The Top-Down Approach to Calculation of the Insurance Premium

Transcription

1 The Top-Dow Approach to Calculatio of the Isurace Premium ojciech Otto, dr hab.; Faculty of Ecoomic Scieces, arsaw Uiversity. Itroductio Commercial price of isurace coverage fluctuates due to chagig market coditios. Premium calculatio produces respose to the questio of a idifferece price for the isurace compay sellig coverage below this price meas makig expected loss, sellig above this price meas makig excessive expected profit. Both situatios ofte happe i practice, but they usually do ot persist for a log time because of actios udertake i respose to them by actors of this game. So the premium calculatio problem ca be read as reflectig the supply side cosideratios. The premium calculatio requires to evaluate a umber of compoets: expected value of claims due to the cotract i questio, risk loadig ad profit margi, margi for expeses (acquisitio, settlemet, ad admiistratio costs), solidarity compoet. The first two compoets are the most specific to the isurace busiess, ad the risk theory focuses o them almost solely. It does ot mea that the margi for expeses is irrelevat. Ofte expese margi is set well above 5% of the commercial premium. However, typical problems with this compoet (allocatio of admiistratio costs to idividual cotracts, allocatio of iitial costs alog the lifetime of a product etc.) are similar to those ecoutered i ay commercial activity, o matter if we sell shoes, cars, or isurace policies. The solidarity compoet is much more specific to isurace it ca be ecoutered whe the isurace coverage is offered at the same price to two groups of risks that systematically differ. A good example is whe, by some reasos, life auities are offered to males ad females by the same premium. The the premium for males cotais the positive solidarity compoet ad that for females the egative solidarity compoet. Thus equal premium leads to redistributio from overpriced males to uderpriced females. Existece of such redistributio sometimes is eforced by law, ad the aim to recocile it with rules of market competitio poses may challegig problems. However, they are beyod the scope of problems cosidered i this article. 96 ekoomia 9

2 The Top-Dow Approach to Calculatio of the Isurace Premium Out of the first two compoets that are cosidered here, oe is o-cotroversial ad obvious it is the expected value of claims. It is the other oe risk loadig ad profit margi that is much less obvious, ad a bulk of literature is devoted to various approaches to it. Sometimes risk loadig ad profit margi are treated separately especially whe various risk measures are directly applied to evaluate idividual risks without takig the cotext of the whole portfolio of risks accepted by the isurace compay ito accout. This approach is especially valid whe we focus o the demad side. The so called top-dow approach explicitly states that the isurer is exposed to risk that stems from a idividual cotract oly as far as the cotract cotributes to the risk of the whole portfolio. Thus the approach etails two basic steps. First, the risk loadig formula for the whole portfolio is chose o the basis of risk ad retur cosideratios o the whole compay level. The the risk loadig has to be allocated to idividual risks. The first step is the phase whe the capital eeded to esure solvecy is assessed, ad so the required retur o that capital is take ito cosideratio. This is why the secod step is ofte called capital allocatio. Despite it, uder this approach it meas essetially the same as risk loadig allocatio. A good example is the case of a large decetralized compay where pricig (ratemakig, gratig rebates etc.) is to some extet delegated to regioal divisios, but the problem that remais to be solved by top maagemet is to allocate capital (expected cotributio to cover costs of capital) amog these divisios. The problem arises whe the pricig criteria applied at compay level lead to o-additive premium formulas. I particular, uder a umber of differet sets of assumptios the pricig formula with risk loadig proportioal to the stadard deviatio is obtaied. Despite the stadard deviatio, the priciple applied to the portfolio does ot lead directly to the balaced allocatio of the risk load to idividual risks, the well kow ad hoc solutio is to allocate the risk load proportioally to variaces. The aim of the article is to show that the game theory provides a ew justificatio to this old solutio.. Pricig the portfolio by the stadard deviatio priciple The stadard deviatio priciple applied to the premium for the whole portfolio reads: Π( ) = µ w + α w () where = is the aggregate amout of claims for the portfolio that cosists of idividual risks,,,, ad µ, are expectatio ad stadard deviatio of the portfolio. The simplest justificatio of this priciple is that it ca be derived from the more geeral quatile priciple: Π( ) = ( ε) F ekoomia 9 97

