special case • Suppose that u is exponential and X is N(m,s2). • In that case, the Arrow-Pratt approximation is exact: • Optimal solution:

26. The impact of more risk aversion • More risk aversion less risk-taking? • u2 more concave than u1a2<a1? V1(a) a1 a

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 f1 and f2. • 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 f1 and f2. • 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 f1(0)=f2(0)=0. Then, the only possible m is m=f'1(0)/f'2(0).

31. A useful tool • Consider two real-valued functions f1 and f2. • 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 f1(0)=f2(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'(w0+x).

34. The impact of an FSD-deterioration in risk • Examine the slope of f(x)=xu'(w0+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'(w0+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: f2(t)/f1(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...