Global Optimization Algorithms with Application to Non-Life Insurance

Transcription

1 Global Optmzaton Algorthms wth Applcaton to Non-Lfe Insurance Problems Ralf Kellner Workng Paper Char for Insurance Economcs Fredrch-Alexander-Unversty of Erlangen-Nürnberg Verson: June 202

2 GLOBAL OPTIMIZATION ALRITHMS WITH APPLICATION TO NON-LIFE INSURANCE PROBLEMS Ralf Kellner ABSTRACT If optmzaton problems cannot be solved through analytcal methods, global optmzaton algorthms consttute useful tools to derve numercal solutons. Global optmzaton methods are often appled to non-lfe nsurance optmzaton problems. The am of ths paper s to analyze the applcaton of four global optmzaton algorthms to nsurance optmzaton problems and compare the results wth respect to the algorthm s ftness, whch reflects the ablty to reach the hghest/lowest objectve functon value, and the computatonal ntensty. The nsurance problems examned n the analyss nclude typcal optmal decsons concernng the nsurer s nvestment choce and rsk management nstruments. The results show that for the problems and methods consdered wthn the smulaton analyss, genetc optmzaton leads to the best objectve functon values, whle partcle swarm optmzaton exhbts the lowest computatonal ntensty.. INTRODUCTION In the past decades, many optmzaton algorthms have been developed wth the purpose to numercally optmze functons for whch analytcal methods are not avalable. Most of these method s functonaltes are derved from prncples of nature and bology, such as evoluton or swarm ntellgence. Analyses comparng the performance of these algorthms for typcal optmzaton problems exst for dfferent felds of scence, e.g. statstcs, engneerng or bonformatcs (see, e.g. Vesterstrøm and Thomson, 2004; Paterln and Krnk, 2006), n whch results regardng the algorthms performance dffer for each feld of scence due to dfferences n optmzaton and objectve functons. However, even though these algorthms are appled to several nsurance optmzaton problems (always only one algorthm to a specfed optmzaton problem), a comparson of global optmzaton algorthms has not been conducted n the feld of non-lfe nsurance. Thus, t seems to be an mportant topc to conduct such an analyss n the feld of non-lfe nsurance. Hence, the am of ths paper s to compare the performance of global optmzaton algorthms wth respect to non-lfe nsurance optmzaton problems. Ralf Kellner s at the Fredrch-Alexander-Unversty (FAU) of Erlangen-Nuremberg, Char for Insurance Economcs, Lange Gasse 20, Nuremberg, Germany, Tel.: , ralf.kellner@wso.un-erlangen.de.

3 2 Early lterature n the feld of optmzaton of non-lfe nsurance problems s often based on assumptons such as normalty for rsk processes to derve analytcal solutons (see, e.g. Kahane and Nye, 975; Cummns and Nye, 98) and focuses on optmzaton problems n a mean-varance framework, smultaneously consderng the asset and lablty sde. Kahane and Nye (975) estmate parameters for the dstrbutons of nneteen lnes of busness of an nsurer and two asset classes as well as the correlaton matrx for the rsk process and follow a mean-varance approach to smultaneously optmze nvestment and underwrtng decsons. As the approach appled n Kahane and Nye (975) s very senstve to parameter estmaton, an alternatve s appled n Kahane (977) by usng Sharpe s Sngle-Index Model, where each of the nsurer s return on nvestment and underwrtng decsons s explaned through a relatonshp wth a market performance ndex. Ths reduces the extent of parameter estmaton, as nstead of estmatng the relatonshps between all of the nsurer s underwrtng and nvestment actvtes, only the relatonshp wth the ndex has to be estmated for each actvty. Cummns and Nye (98) also derve mean-varance effcent fronters for an underwrtng and nvestment decson problem and determne how decson rules, such as run or utlty theory, can be used to select mean-varance effcent ponts on the deduced fronter. Further examples for analyses concernng mean-varance effcent portfolo optmzaton of assets and labltes are amongst others Krouse (970) and Esenberg and Kahane (978). Moreover, n the recent lterature on non-lfe nsurance, makng smplfyng assumptons, e.g. normalty of rsk processes, s usually avoded n order to capture the nsurance optmzaton problems n a more realstc manner. Furthermore, the optmzaton problems objectve functons become more complex than mean-varance effcent problems and often nclude rsk management nstruments wth complex payoff structures as well as tal rsk measures wthn the constrants. As a consequence, analytcal solutons are not easly avalable and solutons have to be analyzed through smulaton analyss and numercal methods. Several applcatons of global optmzaton algorthms can be found n the non-lfe nsurance lterature. Zeng (2003, 2005) uses a genetc algorthm to search for the optmal quantty of ndustry loss warranty contracts, whle Cummns, Lalonde and Phllps (2004) apply a hybrd genetc algorthm to solve an optmzaton problem searchng for the optmal use of call spreads based on catastrophc loss ndces. A pattern search method s employed by Yow and Sherrs (2008), who optmze the frm value n a model allowng market mperfectons, whle Gatzert and Kellner (20b) use dfferental evoluton to search for the optmal usage of alternatve rsk transfer nstruments n order to maxmze the net shareholder value of a non-lfe nsurer. In addton, Le (20) llustrates how genetc algorthms and Monte Carlo smulaton can be used to derve solutons for cost mnmzaton. Even though these methods are appled n the feld of