3 ojciech Otto which meas that we accept a situatio whe the premium will ot suffice to cover claims with (presumably low) probability ε, ad that the cumulative distributio fuctio F ca be well approximated by the ormal distributio. It is importat here to emphasize that the commo additioal assumptio of idepedecy of risks,,, is ot ecessary, as approximated ormality ca be achieved despite some weak depedecies, provided the umber of risks is sufficietly large. The parameter α of formula () equals q ε, the quatile of order ( ε) of the stadard ormal distributio. A more sophisticated justificatio of the stadard deviatio priciple takes ito accout the trade-off betwee risk ad retur o capital. The expected rate of retur o isurace operatios ca be represeted by the postulated equatatio formula: ( ) Π µ = ru where r deotes the risk premium rate (rate of retur i excess of the risk free rate i RF ), ad u deotes the amout of capital that serves to back the isurace risk (it is assumed that this part of capital is ivested i risk-free assets). The willigess of shareholders to accept the isurace risk is expressed i the form of the followig requiremet: { ( ) ( ) ( )} Pr u + i + Π < u + i + r η = ε RF which meas that shareholders admit that the rate of retur could fall dow by η from the expected rate i RF +r with probability ε. Assumig that is ormally distributed we obtai: RF Π( ) = u q u( r ) + ε + η Solvig both equatios i respect of the capital ad the premium we obtai: ad: ( ) Π u = q ε η rq = µ + η that is aother case of the formula (), ideed. The model ca be made more realistic by takig ito accout that a part of the capital is suk i the ifrastructure of the compay ad so the whole retur o this part has to be made o isurace operatios. Just aother ehacemet ca be obtaied by derivig the risk premium rate from the capital market quotatios. This ca be justified by the assumptio that share- 98 ekoomia 9 ε

4 The Top-Dow Approach to Calculatio of the Isurace Premium holders are idifferet whether to ivest i the isurace compay or elsewhere i the capital market. Just aother justificatio of the stadard deviatio priciple comes from the rui theory. The well kow approximatio to the probability of rui i the log ru Ψ give the iitial capital u is provided by the formula: Π( ) µ Ψ( u) = exp For the predetermied level ψ of the rui probability we obtai the premium formula: ( ) Π = + l ψ µ u However, the model assumes that yearly icremets of the compay s capital come from the techical result Π() oly, leavig the cost of capital ot covered (or, allowig to cover the cost implicitly by returs o ivestmet of curret capital i exteral assets). Supplemetig the model by the assumptio that the premium should also cover the costat yearly divided paid out to shareholders (as a reward for beig exposed to the risk of rui), we obtai: ( ) Π = + l ψ µ + du u where d deotes the divided rate. Now the premium loadig cotais two separate compoets pure risk loadig ad the profit margi. Miimizig the premium we fid the most efficiet level of iitial capital: l ψ u = opt d Ad the correspodig premium formula: ( ) Π = µ + dlψ that is agai a case of formula (). The last model follows the semial paper by Bühlma [985], who also promoted usig the term top-dow approach. The approach itself has bee kow much earlier. 3. Margial premium formula ad the balacig problem Provided the whole portfolio is priced by formula (), the idifferece price of the additioal risk (idepedet of risks,,, ) is give by the margial premium formula: ( ) ( ) ( ) ( ) Π arg = Π + Π = µ + α () m + ekoomia 9 99

5 ojciech Otto Assumig that stadard deviatios of the portfolio with ad without the additioal risk do ot differ too much, we ca make a approximatio: ( ) = that leads to the margial premium formula: = Π m ( ) arg = µ + (3) However, the sum of margial premiums calculated for all risks from the portfolio does ot suffice to cover the aggregate risk: Π( ) Π marg ( i ) = α i= ad the deficit equals exactly oe half of the required loadig for the portfolio. The balacig problem is othig strage the excess of average cost over the margial cost is a commo effect of returs to scale, ad i the case of a isurace compay, it is essece of the diversificatio effect obtaied by poolig risks. Nevertheless, the problem of allocatig the missig half of the loadig α remais. A ituitively appealig rule is to allocate the commo safety fud proportioally to the idividual cotributio to the loadig. This leads to the basic pricig formula: Π b ( ) = µ + α (4) However, the followig alterative solutio is also balaced: c k Π b ( ) = +, µ α + w c, k despite it allocates the missig half of the safety loadig proportioally to cumulats of order k, where k = is just a special case. Ay weighted average of differet versios of the above formula is balaced as well, ad we eed to somehow justify our choice. 4. Justificatio of the basic premium formula (4) I order to explai the solutio proposed by Borch [96], let us imagie that all risks are isured sequetially. Uder the particular orderig of risks from the basic portfolio {,, j, j +,, } the risk j + is priced as if the first j risks were isured already. Thus the correspodig margial premium formula reads: ( ) = µ + α[ ] Π m arg j + j + j + j 00 ekoomia 9

