The Merits of Pooling Claims Revisited

Transcription

1 The Merts of Poolg Clams Revsted Nade Gatzert, Hato Schmeser Workg Paper Char for Isurace Ecoomcs Fredrch-Alexader-Uversty of Erlage-Nürberg Verso: August 2011

2 1 THE MERITS OF POOLING CLAIMS REVISITED Nade Gatzert, Hato Schmeser ABSTRACT Deftos of poolg effects surace compaes may covey the mpresso that the acheved rsk reducto effect wll be beefcal for polcyholders, sce typcally a) lower premums are pad for the same safety level wth a creasg umber of sureds, or b) a hgher safety level s acheved for a gve premum level for all pool members. However, ths vew s msleadg ad the purpose of ths paper s to reexame ths apparet mert of poolg from the polcyholder s perspectve. Ths s acheved by comparg several valuato approaches for the polcyholders' clams usg dfferet assumptos of the dvdual polcyholder s ablty to replcate the cotract s cash flows ad clams. The paper shows that the two cosdered deftos of rsk poolg do ot offer sght to the questo of whether poolg s actually beefcal for polcyholders. JEL-Classfcato: D46; G13; G22 Keywords: Rsk Poolg, Theory of Rsk, Rsk Valuato 1. INTRODUCTION Rsk poolg surace compaes s ofte referred to as the producto law of surace. Artcles ad stadard textbooks o surace ad rsk theory may thereby covey the mpresso that rsk poolg surace compaes (group balace cocept) geerates a addtoal value for polcyholders. The reaso for ths ca be outled as follows. Actuarally calculated premums are usually gve by the value of expected losses plus a rsk premum, or safety loadg, to acheve a gve safety level for the portfolo of the sured. Hece, wth a creasg commuty of sureds, the actuarally calculated dvdual premum geerally decreases, whle smultaeously, the surace compay s safety level remas costat. O the other had, for a gve dvdual premum, a creasg umber of pool members mples Nade Gatzert s at the Uversty of Erlage-Nürberg, Char for Isurace Ecoomcs, Lage Gasse 20, D Nürberg. Emal: ade.gatzert@wso.u-erlage.de. Hato Schmeser s at the Isttute of Isurace Ecoomcs, Uversty of St. Galle, Krchlstrasse 2, CH-9010 St. Galle. Emal: hato.schmeser@usg.ch.

3 2 that the ru probablty coverges to zero,.e., a creasg safety level the pool ca be acheved (see e.g., Smth ad Kae (1994), Albrecht (1982, 1990) for both deftos of rsk poolg). The purpose of ths paper s to reexame ths apparet mert of poolg also called group balace cocept for polcyholders by aalyzg ths ssue more depth. To detfy whch stuato poolg s beefcal for polcyholders, we compare several valuato approaches wth dfferet assumptos o a dvdual polcyholder s ablty to replcate the cotract s cash flows ad clams. The key pot ths aalyss s, thus, the questo to what extet dvdual polcyholders ca acheve advatages from dversfcato obtaed at the compay s level. I the lterature, several partcular aspects of rsk poolg have bee cosdered. For stace, cases are examed whch poolg effects, ad thus rsk dversfcato, are acheved depedg o the loss dstrbuto, depedece structure, umber of pool partcpats, rsk measures ad premum calculato schemes (see, e.g., Albrecht, 1982, 1984; Zgehor, 1990; Cumms, 1991). Furthermore, postve safety loadgs for premums the pool are derved based o the ru probablty by argug that the surer wll certaly become solvet f the premum s based o the expected loss oly (see Bühlma, 1996). I a smlar settg to the preset paper, Borch (1990) shows that a chage the premum level a pool wth homogeous rsks does ot fluece the utlty the specfc case of µ/σ prefereces from the surer s perspectve the cotext of poolg. Cumms (1991) llustrates that the case of depedet ad detcally dstrbuted rsks, the surer s total buffer fud, ecessary to esure a gve safety level (e.g. a acceptable ru probablty), goes to fty wth a creasg umber of pool members, whle the requred buffer for each polcy goes to zero. Ths mples that a polcy premum approxmately equal to the expected loss should be suffcet to provde the desred safety level. The buffer fud could be composed of, for example, equty captal or, for a oe-perod mutual surer, be provded by polcyholders. However, Cumms (1991) does ot explctly evaluate the polcyholder ad shareholder clams ad, furthermore, does ot focus o the questo as to whether rsk poolg per se, or the fact that the polcy premum ca be reduced wth a creasg umber of polcyholders, s of value for the sured. The am of ths paper s thus to exted ad combe prevous work by focusg o the merts of poolg clams (usg the two deftos above) from the polcyholder s perspectve usg dfferet valuato approaches. The valuato approaches ad observatos are summarzed as