4 3 non-lfe nsurance, other popular global optmzaton algorthms, e.g. partcle swarm optmzaton, have not been appled. Furthermore, global optmzaton algorthms are usually appled to the optmzaton problem wthout comparng ther effectveness to other global optmzaton methods. However, wth respect to other felds of scence, the comparson of the global optmzaton algorthms performance has been subject to several analyses. Storn and Prce (997) compare dfferental evoluton to other evolutonary algorthms, annealng methods, and methods of stochastc dfferental equatons wthn selected benchmark functons and detect dfferental evoluton to be superor wth respect to fnd the global optmum n a requred number of functon evaluatons. Vesterstrøm and Thomson (2004) and Panduro et al. (2009) compare the performance of partcle swarm optmzaton, genetc optmzaton and dfferental evoluton. Vesterstrøm and Thomson (2004) observe that dfferental evoluton outperforms partcle swarm optmzaton and the genetc algorthm n 32 of 34 benchmark cases, whle Panduro et al. (2009) fnd partcle swarm optmzaton and dfferental evoluton to be better than the genetc algorthm for an optmzaton problem n the feld of electromagnetsm. Moreover, dfferental evoluton also proves to be more effcent n comparson to partcle swarm optmzaton wthn an analyss by Paterln and Krnk (2006), examnng the applcaton to parttonal clusterng. Whle Pham and Wlamowsk (20) refne the Nelder-Mead smplex method and compare results of the newer to the orgnal verson, Bera and Mukherjee (200) detect the latter to be nferor to a hybrd smulated annealng method for a multple response optmzaton problem. However, Katar et al. (2007) detect the Nelder-Mead smplex method to be superor for certan cases of data clusterng problems n comparson to an mproved genetc algorthm. Hence, n ths paper, we expand prevous work n the followng way. We choose four global optmzaton algorthms, whch are frequently used n the optmzaton lterature as lad out above and apply them to varous nsurance optmzaton problems, whch focus on optmal nvestment and rsk management decsons. We frst start wth an optmzaton problem that s expressed n a mean-varance framework, analyzng an nsurer s rsk management decsons on the asset and lablty sde. Ths problem exhbts an analytcal soluton whch can be compared to the solutons provded by the numercal methods of the global optmzaton algorthms. Furthermore, optmzaton problems maxmzng the nsurer s surplus or the net shareholder value under constrants accountng for the nsurer s solvency stuaton are formulated. The optmzaton algorthm s results are compared wth respect to the optmzaton algorthm s ftness, whch descrbes the algorthm s ablty to reach the hghest/lowest objectve

5 4 functon value n case of a maxmzaton/mnmzaton problem (see, e.g. Paterln and Krnk, 2006) and the computatonal ntensty. In addton, senstvty analyses wth respect to dfferent nput parameters of the global optmzaton algorthms and dfferent startng values for the optmzaton algorthms are conducted. Concernng the global optmzaton algorthms, we choose the Nelder-Mead smplex method, partcle swarm optmzaton, dfferental evoluton and genetc optmzaton. The Nelder- Mead smplex s one of the oldest optmzaton algorthms (see Nelder and Mead, 965), but despte ts age s stll often appled (see, e.g. Prce, Coope and Byatt, 2002; Katar et al., 2007). Due to the characterstc of havng just one startng pont nstead of conductng a parallel search technque of multple vectors, t exhbts advantages concernng the computatonal ntensty, but also ncreases the lkelhood of gettng trapped n a local optmum. To avod ths potental drawback, modern optmzaton algorthms lke partcle swarm optmzaton, dfferental evoluton and genetc optmzaton usually apply parallel search technques. Wthn our analyss, our results show that genetc optmzaton leads to the hghest objectve functon values, whle partcle swarm optmzaton exhbts the lowest computatonal ntensty. Overall, our results show that the type of search technque (drect or parallel) and the number of heurstc rules that are appled durng the optmzaton are crucal for the algorthms performance. The remander of ths paper s structured as follows. In Secton 2, the optmzaton algorthms are presented, whle Secton 3 llustrates the optmzaton problems. Secton 4 contans numercal analyses and Secton 5 concludes. 2. GLOBAL OPTIMIZATION ALRITHMS In the followng analyss, we compare the performance of four global optmzaton algorthms, the Nelder-Mead smplex method, dfferental evoluton, genetc optmzaton and partcle swarm optmzaton. The Nelder-Mead smplex s a smplex-based drect search method, whle dfferental evoluton and genetc optmzaton belong to the class of evolutonary algorthms, and partcle swarm optmzaton follows the approach of swarm ntellgence computaton. Whle the Nelder-Mead smplex starts ts search for a global optmum at one startng pont, the remanng algorthms use sets of ntal tral vectors and, thus, exhbtng a parallel search technque. All four optmzaton algorthms are appled to solve an optmzaton problem, whch n general can be formulated as

6 5 mn f x, d x, and, f necessary, be complemented wth addtonal constrants x x g 0 c,,, ncg, j,, n hj 0 where x represents a vector wth d parameters to be optmzed, f to be mnmzed, g x and ch, x the objectve functon h x the constrant functons and n, n the number of constrants (see Mart, 2005). Dependng on the characterstcs and the progresson of f cg ch x, dfferent algorthms may lead to dverse results, whch s based on dfferent heurstc rules that are appled durng the optmzaton. 2. Nelder-Mead smplex method The Nelder-Mead method was frst publshed n 965 (see Nelder and Mead, 965) and s a method to optmze an unconstraned functon f x of d varables,. To fnd an optmum, only functon values are used. Thus, t represents a dervatve-free method. x d Each teraton step starts wth a set of d ponts, buldng a smplex, whch converts tself accordng to operaton rules (see Lagaras et al., 998). These rules are constructed to mprove the best functon s value at each teraton step or to stop f a stoppng condton, e.g. the best functon s value cannot be mproved for more than a predefned amount of teraton steps, s reached. In partcular, at the th teraton step 0, d ponts, gven through d vertces from the smplex, are ordered accordng to ther functon values f x for d x. Based on the order of the ponts x j, j,, d, the operatons reflecton, expanson, contracton and shrnk step as llustrated n Fgure a), whch llustrates the four operatons for a 2-dmensonal problem, are conducted (see Lagaras et al., 998). A smplex s a d-dmensonal polytope, n whch each smplex s defned by d vertces, e.g. a smplex of dmenson two s a trangle (see, e.g. Prce, Coope and Byatt, 2002).

