Optimal Asset Allocation and Technical Efficiency in ...

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Optimal Asset Allocation and Technical Efficiency in European Non-Life Insurance Companies

Abstract Most previews studies investigate optimal asset allocation that maximizes shareholders‟ expected utility. But, in addition to the maximization of shareholders' expected utility, the optimal assets allocation plays additional key roles to insure the insurance company's survival such as building necessary reserves to deal with claims payments. So, illiquidity, excessive risk and inadequate returns have a negative influence on shareholders‟ behavior as well as on customers‟ behavior. Furthermore, the mismanagement of asset allocation by an insurance company will increase excessively its probability of insolvency, this will push the decision maker to increase premium rates and as a consequence the company will lose its competitiveness in the market. This study is original; it intended to investigate the optimal asset allocation of European non-life insurance companies that maximizes technical efficiency. The definition of our objective function is based on the directional output distance function. Two metaheuritics has been used PSO and GA. Achieved results show that proportion allocated to the “alternative investment with high-risk highreturn” is in average lower then these founded in previous studies. However, the percentage allocated to the “risk-free assets” is in average different from zero. Thus, the obtained investment portfolio is relatively more diversified between available assets compared to this proposed in previous studies. This can be explained by the implicit paying attention for competitiveness, survival and long term profitability when one maximizes technical efficiency. So, any insurance company has to pay more attentions to the presence of different stakeholders and resolves the conflicts of interest between different stakeholders. Keywords: Technical efficiency, survival, Asset allocation, PSO, GA, Directional output distance function, Non-life insurance companies. JEL codes: C63; C67; G11; G22; L21; L23

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1. Introduction Usually, researchers rely on operational research methods to investigate available observed information of firms or stocks, and supporting managers in making optimal financial decisions. There are many research works use operational research methods to study optimal asset allocation in insurance companies (Yu et al (2010)). When, the insurance company allocates optimally its assets, this enables it to improve both its solvency and competitiveness. Most previews studies investigate optimal asset allocation that maximizes shareholders‟ expected utility. But, in addition to the maximization of shareholders' expected utility, the optimal assets allocation plays additional key roles to insure the insurance company's survival such as building necessary reserves to deal with claims payments. So, illiquidity, excessive risk or inadequate returns all of them have a negative influence on shareholders‟ behavior as well as on customers‟ behavior. Furthermore, the mismanagement of asset allocation by an insurance company will increase excessively its probability of insolvency, this will push the decision maker to increase premium rates and as a consequence the company will lose its competitiveness in the market. Usually shareholders and other stakeholders of a same company have conflicted objectives and perspectives. Shareholders have interest to the short-term profits and the officer takes all responsibilities in bankruptcy or insolvency cases. However other stakeholders, especially customers, have interest to competitiveness, productivity and efficiency of the insurance company because these insure its solvency, its survival and long-term returns. So in addition to the maximization of shareholders' expected utility, the insurance company has to ensure its competitiveness essentially by maximizing productivity. With more precision an insurance company should maximize its technical efficiency by producing the maximum desirable outputs with the minimum quantities of inputs and undesirable outputs. The main contribution of this paper is in seeking optimal asset allocation while maximizing technical efficiency, while referring to the production process of insurance companies, instead of maximizing shareholders' expected utility. So in this paper, we propose a methodology to investigate the optimal asset allocation of European non-life insurance companies based on technical efficiency. First we simulate assets in which the insurer can invest. Second we specify the objective function that must be used to maximize technical efficiency. So, we use the general functional form of the directional output distance function proposed by Färe et al. 2

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(2005). This function allows a complete characterization of the production process (Färe et al. (2005); Hachicha and Jarraya (2010); Jarraya and Bouri (2013) and Jarraya (2014)). Then, to search the optimal asset allocation, that optimizes our objective we will use two Metaheuristics namely: Genetic algorithm and Particle Swarm optimization algorithm. The first one is classified among the oldest and most known algorithms. By cons the second algorithm is one of the newest techniques within the family of evolutionary optimization algorithms. We apply our model on a sample of 175 non-life insurance companies dispersed in nine European countries over the period 2002-2008. Our research is structured as follows. In the next section we survey some related works focused on asset allocation issues in insurance companies. Section 3 illustrates our methodology. Section 4 describes dataset, variable definitions, simulated assets and empirical results. Finally, section 5 closes the chapter with some concluding remarks.

1. Literature Review 1.1.

Asset Allocation in Insurance Industry

Asset allocation represents an important and delicate issue in insurance industry. Most previous studies focused on the insurance industry designate available funds for investment by capital or surplus. In this paper we investigate the optimal investment of this surplus in European non-life insurance companies. Mayers and Read (2001) suggest that a highest level of surplus represents a guarantee for policyholders. But, on the other hand, these authors emphasized the high costs related to this surplus, as well as, the competitive premiums are influenced by total surplus requirements and their allocation. So, in a competitive market the mismanagement of asset allocation will increase excessively the probability of insolvency, this will push the insurer to increase premium rates and as a consequence the insurance company will lose its competitiveness position in the market. The asset allocation issue in insurance companies is treated in the literature by two principal approaches. The appearance of mean-variance analysis by Markovitz (1952) represents the cornerstone of this first approach followed by many literature extensions. This approach aims principally to generate an efficient frontier in a risk-return space. This frontier is constructed by most efficient portfolios in which investors have to invest. This approach has provided a birth of extension for previous studies interested by the liability side of financial institutions 3