4 3 follows. The artcle cosders a mutual surace compay where polcyholders are both debt ad equty holders. Startg wth actuaral prcg, a premum s calculated for a gve safety level determed by a fxed ru probablty. Whe payg the premum, polcyholders also acqure a clam o the surplus (excess of asset value over clams paymets). Hece, they possess a shareholder stake as well as a debt holder posto. The descrpto of poolg effects such a compay may gve the mpresso that the group balace cocept wll, geeral, be beefcal for polcyholders, sce lower premums are pad for the same safety level wth a creasg umber of sureds. However, ths vew s msleadg as the polcyholder s value of clams wll strogly deped o the model framework, e.g., o dvdual prefereces or polcyholders ablty to replcate the cotract s cash flows. If polcyholders ca fully replcate cash flows, the value of ther clams postos ca be evaluated usg a facal prcg approach assumg that requremets spag, formato, o arbtrage, ad compettvty are satsfed. Wth a creasg umber of sureds, the decreasg premum ca be separated a decreasg equty posto value ad the value of the debt holder posto, whch approaches the preset value of losses. Overall, the value of equty ad debt postos calculated usg preset value wll sum up to the tal premum pad by the polcyholder for all, ad oly the partto betwee equty clams ad debt holder s clams s altered. Thus, ths settg, o addtoal value s geerated through dversfcato o the compay s level as t ca equvaletly be acheved by polcyholders o a dvdual level such a model setup. Poolg may mply addtoal value from the polcyholder s perspectve, f replcato or dversfcato s ot achevable for polcyholders as t s for the surace compay (see Borch, 1990). To exame ths case, we addtoally focus o a dfferet resultat valuato approach ad cosder the case where dvdual polcyholders caot dversfy at all. We use the cocept of utlty fuctos to aalyze whether ths case poolg s of value from the perspectve of the polcyholder. Results the deped o dfferet assumptos of tal wealth ad degree of rsk averso ad llustrate that a creasg umber of pool members ad the aïve dversfcato s cotrast to the facal approach beefcal for the polcyholder. We ca further hghlght that the premum level both model frameworks (.e., wth or wthout the possblty to replcate further cash flows) plays o role possble advatages from rsk poolg. More precsely, the fulfllmet of codtos uder whch the two deftos of rsk poolg hold true fxg the safety level ad reduced premums or fxg

5 4 the premum ad creasg the safety level for a creasg umber of pool members do ot provde ay formato o whether poolg s actually beefcal from the polcyholder perspectve or ot. The paper s structured as follows. Secto 2 provdes a dscusso of prevous lterature ad Secto 3 presets the base case wth the dvdual ad collectve perspectve o rsk poolg clams. A aalyss of the merts of poolg clams from the polcyholder s perspectve s coducted Secto 4, ad Secto 5 cocludes the paper. 2. LITERATURE REVIEW Ths secto provdes a overvew of prevous lterature o poolg clams ad summarzes ther ma results. I partcular, as descrbed above, two deftos of rsk poolg have bee extesvely dscussed the lterature. Both deftos are based o actuarally calculated premums that are gve by the value of expected losses plus a safety loadg to acheve a gve safety level for a pool of cotracts. I Case A, dversfcato effects mply that the dvdual premum decreases wth a creasg commuty of sureds the pool whe fxg the ru probablty. I Case B, the ru probablty decreases f the dvdual premum s fxed. Smth ad Kae (1994) preset ad dscuss both Cases A ad B. They show, usg examples, that poolg s beefcal for polcyholders wth respect to the ru probablty of the pool. However, a evaluato of the polcyholders cotract cash flows s ot coducted ths cotext. I regard to Case A, Wllam, Smth, ad Youg (1995, pp ), as well as Smth ad Kae (1994), further clarfy that poolg effects do ot costtute a ecessary precodto to coduct surace busess, as log as the surace seller holds suffcet equty captal relatve to the maxmum loss. I ths case, dvdual rsks ca also be sured wthout poolg. Dayk, Petkaïe, ad Pesoe (1994, pp ) descrbe Case B, whch s also aalyzed Powers, Veeza, ad Juca (2003), wth a sgle-perod ru probablty model from a regulator perspectve, ad show how to mprove rsk maagemet programs. They derve ecessary ad suffcet codtos for the ru probablty to coverge to zero ad aalyze requremets for the ormal approxmato for dfferet assumptos wth respect to

