Risk Transfer Efficiency Presented by SAIDI Neji University Tunis El-Manar

2  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?.

3  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. oA lottery is noted X=(x 1,p 1 ;… ; x n, p n ) oOn assimilate a decision to risky variable Introduction: Uncertainty Typology

4 a. Expected Utility Model b. Mean-variance model 1. Risk decision models

5  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 risk

6 a/ Insurance Mossin (1968), optimal contract is: i. Full coverage if premium is equitable ii.Co-assurance (agent preserve a lottery portion) if premium is loaded b/ Ask for risky assets (investment): diversification 3. Risk transfer examples

 Allais (1953); Kahenman et Tversky (1979) starting point of rank dependant expected utility model (RDEU)  For X=(x i,p i ) i=1,…,n where x 1 <….<x n, we have :  Particular case: dual theory (Yaari (1987)) 4. EU Model weaknesses

8 RDEU ModelEU ModelYaari Model Weak Aversion  u concave  u convex and f convex (enough) u concavef (p)≤ p (pessimism) Strong Aversion (R&S risk increase ) u concave and f convex u concavef convex Tableau 1:Behavior Characterizations in risk

9 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 antecedents

10  Our analysis is restricted to two economic agents  The first has a composed wealth of a certain part W 1 and of a lottery X  the initial wealth W 2 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 definitions a/ Hypothesis

11  Selling Price p V (W 1,αX) defined by:  Buying Price p A (W 1,αX) defined by: b/Definitions s(α)= p a (W 2, α X) - p v (W 1, α X) is called transfer added value (or social surplus).  If transfer occurs, then the difference:

12 c/ Within Yaari model, risk transfer is possible if and only if : DT 1 (X) ≤ DT 2 (X) 3.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

Added value maximization αα* We have Max s(α ) by a total transfer (α* =1) if: a/ First agent is risk averse and second one is neutral b/ Within Yaari model with DT 1 (X) ≤ DT 2 (X) Proposition 2 Condition: DT1(X) ≤ DT2(X), can be interpreted as first agent is more pessimist than second one

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

15 If agents were risk averse in RDEU model, with concaves and differentials utility functions, then an allocation (W 1,W 2 ) is Pareto optimal if and only if it maximize F If agents have the same probabilities transformation function, then: Proposition 4

16 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

17 w21w21 w11w11 O2O2 O1O1 optimum in EU model optimum in Yaari model Figure 1. Optimal Pareto Allocations

18 agent typeTransfer type Relation between Max F(.) and Max s(.) Added value repartition 1 risk averse and 2 neutral totalequivalentsTo agent 1 Yaari modeltotalequivalents  To first agent if λ<1  To second if λ>1  λ=1, surplus is divided Tableau 2: Relations between welfare and added value

19 1 st application: optimal loading within dual theory (E(X)<0)

20 2eme application : Transfer by intermediate An intermediate who seek to guarantee a spread αЛ 0 by transferring a portion αX The surplus will be divided between agent 2 et intermediate The surplus will be divided between three agents. The surplus will be divided between buyer et intermediate

21 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 preferences His inventory (stock effect) His information's (pessimism degree)

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