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(Chiu and Li 2006; Craft, 2005, Sharpe and Tint 1990). Brennan et al. (1997) have noting that a miss-definition of utility function represents the most striking drawback of this approach. In the other hand Merton (1971) represents the pioneer of the second approach of building an optimal portfolio of investment. The literature extension of this approach considers the asset allocation problem as stochastic and the solutions are illustrated by Hamilton-Jacobi-Bellman partial differential equations (Yu et al. (2010)). Mayers and Read (2001) have stated that surplus returns known a significant magnitude in the net income of the insurance company, in addition a substantial change have influenced the financial investment regulation. As a consequence of these presented changes the authors have recommended the reinvestigation of the asset allocation issue in insurance industry to provide more appropriate solutions for the decision makers. The most recent research work which has investigated with an explicit and simple manner the asset allocation issue in non-life insurance industry is that of Yu et al. (2010). These authors use a simulation model for building assets' prices data. In this work a new evolutionary algorithm has been developed allowing seeking the optimal asset in a dynamic environment while maximizing shareholders' expected utility. Noting also that Kahane and Nye (1975) and Cummins et al. (1981) are the pioneers in the issue of asset allocation in insurance companies and they have investigated this problem for a single period. In the other hand, Browne (1995) is the leading of exploring this issue of asset allocation in a dynamic framework. He has shown in its study an optimal investment strategy must involve a fixed amount invested in risky asset whatever the surplus amount. By against the results of Hipp and Plum (2000) show that optimal amount invested in risky assets should be based on the current surplus. The model of these authors is extended by the paper of Liu and Yang (2004) by taking account of risk-free asset. Mayers and Read (2001) has proposed the model characterization as a principal reason for divergence of results between Hipp and Plum (2000) and Browne (1995). The maximization of shareholders‟ expected utility is the sole objective that has attracted the most of previous research works focused on asset allocation issue in insurance industry. For more information on asset allocation and portfolio optimization problems with Metaheuristics one can refers to Jarraya (2013). As shareholders are generally risk-averse, the focus on utility maximization can be a destruction source of other objectives such as productivity, competitiveness and solvency. So we shouldn't limit our researches to this objective providing only the shareholders satisfaction but we must investigate the asset allocation issue while 4

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taking account of objective of other stakeholders. In this paper we will try to seek the optimal asset allocation allowing the maximization of insurance companies‟ efficiency. Like previous research works in our study will be based on metaheuristic optimization algorithms to find the optimum asset allocation allowing maximization of efficiency for non-life insurance companies. The next paragraph is a theoretical survey of metaheuristic optimization techniques.

1.2.

Metaheuristics Optimization Techniques

Solving optimization problems became a central topic in several research areas. Making decision problems that can be formalized as an optimization problem is growing rapidly. There are many deterministic methods which make possible to resolve some types of optimization problems in a finished time period. Among the most known methods: the linear programming Schrijver (1998), the quadratic programming and the Newton method Nocedal and Al. (1999), the dynamic programming Bertsekas (2000), the Simplex method Nelder and Al. (1965) and the gradient method Avriel (2003). In practice the posed problems are more complex and require too much time to be resolved by these deterministic methods. So, researchers usually resort to the stochastic optimization algorithms, such as the Metaheuristics, which make it possible to find an approached solution, in a reasonable time. Metaheuristics are a family of stochastic algorithms intended for the resolution of the optimization problems. Their key advantage is the adaptability to a great number of problems without significant changes in their algorithms. Their capacity to optimize a problem with minimum information is weakened, because these algorithms cannot offer any guarantee for the optimality of the best found solution. However, from experts' point of view, that cannot be an inevitable disadvantage, since, in practice, it is always preferable to found quickly an approximate optimum that an exact value found in too much of time. Meta-heuristics have usually an iterative behavior. The same pattern is repeated during the optimization until a stopping criterion, specified at the beginning, is met.

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The most known Metaheuristic optimization techniques can be divided into two most used approaches: Evolutionary algorithms and Algorithms based on swarm intelligence. Fraser (1957) is the pioneer of evolutionary algorithms. They represent a family of research algorithms inspired from species' biological evolution, such as: natural selection, mutation, reproduction, and recombination. An evolutionary algorithm has usually three key operators: 5

selection operator, crossover operator and mutation operator. Using these operators in the evolutionary algorithm has favored the emergence of four different approaches Bäck et al. (1997). First the Evolutionary Strategy (ES) represents a family of Metaheuristic optimization, inspired from evolution theory. This model was originally proposed by Rechenberg (1965), it is the first genuine Metaheuristic and the first evolutionary algorithm. Second the Evolutionary programming (EP) introduced by Fogel et al. (1966) and aims to create a Finite-state machine by successions of crossover and mutations. This Metaheuristic serves to predict future events based on previous observations. Noting that, Finite-state machine is a fundamental tool in computer programs. Third the Genetic Algorithm (GA) stochastic search techniques and theoretical foundations were established by Holland (1975). They are inspired from Darwin theory: the natural evolution of living species. There are two mechanisms allowing evolve living species: natural selection and reproduction. Natural selection favors the most adapted population individuals to their environment. The selection is followed by reproduction, performed by crossovers and mutations within individuals' genes. Thus, two parents intersect and transmit some of their genetic heritage to their offspring. In addition some individuals' genes could mutate during the reproductive phase. The combination of these two mechanisms leads to a more adapted population to its environment. Genetic algorithms were conceived as an optimization method to resolve problems with discrete and continuous variables (Holland (1973); Goldberg (1989); Holland (1992)). In their canonical version, Genetic Algorithms suffer most often of slow convergence or premature problems. To overcome these drawbacks several improvements have been proposed such as: organic operators, elitist strategy, etc. Michalewicz (1996). Finally the Genetic Programming (GP) that represents an alternative of genetic algorithms, designed to manipulate programs and implement an automatic learning model Koza (1992). The programs are usually coded by trees that can be viewed as bit strings of variable length. Most techniques and results of genetic algorithms can also be applied to genetic programming. The second approach of Metaheuristics and the most recent is that of Algorithms based on swarm intelligence. Collective intelligence refers to communities' cognitive abilities resulting from multiple interactions between community members also called agents. From a simple behavior, agents could perform complex tasks owing a fundamental mechanism known synergy. Under particular conditions, created synergy through collaboration between individuals emerges some opportunities of representation, creation and learning better than 6