6 5 the surer s captal supply, the uderwrtg proft loadg, ad adverse selecto. 1 I ths cotext, they cosder the case of parameter ucertaty ad errors parameter estmato ad thus exted prevous work by Veeza (1983), where poolg effects ad safety captal are defed depedg o the characterstcs of ucertaty measured by the varace of the estmates, such that the surer acheves a safety level wth a gve relablty. To aalyze these ssues, Veeza (1983) apples a ormal power method ad calculates the surer s facal effcecy, whch s measured by the requred safety captal per sured rsk (correspodg to the safety loadg o the premum) order to acheve a fxed safety level. Case B s also aalyzed Veeza (1984), whereby rsk poolg of heterogeeous (ot detcal but depedet) rsks s derved order to crease the surer s facal effcecy, whch s beefcal by reducg captal requremets. The artcle further focuses o the questo of far premums by meas of prce dscrmato ad the dstrbuto of equty f groups of rsks are ot detcal ad pooled wth oe sgle surer, as compared to surg each group by a dfferet surer. Beard, Petkaïe, ad Pesoe (1984) also study Case B ad show that for a fxed ru probablty, the safety captal decreases for a creasg umber of rsks the pool, thereby partcularly focusg o the et reteto, whch ths settg s c. p. hgher for larger compaes. Based o Housto (1964), Cumms (1974) exames depedet ad detcally (ormally) dstrbuted rsks ad a buffer fud, whch serves to cover losses that exceed the expected losses ad to esure a gve safety level (e.g. a acceptable ru probablty). The buffer fud per sured rsk s show to decrease for a creasg umber of rsks the pool, whch s also the case for o-homogeous, postvely correlated rsks. Cumms (1991) exteds ths work ad for Case B demostrates that the case of depedet ad detcally dstrbuted rsks, the total buffer fud goes to fty wth a creasg umber of rsks, whle the requred buffer for each dvdual goes to zero, whch, as smlarly stated by Veeza (1983, 1984), s geerally cosdered as beefcal. I the cotext of tax aspects of captve surace, Porat ad Powers (1999) defe both Cases A ad B ad clarfy respect to Case A that premums ad captalzato are forced by market codtos rather tha beg drve by the classcal deftos of rsk poolg. Other work cludes Helma (1988), where wth a fte plag horzo, a gve fte reserve requres the premum to have a postve loadg to avod a ru probablty of 1 The authors also pot out that the sole cosderato of the ru probablty oly accouts for the uderwrtg proftablty, thus eglectg the asset sde.

7 6 100 %. The approach s based o Cramér (1955) (see also Bühlma (1996, pp. 141 ff.) ad Straub (1997, S. 37 ff.)). Furthermore, Powers (2006) dscusses the emprcal observato that rsk poolg s ot beefcal for larger compaes (measured by premum volume), eve though, theoretcally, the premum-surplus-rato should crease accordg to the law of large umbers. Powers (2006) argues that ths effect mght be due to dsadvatages rsk selecto, as growth s accompaed wth a uderwrtg of bad rsks. Brockett (1983) emphaszes the correct use of the cetral lmt theorem justfcato whe calculatg the ru probablty of a creasg umber of depedet ad detcally dstrbuted rsks a pool, whch represets a large devato probablty problem. I partcular, the sum of depedet ad detcally dstrbuted rsks s approxmately ormally dstrbuted for a fxed large umber of rsks oly uder certa codtos for a creasg umber of rsks. Wth respect to the beefts of rsk poolg from the surer s perspectve, Damod (1984) shows uder whch codtos a crease the umber of rsks rases the surer s utlty, thereby also demostratg that the rsk premum per sured decreases. Furthermore, Borch (1990) demostrates that the case of rsk poolg, the premum calculato does ot fluece the surer s utlty case of µ/σ prefereces. Deut, Eeckhoudt, ad Meegatt (2010) exame the beefts of rsk poolg accordg to Case A, referrg to Smth ad Kae (1994). They assume that the surer s Beroull rsk-averse ad, thus, requres a premum loadg o the expected loss. Cosequetly, prevous work focuses maly o the deftos ad ecessary codtos for rsk poolg wth respect to the surer s safety level as well as the surer s utlty of poolg. Some papers suggest that poolg s beefcal for the surer sce the requred safety captal per sured rsk decreases for a creasg umber of pool members. Whle poolg ca geerally be beefcal order to satsfy regulatory requremets, for stace, ths terpretato caot be affrmed from the fdgs of captal market theory, as costs of captal oly arse due to o-dversfable rsks, whle dversfable surace rsks are ot relevat for prcg. Thus, to assess whether poolg s beefcal for stakeholders ad partcularly polcyholders, future cash flows should be explctly evaluated depedg o assumptos o the polcyholder s ablty to replcate ad dversfy. I ths cotext, both polcyholder ad shareholder postos must be take to cosderato.

8 7 Clearly, f polcyholders ca fully dversfy ad replcate ther rsks, a facal termedary s ot ecessary ad o addtoal value ca be geerated from poolg. Ths may dffer f replcato s ot fully possble, thus usg a preferece-depedet valuato method. Furthermore, the valuato process, the shareholder stake must be cosdered to esure a far stuato (see Veeza, 1984) wthout arbtrage opportutes. Whle the kd of premum calculato scheme (pre-, post-, or mxed fudg) ad the amout of the premum do play a role wth respect to the deftos of poolg (Cases A ad B), t has o effect o the value of the cash flows, f polcyholders receve ther clams ad the remader s dstrbuted equally amog all homogeous rsks ( case of a mutual surer; smlar argumets hold for a stock surer). I summary, the two deftos of poolg do ot provde a clear dcato, or ecessary codtos, o whether poolg (ad surace) s actually beefcal for polcyholders. To aswer ths questo, the cash flows eed to be comprehesvely evaluated. 3. POOLING CLAIMS: THE BASE CASE 3.1 Clams ad premums the base case: Idvdual ad collectve perspectve As a frst step, we cosder the base case, where rsks ( exposure uts) resultg from polcyholders are pooled a portfolo wth a specfed reportg perod (e.g., oe year wth t = 0, 1). 2 The umber of rsks wth the portfolo s determstc. A cetral prerequste the aalyss of poolg effects are the assumptos o the dstrbuto ad depedece structure of rsks to assess the dstrbuto of a sum of rsks. As t s doe ths cotext by, e.g., Cumms (1991), we assume that the clam szes of the rsks are depedet ad detcally dstrbuted (..d.), followg a ormal dstrbuto. Let X deote the ormal dstrbuted clam sze of rsk (wth = 1,,) at tme t = 1. The stochastc total clam amout S at tme t = 1 the pool cosstg of rsks s ormally dstrbuted ad gve by = =1 S X. 2 For the base case ad actuaral rsk poolg geeral, see, e.g., Beard et al. (1984), Kaas et al. (2001), Straub (1988).

