EFFICIENT, WEAK EFFICIENT, AND PROPER EFFICIENT PORTFOLIOS

AND PROPER EFFICIENT PORTFOLIOS

MANUELA GHICA

We give some necessary conditions for weak efficiency and proper efficiency port- folios for a reinsurance market in nonconvex optimization problems.

AMS 2000 Subject Classification: 62P05, 91B30, 90A12.

Key words: reinsurance market, portfolio, efficiency.

1. INTRODUCTION

The reinsurance problem appears at first sight to be a problem which can be analyzed in terms of classical economic theory, if the objectives of the companies have been formulated in an operational manner by means of Bernoulli’s utility concept: the expected gain must not be maximized, but the expected utility of the gain [3]. However, closer investigations show that the economic theory is relevant only part of the way. Then the problem becomes a problem of cooperation between parties that have conflicting interests and are free to form and break any coalitions which may serve their particular interests [12]. Classical economic theory is powerless when it comes to analyze such problems. In the last decades it was shown that there are many possibilities to study and explain the apparently chaotic situation by using game theory or convex analysis.

In this paper we use and extend results from convex analysis established by Rockafellar [13] and Kaliszewski [11]. A basic result of convex analysis is the Fundamental Theorem on Convex Functions [4], [13] which states that an efficient solution of a convex vector optimization problem necessarily min- imizes a linear combination of objective functions. Kaliszewski characterized efficient solutions without assuming convexity. All these results help us to establish some new results for portfolios in a reinsurance market in nonconvex optimization problems.

If we seeN ={1,2, . . . , n}as a group ofnreinsurers, having preferences

≥_i, i∈N, over a suitable set of random variables denoted by R, or gambles with realizations (outcomes) in some A ⊆ R, we represent these preferences

MATH. REPORTS10(60),4 (2008), 309–315

by von Neumann-Morgenstern expected utility, meaning that there is a set of continuous utility functions u_i : R → R such that X ≥_i Y if and only if Eui(X) ≥ Eui(Y), where E stands for the mean operator. We assume monotonic preferences, and risk aversion, so that we haveu⁰_i(w)>0, u⁰⁰_i(w)≤0 for allw in the relevant domains [5]. In some cases we shall also require strict risk aversion, meaning strict concavity for someui.For a better understanding we assume that each agent is endowed with a random variable payoffX_icalled initial portfolio. More precisely, there exists a probability space (Ω,K, P) such that we have the payoff X_i(ω) when ω ∈ Ω occurs and both expected values and variances exist for all these initial portfolios, which means that all Xi ∈ L²(Ω,K, P) [6]. Because every agent can negotiate any affordable contracts, we will have a new set of random variables Y_i, i∈N,representing the final portfolios. We say if Pn

i=1Zi ≤ Pn

i=1Xi =XN then the allocation Z = (Z₁, Z₂, . . . , Z_n) is said to be feasible [1].

The paper is organized as follows. In Section 2 we give an alternative theorem for a basic result in the nonconvex framework. In Sections 3 and 4 we characterize efficiency, weak efficiency and proper efficiency of portfolios for a reinsurance market in nonconvex optimization problems by different necessary conditions related to the alternative theorem given in Section 2.

2. AN ALTERNATIVE TYPE THEOREM

In this section we extend a fundamental theorem on convex functions to the case when the convexity assumption does not hold. This alternative type theorem enables us to characterize the efficiency of portfolios for a reinsurance market.

Theorem1. LetZ be the set of feasible allocations. Then one and only one of the following alternative holds:

(a)∃Z_i∈ Z such that Eui(Zi)<0, i∈N;

(b) for any negative numbers δ₁, δ₂, . . . , δ_n there exist positive numbers λ1, λ2, . . . , λn such that max

i λi(Eui(Zi)−δi)≥1, ∀Z ∈ Z.

Proof. Suppose (a) holds. Let Z ∈ Z such that Eu_i(Z_i) <0 fori∈N.

Take δi ∈R^∗−, i∈N, with

δ_i ≥max

i Eu_i(Z_i).

Thus,Eu_i(Z_i)−δ_i <0,for any i∈1, n,and so (b) cannot hold.

Now, suppose (a) does not hold. Let Z ∈ Z. Then there exists at least onej ∈N such thatEu_j(Z_j)≥0.There fore, we have

maxj Euj(Zj)≥0.

Letδ1, δ2, . . . , δnbe any negative numbers and ¯λi= (δi)⁻¹,i∈N.Hence max

i=1,n

¯λ_iEu_i(Z_i)≥0.This inequality is equivalent to max

i=1,n

[¯λ_i(Eu_i(Z_i)−δ_i)−1]≥0, i.e., max

i=1,n

λ¯_i(Eu_i(Z_i)−δ_i)≥1.

Thus, (b) holds.

