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?

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Evaluate your own degree of risk aversion

• Utility function:• Certainty equivalent :

1( ) /(1 )u c c 1 1 1(100(1 )) 1 (90) 1 (110)

1 2 1 2 1

RRA =10% =30%

=0.5 =0.3% =2.3%

=1 =0.5% =4.6%

=4 =2.0% =16.0%

=10 =4.4% =24.4%

=40 =8.4% =28.7%

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Complete insurance markets

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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.

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Two interpretations

• Interpretation for individual risks: Optimal insurance.

• Interpretation for macroeconomic risk: Optimal asset portfolio.

12

The decision problem

• Select (c(.)) such that it

1

1 1

max ( ) ( ( ))

. . ( ) ( ) ( ) ( ) ( ) ( )

S

s

S S

s s

p s u c s

s t p s s c s p s s s

: '( ( )) ( ).FOC u c s s

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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())=.

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The optimal exposure to risk

• C’( ) measures the exposure to risk (locally).

• C’()=-T(C())/ <0.

• If u1 is more risk-averse than u2, then C1 single-crosses C2 from below.

C1

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A complete Risk Pricing Model

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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.

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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

1

1 1

'( )( ) '( ( ))S

s

Eup s u s

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The equity premium

• Price of one share of the entire economy ("equity"):• The equity premium is equal to

• If CRRA + LogNormal distribution:

( ) / '( )P E Eu

'( )

'( )

E E Eu

P E u

2

( )

( )

Var

E

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The equity premium puzzle

• USA 1963-1992:

• 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

2.41%; 1.0186.

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Markets for coinsurance

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Chapter 4: The standard portfolio problem

The simplest model of decision under risk.

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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: dollars.

0max ( ) ( ).V Eu w X

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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.

(1 )( )z L bL bP z P b P L

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FOC and SOC

V()

* *0: '( ) '( ) 0FOC V EXu w X

20: ''( ) ''( ) 0SOC V EX u w X

*00 '(0) '( ) 0 0EX V u w EX

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A special case

• Suppose that u is exponential and X is N(2).

• In that case, the Arrow-Pratt approximation is exact:

• Optimal solution:

2 20( ) ( 0.5 ).V u w A

*2A

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The impact of more risk aversion

• More risk aversion less risk-taking?

• u2 more concave than u1

V1()

1

' '1 0 1 2 0 1( ) 0 ( ) 0?EXu w X EXu w X

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A useful tool

• Consider two real-valued functions f1 and f2.

• Under which conditions on these functions is it always true that 1 2( ) 0 ( ) 0 ?Ef X Ef X

' '1 0 1 2 0 1( ) 0 ( ) 0?EXu w X EXu w X

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Searching for the best lottery

• Search for the r.v. that is the most likely to violate the property.

1 ,..., 21

11

1

max ( )

( ) 0

. . 1

0, 1,...,

S

S

p p s ss

S

s ss

S

ss

s

p f x

p f x

s t p

p s S

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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.

1 2( ) 0 ( ) 0 ?Ef X Ef X

2 1( ) ( )f x mf x

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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).

1 2( ) 0 ( ) 0 ?Ef X Ef X

2 1( ) ( )f x mf x

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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

1 2( ) 0 ( ) 0 ?Ef X Ef X

2 1( ) ( )f x mf x

'' ''2 1(0) (0).f mf

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The impact of risk aversion on the optimal risk exposure

2 0 1 0

2 0 1 0

2 0 1 0

2 0 1 0

' ( ) ' ( ):

' ( ) ' ( )

'' ( ) '' ( ):

' ( ) ' ( )

u w x u w xNSC x x x

u w u w

u w u wNC

u w u w

Conclusion: More risk-averse agents purchase less stocks. more insurance.

' '1 0 2 0( ) 0 ( ) 0?EXu w X EXu w X

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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).

2 1 2 1( ) ( ) for all increasing .FSDX X Ef X Ef X f

2 1 2 1( ) ( ) for all increasing and concave .SSDX X Ef X Ef X f

1 0 1 1 2 0 1 2'( ) 0 '( ) 0.EX u w X EX u w X

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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.

0 0

0 0 0 0 0

0 0

'( ) '( ) ''( )

'( ) '( ) ( ) ''( ) ''( )

'( ) '( ) 1

f x u w x xu w x

f x u w x w x u w x w u w x

f x u w x R w A

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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.

0 0

0 0

0 0 0 0 0

0 0

'( ) '( ) ''( )

''( ) 2 ''( ) '''( )

''( ) 2 ''( ) ( ) '''( ) '''( )

''( ) ''( ) 2 r a

f x u w x xu w x

f x u w x xu w x

f x u w x w x u w x w u w x

f x u w x P w P

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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.

1 0 1 1 2 0 1 21. For any concave : '( ) 0 '( ) 0.u EX u w X EX u w X

2 12. : : ( ) ( ).x x

m x tdF t m tdF t

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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...