9 8 To cover the clams wth the pool, the surer collects premums at tme t = 0 for each rsk take. The dvdual premum s calculated based o a actuaral premum prcple ad, thus, the premum for rsk s gve by the expected clam per rsk ad a safety loadg c > 0: ( ) π = E X + c, where E(. ) stads for the expected value. I what follows, we assume that tal cotrbutos the pool are compouded wth a rsk-free rate of r = 0 %. The collectve premum of the pool ca be calculated by ( ) ( ) π π = E X + c = E S + c =, = 1 whch, due the assumpto of detcally dstrbuted rsks, correspods to the umber of rsks tmes the dvdual premum. 3.2 The effect of poolg clams Whe studyg the effects of poolg clams for a creasg umber of rsks, the choce of a sutable rsk measure s crucal. I the followg, we cosder the ru probablty,.e., the probablty that the total premums collected the pool are ot suffcet to cover the total clams occurred at tme t = 1. I geeral, oe ca dstgush two dfferet approaches whe aalyzg poolg effects for a creasg umber of rsks the pool. Dversfcato effects ca ether arse wth a reduced premum for a gve safety level of the pool (Case A, hereafter), or case premums are fxed ex ate wth c > 0 wth a reducto the ru probablty wth the pool (Case B). Case A Fxed ru probablty I the frst case, the effect o the dvdual premum ecessary to esure the desred ru probablty ca be studed whle requrg that the ru probablty R of the pool remas fxed at a gve level R = ε:!! ( ) ( ) ( ) ( ) R = P S > π = ε P S > E S + c = ε.

10 9 Uder the gve assumptos, the total clam amout S the pool follows a ormal dstrbuto ad, thus, the ru probablty ca be wrtte as ( ) + ( ) ( ) σ ( S ) ( ) ( X ) E S c E S c! R = 1 N = 1 N = ε, σ where N deotes the dstrbuto fucto of a stadard ormal dstrbuto. Ths s equvalet to dervg c for a gve umber of rsks from σ ( ) ( X ) c ( ) ( X ) 1 ε = z1 ε c =, z σ where z1 ε deotes the (1 ε)-quatle of the stadard ormal dstrbuto. For a creasg umber of rsks the pool, the dvdual premum coverges towards the expected clam per rsk E ( X ) for a fxed ru probablty ε. I case c s postve, 3 creases wth a creasg umber of rsks the pool, ad hece c ad thus π ca be lowered. 4 Thus, gve the assumpto used ths secto, potetal merts of poolg are ofte formulated the followg way: gvg a costat safety level (1 ε), surace ca be provded for each pool partcpat at a cheaper rate f the umber of partcpats the pool creases. Case B Fxed premum Alteratvely, oe ca ex ate fx the dvdual premum wth some postve safety loadg c > 0. I ths case, the ru probablty coverges to zero for a creasg umber of rsks the pool: ( ) + ( ) ( S ) ( X ) E S c E S c R = 1 N = 1 0 σ N. σ 3 4 Uder the assumed ormal dstrbuto for X, the ru probablty ε s lower tha 50% for all. See also Cumms (1991, p. 268).

11 Dscusso regardg Case A ad Case B Besde the dscussed cases, hybrd forms of Cases A ad B ca be foud (see, e.g., Zgehor, 1990). I partcular, premum prcples ca be derved that lead to decreasg dvdual premums (covergg towards the expected clam per rsk) for a creasg umber of rsks ad to a cotuous mprovemet of the ru probablty R wth R 0. The poolg cocept suggests that uder the gve assumptos, polcyholders beeft from poolg clams surace compaes. More precsely, gve a fxed probablty ε < 50% for the cotract fulfllmet, the premum decreases wth a creasg umber of polcyholders (see Case A). Or, as Case B, the probablty of a fulfllmet of the cotract creases gve a fxed premum (wth c > 0) ad a creasg umber of rsks the portfolo. However, ths cotext, the possblty that the customers may ot wat to purchase surace (here: partcpate a homogeous pool wth 2) the frst place s ot cosdered. I addto, the shareholder posto s ot take to accout ad s, hece, ot evaluated, eve though the pool s solvet wth probablty 1-ε. I the case of a mutual surer, the remag surplus s, our case, owed by polcyholders ad, hece, should be dstrbuted to them. I the case of a stock surer, a group that partcpates the surplus wthout tal cotrbuto s barely cocevable from a ecoomc pot of vew, as ths would mply a clear arbtrage opportuty. 5 Furthermore, poolg effects as descrbed above suggest that the type of premum calculato s mportat for poolg. For stace, the safety level c eeds to be postve to acheve the poolg effects descrbed above. However, what follows, we compare dfferet valuato schemes to llustrate that merts of poolg from the polcyholders perspectve uder the gve assumptos do ot deped o the way premums are calculated. Eve more mportatly, the merts of poolg clams as defed Case A ad B do ot provde ay formato regardg possble advatages polcyholders may face whe rsk poolg s coducted. Thus, the followg secto ams to study ths ssue by evaluatg the respectve clams. 5 See Cumms (1991, pp ).