6 Obviously the sum of premiums calculated this way for all risks i the portfolio is equal to the portfolio premium, but the loadig is allocated very uequally, chargig heavily those risks that have bee isured at first, ad oly slightly those oes that have bee isured the latest. Borch suggested settig the price of a give risk as a average of prices calculated accordig to this rule uder all! possible orderigs, showig that this rule is just equivalet to the equilibrium solutio of a -perso game (derived first by Shapley i 953). Borch recommeded this solutio as suitable to the problem of allocatig the loadig betwee few groups of risks rather tha to a large umber of idividual risks. Ideed, oe ca hardly imagie calculatios for large. This would require averagig over possibly differet values, that s how may subsets the set of remaiig risks may have, whe oe of risks is priced. However, i a case of large a approximate solutio ca be derived. Let us cosider agai the problem of pricig a additioal risk whe each of its possible ( + ) positios i the sequece {,, j,, j+,, } is treated as equally probable, ad all! orderigs of remaiig risks are equally probable, too. Let us deote the set of basic risks precedig the risk i a particular orderig by PRE. The derivatio is based o the remark that uder certai coditios the ratio of the sum of variaces of precedig risks (elemets of PRE) i the aggregate variace of all risks i the portfolio is approximately uiformly distributed over the uit iterval: u [ 0, ] Pr j u j u (5) j PRE j= Let us make use of this approximatio ow, leavig its justificatio to the ext sectios. Accordig to the approximatio we ca write: E j + i ( u + u ) du j PRE i PRE 0 Usig the abbreviated otatio c = we ca express the above itegral as: ( + ) = ( + ) 0 u u du u c u du Multiplyig ad dividig the itegrated fuctio by c, ad takig ito accout that c is small we get the result: The Top-Dow Approach to Calculatio of the Isurace Premium 0 u udu = c = u+ c u c du c 0 c 0 that ideed leads to the balaced pricig formula (4). The above derivatio was proposed i the book writte i Polish [004] ad aother oe writte i Eglish [005, Chapter 0]. However, i both cases the approximatio (5) has bee preseted as ituitively appealig. Next ekoomia 9 0

7 ojciech Otto sectios of this article cotai the rigorous justificatio i terms of the theorem o covergece i distributio ad its refiemets. 5. Covergece to the uiform distributio Approximatio (5) ca be justified by the covergece i distributio of the share of risks precedig a give risk i the variace of the portfolio to the uiform distributio. The covergece requires that uder a icreasig umber of risks i the portfolio the share of maximal variace i the aggregate variace of the whole portfolio vaishes: { } lim max,,, = (6) I order to prove sufficiecy of the above requiremet some more precise defiitios, otatios ad assumptios are eeded. Risks will be treated as elemets of the set (portfolio), with the share of their variaces i the variace of the whole portfolio assiged to each of them. Notatios ad assumptios are summarized below: (A) E= { e, e,, e } is a basic set of elemets; (B) ye : [ 0is, ] a fuctio that assigs the real oegative umber y j =y(e j ) to each elemet of the basic set, such that y + y + + y = ; (C) M = max {y, y,,y }deotes the maximum of these umbers; j =,, (D) E* =E {e*}is a basic set E supplemeted by the special elemet e*; (E) U is defied as a sum of y j characterisig these elemets e j that precede special elemet e* for a give orderig of elemets of set E*; (F) the probability fuctio assigs to each orderig of the set E* the probability /( + ). Uder assumptios (A) (F) the radom variable U has a discrete distributio o the support [0, ] that is uiquely determied by the set of real umbers {y, y,,y }. Let us deote the correspodig cdf by F U. Now the theorem ca be formulated as follows: Theorem. Uder assumptios (A) (F) the cdf F U is uiformly bouded from both sides: + um F u u + M U + M u [ 0, ) ( ) ad equals zero for u < 0, ad oe for u. I lights of the theorem covergece of M to zero meas that the distributio coverges to the uiform distributio. Covergece of M to zero implies that teds to ifiity, as obviously M. For a give M the umber is ubouded from above, of course. 0 ekoomia 9