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isolated individuals. The collective intelligence forms are various according to community types and members that met. Collective systems are more or less sophisticated. Noting that, human societies do not obey to rules as mechanical like other natural systems, such as animal world. Collective intelligence is seen principally in social insects (ants, bees...) and animals in movement (migrating birds, fish schools). Therefore, several algorithms based on collective intelligence phenomenon have been introduced, such as ant colonies and particle swarm (Hoffmeyer (1994); Ramos et al. (2005); Nickabadi et al. (2011); Wang et al (2011); Soares et al. (2012); Wang et al. (2013); Liu et al. (2013); …). Ant colony algorithms are born from a simple observation. Insects, particularly ants, solve naturally complex problems. The principal factor that facilitates this behavior is that ants communicate with each other indirectly owing to deposit of chemicals substances called pheromones. This indirect communication type is called stigmergy. According Goss et al. (1989) if an obstacle is introduced in ants path, they will all tend, after a research phase, to follow the shortest way between nest and obstacle. They are more attracted to the area where the pheromone substance rate is highest. Ants that passed by food source and arrived too quickly to the nest are those that have taken the shorter way. Therefore, the pheromone quantity in this way will be more important than the longer distance. Thereby, eventually the shortest way has a greater probability to be used by all ants than other ways. First algorithms inspired from this analogy were proposed by Colorni et al. (1992) and Dorigo et al. (1996) to resolve problem of business traveler. In these algorithms each solution is considered as an ant moving in the search space. Ants mark better solutions and take account of previous markings to optimize their research. Ant colony algorithms use an implicit probability distribution to perform the transition between iterations. In their adapted version to combinatorial problems these algorithms use an iterative construction of solutions. Particulate Swarm Optimization is a Metaheuristic proposed by Kennedy et al. (1995). This method inspired from animals social behavior in their moving in swarms. The most used example is the behavior of fish school (Wilson (1975) and ReynoldsBilel (1987)). Indeed, these JARRAYA animals are characterized by a movement dynamics relatively complex, while individually each one has a limited intelligence and local knowledge focused on its position in the swarm. Thereby, each individual has knowledge only of the position and speed of its nearest 7

neighbors. It therefore uses not only its own memory, but also local information of nearest neighbors to decide its own movement. Simple rules, such as "go in same speed as others", "moving in the same direction" or "stay close neighbors" are among key behaviors that maintain cohesion of the swarm and allow the implementation of complex and adaptive collectives behaviors. The swarm's global intelligence is a direct consequence of local interactions between different particles. Therefore, system performance as whole is greater than the performance sum of its different parts. The particle swarm optimization is inspired from these socio-psychological behaviors by Kennedy et al. (1995). Potential solutions of the optimization problem are presented by particles swarm that flies over search space to seek the global optimum. Particle‟s movement Bilel JARRAYA is influenced by three components:  Physical component: the particle follows the current movement direction.  Cognitive component: the particle moves towards the best place by which it has already past.  Social component: the particle relies on congener‟s experience, and thus move towards the best already reached area by its neighbors. A point's quality in the search space is determined by the objective function value at this point. It is possible in some cases a particle moves out of the search space. In these cases it could have a positive feedback amplification which leads to a system divergence. To overcome this problem, Eberhart et al. (1996) propose to introduce the maximum velocity as a new parameter, which allows controlling the system's explosion. A study of the PSO behavior according the maximum velocity is available in Fan et al. (2001). In addition, a particle confinement strategy can be established. This strategy can bring back a particle moved away from the search space to inside of it. Then, several methods can be employed:  The particle is left outside the search space without assess of its objective function. Thus, it can't attract other particles outside the search space.  The particle is stopped at the border and the corresponding components of velocity are canceled.  The particle bounces on the border, but the corresponding velocity components are multiplied by a random coefficient fired from  1,0 . 8

In the next section we develop our methodology.

2. Model This section includes three parts. In the first part, is allocated to present our model design. In the second part, based on the directional output distance function, we set up the objective function which will be optimized. The last part is conserved to present the used Metaheuristic algorithms to optimize this objective function and providing the optimal asset allocation that maximize efficiency of the European non-life insurance companies.

2.1.

Model Design

Our modeling is divided into two principal steps. First step we specify the technology frontier T by using the directional output distance function. So based on the stochastic estimation

method we assess the parameters of the technology frontier, the inefficiency scores and the error term. In the second step, we specify the objective function from which we investigate the optimal asset allocation that minimizes technical inefficiency. So, we express technology frontier just function of available funds for investment and other variables will be replaced by their observed values. In the following figure we will present a simplified illustration our model in a two axes plan. The abscissa axe represents the undesirable output and the ordinate Bilel JARRAYA

axe symbolizes the desirable output. [Insert Figure 1]

In this figure A represents an observed insurance company with the coordinate (bA , YA). Were YA and bA are the observed quantities of desirable and undesirable outputs, respectively, of this company. μ is the inefficiency score attributed to this company regarding to the technology frontier T. After an investigation of the optimal asset allocation that maximizes technical efficiency, based on the available funds for investment, the insurance company will remove from point A to the point A’. In this new point the insurance company keeps the same quantity of undesirable output (as well as for input) and only the quantity of desirable output will be enhanced. So the new place A’ will be closer to the technology frontier and obviously the insurance company will be technically more efficient μ> μ’.

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2.2.

Technology frontier

To model the insurance companies‟ production process and measure their technical efficiency we use the directional output distance function. We assume there are k  1,2,..., K insurance companies which use a vector of inputs x  ( x1 , x2 ...xn )  desirable

outputs y  ( y1 , y2 ... ym )  M and

a



vector

N



of

to produce a vector of undesirable

outputs

b  (b1 , b2 ...bh )   . The production possibility set under a technology (T ) is defined as H

the set of all feasible inputs, desirable outputs and undesirable outputs vectors:



T  ( x, y, b) : x   , y   , b   , x can produce( y, b) N

M

H

(1)

the directional output distance function allows a complete characterization of above defined technology (1) in presence of undesirable outputs. This function seeks simultaneously the maximum technologically feasible expansion of desirable outputs and the maximum possible contraction of both inputs and undesirable outputs. Let g  ( g x , g y , g b ) , g x 



N



,

g y   and g b   denote the directional vector that define these possible expansions M

H

and contractions. The directional output distance function can be defined as follows:  D( x, y, b; g x , g y , g b )  max : ( x   g x , y  g y , b  g b )  T 

(2)

We assume that g  ( g x , g y , g b )  (1,1,1) which provides the same weight to inputs, desirable outputs and undesirable outputs. The directional output distance function takes zero value for efficient company insurance. Otherwise this function will be strictly greater than zero and in this case the insurance company is classified inefficient. Following Chambers et al. (1996, 1998) the directional output distance function should satisfy many properties. The well known Bilel JARRAYA

important property is the translation property expressed as follows:   D( x, y, b; g x , g y , g b )    D( x  g x , y  g y , b  g b ; g x , g y , g b )

