EFFICIENT, WEAK EFFICIENT, AND PROPER EFFICIENT PORTFOLIOS MANUELA GHICA We give some necessary conditions for weak efficiency and proper efficiency portfolios for a reinsurance market in nonconvex optimization problems. AMS 2000 Subect 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 obectives 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 minimizes a linear combination of obective 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 see N = {1, 2,..., n} as a group of n reinsurers, 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. REPORTS 10(60), 4 (2008), 309 315
310 Manuela Ghica 2 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 Eu i (X) Eu i (Y ), where E stands for the mean operator. We assume monotonic preferences, and risk aversion, so that we have u i (w) > 0, u i (w) 0 for all w in the relevant domains [5]. In some cases we shall also require strict risk aversion, meaning strict concavity for some u i. For a better understanding we assume that each agent is endowed with a random variable payoff X i called 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 X i L 2 (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 n i=1 Z i n i=1 X i = X N then the allocation Z = (Z 1, Z 2,..., 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. Theorem 1. Let Z be the set of feasible allocations. Then one and only one of the following alternative holds: (a) Z i Z such that Eu i (Z i ) < 0, i N; (b) for any negative numbers δ 1, δ 2,..., δ n there exist positive numbers λ 1, λ 2,..., λ n such that maxλ i (Eu i (Z i ) δ i ) 1, Z Z. i Proof. Suppose (a) holds. Let Z Z such that Eu i (Z i ) < 0 for i N. Take δ i R, i N, with δ i max Eu i (Z i ). 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 one N such that Eu (Z ) 0. There fore, we have max Eu (Z ) 0.
3 Efficient, weak efficient, and proper efficient portfolios 311 max Let δ 1, δ 2,..., δ n be any negative numbers and λ i = (δ i ) 1, i N. Hence λ i Eu i (Z i ) 0. This inequality is equivalent to max [ λ i (Eu i (Z i ) δ i ) 1] 0, i.e., max λ i (Eu i (Z i ) δ i ) 1. Thus, (b) holds. Consider the vector problem (1) min Z Z (Eu 1(Z 1 ), Eu 2 (Z 2 ),..., 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. Let Z 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 Eu i (Z i ) Eu i (Y i ), 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 that Eu i (Z i ) < Eu i (Y i ), i N. Theorem 4. Let Y Z be a weakly efficient portfolio. Then it solves the problem min ϕ(z), Z Z where ϕ(z) = max λ i (Eu i (Z i ) yi ), with λ i = (Eu i (Y i ) yi i N ) 1, i N and yi is any real number such that λ i > 0 for every i N. Proof. Let Y Z be a weakly efficient portfolio. Then the system of inequalities Eu i (Z i ) < Eu i (Y i ), i N, Z Z has no solution. Now, by Theorem 1, for any negative numbers ε 1, ε 2,..., ε n we have maxλ i (Eu i (Z i ) Eu i (Y i ) δ i ) 1, Z Z, i where λ i = ( δ i ) 1, i N. Take δ 1, δ 2,..., δ n and y1, y 2,..., y n such that δ i = yi Eu i (Y i ) < 0, i N. Then λ i = (Eu i (Y i ) yi ) 1 > 0 and max λ i (Eu i (Z i ) yi ) 1 for every Z Z. i N
312 Manuela Ghica 4 We see that λ i (Eu i (Y i ) yi ) = 1 for any i N. Hence max λ i (Eu i (Z i ) yi ) max λ i (Eu i (Y i ) yi ), Z Z, i N i N i.e., Y solves the problem min max λ i (Eu i (Z i ) yi ) Z Z i N 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 Eu i (Y i ) Eu i (Z i ) Eu (Z ) Eu (Y ) M for some such that Eu (Y ) > Eu (Z ) whenever Z Z. Lemma 6. Let Y be a properly efficient portfolio for (1). Then the system of inequalities (2) α i Eu i (Z i ) + ρ N Eu (Z ) < α i Eu i (Y i ) + ρ N Eu (Y ), where α 1, α 2,..., α n R +, has no solutions in Z for some ρ > 0. Proof. Suppose Y 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 α 1, α 2,..., α n R +. Then the system α i Eu i (Z i ) < α i Eu i (Y i ), i N, Z Z has no solution. Suppose that system (2) is consistent. Let Ẑ Z, Ẑ Y with α i Eu i (Ẑi) + ρ i We consider two cases. Eu i (Ẑi) < α i Eu i (Y i ) + ρ i Eu i (Y i ).