8 The Top-Dow Approach to Calculatio of the Isurace Premium 5.. Proof of the Theorem It is obvious that each particular orderig of set E* correspods to the iverted orderig. Hece, both variables U ad ( U) must have the same distributio. That is why the upper boud for F U (u) i the theorem equals the lower boud for F U ( u) ad vice-versa. So it suffices to prove that oe of the bouds is true. Below, the proof for the upper boud is preseted. The proof is based o iductio, ad shows that the upper boud assumed to be correct for a radom variable U determied by ay admissible sequece {y j } cotaiig ( )elemets, implies that it is also correct for a variable U determied by ay admissible sequece {y j } that cotais elemets. The first step the, is to show that the upper boud is correct for =. st step of the proof For = the value y equals oe, ad so M =. Variable U ca take oly values 0 or, each of them with probability /. So the cdf F U (u) takes a value / o the iterval u [0, ) ad the value of at the right edpoit u =. Thus the cdf obviously satisfies the iequality F U (u) (u + )/ for all u [0, ]. The same distributio of variable U cocers also the case whe set {y, y,,y } cotais oly oe positive elemet so that y j = M = for a give j. This is because i all orderigs the value of the variable U depeds oly o the mutual positios of e j ad e*. Thus we ca restrict further cosideratios to the case whe ad M <. d step of the proof For the purpose of this step of the proof let us deote by U the variable determied by a set of elemets {e, e,,e } ad a correspodig set of values {y, y,,y }, ad assume that ad M <. Let us also assume, for simplicity, that there are o larger values of y j tha the first oe, so y + M, ad let us deote the maximum out of the subsequece {y,,y }bym. Summarizig we have 0 < M M <. Let us also defie the correspodig family of radom variables U j, for each j =,,, beig determied by the set Ej = E\{ e j} ad the same fuctio y. Of course, assumptios (A) (F) are satisfied for variables U j /( y j ), ad ot for variables U j themselves. So, assumig that the upper boud stated i the theorem is correct for all variables U j /( y j ), meas that for all u [0, ] the followig iequalities hold: u+ M Pr( U u) (7) + M M u+ M Pr ( Uj u) j= 3,,, (8) + M yj The above upper bouds could be easily tighteed, makig use of simple remark that a probability is ever greater tha. However, for our purposes ekoomia 9 03

9 ojciech Otto it suffices to refie the upper boud for the variable U oly. The refiemet takes a form: Pr( U u) u + M (9) For all u > Mthe above iequality is correct as a probability is ever greater tha. I the case whe u Mits correctess could be deduced as follows: u M implies that: (M + u); but by defiitio M M, so multiplyig both sides of the last iequality by (M M ) we obtai: (M M) (M M) (M + u), addig the term M + u to both sides we obtai: (M M) ( + M M)(M + u), fially, dividig both sides by ( + M M) ad combiig the result with (7) we obtai the cofirmatio that iequality (9) holds. Now we ca retur to the distributio of variable U. Let us defie i the space of all! orderigs of elemets of the set E the evet A j, that the elemet e j has take the last positio i the orderig. The evets A, A,,A defied this way are separate, equally probable, ad altogether they cover the etire space. Hece, we ca write: Pr( U u) = Pr( U ua ) j (0) j= However, coditioal probabilities appearig o the RHS cocerig evets expressed i terms of values of variable U, could be expressed i terms of values of variables U j as well. I order to do that, let us otice that for ay give orderig of elemets of set E there are ( + ) correspodig orderigs of elemets of the exteded set that oly differ by the positio of special elemet e*. he a arbitrary orderig of set E ecompassed by evet A j takes place, the almost always (with oe exceptio) variable U is idetical to variable U j. This exceptioal case happes whe special elemet e* takes the last positio, but the obviously U =. That is why we ca write: u [ 0, ) Pr( U uaj) = Pr + ( Uj u) j=,,, () Combiig ow (0) ad () we obtai: u [ 0, ) Pr( U u) = + Pr [ Uj u] () j= Makig use ow of upper bouds (8) ad (9) we coclude that: M+ u u [ 0, ) Pr ( U u) M+ u+ + (3) j M+ y = j Cosiderig ito accout that for ay j the followig equality holds: 04 ekoomia 9