 

(4)

The development of our model will be based on a quadratic functional form of the directional output distance function. As a reference we will use Färe et al. (2005) to parameterize this function:

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N M N N M M  D( x, y, b; g x , g y , g b , t , )   0    n xn    m y m  1 / 2  nn' xn xn '  1 / 2  mm ' y m y m ' n 1

N

M

H

N

m 1

H

n 1 n '1

M

H

m 1 m '1

H

H

   mn y m xn   h bh    nh xn bh    mh y m bh  1 / 2 hh'bh bh ' n 1 m 1

h 1

n 1 h 1

m 1 h 1

N

M

H

n 1

m 1

h 1

h 1 h '1

  1t  1 / 2 2 t 2   n tx n   m ty m    h tbh





(5)

Symmetric restrictions  nn'   n'n

n  n'

 mm'   m'm

m  m'

 hh'   h 'h

h  h'

(6)

Mathematical constraint defined from the translation property N

H

M

       n 1

n

h 1

h

m1

m

M

N

H

m1

n 1

h 1

M

N

H

m1

n 1

h 1

M

N

H

m1

n 1

h 1

1

  mn    nn'    nh  0   mm'   mn mh  0 mh    nh   hh'  0 M

N

H

     m 1

m

n 1

n

h 1

h

 n  1,...., N  m  1,...., M

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 h  1,...., H

0

(7)

Where   ( ,  ,  ,  ,  ,  , ,  , , ,  ) is the parameters vector which should be calculated. These parameters can be computed via either linear programming (LP) proposed by Aigner and Chu‟s (1968) or stochastic frontier techniques suggested by Aigner, Lovell and Schmidt (1977) and Meeusen and Vanden-Broek (1977).

The last approach has some

advantages over the first (LP). The most known advantages is the suitable processing of stochastic shocks and random error, in addition the econometric approach allows testing several statistical assumptions. In our model this stochastic approach can be specified as follows:

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 Dtk ( xtk , Stk , btk ; g x , g y , g b , t , )   tk  0

(8) iid

Noting that  tk  tk  tk where the error term is expressed as follows tk  N (0, 2 ) and the iid

one sided error term tk  N (0, 2 ) expresses the technical inefficiency. In a second step we specify the objective function from which we seek the optimal asset allocation allowing the minimization of technical inefficiency. So we express estimated k

directional output distance function just with surplus S t , and the other variables will be replaced by their estimated or observed (parameters or variables respectively) values:

  Dtk ( xtk , Stk , btk , t ,ˆ)  Dtk (Stk ,ˆ) (9) Finally we have to search the optimal asset allocation that minimizes the average of technical inefficiency for our sample of European non-life insurance companies, expressed as follows:

1 K 1 T k k ˆ   Dt (St , ) K k 1 T t 1 L

5

l 1

j 2

S kt   P1lt Z1lt   Pjt Z jt

(10) s.t.

l  1,...,15 and 1  L  15

Where Z jt and Pjt represent the quantities and unit‟s price, respectively, of asset j at time t . On the other hand Z 1lt and P1lt represent the quantities and unit price, respectively, of a bond with l year maturity at time t .

2.3.

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Optimal Asset Allocation

To find the optimal asset allocation while maximizing technical efficiency we need to use a metaheuristic optimization algorithm. In this work we use two algorithms PSO and GA to seek the optimal asset allocation for European nonlife insurance companies. The first algorithm it's among Algorithms based on swarm intelligence, it's newly developed and judged the most efficient in its results. The second algorithm forms part of evolutionary algorithms, it's ranked among the oldest algorithms and it's the most famous in the field of Metaheuristics. However, before starting the description of processes linked to each one of these two algorithms we have to present the simulated assets in which insurance companies can invest.

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2.3.1. Assets Definitions Like Yu et al. (2010) we propose five asset types in which each insurer can invest, such as stock index, three alternative investments and bonds knowing that maturity years, of this last asset, range from one to fifteen years. Noting dW the differential vector of five wiener processes:



dW  dWr dWs dWAI hh dWAI ll dWAI hl



'

(11)

Where dWr represents the process of one year spot rate (r), dWs the process of the stock

dWAI ll and dWAI hl the alternative investment (AI ) processes

index return, dWAI hh ,

characterized respectively by high-risk high-return (hh) , low-risk low-return (ll ) and highrisk low-return (hl ) . We use the correlation matrix

R proposed by Yu et al. (2010) to estimate

these wiener processes (see Appendix 1).

2.3.1.1. Bond markets To simulate one-year spot interest rate we use the model proposed by Cox et al. (1985) also called CIR process. This model assumes that the instantaneous interest rate follows the stochastic differential equation:

drt  a(b  r )dt   r r dWt (12) Where a(b  r ) is a drift factor that ensures the mean reversion of the spot interest rate, a is a positive parameter that represents the speed of adjustment, b denotes the long term level of the spot interest rate and  r r represents the standard deviation factor. According to Cox et al. (1985) we obtain the bond prices at time t as follows:

P(r, t , T )  A(t , T )e rB(t ,T )

(13)

Where 2 ab

A(t , T ) 

B(t , T ) 

(T t )( b  )   2  ( T t ) 2 2  e ( b   )( e  1 )  2     





(14)





(15)

2(e (T t )  1) (b   )(e (T t )  1)  2

  b²  2 ²

(16)

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2.3.1.2. The stock index Like Yu et al. (2010) we suppose that the stock index behavior is a continuous time stochastic process. This process follows an interest rate adjusted geometric Brownian motion: dS  (r  rs )dt   S dWt (17) S Where dS is the change in the market index‟ prices, rs represents the risk premium and  S

denotes the index price volatility.