12 11 4. AN ANALYSIS OF MERITS OF POOLING CLAIMS FROM A POLICYHOLDER'S PERSPECTIVE 4.1 The polcyholder s startg pot The value of poolg from the polcyholder s perspectve ca be derved dfferet ways ad depeds o uderlyg assumptos such as the polcyholder s dversfcato opportutes. I the followg, we cosder the settg of a mutual surer, where polcyholders are also owers of the surace frm. 6 The polcyholder s wealth be descrbed as follows: ( 1 ) π ( 1 ) W, = 1,,, at tme t = 1 ca thus W = A + r X + r + I + E. (1) Here, A represets the tal captal of the polcyholder at tme t = 0, whch s compouded wth the rsk-free rate of retur. For reasos of smplfcato, we aga assume r = 0 % ad, thus, omt r what follows. The value of the vestmet at tme 1 s reduced by the stochastc clam X ad by the compouded premum addto, the wealth s creased by the demty paymet as well as by the shareholder stake the rght had sde of Equato (1) are stochastc. π pad at tme zero to sure agast losses. I I gve by the surace cotract E for the surplus clam. The varables X, I, ad E o I the followg, we aalyze the effect of poolg o the wealth posto of the polcyholder. A polcyholder also has the choce ot to partcpate poolg ad, hece, ot to purchase surace. I ths case, the wealth at tme t = 1 s gve by W = A X. 4.2 Cosderatos a frctoless ad effcet market I a frctoless ad effcet captal market, the dvdual s able to replcate all future cash flows by meas of captal market strumets. I ths settg, poolg effects may occur as descrbed by the crtera A ad B ad examples lad out Secto 3. However, for the wealth posto of the polcyholder a far settg ( the sese of a arbtrage-free valuato), rsk poolg effects more precsely, the reducto of usystematc rsk have o relevace. Far meas, ths cotext, that the tal surace premum s equal to the preset value 6 Note that the followg argumets aalogously hold for a stock surer.

13 12 PV of future cash flows cosstg of the stochastc demty paymet I ad the shareholder clam E : ( E ) π = PV I +. (2) Whe determg the polcyholder s demty paymets, possble default has to be take to accout by subtractg the default opto value from the preset value of the clams to accout for the cases where the surer s ot solvet ad thus ot able to fully cover all labltes of the polcyholders the pool. As Secto 3, default s defed as the case where the total premum come π s ot suffcet to cover the total losses S the pool. Hece, uder far codtos, for the polcyholder s demty paymet t holds that 1 PV ( I ) = PV ( X ) PV ( max [ S π,0]). (3) Thus, for every partcpat the pool, the preset value of the dvdual clams s reduced by oe-th of the preset value of the default opto the pool. X Regardg the shareholder clam, the remag surplus (premums less clams) s also dstrbuted equally to the pool members f the pool s solvet: 1 PV ( E ) = PV ( max [ π S,0] ). (4) The requremet of a far valuato descrbed Equato (2) s fulflled, as ca be see whe usg the fact that max ( S, π ) S max ( π S, 0) π max ( S π,0) = + = + to obta ( + ) = ( ) + ( ) PV I E PV I PV E 1 1 = PV X PV S + PV S 1 = PV ( X ) + PV ( π S ) 1 = PV X + = π. ( ) ( max [ π,0]) ( max [ π,0]) ( ) π PV ( X ) = 1 (5) The last trasformato Equato (5) leads to the dvdual premum pad by all polcyholders ad s vald wheever the same preset value calculato s used for all rsks.