7 6 Fgure : Operator functons and procedure for the Nelder-Mead smplex method a) Operator functons for the Nelder-Mead smplex method () Reflecton Expanson Contracton (4) Shrnk Step x3 x3 Outsde Insde x3 x3 x x c x2 x x c x2 x x c x2 x c, n v3 x v2 x r x x c x2 x r x c, out x r b) Procedure for the Nelder-Mead smplex method f x k f x k 2 f x k d x e Reflecton (see ()) k r f x f x k k r d f x f x f x k r d f x f x Expanson (see ) accept x r and Contracton (see ) termnate teraton f xe f xr f x f x e r c, out r / c, n d f x f x f x f x c, out r / c, n d f x f x f x f x accept x e and termnate teraton accept x r and termnate teraton accept x c,out /x c,n and termnate teraton perform a shrnk step (see (4))

8 7 Accordng to the order of functon values f x f x 2 f x d, x denotes the best pont, x 2 the second best pont and xd the worst pont, f the am s to mnmze f. After sortng the functon values, reflecton s appled, whereas the reflecton pont x r s gven through x x x x, r c c d where x c s the centrod 2 of the d best ponts (see Lagras et al., 998) and represents the reflecton factor of the Nelder-Mead smplex method, whch satsfes 0. The hgher the value for s chosen, the further away moves the new reflecton pont from the centrod. Fgure b) llustrates the three possbltes for the further development of the teraton. If the functon value f x r of x r les between the best and the second worst functon values f x f xr f xd, r termnated. In case of f x r f x x x x x, e c r c x s accepted to replace xd and the frst teraton step s, the expanson pont x e s calculated usng wth as the expanson factor, satsfyng and (see, e.g. Prce, Coope and Byatt, 2002). If the functon value of x e s hgher or equal to the reflecton pont s functon value, x r s chosen to be the new vertex, otherwse x e replaces xd. For the thrd possble case f xr f xd, an outsde contracton s performed f f xd f xr f xd and the outsde contracton pont x c, out s calculated accordng to x x x x, c, out c r c or an nsde contracton s conducted, determnng the nsde contracton pont x c, n x x x x, c, n c c d f f xr f xd, wth f xc, out f xr or f xc, n f xd representng the contracton factor, satsfyng 0. If, respectvely, the contracton pont x c, out or x c, n, respectvely, s accepted to be the new vertex for the next teraton step (see Lagras et al., 998). Otherwse, a shrnk step s performed and the vertces for the smplex n the next teraton,, x v2,, vd, are calculated accordng to 2 The centrod s the center of mass and n the example llustrated n Fgure a) the pont whose coordnates are the weghted means of the x and x 2 coordnates, respectvely.

9 8 v x x x, wth j beng the shrnk step factor, satsfyng 0. Summng up, each teraton step starts through reflectng the worst pont at the centrod of the remanng ponts. If the functon value of the new reflecton pont falls between the functon values of the remanng pont, the teraton step s closed. If the functon value of the new reflecton pont leads to better results than the results of the remanng ponts, the search s extended through expanson, otherwse, f the functon value of the reflecton pont s stll worse than the functon values of the remanng ponts, the search s narrowed down through contracton. The algorthm explaned above s repeated untl a stoppng crteron s satsfed, e.g. a maxmum number of teratons s reached or the best objectve functon s value could not be mproved for a predefned absolute or relatve value. Accordngly, the algorthm must not lead to the global optmum, but can lead to results close to the global optmum n a lower computatonal tme than, e.g. exhaustve testng n the search space. The method s qualty depends on the choce of startng values, whch must be set wth the ntalzaton of the method, operaton factors,,, and stoppng condtons. Due to ts user-frendly applcaton, the Nelder-Mead algorthm s frequently appled n the felds of statstcs, engneerng, physcal and medcal scences (see Prce, Coope and Bryatt, 2002). Nevertheless, a drawback of the Nelder-Mead algorthm s that ts search for the global optmum could be trapped n a local optmum. Ths can be solved through tryng dfferent startng ponts for the algorthm. As an alternatve, other optmzaton algorthms avod ths problem through smultaneously runnng several tral vectors. Examples are dfferental evoluton, genetc optmzaton and partcle swarm optmzaton, whch are explaned n detal n the next paragraphs. 2.2 Dfferental evoluton Dfferental evoluton (DE) belongs to the class of evolutonary algorthms and s a parallel drect search method (see Storn and Prce, 997). It starts wth a set of p d - dmensonal DE tral vectors x g,,,2,, p, representng the populaton, and modfes ts p tral vectors through the operaton functons mutaton, crossover and selecton for a number of generatons g, untl a stoppng condton s satsfed. Before the ntalzaton, lower and upper bounds for each parameter, whch should be used for the optmzaton, must be defned. The ntal set of

10 9 tral vectors s then randomly drawn on a unform probablty space. Subsequently, for each DE DE tral vector x g,, a mutant vector v g, s created through v x F x x, DE DE DE DE, g r0, g r, g r2, g wth random ndces r, r, r,2,, p 0 2, whch have to be dfferent to, and F as the step sze, satsfyng F 0 (see Prce, Storn and Lampnen, 2006). 3 In the next step, an ntal gen- DE DE eraton s created through crossng over the tral vector x g, and the mutant vector v g,, DE DE DE DE u u, u,, u are created accordng to whereas the tral vectors, g, g 2, g d, g u DE j, g DE v j, g f randb j CR or j rnbr DE x j, g f randb j CR and j rnbr, j,2,, d, where CR 0, s the predefned crossover constant, 0, number generator and rnbr,2,, randb j a unform random d a random ndex, ensurng that the tral vector u, DE gets at least one parameter of the mutant vector v g, (see Storn and Prce, 997). The next DE generaton of tral vectors xg, s then generated by comparng the objectve functon values of the ntal vectors f x DE g, and the tral vectors DE g, x DE g, DE DE DE u, g f f u, g f x, g, DE xg, otherwse DE f u. Hence, xg, s created through DE g (see Prce, Storn and Lampnen, 2006). The process of mutaton, crossover and selecton s repeated untl a stoppng condton s satsfed. Through the parallel search technque, the dstress of gettng trapped n a local optmum s sgnfcantly reduced. However, at the same tme ths search technque ncreases the computatonal ntensty n comparson to the Nelder-Mead smplex. Whle the mutaton and crossover operators cause tral solutons to scan the objectve functon s surface, the selecton operator memorzes good solutons from the prevous generaton. 2.3 Genetc optmzaton The functonalty of genetc optmzaton () s derved from the mechancs of natural selectons and natural genetcs and proceeds over a number of generatons g to reach the objec- 3 As the ndex has to be dfferent to r, r, r,2,, p 0 2, p must be greater or equal to four.