2.3.1.3. Alternative investment markets The price process of the three proposed alternative investments supposed to be geometric Brownian motions like Yu et al. (2010):

dAI ij   AI ij dt   AI ij dWt AI ij

i   l,h  and j   l,h 

(18)

Where  AI ij and  AI ij denote, respectively, expected return and standard deviation of the alternative investment prices characterized by a risk i and a return j . 2.3.2. Particle Swarm Optimization Algorithm Kennedy and Eberhart (1995) develop a parallel evolutionary algorithm based on the social behavior metaphor called particle swarm optimization (PSO). Kennedy et al., (2001) Represents a reference exploring social and computational paradigms of PSO. Like evolutionary algorithms an initial population is defined for the PSO algorithm. This population is composed by a set of random candidate solutions called particles. A defined number of particles are considered as a population P(t ) at the generation t . Each particle i characterized by a position yti and a velocity vti . In each generation t all particles are evaluated and the non-dominated particles are archived in the external repository, while dominated particles are removed. In the next step, the position and the velocity of each particle are updated by equations similar to the following: vti1  w.vti  c1.r1.( Pt i  yti )  c2 .r2 .( Rth  yti )

y

i t 1

(19)

 y v i t

i t 1

(20)

Where w called inertia coefficient that controls previous history impact of velocities, it can 14

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taken large or small value for the global or local search respectively (Mendes et al. (2004), Elbeltagi et al. (2005) and Jung and Karney (2006)). The two constants c1 and c2 are positive, r1 and r2 are uniform random values in the range 0, 1 . Rth is a particle solution chosen from the repository in each iteration t and guides the movement of particles toward optimum. Pt i represents the most excellent position vector of particle i , in a first time is set equal to the initial position corresponding to this particle. The iterative process of the PSO technique can be resumed in a five steps as follows: 1- An initial population of particles is generated. For each particle is attributed a position y 0i and a velocity v0i . The current position of each one is recorded as P0i . The best value

of P0i is chosen as R0h that will be saved. 2- Then new positions will be generated for particles using the two equations (19) and (20) presented above. 3- The objective function is evaluated for each new particle‟s position. The objective function is evaluated for each newest position of particle i. if this particle has achieved a better position its previews Pt i value will be replaced by the current value yti 1 . 4- Similarly to the first step the Rth value must be updated from the new stored values of Pt i1 . So if Rth  Rth1 nothing will change, otherwise the Rth will be replaced by the new

value Rth1 . 5- The steps 2, 3 and 4 are repeated until the number of iterations reaches a predetermined value. 2.3.3. Genetic Algorithm The concept of „survival of the fittest‟, developed by Darwin (Davis, 1991), represents the corner stone of genetic algorithms. These algorithms are inspired from the natural law of evolution. The key idea of this law is that species most suited to their environment and which have the best fitness, will dominate the world. According to Maturana and Nauann (1977) and Goldberg (1989) genetic algorithms inspired from biological principles allow the fitness improvement of a chromosomes population and providing the set of solutions linked to an optimization problem. For an appropriate use of genetic algorithms, Chang (1998) suggest the transformation of any optimization problem into parameters appropriated to this problem. The 15

optimization goal is to find the most appropriate combination of these parameters to achieve the optimality (minimum or maximum). The iterative steps of the genetic algorithm can be resumed in the following steps: 1- Generate an initial population of solutions. The elements building these solutions represent the decision variables. These latter are symbolized by the proportions of total investments allocated to five types of asset. The number of our decision variables is nineteen (since maturity years of bonds range from one to fifteen years). 2- In this selection step a proportion of individuals are extracted from the last generated population. Indeed, a fitness evaluation, based on the defined objective function, is applied on each individual in the existing population. The best solutions have more chance to be selected. 3- In this step a set of genetic operators (crossover, mutation, regrouping, colonizationextinction and migration) are applied on the selected individuals to generate a new population. The most known and used operators are crossover and mutation. Each new off-spring is produced by at least a pair of "parent" solutions selected from extracted individuals in the previous step. This new produced child share many characteristics like its parent. This process continues until the appropriate population size is achieved. 4- Checking the termination criterion is reached. If this condition is satisfied the generation process will be stopped, otherwise the steps two and three will be repeated until the termination condition is satisfied. Bilel JARRAYA

3. Empirical Implications 3.1.

Dataset

To validate empirically our model we use a sample of 175non-life insurance companies dispersed in nine European countries (United-Kingdom, Sweden, Denmark, Belgium, Netherlands, French, Spain, Italy and Germany). In constructing our sample, we first use a list and information of European non-life insurance companies published in the “Thomson ONE Banker” database. Then we establish needed information of our sample referring to annual reports published by non-life insurance companies in their official websites.

16

3.2.

Inputs-Outputs Definitions

The definition of inputs and outputs is a key step in efficiency investigation. A poor definition of these quantities may lead to misleading or meaningless results. This problem is particularly keen in the service sector, where most outputs are intangible and some basic input's data are not publicly available, such as the employees' number or the worked hours. In this part we specify the followed approach to define inputs and outputs in the European non-life insurance Bilel JARRAYA

companies.

Let beginning by the definition of inputs. There is a well-known agreement about the definitions of these variables in most previous studies focused on efficiency issue in the insurance industry (Cummins et al., 2004). So we will use four inputs namely: operating k

k

k

expenses ( x1t ) , equity capital and reserves ( x2t ) , financial debt capital ( x3t ) and technical k

provisions ( x4t ) as inputs. Where the Net operating expenses are used as a proxy to business services, labor force, used materials and products distribution. The Equity capital and reserves involve any balance sheet item linked with shareholders‟ capital or reserves (minority interests, participating rights capital…). The Debt capital illustrates all funds borrowed from creditors. Finally, the technical provisions take account of both loss and unearned premium reserves. In the other hand, previous studies have been labeled by an acute divergence concerning the suitable definition of outputs. There are two main approaches to specify outputs in insurance industry: the intermediation or asset approach and the value added approach also called production approach. In the first one, the insurance company is considered as a pure financial intermediary. In a first step it receives funds from policyholders, paying out claims and all other additional loads. Then the remaining funds are invested in capital markets. Cummins and Weiss (2000) affirm that this approach is inappropriate to specify outputs for insurance industry and especially for non-life insurance companies. The second approach is considered, by many authors as the most suitable to specify outputs for studying efficiency issue in insurance industry (Berger et al., 1999; Cummins and Weiss, 2000). This approach considers

output as significant if it contributes a substantial added value, as judged using operating cost allocations (Berger et al., 2000). In this approach we distinguish between three principal services provided by non-life insurance companies: risk17

pooling and risk-bearing, financial services and intermediation. The total investment is considered as the best proxy for intermediation function. While the best proxy for the two remaining services, is the net incurred claims plus additions to reserves (Eling and Luhnen (2009)). In practice, insurance companies search to minimize claims. However, when considering net incurred claims plus additions to reserves as output, this latter must be maximized in efficiency analysis. This paradox represents the main drawback of this approach. To resolve this problem, in this chapter, we use the directional output distance function technique. So this output will be considered as undesirable output and will be minimized. Thus, in our efficiency k analysis, the total investment (surplus) express the desirable output noted ( S t ) and net k incurred claims plus additions to reserves represents undesirable output noted (bt ) .