Consider the vector problem

(1) ∨min

Z∈Z (Eu₁(Z₁), Eu₂(Z₂), . . . , Eu_n(Z_n)),

where ∨min denotes the operation of deriving all efficient portfolios.

3. EFFICIENT AND WEAK EFFICIENT PORTFOLIOS

In this section we first define efficiency and weak efficiency of portfolios for a reinsurance market in nonconvex optimization problems. Then we give a necessary condition for weak efficiency and show that it is a consequence of Theorem 1.

LetZ be the set of feasible allocations.

Definition 2. Y ∈ Z is said to be an efficient portfolio (or a Pareto portfolio) if for Z ∈ Z, it follows from Eui(Zi) ≤ Eui(Yi), i ∈ N, that Eu_i(Z_i) =Eu_i(Y_i), i∈N.

Definition 3. Y ∈ Z is a weakly efficient portfolio (or a Slater portfolio) if there is no Z∈Z such thatEui(Zi)< Eui(Yi), i∈N.

Theorem 4. Let Y ∈ Z be a weakly efficient portfolio. Then it solves the problem

minZ∈Z ϕ(Z), where ϕ(Z) = max

i∈N λ_i(Eu_i(Z_i)−y_i^∗),with λ_i = (Eu_i(Y_i)−y_i^∗)⁻¹, i∈N and y_i^∗ is any real number such that λi>0 for everyi∈N.

Proof. Let Y ∈ Z be a weakly efficient portfolio. Then the system of inequalities

Eui(Zi)< Eui(Yi), i∈N, Z∈ Z

has no solution. Now, by Theorem 1, for any negative numbers ε₁, ε₂, . . . , ε_n we have

maxi λ_i(Eu_i(Z_i)−Eu_i(Y_i)−δ_i)≥1, Z ∈ Z,

where λi = (−δ_i)⁻¹, i ∈ N. Take δ1, δ2, . . . , δn and y₁^∗, y₂^∗, . . . , y_n^∗ such that δi = y_i^∗ −Eui(Yi) < 0, i ∈ N. Then λi = (Eui(Yi)−y^∗_i)⁻¹ > 0 and maxi∈Nλ_i(Eu_i(Z_i)−y^∗_i)≥1 for every Z∈ Z.

We see thatλi(Eui(Yi)−y_i^∗) = 1 for any i∈N. Hence maxi∈N λ_i(Eu_i(Z_i)−y^∗_i)≥max

i∈N λ_i(Eu_i(Y_i)−y^∗_i), Z ∈ Z, i.e., Y solves the problem

minZ∈Zmax

i∈N λ_i(Eu_i(Z_i)−y_i^∗) and the theorem is proved.

4. PROPERLY EFFICIENT PORTFOLIOS

In this section we define properly efficient portfolios and then give two characterization theorems. First we prove that a properly efficient portfolio is a solution of problem (4), as a consequence of Theorem 1. The last theorem, closely related to Theorem 5, cannot be derived from Theorem 1, but follows directly from Lemma 6.

Definition 5. Y ∈ Z is said to be a properly efficient (or a Geoffrian portfolio) portfolio for (1) if Y is an efficient portfolio for (1) and there exists a real number M >0 such that for each i∈N we have

Eui(Yi)−Eui(Zi) Eu_j(Z_j)−Eu_j(Y_j) ≤M

for some j such thatEuj(Yj)> Euj(Zj) wheneverZ ∈ Z.

Lemma6. LetY be a properly efficient portfolio for(1). Then the system of inequalities

(2) αiEui(Zi) +ρX

j∈N

Euj(Zj)< αiEui(Yi) +ρX

j∈N

Euj(Yj), where α1, α2, . . . , αn∈R^∗₊,has no solutions in Z for some ρ >0.

Proof. SupposeY is a properly efficient portfolio for (1). Then Y also is a weakly efficient portfolio for (1). Hence the system of inequalities

Eu_i(Z_i)< Eu_i(Y_i), i∈N, Z∈ Z, has no solution. Let α1, α2, . . . , αn∈R^∗₊.Then the system

α_iEu_i(Z_i)< α_iEu_i(Y_i), i∈N, Z ∈ Z has no solution.

Suppose that system (2) is consistent. Let ˆZ ∈ Z, ˆZ 6=Y with α_iEu_i( ˆZ_i) +ρX

i

Eu_i( ˆZ_i)< α_iEu_i(Y_i) +ρX

i

Eu_i(Y_i).

We consider two cases.

Case1. P

j

Eu_j(Y_j)≤P

j

Eu_j( ˆZ_j).

If we compare the last two inequalities, we deduce that Eu_i( ˆZ_i) +ρX

j∈N

Eu_j( ˆZ_j)< Eu_i(Y_i) +ρX

j∈N

Eu_j(Y_j), i∈N has no solutions.