11 0 tve functon s maxmum (see Goldberg, 989). Smlar to dfferental evoluton, a genetc algorthm starts wth a set of d - dmensonal ntal vectors x g,,,2,, p, representng the populaton. x, the objectve functon s value Wth each vector g, f x g, s calculated. The populaton n the followng generaton g s created accordng to operator functons, whch use data from the former generaton to generate a new populaton and adopt rules to ncrease each generaton s average objectve functon s value (see, e.g. Panduro et al., 2009). Thus, n contrast to dfferental evoluton, the ntal tral vectors are not passed to the next generaton. 4 Operator functons among the genetc algorthm can dffer (see, e.g. Flho, Alpp and Treleaven, 994). In our analyss, we conduct a genetc algorthm based on Mebane and Sekhon (20). The operator functons are dsplayed n Table. Even though the general procedure appled n dfferental evoluton and genetc optmzaton s dentcal (parallel search of ntal tral vectors, representng the populaton, s evolved over a number of generatons through operator functons), results whch are obtaned may dffer due to the dfferent operator functons. Especally the usage of an ntal generaton and the selecton operaton wthn dfferental DE DE evoluton leads to new generatons wth dentcal vectors x, g x, g, thus memorzng solutons from prevous generatons, whereas wthn the genetc algorthm, only a predefned number of vectors, whch lead to the best objectve functon s values, s kept for the next generaton through the clonng operaton (see Mebane and Sekhon, 20). For both methods, the choce of the populaton sze p and the number of generatons g s crucal for the optmzaton s qualty. Nevertheless, due to the dfferent operator functons, the optmal choces for p and g may also dffer between dfferental evoluton and genetc optmzaton. In addton, a tradeoff between a hgher optmzaton s qualty and a longer runnng tme, comng along wth an ncrease n p and g, exsts. 4 The clonng operator (see Table ) consttutes an excepton, as t copes a predefned amount of the best tral vector solutons from the prevous generaton nto the next generaton. Usually, only a few populaton members are generated through clonng,.e. Mebane and Sekhon (20) recommend to solely copy the current generaton s best tral vector n the next generaton.

12 Table : Operator functons for the genetc algorthm by Mebane and Sekhon (20) Clonng Copy x g, n the next generaton x g, Unform mutaton Boundary mutaton Non-unform mutaton Polytope crossover Smple crossover Whole nonunform mutaton Heurstc crossover Local-Mnmum crossover Randomly choose j,2,, d. Select a value accordng to x ~ U x, x, where,~,, j, g j, g j, g x and, lower j, g x represent the lower, upper j, g,~ and upper bounds for the values of x j, g. Set xj, g xj, g. Randomly choose j,2,, d. Set ether x wth probablty 0.50 of usng each value. Randomly choose j,2,, d x or, upper x j, g x, j, g, lower j, g j, g gen current max. Compute / current g representng the current generaton number, max g the maxmum generaton number, 0 B a tunng parameter and ~ 0,,, gen gen lower x j, g q x j, g q x or j, g gen gen upper j, g j, g j, g q g g u, wth u U. Set ether x q x q x wth probablty 0.50 of usng each value. Usng m max 2, d numbers s 0, vectors x g, from the current populaton and m random m m l, so that sl, set x, g sl xl, g. l Pck a sngle nteger j from d. Usng two parameter vectors, x, g and x 2, l B g, set and x j2, g t x j2, g t x, where t 0, j, g x t x t x j, g j, g j2, g s a fxed number Do non-unform mutaton for all the elements of x j, g Pck ~ 0, u U. Usng two parameter vectors, x, g and x 2, g, compute x, g u x, g x 2, g x. If, g x, g satsfes all constrants, t can be used, otherwse pck another u value and repeat untl a preset number of attempts. If x value s found, set x, g equal to the better value of x, g and x 2, g. no, g Two x, g values must be produced n ths manner Run a BFGS 5 optmzaton wth a vector x, g up to a preset number of teratons to produce If g, shrnk u,* x. Pck u~ U 0, and calculate,* g, x, g u x, g u x., g x satsfes all constrants, t can be used, otherwse shrnk u by settng g, u/2 and recomputed xg, wth shrnk u. If no satsfactory value for x can be found untl a preset number of attempts s reached, return x,. g 5 BFGS refers to the Broyden-Fletcher-Goldfarb-Shanno quas-newton method. In the algorthm of Mebane and Sekhon (20), the possblty exsts to nclude and combne ths method wth the genetc algorthm. As our analyss ntends to focus on the optmzaton through the genetc algorthm, we calbrate the algorthm such that no addtonal nformaton derved through the BFGS method wll be nvolved n the optmzaton process.

13 2 2.4 Partcle swarm optmzaton Partcle swarm optmzaton (PSO) s a parallel drect search method, whose methodology s derved from the behavor of organsms such as brd flockng and fsh schoolng (see Panduro PSO PSO PSO PSO et al., 2009). At the begnnng at tme t, a set of partcles x, t x, t x2, t xd, t,,,, PSO,, s representng the swarm, s randomly placed n the search space whle assocated functon values are determned. 6 Subsequently, the partcles move around n the search space PSO PSO PSO PSO wth velocty v, t v, t, v2, t,, vd, t, where ther movement depends on the hstory of ther own and of the swarm members functon values,.e. over t, 2,, T, the partcles are gravtated n the drectons of ther own and the swarm s best functon value acheved so far PSO (see Pol, 2007). The velocty for each component n t, v j, t, j,2,, d, s calculated through PSO PSO r PSO PSO r PSO PSO v j, t w v j, t c u x j, t pbest x j, t c 2 u 2 x j, t sbest x, j, t where w s the nerta weght, c, c 2 are the acceleraton constants, r r u, u 2 are unform dstrbuted random numbers n 0,, x PSO j, t pbest s the j th component from the best soluton, PSO x sbest represents the j th component whch s derved through the partcle so far, PSO from the best soluton, whch s derved through the swarm and x j, t s the j th component of the current partcle. Correspondng to the velocty n t, the new poston of the partcle s component s adjusted accordng to j, t x x v. PSO PSO PSO j, t j, t j, t After the movement of all partcles, the frst teraton step s closed. In the next step, ths procedure s repeated untl a stoppng condton s satsfed. Thus, n contrast to dfferental evoluton and genetc optmzaton, partcle swarm optmzaton memorzes prevous results through retanng all prevous results for a possble soluton to the optmzaton problem, whle wthn dfferental evoluton and genetc optmzaton, only results wthn the present generaton are possble solutons for the optmzaton problem. On the one hand, ths characterstc mght enhance the computatonal ntensty but on the other hand, t mght also ncrease the probablty of fndng the global optmum. Furthermore, partcle swarm optmzaton does not feature a set of dfferent operator functons as they can be found for dfferental evoluton 6 At the begnnng, the ntal partcles are comparable to the ntal populaton members n dfferental evoluton and genetc optmzaton.