Table 1 presents descriptive statistics, the mean and the standard deviation, of each above defined variable by country over the period 2002-2008. [Insert Table 1]

3.3.

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Results and interpretations

This subsection illustrates our principal achieved results and their interpretations. It contains three parts. 3.3.1. Frontier Estimation As previously mentioned, we use the maximum likelihood technique for the estimation of the directional output distance function, and the one-sided error term is supposed to be independently and identically distributed. Table 2 presents estimated parameters of the technology frontier. The result of the one-sided generalized likelihood ratio test is LR=987. From an econometric point of view this result indicates that the model is statistically significant. In other hand from an economic point of view this result confirms the importance of technical inefficiency effects. From the same table we show that first order estimated coefficients of inputs and outputs have the expected values regarding economic behavior. The estimated parameters of the stochastic directional output distance function validate that most of the maximum likelihood coefficients are statistically significant. The majority of variables are significant at the 1 to 10% levels. Using the estimated parameters in Table 2, we verify 18

that resulting directional output distance function ensures the convexity conditions on inputs and concavity conditions on outputs for most observations. [Insert Table 2] 3.3.2. Inefficiency scores before asset allocation Country-specific technical inefficiency is estimated for each country in each year over the period 2002-2008. From table 3, we show that inefficiency scores of European non-life insurance companies are on average waiting of 43.02%. This implies that insurance companies operating around the net-puts average values have the potential to increase total investment (desirable output), and simultaneously decrease the quantities of net incurred claims plus additions to reserves (undesirable output) and inputs by about 43%. The most efficient non-life insurance system is that of Belgium with an average score inefficiency of 22.21%. However, the most inefficient non-life insurance system is that of Sweden with an average score inefficiency of 51.63%. So the non-life insurance system of Belgium and Sweden have to increase total investment and simultaneously decrease the net incurred claims plus addition to reserves and all inputs by about 22.21% and 51.63%, respectively. The European non-life insurance companies are invited to minimize incurred claims by revising offered premium rates and ensure a severely records control of insured customers so as to minimize asymmetric information. Also they are invited to put more funds for investment and seek the optimal manner to allocate them. [Insert Table 3]

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3.3.3. Inefficiency scores after asset reallocation Simulated assets have been used for seeking the optimal assets allocation allowing the maximization of technical efficiency for European non-life insurance industry. The simulation process is based on the correlations between these assets like in the work of Yu et al. (2010). Achieved results by Genetic algorithm and Particle Swarm Optimization algorithm are presented in the table 4. This table shows that the obtained results during the period 20022008 are in average quite similar for both algorithms. For 2002, 2005 and 2006 the both algorithms have reached the same optimal solution with negligible or inexistent differences. The maximum of technical efficiency can be reached by European non life insurance companies is 91.5%, 89.6% and 91.1%, respectively. Therefore, highly likely these points represent relative global optima for the experimented scatter plot. By cons there is a slight 19

difference between results obtained by these two algorithms for the remaining years (2003, 2004, 2007 and 2008). So, the GA reaches, for those years, a local optimum very close to the optimum achieved by the PSO algorithm. Maximum efficiency levels achieved by the latter algorithm are 93.5%, 93.2%, 86.4% and 85.8% respectively to aforesaid years. However efficiency levels achieved by the GA are 92.1%, 91.8%, 84.9% and 86%, respectively. Table 4 shows that the optimal asset allocation allowing the maximization of technical efficiency for European non-life insurance companies. First, the PSO algorithm provides the following average percentages of assets allocation for the period 2002-2008: 55.6% of surplus in stock index, 13% in alternative investment with high-risk high-return, 11.3% in alternative investment with low-risk low-return, 13.4% in alternative investment with high-risk lowreturn and 5.8% in risk-free assets. In other hand, the average percentages provided by the GA for the same period are as follows: 57.2% of surplus in stock index, 16.1% in alternative investment with high-risk high-return, 9.1% in alternative investment with low-risk lowreturn, 13.3% in alternative investment with high-risk low-return and 4.3% in risk-free assets. So, the PSO algorithm offers an average of 32% as amelioration of technical efficiency after assets reallocation. By cons, the GA achieves only 31.4% of amelioration on technical efficiency. This slight difference is explained above, since the GA is stopped for some years Bilel JARRAYA

in a relatively local optimum.

Comparing achieved results with those obtained by previous research works especially this of Yu et al (2010). This paper seeks the best asset allocation allowing maximization of shareholders‟ expected utility function while using a simulation approach like in our model. The strike difference between this study and our model is the objective function since we maximize technical efficiency of non-life insurance companies. In average Yu et al. (2010) show that surplus of non life insurance companies must be allocated as follows: 39% of surplus in stock index, 34.6% in alternative investment with high-risk high-return, 8.2% in alternative investment with low-risk low-return, 18.2% in alternative investment with highrisk low-return and 0% in risk-free assets. From a financial perspective, managers are the first responsible of losses or bankruptcy events by cons shareholders seek short-term return. For this reason we find a relatively great part of investment is allocated into “alternative investment with high-risk high-return” and zero percent for “risk-free assets” when we are interested by the shareholders utility function. In other hand, while maximizing technical efficiency the average percentage founded for “alternative investment with high-risk high20

return” is 13% by the PSO algorithm and 5.8% for “risk-free assets”. So, the investment portfolio is relatively more diversified between available assets. This can be explained by the implicit paying attention for competitiveness, survival and long term profitability when one maximizes technical efficiency. [Insert Table 4]