14 13 Equato (5) further represets the specal case whch the polcyholders are rsk-eutral ad oly expected values matter. More precsely, ths case we have ( ) ( ) PV I + E = E I + E = π. (6) Gve Equatos (5) ad (6) respectvely, the polcyholder s dfferet as to whether he or π = + c ) does she purchases surace or ot. I partcular, the safety loadg c (wth E ( X ) ot fluece the wealth posto of the polcyholder. The polcyholder stays dfferet for postve or egatve values of c. Recosderg Case A Fxed ru probablty Sce the ru probablty, R, of the pool s fxed to ε ad the safety loadg for each polcyholder s postve ( c( ) > 0), the dvdual premum π wll decrease for a creasg umber of rsks the pool. Hece, the dvdual premum s a fucto of. Replacg the total premums ad aggregate losses Equato (3) leads to 1 1 PV ( I ) = PV ( X ) PV max X ( E ( X ) + c( ) ),0 = 1. (7) For a creasg umber of..d. rsks (a radom sample from a probablty dstrbuto X wth fte mea ad varace), the law of large umbers mples that 1 lm P X µ < ε = 1 = 1 for E ( X ) = µ, = 1,, (see, e.g., Cumms, 1991). Sce c depeds o Case A ad s decreasg to zero for large, the secod part of the rght had sde of Equato (7).e., the polcyholder s part of default put opto value of the pool coverges to zero. The preset value of the demty paymet wll reach the preset value of the dvdual clam as. Aalogously, the preset value of the shareholder clam, 1 1 PV ( E ) = PV max ( E ( X ) + c( ) ) X,0 = 1, (8)

15 14 coverges to the loadg c(), whch tur decreases to zero as the umber of rsks the pool creases for a gve ru probablty. Therefore, case of defto A, the preset value of the demty paymet coverges to the preset value of the dvdual clam ad the preset value of the shareholder clam becomes worthless for the polcyholder for large. Hece, for large, the wealth posto of the polcyholders the pool becomes rsk-free. To llustrate ths theoretcal observato, Table 1 cotas a umercal example. Let E ( X ) c( ) s fxed to 99 % (hece, R P( S π ) 1% dstrbuted wth E ( X ) = 30 ad ( X ) hece, PV ( ) = E ( ). π = + deote the premum for each pool member. The safety level of the pool Table 1: Premums = > = ). Clams are depedet ad ormally σ = 10, r = 0, ad we assume a rsk-eutral market; π ad preset values of payouts PV ( I ) ad ( ) clams for a gve ru probablty of 1 % (Case A fxed ru probablty) PV E for poolg π c( ) PV ( I ) PV ( E ) ( + E ) PV I Table 1 shows that the premum per partcpat pad to the pool decreases for a creasg umber of partcpats. 7 Wth c( ) > 0, the poolg effect of Case A descrbed Secto 3 s fulflled. Nevertheless, o addtoal value s created for the polcyholders through a creasg umber of pool partcpats sce PV ( I E ) π = + holds true for ay value of. Recosderg Case B Fxed premum I Case B, the premum ad thus c s fxed ad does ot deped o. Hece, for a creasg umber of rsks, the default opto value Equato (7) coverges to zero, sce the loadg c s assumed to be postve. I addto, the value of the shareholder clam coverges to c. 7 See also Cumms (1991) for a smlar example.

16 15 Table 2 gves a llustrato for a fxed loadg of c = 0.5 for each polcyholder. As Table 1, clams are depedet ad ormally dstrbuted wth E ( X ) = 30 ad ( X ) rsk-free rate of retur r s aga set to zero ad a rsk-eutral market s assumed. σ = 10. The Table 2: Ru probablty of the pool R, preset values of payouts PV ( I ) ( ) case of poolg clams for a gve safety loadg c = 0.5 (Case B fxed premum) PV E for the π c R = P( S > π ) % % % % % % PV ( I ) PV ( E ) ( + E ) PV I Comparso of Cases A ad B I both Cases A ad B, the preset value of the polcyholder s total wealth at tme t = 1, PV ( W ), remas uchaged for all premum levels as log as PV ( I E ) for all polcyholders the pool. Ths results from Equato (1), where PV ( π ( 1 r) ) cacels PV ( I E ) π = + holds true + = π π = + out, ad A ad X rema uchaged. The wealth posto of a polcyholder could oly be mproved f the premum s below the preset value of future ( ) payouts, ( ) ( ) π < PV I + E E X + c. However, such a premum prcple for some polcyholders would be a dsadvatage for other polcyholders the pool. Sce uder ths valuato prcple, dvduals ca perfectly dversfy ad replcate future cash flows, poolg, or, more precsely, dversfcato of usystematc rsk, does ot offer ay addtoal beeft. Clearly, o addtoal value ca be geerated by meas of poolg ad thus, o reaso for the exstece of surace sttutos ca be establshed such a cotext. I partcular, t s ot relevat for polcyholders whether poolg effects as defed Case A or Case B exst uder the assumed settg. For stace, a calculato of a premum accordg π = + c wth c < 0 s stll far from the polcyholder s perspectve as log as to E ( X ) PV ( I E ) π = + holds (see Equato (5)), eve though t does ot lead to a poolg effect as descrbed Case B. Ths s llustrated a thrd umercal example provded Table 3 wth the put data from Table 2 ad a safety loadg of c = 1.