14 3 and genetc optmzaton, whch could adversely affect the search for the global optmum and decrease the computatonal ntensty. 2.5 Comparson and evaluaton Summng up, the Nelder-Mead smplex method s the only algorthm that does not apply a parallel search technque, whch reduces the computatonal ntensty n comparson to the remanng algorthms appled n the analyss but enhances the probablty to be trapped n a local optmum. Partcle swarm optmzaton s smlar to the genetc algorthm and dfferental evoluton through applyng a heurstc rule (determnaton of the velocty) to the partcles n the swarm to mprove the algorthm s results, but dffers n that all prevous solutons are memorzed over the procedure, thus causng a hgh memory effort. The genetc algorthm creates a new set of vectors for each generaton, solely contanng a few of the prevous generatons best tral solutons, whle n dfferental evoluton, vectors are retaned, f they lead to better results than the new mutated tral vectors. Partcle swarm optmzaton uses the partcle personal s best result as well as the swarm s best result of the prevous search to move new partcles n the most promsng area wth respect to the best functon s value. Otherwse, only the velocty s used to determne the evolvement of new tral solutons, whereas dfferental evoluton and genetc optmzaton apply several operaton functons. To evaluate the optmzaton algorthm s qualty wthn our analyss, two crtera are consdered. In a frst step, we examne whch optmzaton algorthm leads to the best results,.e. leads to the hghest (lowest) value of the functon to be maxmzed (mnmzed). The algorthm s ablty to acheve the hghest (lowest) objectve functon value s denoted as the optmzaton algorthm s ftness (see, e.g. Paterln and Krnk, 2006). Concernng the optmzaton methods appled n our analyss, an enhancement n the populaton and swarm sze for dfferental evoluton, genetc and partcle swarm optmzaton, respectvely, as well as n dfferent startng values for the Nelder-Mead method ncreases the lkelhood to fnd the hghest (lowest) objectve functon value, whle at the same tme the computatonal ntensty s rased. Ths tradeoff has to be kept n mnd when analyzng the optmzaton methods effcency. Therefore, n a second step, the computatonal ntensty s examned by comparng the runnng tme of each optmzaton run. Besdes the analyss of the algorthms performance through the crtera ftness and computatonal ntensty, an analyss s conducted whch focuses on the mpact of dfferent random numbers for each of the algorthms startng values. More precsely, the optmzaton algo-

15 4 rthms start wth a random draw of an ntal populaton (dfferental evoluton and genetc optmzaton), swarm (partcle swarm optmzaton) or startng pont (Nelder-Mead smplex), whch nfluences the optmzaton s procedure by the ntal seeds that are used for ths random draw,.e. dfferent values for the ntal populaton, swarm or startng pont mght lead to dfferent optmzaton results. To analyze the mpact of a dfferent ntal generaton, swarm or startng pont n more depth, we analyze the optmzaton algorthm s robustness through the average relatve devaton n modulus, whch s determned accordng to x opt orgn dff. seed n f x f x orgn dev f f x,,, n opt n opt, () where, startng pont, ( orgn ) and opt n s the number of optmzaton runs wth a dfferent ntal populaton, swarm or orgn. f x s the hghest objectve functon value of a predefned standard case dff seed f x s the result of the th optmzaton run wth dfferent ntal seeds n random number generaton ( dff.seed ) INSURANCE OPTIMIZATION PROBLEMS To examne whether the optmzaton algorthms llustrated n Secton 2 are sutable for optmzng nsurance problems and to determne whch algorthm s most effcent, we consder relevant optmzaton problems from the feld of non-lfe nsurance and alternatve rsk transfer, where each optmzaton problem ntends to fnd optmal rsk management strateges. The frst s taken from Cummns and Song (2008) and exhbts a closed-form soluton. Ths optmzaton problem s analyzed to obtan an mpresson how close the algorthms can get to the analytcal optmum. The further optmzaton problems are formulated on the bass of Gatzert and Kellner (20a) and Gatzert and Kellner (20b), respectvely, and do not exhbt closedform solutons and thus requre smulaton technques. The llustraton of the optmzaton problems n ths secton s conducted wth the purpose to clarfy the optmzatons dea and notaton. A detaled llustraton of the model frameworks behnd the optmzaton problems can be found n Cummns and Song (2008), Gatzert and Kellner (20a), and Gatzert and Kellner (20b). 7 Cummns, Lalonde and Phllps (2004) apply a smlar procedure to determne whether the optmzaton results are stable wth respect to dfferent ntervals for startng values of the optmzaton.