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4. Conclusion Optimal asset allocation is a key issue for any insurance company and it has a great influence on its solvency and competitiveness. Most previous studies investigate optimal asset allocation that maximizes shareholders‟ expected utility. But these studies have overlooked the objectives of other stakeholders of the insurance company. So, illiquidity, excessive risk or inadequate returns all of them have a negative influence on all stakeholders of the company especially on the customers‟ behavior. Such risks can influence dramatically the company‟s survival and the building of necessary reserves to deal with claims payments. Furthermore, the mismanagement of asset allocation by an insurance company will increase excessively its probability of insolvency, this will push the decision maker to increase premium rates and as a consequence the company will lose its competitiveness in the market. The suggested approach in this chapter attempts to seek the optimal asset allocation that maximizes technical efficiency of the nonlife insurance company. So, the directional output distance function has been used as an objective function. Two metaheuristics have been applied in our model (PSO and GA) to find the optimal asset allocation allowing the maximization of technical efficiency. Used sample includes 175 non-life insurance companies dispersed in nine European countries over the period 2002-2008. The empirical application of the model led us to the following results and recommendations. At the beginning, results show that estimated production frontier is globally meaningful. Then, generated efficiency scores implies that European nonlife insurance companies should decrease incurred claims by revising offered premium rates. Also, they must increase the available funds for investment and seeking the optimal manner to allocate them. In other hand, applying metaheuristics algorithms to seek optimal proportions of asset allocation we find more diversified portfolio. Indeed, the proportion allocated to the “alternative investment with high-risk high-return” is in average lower then these founded in previous studies. However, the percentage allocated to the “risk-free assets” is in average different from zero. Thus, the obtained investment 21

Bilel JARRAYA

portfolio is relatively more diversified between available assets compared to this proposed in previous studies. This can be explained by the implicit paying attention for competitiveness, survival and long term profitability when one maximizes technical efficiency. So, any insurance company has to pay more attentions to the presence of different stakeholders and resolves the conflicts of interest between different stakeholders. Bilel JARRAYA

22

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27

Figure

T

Y μ’ YA’

μ

A’ Bilel JARRAYA

YA

A

bA=bA’

b

Figure 1 : Optimal asset allocation and inefficiency scores

1

Bilel J

Table

Table 1 : Descriptive statistics of variables by country U-K

Swe

Den

Bel

Neth

Fr

Sp

Ita

Ger

( x1kt )

Mean

315,31

49,486

30,696

333,797

77,809

929,481

99,313

102,698

108,893

S.D.

314,439

37,202

28,651

239,581

106,833 1 141,03

81,007

108,16

105,809

( x2kt )

Mean

752,539

2 778,86

54,779

629,632

66,475JARRAYA 22 303,06 432,836 Bilel

38,894

348,53

S.D.

519,46

2 876,25

84,846

581,739

104,566 35 856,52 475,381

45,757

396,641

( x3kt )

Mean

653,017

1 776,39

301,323

743,71

173,241 2 490,92

216,772

28,238

231,587

S.D.

535,933

2 138,42

375,221

497,091

247,307 2 825,76

202,927

32,395

254,532

( x 4kt )

Mean

2 843,80

6 697,98

2 748,79

5 574,59

352,53 19 785,38 1 719,06

99,326

2 917,97

S.D.

1 682,70

6 162,97

4 929,02

5 318,41

461,597 26 257,70 2 165,05

127,004

3 757,28

Inputs

Outputs

( y1kt )

Mean

2 241,30

8 645,74

2 871,74

6 098,80

252,611 25 549,76 1 725,24

101,225

3 239,45

S.D.

1 482,40

8 210,86

4 830,80

6 204,66

309,159 34 434,26 2 070,11

124,262

4 041,19

(btk )

Mean

493,111

484,136

210,04

1 255,70

113,85

83,285

346,665

3 843,42

266,56

338,344 384,547 313,25 1 326,23 141,965 4 828,09 257,556 56,33 371,499 S.D. Notes: The table reports the mean and the cross-sectional standard deviation (SD) of each variable by country. Different notations used in the table are defined as follows: x1= Operating expenses; x2= Equity capital and reserves; x3= Financial debt capital; x4= Technical provisions; b = Incurred claims plus additions to reserves; y1= Total investment (surplus). All variables are expressed on € million.

1

Table 2 : Frontier estimation Par.

C X1 X2 X3 X4 B S t X1*X1 X1*X2 X1*X3 X1*X4 X1*b X1*S X1*t X2*X2 X2*X3 X2*X4

α0 α1 α2 α3 α4 λ1 β1 δ1 α11 α12 α13 α14 χ11 ɣ11 ψ1 α22 α23 α24

Par. Est. -1,25E+02

SD 3,46E+01

X2*b

Par.

8,42E-01

4,35E-05

X2*S

1,48E-02

1,61E-08

X2*t

7,20E-02

4,02E-06

X3*X3

4,90E-02

7,54E-08

X3*X4

-3,56E-02

Bilel JARRAYA 2,66E-06 X3*b

-5,79E-02

4,85E-08

X3*S

2,19E+01

8,63E+00

X3*t

2,63E-04

3,12E-12

b*b

-1,49E-06

2,17E-16

X4*b

-2,24E-04

1,60E-13

S*b

-3,79E-06

6,29E-15

b*t

-7,16E-05

1,91E-13

X4*X4

1,25E-05

1,82E-15

X4*S

-2,73E-03

2,53E-06

X4*t

-3,86E-07

1,54E-19

S*S

3,32E-06

2,91E-17

S*t

-2,36E-06

4,32E-18

t*t

χ21 ɣ12 ψ2 α33 α34 χ31 ɣ13 ψ3 τ11 χ41 ⱷ11 Ф1 α44 ɣ14 ψ4 β11 η1 δ11

Par. Est. 8,86E-06

SD 9,54E-17

2,91E-07

4,56E-19

-8,68E-05

7,84E-10

-8,39E-06

1,61E-14

-6,88E-06

6,65E-16

5,14E-05

2,41E-14

4,67E-06

2,03E-16

-1,59E-02

1,25E-07

-2,57E-05

1,40E-14

1,93E-05

8,91E-17

-1,78E-05

7,21E-17

4,12E-04

1,39E-07

-6,79E-06

7,79E-18

2,28E-06

1,65E-18

1,01E-02

3,48E-09

1,94E-06

4,61E-18

-8,22E-03

2,28E-09

-4,70E+00

5,15E-01

LR=987 Notes: This table presents the estimated parameters and the standard deviation for each one of the estimated directional output distance function. Different notations used in the table are defined as follows: X1= Operating expenses; X2= Equity capital and reserves; X3= Financial debt capital; X4= Technical provisions; b= Incurred claims plus additions to reserves; S= Total investment (surplus); t= trend time variable that explains technical progress; LR: the one-sided generalized likelihood ratio; Par.: estimated parameters.