17 16 Table 3: Premums π ad payouts ( I premum level per of π = (Case B fxed premum) + E ) for the case of poolg clams for a fxed π c R = P( S > π ) % % % % % % PV ( I ) PV ( E ) ( + E ) PV I Because the safety loadg c s ot postve, poolg effects accordg to Case B as defed Secto 3 do ot exst ad the surace compay becomes solvet wth certaty for large. 8 However, a polcyholder usg the preset value s stll dfferet wth respect to a partcpato the pool. Clearly, the level of c does ot play a role as log as the codtos are the same for all partcpats ad hece, all payoffs after payg the clams are homogeously dstrbuted to the polcyholders t = The case of rsk-averse polcyholders I cotrast to the prevous secto, what follows, we assume a complete market settg whch polcyholders are ot able to replcate future cash flows wth gve market strumets. Hece, a preferece-depedet valuato s requred. Preferece-depedet valuato I the followg, we assume that the polcyholder has µ / σ -prefereces. The preferece fucto Φ of the polcyholder s wealth posto a 2 Φ = E ( W ) σ ( W ), 2 W at tme t = 1 ca be wrtte as wth a > 0 deotg the rsk averso parameter. I the followg, we ca see that ths settg, t s ot ecessary to dstgush betwee the rsk poolg deftos accordg to Case A ad Case B. 8 See also Bühlma (1996) ad hs reasog of a postve safety loadg.

18 17 As ca see from Equatos (5) ad (6), the premum prcple E ( X ) mples E ( I E ) π = + c (c R ) π = +. 9 Hece, the expected wealth t = 1 of the polcyholder depeds ether o the value of the safety loadg c or o the umber of the poolg partcpats: ( ) = ( ) π + ( + ) = ( ). E W A E X E I E A E X Thus, the expected wealth of the polcyholder s ot flueced by the purchase (or ot) of surace. Hece, for the questo of whether the utlty level Φ ca be creased va rsk W 2 poolg, a aalyss of ( ) σ s suffcet. 2 2 If o surace s purchased, σ ( W ) σ ( X ) oe obtas σ ( W ) σ ( X I E ) = W 2 I ay case, ( ) =. If rsk poolg (ad surace) s chose, σ does deped o the umber of pool members but ot o the premum prcple. Ths ca be show by usg the result Equato (5). Sce we have 1 I + E = X + ( S ) π, the varace of the wealth of the polcyholder t = 1 ca be wrtte as σ σ σ π ( W ) = ( X + I + E ) = X + X + ( S ) = σ ( π S ) = σ E ( S ) + c X = = σ = 1 2 = σ ( X ). 2 2 X 2 σ = 1 = 1 ( X ) (10) 9 Ths does ot oly hold true case of a rsk-eutral polcyholder, as the premum ca be rewrtte usg the rsk adjustmet R adj (that s ot flueced by chages c) as follows:! ( + ) = π = ( ) + ( ) adj ( ) * E ( I E ) E ( X ) c R E ( X ) c PV I E E X c E I + E + R = E X + c + = + = +. adj For stace, usg the Captal Asset Prcg Model, the rsk adjustmet R adj s gve by ( λ stads for the market prce of rsk, M1 deotes the value of the market portfolo t = 1) ( 1 ) R = λ cov I + E, M. adj A chage the safety loadg c wll chage c* to the same amout; however, R adj s ot affected.

19 18 Wheever rsk poolg s coducted, t holds that σ 2 ( X ) 2 I E σ ( X ) 2 dversfcato (for 2 ). For large, ( ) the polcyholder becomes determstc. 10 W + + < due to aïve σ coverges to zero, ad the wealth posto of Thus, the utlty level ca be creased wth a creasg umber of pool partcpats for rsk-averse polcyholders. Aga, the exstece of rsk poolg effects accordg to Case A ad Case B as defed Secto 3 do ot provde a ht wth regard to possble merts of poolg, sce the way premums are defed do ot play a role, ceters parbus, wth respect to the polcyholder s utlty Φ. The followg example llustrates ths pot. We employ the same put data as used the umercal example before (see Tables 1 to 3, E ( X ) = 30, ( X ) σ = 10, r = 0) ad set the polcyholder s rsk averso parameter to a = 2. The tal wealth of the polcyholders s gve by A = 500. Table 4: The polcyholder s utlty Φ depedg o the umber of rsks the pool Φ Table 4 shows that for large, the dvdual polcyholder s wealth posto t = 1 almost ( ) becomes rsk-free ad coverges to the maxmum possble utlty of 470 A E ( X ) =. The utlty s ot flueced by the premum paymets π ad eve remas uchaged the extreme case where o upfrot premums paymets are requred ( π = 0 wth = = ). For postve values of c, the requremets of Case B ca be fulflled; c E ( X ) 30 however, ths has o relevace from the perspectve of the polcyholder ths cotext. I summary, t ca be show that at least the model frameworks cosdered, the classcal defto of merts of poolg s ether a suffcet or a ecessary codto for the exstece of postve effects va rsk poolg for the polcyholder. 10 Borch (1990, p. 85) shows a resurace cotext that, ceters parbus, a chage the premum level does ot fluece the utlty of a surace frm poolg rsks.