16 5 3. Optmzng nvestment and rensurance decsons Consderng a holstc rsk management approach regardng the asset and lablty sde, a tradeoff between hedgng decsons on both sdes exsts. If rsky nvestment decsons on the asset sde are made, hgher expenses for rsk management have to be spent and/or less rsky underwrtng decsons have to be done on the lablty sde to acheve a desred overall rsk level, whereas less rsky nvestment decsons on the asset sde allow rsker underwrtng decsons on the lablty sde. Cummns and Song (2008) analyze substtuton effects between an nsurer s nvestment decson and ts rsk management decson on the lablty sde. Therefore, they set up a mean-varance optmzaton problem that s optmzed over the fracton nvested n rsk-free assets and the amount of a proportonal (quota share) rensurance contract. In ther model, n t 0, an nsurer nvests a fracton of ts ntal captal IC 0 n rskfree assets, whle the remanng fracton s nvested n hgh-rsk assets. The ntal captal con- ssts of shareholder s ntal contrbuton EC 0, premum ncome S, for nsurng possble re, losses at t and premum expenses for proportonal rensurance, whch pays a fracton of the nsurer s loss. 8 Hence, the ntal captal s gven by IC EC S, re, The superscrpt denotes the frst optmzaton problem, whch we denote as the nvestment and rensurance optmzaton problem. In t, the nsurance company faces possble losses S and receves payments from the proportonal rensurance contract. Hgh-rsk assets are assumed to be normally dstrbuted and ndependent from the nsurer s loss, where H and 2 H represent the expected value and varance of the hghrsk assets return. In ths settng, the nsurer s task s to maxmze the expected wealth W x wth x 2, 8 9 In Cummns and Song (2008), e.g. equty captal s denoted as nternal wealth and hgh-rsk assets are named rsky assets. To retan a consstent notaton wth the subsequent optmzaton problems, expressons and notatons are adapted. n the superscrpt clarfes that the notaton s not vald for the followng optmzaton problems, the nvestment n rsk-free assets n the frst optmzaton problem s not the same as the nvestment n low-rsk assets n Secton 3.2 and Secton 3.3, e.g.. Contrarwse, no- RBC RBC tatons wthout superscrpt are consstent for the three optmzaton problems.

17 6 whch conssts of the expected return on nvestments and underwrtng operatons E R x S re, 0 H 0 f E R x IC IC r E S, 2 mnus the varance of the nsurer s poston Rx Rx IC 0 H S 2 2 2, weghted wth a coeffcent of rsk averson the nsurer wthn ts premum calculaton, S 2 and S q, where, gven by denotes the loadng charged by re, the rensurance premum loadng, ES the expected value and varance of the nsurance company s losses. Hence, the optmzaton problem s expressed as max x q f x W x max E Rx R x x 2,* 2, such that the closed-form soluton for the optmal fracton nvested n rsk-free assets,* can be calculated through takng the frst order dervatve and settng the equaton equal to zero, leadng to,* S H rf 2 E S, 2 2 S S 2 re, 2 re, re, re, q I EC 2 0 E S I rf ES,* whle the optmal fracton of proportonal rensurance s gven through 2,* r q re, re, f 2 S E S E S 2, (see Cummns and Song, 2008).

18 7 3.2 Maxmzng an nsurer s surplus under solvency constrants For an nsurance company, an mportant task s to generate new earnngs and at the same tme meet requrements concernng the solvency stuaton. Hence, we express optmzaton problems that try to maxmze the dfference between the nsurer s assets and labltes at tme T under certan constrants wth respect to the nsurer s solvency stuaton. The optmzaton problems are based on the model framework presented n Gatzert and Kellner (20a), n whch the mpact of non-lnear dependences on the bass rsk of ndustry loss warrantes s examned n the context of solvency captal requrements and the nsurer s free surplus. Wthn the model framework, n t 0, the nsurance company collects ts premum ncome for nsurng possble losses n t and decdes to purchase an ndustry loss warranty or an aggregate excess of loss rensurance contract to manage ts underwrtng rsk. 0 Furthermore, the ntal captal, whch conssts of shareholder s ntal contrbuton E 0, premum ncome S,, whch s calculated on the bass of the expected value for the nsurer s loss and an addtonal constant loadng, and premum expenses ILW,, s nvested n hgh- and low-rsk assets. The superscrpt denotes the surplus optmzaton problem, whch s llustrated n ths subsecton. In t, the nsurer faces possble losses S and receves payments from the ILW, ndustry loss warranty contract X. The ndustry loss warranty s payoff s gven by X A, L, Y mn max S A,0, L I Y, ILW, ILW, ILW ILW, ILW ILW, ILW where A s the contract s attachment pont, L the ILW s layer lmt that consttutes the maxmum payment and I Y represents the trgger functon, whch s equal to, f the ndustry loss I n t exceeds a predefned trgger level Y and 0 otherwse (see Gatzert, Schmeser and Toplek, 20). The premum for the ndustry loss warranty contract s based on the expected value prncple, such that ts prce s determned accordng to S, I ILW, ILW, ILW, E X. ILW, Gatzert and Kellner (20a) assume a constant loadng for the ndustry loss warranty contract. In general, ndces for ndex-lnked nstruments can be structured accordng to the nsurer s needs on the bass of zp-codes (see, e.g. Major, 999; Cummns, Lalonde and Phl- 0 In our analyss, the nsurance company exclusvely purchases an ndustry loss warranty contract, such that the case wthout rsk management or wth an excess of loss rensurance contract s not explaned n detal n the followng. Furthermore, see Gatzert and Schmeser (20) for a detaled dscusson of ndustry loss warranty s key characterstcs.

19 8 lps, 2004), whch can be done wth the purpose to decrease potental bass rsk that n general decreases for hgher values of S, I. However, more ndvdually structured ndces are less transparent, whch mght ncrease the costs assocated wth these contracts (see, e.g. ILW, SwssRe, 2006). One way to ntegrate ths behavor s to vary subject to Kendall s tau S, I ILW, S, I. Hence, we assume that the costs, whch are represented through S, I, are calculated through a convex cost functon that depends on, and can be determned through S, I S, I ILW, 2. The extenson of the cost functon represents a tradeoff between lower potental bass rsk and hgher costs due to a more ndvdually structured ndex. The nsurer s losses as well as the assets are assumed to be lognormal dstrbuted, whle the dependence structure among assets and labltes s modeled through copulas (see Gatzert and Kellner, 20a). Consderng the optmzaton problem, the nsurer s am s to maxmze the nsurer s surplus n t wth respect to the nsurer s solvency stuaton. The surplus s gven through L x EC e e X S, S, ILW, r r H ILW, 0 x A S, I ILW, 3 where ILW, A descrbes an ex ante chosen attachment pont of the ndustry loss warranty contract, 2 EC 0 s the ntal captal, denotes the fracton nvested n low-rsk assets and rl / r H are the normally dstrbuted contnuous one-perod returns for low- L and hgh-rsk assets H. The solvency stuaton s represented through the shortfall probablty SP that can be determned accordng to SP P x. 0 2 In practce, addtonal costs for the ndvdually structured ndex mght depend on the relatonshp between the protecton buyer and seller,.e. f a long term relatonshp between both counterpartes exsts, such that an ndvdual ndex can be used for more than one transacton, t s lkely that no addtonal costs occur for the ndvdual ndex. Otherwse, f the transacton s conducted only once, ndvdual ndces mght be more expensve. Further contract parameters lke the layer lmt or the ILW s trgger level can be chosen as well. In the followng analyss only the attachment pont wll be vared wthn the optmzaton.