Table 3: Inefficiency scores by country 2004

2005

2006

2007

2008

20022008

0,3831 0,4972 0,3632 0,6894 0,5837 0,5028 0,3516 0,3525 0,4101 Bilel JARRAYA 0,1279 0,0713 0,0772 0,3813 0,1784 0,2541 0,6121 0,5171 0,5034 0,4728 0,5187 0,5179 0,2671 0,3836 0,6094 0,5384 0,4186 0,5469 0,6894 0,5837 0,5028 0,3516 0,3525 0,4101 0,1279 0,0713 0,0772

0,3737 0,5266 0,4433 0,2545 0,4062 0,4599 0,4271 0,4502 0,4809 0,5266 0,4433 0,2545

0,5772 0,5787 0,4045 0,2962 0,3881 0,4928 0,4847 0,5142 0,4816 0,5787 0,4045 0,2962

0,3126 0,3794 0,4300 0,2202 0,3817 0,4851 0,5116 0,4711 0,4897 0,3794 0,4300 0,2202

0,6646 0,3537 0,4681 0,5071 0,3821 0,3909 0,4008 0,5937 0,4888 0,3537 0,4681 0,5071

0,4531 0,5163 0,4086 0,2221 0,3388 0,4944 0,4762 0,4699 0,4921 0,5163 0,4086 0,2221

2002 U-K Swe Den Bel Den Neth Fr Sp Neth Ita Fr Ger

2003

Notes: This table reports a comparison of the average annual inefficiency scores for each country reported by year and for all the studied period.

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Table 4 : Optimization of the nonlife insurer’s asset allocation Panel A:

PSO Algorithm Stock

AIhh

AIll

AIlh

Bond

Ω

Γ

2002

0,398

0,302

0,158

0,137

0,005

0,085

33,1%

2003

0,377

0.244

0,234

0,133

0,012

0,065

31,2%

2004

0,378

0,245

0,235

0,133

0.008

0,068

33,0%

0,104

31,7%

2005

0,516

0,034

0,030

0,333Bilel JARRAYA 0,087

2006

0,466

0,199

0,135

0,200

0,000

0,089

36,9%

2007

0,909

0,000

0,000

0,000

0,091

0,136

25,7%

2008

0,847

0,000

0,000

0,000

0,153

0,142

32,3%

02-08 Panel B:

0,556

0,130

0,113

0,134

0,058

0,098

32,0%

Genetic Algorithm Stock

AIhh

AIll

AIlh

Bond

Ω

Γ

2002

0,398

0,302

0,158

0,137

0,005

0,085

33,1%

2003

0,414

0,297

0,157

0,132

0,000

0,079

29,8%

2004

0,420

0,294

0,156

0,130

0,000

0,082

31,6%

2005

0,516

0,034

0,030

0,333

0,087

0,104

31,7%

2006

0,466

0,200

0,134

0,200

0,000

0,090

36,8%

2007

0,862

0,000

0,000

0,000

0,138

0,151

24,2%

2008

0,928

0,000

0,000

0,000

0,072

0,140

32,5%

02-08

0,572

0,161

0,091

0,133

0,043

0,104

31,4%

Notes: This table reports the proportions allocated for different assets and the new achieved inefficiency scores while using Genetic Algorithm and Particle Swarm Optimization algorithm over the period 2002-2008. Noting that AIhh: Alternative Investment with high-risk high-return; AIll: Alternative Investment with low-risk low-return; AIhl: Alternative Investment with high-risk low-return; Ω: the annual average of the new inefficiency score after assets reallocation; Γ: the won percentage of efficiency after assets reallocation.

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*1st page

Optimal Asset Allocation and Technical Efficiency in European Non-Life Insurance Companies Bilel Jarrayaa Abdelfettah Bouri b a

Ph D Student, Roud Aeroport FSEG, Sfax 3048, Tunisia

b

Professor, Roud Aeroport FSEG, Sfax 3048, Tunisia

Abstract Most previews studies investigate optimal asset allocation that maximizes shareholders’ expected utility. But, in addition to the maximization of shareholders' expected utility, the optimal assets allocation plays additional key roles to insure the insurance company's survival such as building necessary reserves to deal with claims payments. So, illiquidity, excessive risk and inadequate returns have a negative influence on shareholders’ behavior as well as on customers’ behavior. Furthermore, the mismanagement of asset allocation by an insurance company will increase excessively its probability of insolvency, this will push the decision maker to increase premium rates and as a consequence the company will lose its competitiveness in the market. This study is original; it intended to investigate the optimal asset allocation of European non-life insurance companies that maximizes technical efficiency. The definition of our objective function is based on the directional output distance function. Two metaheuritics has been used PSO and GA. Achieved results show that proportion allocated to the “alternative investment with high-risk highreturn” is in average lower then these founded in previous studies. However, the percentage allocated to the “risk-free assets” is in average different from zero. Thus, the obtained investment portfolio is relatively more diversified between available assets compared to this proposed in previous studies. This can be explained by the implicit paying attention for competitiveness, survival and long term profitability when one maximizes technical efficiency. So, any insurance company has to pay more attentions to the presence of different stakeholders and resolves the conflicts of interest between different stakeholders. Keywords: Technical efficiency, survival, Asset allocation, PSO, GA, Directional output distance function, Non-life insurance companies. JEL codes: C63; C67; G11; G22; L21; L23

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