5 Efficient, weak efficient, and proper efficient portfolios 313 Case 1. Eu (Y ) Eu (Ẑ). If we compare the last two inequalities, we deduce that Eu i (Ẑi) + ρ N Eu (Ẑ) < Eu i (Y i ) + ρ NEu (Y ), i N has no solutions. Case 2. Eu (Y ) > Eu (Ẑ). We have an efficient portfolio Y. Then there exists l, 1 l n, such that Eu l (Y l ) Eu l (Ẑl) and Eu l (Ẑl) Eu l (Y l ) = max (Eu i(ẑi) Eu i (Y i )). Because i N Y is a properly efficient portfolio, there exists a number M > 0 such that for any i, 1 i n, we have Eu i (Y i ) > Eu i (Ẑi), Eu i (Y i ) Eu i (Ẑi) M ( Eu (Ẑ) Eu (Y ) ) for some such that Eu (Ẑ) > Eu (Y ). When Eu i (Y i ) Eu i (Ẑi), we have Eu i (Y i ) Eu i (Ẑi) M ( Eu l (Ẑl) Eu l (Y l ) ). Therefore, ( Eui (Y i ) Eu i (Ẑi) )( Eu l (Ẑl) Eu l (Y l ) ) 1 M(k 1), where is taken over all i such that Eu i (Y i ) Eu i (Ẑi) > 0. Let 0 < ρ ρ 0, [( ) 1. where ρ 0 = min α i M(k 1)] Then we have 1 i n [ ) ( ) ] 1 1 ρ α l (Eu i (Y i ) Eu i (Ẑi) Eu l (Ẑl) Eu l (Y l ) i.e., ) < α l (Eu [ ( ) ] 1 l (Ẑl) Eu l (Y l ) Eu (Y ) Eu (Ẑ), ρ ( ) ( ) Eu (Y ) Eu (Ẑ) < α l Eu l (Ẑl) Eu l (Y l ) and α l Eu l (Y l ) + ρ Eu (Y ) < α l Eu l (Ẑl) + ρ Eu (Ẑ). This means that the system of inequalities (3) α i Eu i (Ẑi) + ρ Eu (Ẑ) < α i Eu i (Y i ) + ρ (Y ) N NEu [( ) 1. is inconsistent for 0 < ρ min α i M(k 1)] 1 i n
314 Manuela Ghica 6 Theorem 7. Let Y be a properly efficient portfolio. Then Y is an optimal solution for the problem (4) min Z Z ϕ(z) (Eui for some ρ > 0, where ϕ(z) = max i{ λ (Z i ) yi i N with real numbers yi such that { (Eui λ i = (Z i ) yi ) ( + ρ Eu (Z ) y ) } 1 ) ( + ρ Eu (Z ) y ) } > 0, i = 1, k. Proof. For a properly efficient portfolio Y and α 1 = α 2 = = α n = 1, by Lemma 6, system (2) has no solution for some ρ > 0. Hence, by Theorem 1, for any negative numbers ε 1, ε 2,..., ε n we have max λ i {Eu i (Z i ) Eu i (Y i ) + ρ [(Eu (Z ) Eu (Y )) δ i ] for all Z Z, where λ i = ( δ i ) 1, i = 1, n, and { δ i = yi Eu i (Y i ) + ρ ( Eu (Y ) y ) } 1, i = 1, n. Hence ϕ(z) 1 for all Z Z. Since for any i = 1, n { (Eui λ i (Y i ) yi ) ( + ρ Eu (Y ) y ) } = 1, we obtain ϕ(z) ϕ(y ) for all Z Z, i.e., Y is an optimal solution for (4). } 1, Theorem 8. If Y is a properly efficient portfolio for (1) then Y solves (4) for some ρ > 0, where λ i = (Eu i (Y i ) y i ) 1, i = 1, k, and y i are real numbers such that λ i > 0, i = 1, n. Proof. Let α i = (Eu i (Y i ) yi ) 1, i = 1, n. Since systems (2) and (3) have no solutions and Ẑ is arbitrary we have ϕ (Z) ϕ (Y ), Z Z, where [ ϕ (Z) = max α i (Eu i (Z i ) yi ) + ρ ( Eu (Z ) y ) ], since ( Eu (Y ) y ). ϕ (Y ) = 1 + ρ The theorem holds with λ i = α i, i N.
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