20 9 In general, the solvency stuaton can be accounted for n two dfferent ways n regard to the optmzaton procedure. Frst, the objectve functon (here: x ) can be dvded by the rsk measure (here: SP ), or, second, the solvency stuaton can be added to the constrants of the optmzaton problem, such that a very low value of the objectve functon s passed to the optmzaton algorthm f the constrant s broken (see Zeng, 2003). For our analyss, ths leads to two dfferent types of optmzaton functons. Frst, max x f x * x, SP subject to the constrants c, that the fracton nvested n low-rsk assets as well as Kendall s tau between the nsurer s loss and the ndex have to be between zero and one and that the attachment pont of the ndustry loss warranty contract, ILW, a predefned barrer ILW A, A c 0 0 S, I ILW, ILW, A A A ILW,, ILW, A s set to be n the range of and second, max x * f x x, (4) subject to the constrants c 2, whch extend c n a way that the shortfall probablty s not allowed to exceed a certan predefned level SP, c 0 2 ILW, ILW, 0 S, I A A A SP SP ILW,. In the numercal analyss n Secton 4, dfferent parameter combnatons for the optmzaton problems accordng to Equaton and (4) are analyzed,.e. frst, we optmze over one of ILW, the varables, A, S, I, whle the remanng varables are kept constant, and, second we optmze over two varables, whle the remanng varable s kept constant. Thrd,

21 20 the optmzaton s done over all three varables. Hence, seven dfferent combnatons for the optmzaton are taken nto account, whch overall leads to 4 dfferent optmzaton problems wth respect to the two optmzaton functons accordng to Equaton and (4). In partcular, the combnatons for each optmzaton problem are gven through ILW, ILW, ILW, ILW,, A, S, I,, A,, S, I, A, S, I,, A, S, I. 3.3 Maxmzng shareholder value Concernng the next type of optmzaton problem, we choose a problem, whch s expressed from a shareholder s perspectve. In Gatzert and Kellner (20b) a rsk management strategy combnng ndex-lnked and ndemnty based rsk management nstruments s consdered wth respect to maxmzng the net shareholder value under safety constrants. The ndex-lnked nstrument s gven through a bnary ndustry loss warranty contract, payng a fracton of a layer lmt, f the ndex exceeds a predefned trgger, whereas gap nsurance represents an ndemnty based nstrument and pays a fracton of the gap between the nsurer s loss and the ndex-lnked nstrument s payment. In t 0, the nsurance company nvests ts ntal captal, consstng of shareholder s ntal contrbuton EC 0 plus premum ncome S, mnus pre-, mum expenses, n hgh- and low-rsk assets, ILW, Gap, where ILW denotes the bnary ndustry loss warranty contract and Gap the gap nsurance contract. n superscrpt denotes the optmzaton problem as the shareholder value optmzaton problem. In t, the nsurer has to pay for the nsured losses and receves payments from rsk manage- ILW, ment nstruments. The payment of the bnary loss warranty contract by X L I Y, ILW, ILW gap, whereas the gap nsurance s payoff X max S X,0, gap, ILW, where and X s gven X can be calculated accordng to are the fractons of the ndustry loss warranty and the gap nsurance contract, respectvely. The net shareholder value s the today s value of equty captal n t mnus the ntal contrbuton of shareholders and should be maxmzed over the optmal fractons of rsk management nstruments. Hence, the optmzaton problem s expressed as

22 * 0 max f x x x max x ILW, gap S,, rf Q, S0 DPO x e E X 2, (5) x 2, wth S0 DPO x representng the value of payments to polcyholders at t 0, DPO the default put opton, whch represents the loss n case of the nsurer s default, S, the, loadng polcyholder s are wllng to pay, the loadng demanded wthn the premum rf Q, calculaton for the rsk management nstruments and e E X the dscounted expected value of the rsk managements payments under the rsk-neutral prcng measure Q. 3 The optmzaton problem s solved subject to the constrants c that the nsurer s shortfall probablty s not allowed to exceed a predefned level SP, the fractons and are n the range between zero and one and that the premum payment s consstent wth the shortfall probablty mpled by the rsk management strategy, as the loadng, polcyholders are wllng to pay n t 0, depends on the nsurer s shortfall probablty n t. Thus, the shortfall probablty, whch s promsed SP 0 and appled to determne the nsurer s pre- promsed mum ncome n t 0 has to be equal to the shortfall probablty, realzed n t, 4 c SP SP 0 0 S S0 DPO x! S,,. Analogously to the optmzaton problems gven n Equaton and (4), the optmzaton s done for the varable combnatons,,,. Thus, n total 8 dfferent optmzaton problems are examned and compared n the numercal analyss n Secton See Gatzert and Kellner (20b) for the dervaton of Equaton (5). For a detaled explanaton, see Gatzert and Kellner (20b).

23 Graphcal llustraton of the optmzaton problems Fgure 2 llustrates some examples for the objectve functons descrbed n ths secton. Fgure 2a) and 2b) dsplay the functons for the expected wealth nsurer s surplus dvded through the shortfall probablty ), dependent on, and, A,. W (see Equaton ) and the x / SP (see Equaton Fgure 2: Examples for Functons accordng to Equaton, and (5) a) b) c) Furthermore, Fgure 2c) dsplays the case for the net shareholder value accordng to Equaton ILW (5) n dependence of the ndustry loss warranty fracton and the shortfall probablty,