20 19 5. SUMMARY I ths paper, we studed two specfc deftos of rsk poolg that are extesvely examed the lterature. Frstly, we cosdered the case where, whe fxg the safety level usg the ru probablty, actuarally calculated premums ca be reduced for a creasg umber of pool members, ad, secodly, the case where the premum level s fxed ad the ru probablty goes to zero wth a creasg pool sze. Both deftos mply a seemg beeft of rsk poolg for the polcyholder, whch ca be msleadg. Therefore, ths paper, we revst the merts of poolg by focusg o the polcyholder s perspectve the case of a mutual surer usg dfferet valuato approaches, thereby also takg to accout both stakes of the polcyholder (shareholder ad debt holder posto). The fudametal dfferece the valuato approaches s ther assumpto of a dvdual polcyholder s ablty to replcate the cotract s future cash flows. We pot out that f polcyholders ca fully replcate cash flows, the value of ther clams postos ca be evaluated usg a preset value approach whch the decreasg premum for a creasg pool sze ca be separated a decreasg preset value of the shareholder clam ad the preset value of the demty paymet, whch decreases towards the preset value of the loss. Overall, however, the value of equty ad debt postos always sums up to the tal cotrbuto by the polcyholder ad oly the partto betwee shareholder ad debt holder s clams s altered. Furthermore, the premum level for each partcpat to be pad up-frot the pool does ot play a role for the polcyholder s wealth posto. Hece, ths valuato framework, o addtoal value s geerated through dversfcato o the compay s level as t ca be acheved equvaletly by polcyholders o a dvdual level. However, poolg does mply addtoal value from the polcyholder s perspectve f replcato or dversfcato s ot achevable for polcyholders as t s for the surace compay ad valuato ca be coducted based o utlty fuctos. I ths case, poolg s beefcal for rsk-averse polcyholders sce the expected wealth remas uchaged, o matter whether surace s purchased or ot, whle the rsk decreases for a creasg umber of pool members. Aga, the premum level ad thus the deftos of poolg A ad B do ot play a role,.e. ths cotext, the polcyholder's utlty level s ot flueced by the amout of premums pad upfrot the pool as log as homogeous rsks are treated detcally.

21 20 I summary, the paper shows that the two cosdered deftos of rsk poolg caot provde sght regardg possble beefts for polcyholders ad that the merts of poolg eed to be aalyzed uder dfferet assumptos o polcyholders ablty of replcatg clams. Followg ths le of reasog, t s ot per se clear for whch ecoomcally relevat questo wth respect to surer/polcyholder decso makg the classcal deftos of rsk poolg allow a clear aswer. The cetral reaso for the cocluso that the classcal deftos of rsk poolg do ot provde formato whether poolg s beefcal s that the premum level ad, thus, the premum s safety loadg s ot oly rrelevat for the utlty of pool members the specfc case of µ/σ prefereces, as show by Borch (1990), but the case of all frameworks cosdered ths paper. Rather, premums ad captal structure ca geerally be cosdered to be drve by market forces (Porat ad Powers, 1999). Furthermore, the case of preset values, the polcyholder s dfferet as regards the solvecy level of the surer, as log as the premum pad correspods to the preset value of future payoffs. However, the reducto of solvecy rsk by meas of poolg ca be beefcal other cotexts, as a crease the safety level ca help surers satsfy regulatory requremets (see Powers, Veeza, ad Juca (2004), Veeza (1984)). REFERENCES Albrecht, P. (1982): Gesetze der grosse Zahl ud Ausglech m Kollektv Bemerkuge zu Grudlage der Verscherugsprodukto. Zetschrft für de gesamt Verscherugswsseschaft, 71: Albrecht, P. (1984): Ausglech m Kollektv ud Prämeprzpe. Zetschrft für de gesamt Verscherugswsseschaft, 73: Albrecht, P. (1990): Premum Calculato wthout Arbtrage?, ASTIN Bullet, 22: Beard, R. E., Petkäe, T., Pesoe, E. (1984): Rsk Theory. Lodo - New York: Chapma ad Hall. Brokett, P. L. (1983): O The Msuse of the Cetral Lmt Theorem Some Rsk Calculatos. The Joural of Rsk ad Isurace, 50: Björk, T. (2004): Arbtrage Theory Cotuous Tme. New York: Oxford Uversty Press. Borch, K. (1990): Ecoomcs of Isurace. New York: North-Hollad Press.

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23 22 Smth, M. L., Kae, S. A. (1994): The Law of Large Numbers ad the Stregth of Isurace. Isurace, Rsk Maagemet ad Publc Polcy: Essays Hoor of Robert I. Mehr (S. G. Gustavso ad S. E. Harrgto, eds.). Bosto: Kluwer: Straub, E. (1997): No-lfe Isurace Mathematcs. Berl: Sprger. Veeza, E. (1983): Isurer Captal Needs uder Parameter Ucertaty. Joural of Rsk ad Isurace, 50: Veeza, E. (1984): Effcecy ad Equty Isurace. Joural of Rsk ad Isurace, 51: Wllam, C. A., Smth, M. L., Youg, P. C. (1995): Rsk Maagemet ad Isurace, 8 th Edto. Bosto: McGray-Hll. Zgehor, U. (1990): Zur Modellerug des Ausglechs m Kollektv. Blätter der DGVFM, 19: