Risk Transfer Efficiency

Presented by SAIDI Neji University Tunis El-Manar

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What are impacts of attitudes toward risk on transfer possibilities  ?;

Problematical

How can we characterize added value and Pareto Optimum allocations?;

What tie can we establish between added value maximization and optimum de Pareto within RDEU model?. 

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Typology from source ( Nature or adversaries) Typology from information degree: If we have de probability distribution P over (S=Ω,Γ), we have risk

situation, else we are in uncertainty.

o A lottery is noted X=(x1,p1 ;… ; xn, pn) o On assimilate a decision to risky variable

Introduction: Uncertainty Typology

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a. Expected Utility Model

ni

1iii x up XUXu E

b. Mean-variance model

)X(Var.k)X(E)X(EV

1. Risk decision models 1. Risk decision models

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Risk Aversion :vN-M function is concave Risk lover : vN-M function is convex. Risk neutrality: vN-M function is linear

Pratt(1964) and Arrow (1965) proposed risk aversion measures in order to compare individual behaviors.

They conclude that some agents are more able to bear risk

2. Attitudes toward risk2. Attitudes toward risk

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a/ Insurance Mossin (1968), optimal contract is:

i. Full coverage if premium is equitableii. Co-assurance (agent preserve a lottery portion) if

premium is loaded

b/ Ask for risky assets (investment): diversification 

0)X(E)1()(

3. Risk transfer examples3. Risk transfer examples

Allais (1953); Kahenman et Tversky (1979)

starting point of rank dependant expected utility model (RDEU)

For X=(xi,pi)i=1,…,n where x1 <….<xn , we have :

nk

2k1kk

nj

kjj1 )x(u)x(upf)x(u V X

Particular case: dual theory (Yaari (1987))Particular case: dual theory (Yaari (1987))

n

2i

nj

1j1iij1 xx)p(fx)X(V

4. EU Model weaknesses4. EU Model weaknesses

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RDEU Model EU Model Yaari Model

Weak Aversion u concave u convex and f convex (enough)

u concave f (p)≤ p(pessimism)

Strong Aversion (R&S risk increase)

u concave and f convex

u concave f convex

Tableau 1:Behavior Characterizations in risk

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• Risk sharing (P.O) in EU setting: Borch(1962),

Arrow(1971), Raiffa(1970), Eeckhoudt and Gollier(1992). • Eeckhoudt and Roger (1994,1998): added value

1. Theoretical antecedents1. Theoretical antecedents

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Our analysis is restricted to two economic agents

The first has a composed wealth of a certain part W1 and of a lottery X

the initial wealth W2 of the second agent is certain

Every agent i (i=1,2) characterizes himself by a relation of preferences on the set of lotteries noted.

X admits an equivalent certain unique EC(X)

2. Analyze framework and definitions2. Analyze framework and definitions

a/ Hypothesis

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Selling Price pV (W1,αX) defined by:

Buying Price pA (W1,αX) defined by:

X) - 1()X ,W(pW~XW 1V111

X)X ,W(p - W ~ W 2a222

b/Definitions

s(α)= pa (W2, α X) - pv (W1, α X)

is called transfer added value (or social surplus).

If transfer occurs, then the difference:

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c/ Within Yaari model, risk transfer is possible if and only if  : DT1(X) ≤ DT2(X)

3.Main results3.Main results

3.1. Transfer possibilities

a/ If E(X) ≤0 and DARA agents with W1≤W2, then transfer occurs

b/ If first agent is risk averse and second one is neutral, then proportional transfer is realizable.

Proposition 1

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3.2. Added value maximization

We have Max s(αα ) by a total transfer (α*α* =1) if:

a/ First agent is risk averse and second one is neutral

b/ Within Yaari model with DT1(X) ≤ DT2(X)

Proposition 2

Condition: DT1(X) ≤ DT2(X), can be interpreted as first agent is more pessimist than second one

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2i

1iifii21 WV(.)V,.VF

1

2

)W(V)W(VF f22f11

3.3. Relation between social welfare et Pareto optimum

2. If Utility possibilities set of two agents is convex, then an

allocation is Pareto optimal if and only if it maximize F

with

1. An allocation maximizing F is Pareto efficient

Social utility function

Proposition 3

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If agents were risk averse in RDEU model, with concaves and differentials utility functions, then an allocation (W1,W2) is Pareto optimal if and only if it maximize F

f

t2,

f2

fs2

,f2

ft1

,1

fs1

,1

Wu

Wu

Wu

Wu

If agents have the same probabilities transformation function, then:

2t,s

1t,s TMSTMS

Proposition 4

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In Yaari model, Pareto optimal allocations were realized by assets total transfer and respond to following conditions  :

ii/ If λ=1 then price [DT1(X), DT2(X)] ;

iii/ If λ<1 then price is DT1(X) DT2(X).

i/ If λ>1 then price is DT1(X) ;

Proposition 5

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w21

w11

O2

O1

optimum in EU model

optimum in Yaari model

Figure 1. Optimal Pareto AllocationsFigure 1. Optimal Pareto Allocations

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agent type Transfer type

Relation between Max F(.) and Max s(.)

Added value repartition

1 risk averse and 2 neutral

total equivalents To agent 1

Yaari model total equivalents

To first agent if λ<1

To second if λ>1

λ=1, surplus is divided

Tableau 2: Relations between welfare and added valueTableau 2: Relations between welfare and added value

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)X(E

)X(E)X(E - )X(DT

*1f1

1st application: optimal loading within dual theory (E(X)<0)1st application: optimal loading within dual theory (E(X)<0)

1

1)X(E

)X(E

)X(E - )X(DT *

2f2

1 ])(

)( ,

)(

)( - )( [ 12

XE

XEDT

XE

XEXDT

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2eme application : Transfer by intermediate 2eme application : Transfer by intermediate

An intermediate who seek to guarantee a spread αЛ0 by transferring a portion αX

The surplus will be divided between agent 2 et intermediate

The surplus will be divided between three agents. 1

1

The surplus will be divided between buyer et intermediate

1

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Intermediate provide new complication to transfer study. In fact, in addition to agents preferences, exchange possibility will be affected by market maker behavior. Thus, fixed spread by intermediate depend:

intermediate preferencesHis inventory (stock effect)His information's (pessimism degree)

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