24 23 promsed whch s used for the premum ncome calculaton SP 0. In general, constrants are ntegrated n the optmzaton problems n two ways. Frst, bounds for the varables, whch are used wthn the optmzaton, are enforced by calbratng the optmzaton, such that the ntal tral solutons are only drawn n the permtted search space. Second, constrants, whch affect quanttes that are calculated durng the optmzaton process, e.g. the shortfall probablty, are ncorporated n a way that n case of maxmzaton (mnmzaton), a very low (hgh) value of the objectve functon s passed to the optmzaton algorthm f the constrant s not satsfed (see, e.g. Mullen et al., 20; Zeng, 2003). Ths can lead to a non-contnuous surface, as t s the case for the objectve functon accordng to Equaton (5), whch s dsplayed n Fgure 2c). The algorthms llustrated n Secton 2 usually are ntended for mnmzaton, whereas n our analyss, maxmzaton problems are consdered. Thus, each objectve functon s calbrated n a way that the negatve functon value s passed to the optmzaton algorthm. Moreover, further approaches for the optmzaton, e.g. robust optmzaton, can exst, whch however, mght be assocated wth the restructurng and dfferent llustratons of Equatons (5) and further assumptons concernng the optmzatons method. Furthermore, the analyss focus les on comparng global optmzaton algorthms. 4. NUMERICAL ANALYSIS Ths secton studes the applcaton of the four optmzaton algorthms descrbed n Secton 2 the Nelder-Mead smplex method, dfferental evoluton, genetc optmzaton and partcle swarm optmzaton to the 8 optmzaton problems llustrated n Secton 3. The analyss examnes whch algorthm s the most effcent when beng appled to the optmzaton problems wth respect to the algorthms ftness and computatonal ntensty. Furthermore, senstvty analyss concernng the number of populaton (dfferental evoluton and genetc optmzaton), swarm sze (partcle swarm optmzaton) and dfferent startng ponts (Nelder- Mead smplex) as well as dfferent ntal populatons, swarms or startng ponts for the optmzaton are conducted (see Secton 2.5, Equaton ()).

25 24 4. Input parameters The nput data for the optmzaton problems, llustrated n Equatons, (4) and (5), are gven n Gatzert and Kellner (20a) and Gatzert and Kellner (20b). 5 For the optmzaton problem dsplayed n Equaton, the same nput parameters as n Gatzert and Kellner (20a) are taken wth the excepton that the loadng of the rensurance contract s assumed to re, be 20%, 0.20, nstead of 0%, and the rsk averson parameter, whch s not neces- sary for the analyss n Gatzert and Kellner (20a), equals q 0.0. Furthermore, for the optmzaton problems accordng to Equaton and (4), the attachment pont of the ILW has ILW, to be n the range of 0 and 300, 0,300 A and for the optmzaton problems accordng to Equaton (4) and (5) the upper lmt for the shortfall probablty s set to 5% (see Gatzert and Kellner, 20b). Moreover, a Gauss copula s used to model the dependence structure for the optmzaton problems accordng to Equatons, (4) and (5), thus, assumng lnear dependences among all relevant rsk processes. Table 2 dsplays the nput parameters for each optmzaton algorthm. Whle the populaton or swarm sze, respectvely, for dfferental evoluton, genetc optmzaton and partcle swarm optmzaton s set to 50, we ntalze the Nelder-Mead smplex method wth 50 dfferent startng values, whch are drawn from a unform dstrbuton space. The number of generatons s set to 200 for dfferental evoluton and genetc optmzaton, whereas 2000 teraton steps are performed wthn the Nelder-Mead smplex method and partcle swarm optmzaton. The numbers for swarm and populaton sze, number of startng values, generatons and teraton steps are chosen wth the purpose to consttute comparable condtons for the analyss. For the Nelder-Mead method, the reflecton, expanson, contracton, and shrnk step factors are adopted from Lagaras et al. (998). A comparably small number 0.0 s chosen for the step sze wthn dfferental evoluton, as accordng to Prce, Storn and Lampnen (2006), effectve values are rarely hgher than.0, even f all postve values are permtted. The crossover constant s chosen based on Mullen et al. (20), whle the acceleraton constants for partcle swarm optmzaton are fxed followng Eberhardt and Sh (200). In case of the frst optmzaton problem (see Equaton ), n whch the analytcal soluton s avalable, we wrte a functon to calculate the objectve functon values, usng the statstcal software R, and apply the optmzaton methods to the correspondng functon. 5 The nput data for the correspondng reference contracts are gven n Table 6 and Table 7 n the Appendx.

26 25 Table 2: Input parameters for the optmzaton algorthms Nelder-Mead Smplex Method Number of startng ponts 50 Reflecton factor Expanson factor Contracton factor Shrnk step factor Number of teraton steps 2000 Dfferental Evoluton Populaton sze p 50 Number of generatons g 200 Step sze F 0.0 Crossover constant CR 0.50 Genetc Optmzaton Populaton sze p 50 Number of generatons g 200 Partcle Swarm Optmzaton Swarm sze pso s 50 Acceleraton constants c, c2 2.0, 2.0 Number of teraton steps 2000 For the optmzaton problems accordng to Equaton, (4) and (5), the numercal results are obtaned usng Monte Carlo smulaton (see Glasserman, 2008). For each analyss, 200,000 sample paths are used and concernng the analyss of the optmzaton algorthm s robustness, 20 dfferent sets of an ntal populaton, swarm or startng ponts, respectvely, are taken nto account (see Secton 2.5, Equaton ()). Each optmzaton run s calculated on an AMD Opteron Istanbul processor wth 2.60 GHz. 4.2 Results In a frst step, we analyze performance profles for the algorthms, whch llustrate the development over tme for the hghest objectve functon values found durng each optmzaton process,.e. when the hghest objectve functon value ncreases most durng the genera-

27 26 ton/teraton steps. Therefore, besdes the fxed number of generatons or teratons, respectvely, no further stoppng condtons are ncluded n the optmzaton procedure and each algorthm searches for the hghest objectve functon value untl the maxmum number of generatons or teratons s reached. Fgure 3: Performance profle plots for each algorthm n case of maxmzng / SP Dfferental Evoluton Genetc Optmzaton SP SP Tme(%) Tme(%) Nelder-Mead Smplex Partcle Swarm Optmzaton SP SP Tme(%) Tme(%) Note: The performance profle plots llustrate the development of the hghest objectve functon values found durng the tme that s needed overall for the maxmum number of teratons or generatons. Fgure 3 exemplarly llustrates the performance plots for the maxmzaton of / SP dependent on. 6 Even though genetc optmzaton seems to be faster wth respect to RBC 6 In the followng, the varable(s) used for the optmzaton are gven n superscrpt of the objectve functon.