10 The Top-Dow Approach to Calculatio of the Isurace Premium M+ u M u y j M+ y = + + j M+ M + yj we ca trasform (3) to the form: [ ) ( ) + M u y j u 0 U u , Pr (4) M M+ y j= j Now we ca make two remarks: that each compoet of the sum appearig o RHS of iequality (4) ca be bouded from above: y j /(M + y j ) y j, ad: that the sum (y + y 3 + +y ) equals ( M). These remarks allow for simplificatio of the upper boud (4) to the form: M [ ) ( ) + u u 0 U u + [ + ( M) ] +, Pr (5) M + Simple maipulatios ow allow for trasformig the RHS to the desired form of the upper boud (M + u) (M ), ad to exted this result to the right edpoit of iterval [0, ]. 5.. Are the bouds for the cdf tight eough? The aswer is that bouds stated i the theorem caot be sigificatly tighteed uless we impose additioal restrictios o the sequece {y, y,, y }. This ca be show by cosiderig the worst case whe our portfolio cosists of risks with equal variaces so that all umbers y j =/ for j =,,,. The the cdf takes o the iterval the form: k k k u, F ( ) U u = k=,,, + O each subiterval the upper boud is attaied at the left edpoit ad the lower boud is almost attaied ear the right edpoit. Hece, we caot tighte the bouds except by replacig the iequality symbol cocerig the lower boud by the strict iequality symbol <. This is ot a sigificat improvemet, of course. O the other had, ay additioal restrictio imposed o the sequece {y, y,,y } may lead to more sigificat improvemets of the bouds stated by the theorem. However, the variety of possible cases is ulimited, so there is a eed to focus o these cases whe the bouds stated by the theorem are usatisfactory. 6. Boudig the premium loadig Let us ispect ow how the theorem works whe used to boud the premium loadig. The loadig for a additioal risk equals the expectatio E[h(U)], where the fuctio h is defied as: ekoomia 9 05

11 ojciech Otto hu ( ) = u + u (for simplicity we assume α =) ad the expectatio is calculated i respect to the true distributio of variable U. Let us ow deote the bouds for the cdf F U (u) stated by the theorem for u [0, ) by: ( ) Fu u =, ad Fu ( ) + M u = + + M where both bouds reach at the right edpoit u =. As cdf s F, F U ad F are stochastically ordered ad the fuctio h(u) is decreasig o iterval u [0, ], we ca express the bouds for the premium loadig as follows: M, hudfu ( ) ( ) E( hu ( ) hudfu ( ) ( ). [ 0. ] [ 0, ] I order to justify approximatio of the loadig by the portfolio loadig + multiplied by the share of the priced risk i the variace of the whole portfolio +, we should ispect whether the bouds divided by the expressio + are close to uity. Deotig (as before) the ratio by c ad executig all ecessary calculatios, we obtai: 3 3 ( ) ( ) M + c + [ + c c ] + hudfu ( ) ( ) = + c 3 (6) [ ] c( + M) 0, for the lower boud ad: 3 3 ( ) M c+ [ + c c ] + hudfu ( ) ( ) = + c 3 (7) [ ] c( + M) 0, for the upper boud. Now we ca cosider various scearios. i. The sceario whe we price a sigle large risk o the backgroud of the portfolio of very umerous small risks ca be ispected by assumig fixed c ad M 0. The, both bouds ted to the same fuctio + c [( + c) 3 ] c 3 that is close to for a reasoably small c. 3c ii. The sceario whe the size of the priced risk is comparable to the size of the largest risk, so that c= cost M. There is still o problem whe we assume ow that M teds to zero, as i this case both bouds ted to. iii. The sceario whe M is fixed ad c teds to zero correspods to the case whe we allow for some large risks i the portfolio ad try to price a risk that is icomparably smaller. I this case the lower boud teds to ( + )( ) M + M, that is still acceptable (at least for small M), but the upper boud teds to ifiity, that is o loger acceptable. 06 ekoomia 9