Case2. P

j

Euj(Yj)>P

j

Euj( ˆZj).

We have an efficient portfolio Y. Then there exists l, 1 ≤ l ≤ n, such that Eul(Yl)≤Eul( ˆZl) andEul( ˆZl)−Eul(Yl) = max

i∈N(Eui( ˆZi)−Eui(Yi)).Because Y is a properly efficient portfolio, there exists a numberM >0 such that for any i, 1≤i≤n, we have

Eui(Yi)> Eui( ˆZi),

Eu_i(Y_i)−Eu_i( ˆZ_i)≤M Eu_j( ˆZ_j)−Eu_j(Y_j)

for some j such that Eu_j( ˆZ_j)> Eu_j(Y_j).When Eu_i(Y_i)≥Eu_i( ˆZ_i), we have Eu_i(Y_i)−Eu_i( ˆZ_i)≤M Eu_l( ˆZ_l)−Eu_l(Y_l)

.Therefore, X0

Eu_i(Y_i)−Eu_i( ˆZ_i)

Eu_l( ˆZ_l)−Eu_l(Y_l)−1

≤M(k−1), whereP0

is taken over all isuch thatEui(Yi)−Eui( ˆZi)>0.Let 0< ρ≤ρ0, where ρ₀ =h

1≤i≤nmin α_i

M(k−1)i−1

.Then we have

ρ≤α_l

X0

Eu_i(Y_i)−Eu_i( ˆZ_i) Eu_l( ˆZ_l)−Eu_l(Y_l)−1−1

< α_l

Eu_l( ˆZ_l)−Eu_l(Y_l)

X

j

Euj(Yj)−Euj( ˆZj) −1

, i.e.,

ρX

j

Euj(Yj)−Euj( ˆZj)

< αl

Eul( ˆZl)−Eul(Yl)

and

α_lEu_l(Y_l) +ρX

j

Eu_j(Y_j)< α_lEu_l( ˆZ_l) +ρX

j

Eu_j( ˆZ_j).

This means that the system of inequalities (3) α_iEu_i( ˆZ_i) +ρX

j∈N

Eu_j( ˆZ_j)< α_iEu_i(Y_i) +ρX

j∈N

Eu_j(Y_j)

is inconsistent for 0< ρ≤h

1≤i≤nmin α_i

M(k−1)i−1

.

Theorem7. LetY be a properly efficient portfolio. ThenY is an optimal solution for the problem

(4) min

Z∈Zϕ(Z)

for some ρ >0, where ϕ(Z) = max

i∈N λ_in

Eu_i(Z_i)−y_i^∗ +ρP

j

Eu_j(Z_j)−y_j^∗o with real numbers y_i^∗ such that

λ_i=

Eu_i(Z_i)−y^∗_i

+ρX

j

Eu_j(Z_j)−y^∗_j −1

>0, i= 1, k.

Proof. For a properly efficient portfolioY and α₁ =α₂ =· · ·=α_n = 1, by Lemma 6, system (2) has no solution for someρ >0. Hence, by Theorem 1, for any negative numbers ε1, ε2, . . . , εn we have

max

i=1,n

λ_in

Eu_i(Z_i)−Eu_i(Y_i) +ρX

[(Eu_j(Z_j)−Eu_j(Y_j))−δ_i]o

≥1, for all Z ∈ Z,whereλ_i= (−δ_i)⁻¹, i= 1, n,and

δ_i=n

y_i^∗−Eu_i(Y_i) +ρX

Eu_j(Y_j)−y^∗_jo−1

, i= 1, n.

Hence ϕ(Z)≥1 for allZ ∈ Z.Since for anyi= 1, n λi

Eui(Yi)−y_i^∗

+ρX

j

Euj(Yj)−y_j^∗

= 1, we obtain

ϕ(Z)≥ϕ(Y)

for all Z ∈ Z, i.e., Y is an optimal solution for (4).

Theorem 8. If Y is a properly efficient portfolio for (1) then Y solves (4) for some ρ > 0, where λi = (Eui(Yi)−y^∗_i)⁻¹, i = 1, k, and y_i^∗ are real numbers such that λi>0, i= 1, n.

Proof. Let αi = (Eui(Yi)−y^∗_i)⁻¹, i = 1, n. Since systems (2) and (3) have no solutions and ˆZ is arbitrary we have ϕ⁰(Z)≥ϕ⁰(Y),∀Z ∈ Z, where

ϕ⁰(Z) = max

i=1,n

α_i(Eu_i(Z_i)−y_i^∗) +ρX

j

Eu_j(Z_j)−y_j^∗

, since

ϕ⁰(Y) = 1 +ρX

j

Eu_j(Y_j)−y_j^∗ . The theorem holds withλi=αi,∀i∈N.

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Received 12 May 2008 “Spiru Haret” University

Faculty of Mathematics and Computer Science [email protected]