12 The Top-Dow Approach to Calculatio of the Isurace Premium Divergig the upper boud i sceario iii. is ot due to poor boudig. Ideed, i the case of portfolio of risks of equal size such that M = ad a additioal risk characterized by c the exact result reads: + c k hudf ( ) ( ) U u c [ ] c c k = , k= ad for ay fixed ad c 0 diverges to ifiity. However, the assumptio that the compay basically specializes i isurig large risks, ad cosiders pricig oly oe icomparably small risk, is urealistic. I practice, compaies do have portfolios that might be composed of some large risks ad a large umber of small risks. Eve whe each small risk is icomparably smaller tha the largest oe, the aggregate variace of all small risks usually cotributes substatially to the variace of the whole portfolio. I the ext sectio a sceario of this kid is cosidered i details. 7. Pricig small risks i the presece of large risks Our cosideratios are restricted here oly to the special case of a portfolio composed both of large risks ad small risks. For simplicity we assume that there are large risks of the same share i the variace of the whole portfolio m such that their aggregate share S = m is less tha. The lastig ( S) is the share of the aggregate variace of a very large umber of very small risks. Variace of each small risk is assumed to be ifiitesimally smaller tha the variace of a large risk. Now the aggregate premium loadig allocated to small risks ca be bouded from above by the total portfolio loadig less the lower boud for the aggregate loadig allocated to large risks. More precisely, if we deote the ratio of the Shapley loadig to the loadig proportioal to variace for the large risk by G ad the aalogous ratio for the aggregate of small risks by g, the the balacig equatio holds: ( S)g + SG =. Thus, if we kow that a lower boud G such that G Gexists, the the resultig upper boud g such that g gequals: SG g = S (8) I order to derive the lower boudg we ca apply formula (6) to price oe of large risks o the backgroud of the portfolio composed of lastig ( ) large risks ad all small risks. This meas that we should make the followig replacemets o the RHS of (6): for = we replace c by m( m) ad M by zero; this leads to the exact result (as i this case lower ad upper bouds coicide): 3 3 [ ( m) m ] G= G= 3m( m) ekoomia 9 07

13 ojciech Otto for > we replace both c ad M by m( m) ; the the lower boud for G reads: 3 3 [ ( m) m ] G= m+ 3m Combiig the last results with (8) ad replacig m by (S/) we obtai the upper boud for ratio g as a fuctio of S ad : [ ] 3 3 S ( S) + S gs (, ) = gs (, ) = + + for =, ad: (9) S 3 ( S) s S S S gs (, ) = + s + 3 S for > (0) Simple calculatios (employig de Hospital s rule etc. to formulas (9) ad (0)) lead to the coclusio that: lim gs (, ) =+ for ay fixed positive iteger () S The result cofirms that i order to have a upper limit for the ratio of the Shapley loadig to the loadig proportioal to variace for small risks, we have to assume that the share of small risks i the portfolio is o- -eglectible. I order to review i tur the case of fixed S ad large the approximatio S S( ) ca be used. This yields the approximated upper boud: 3 S gs (, ) + 3 ( S) where eglected terms are of order, 3/, etc. This obviously meas that: lim gs (, ) = for ay fixed S ( 0, ) () The last result meas that whe we elarge ulimitedly the heterogeeous portfolio by acceptig ew large as well as ew small risks (but keepig the proportio S fixed), the the ratio teds to. I order to illustrate how the upper bouds give by formulas (9) ad (0) work i o-extreme cases, some results are calculated ad preseted i Table below. Table. Upper bouds gs (, ) for the fuctio gs (, ) S = 0.5 S = 0.50 S = 0.75 S = 0.90 Exact values 06.6%.9% 59.5% 36.3% 08 ekoomia 9

14 The Top-Dow Approach to Calculatio of the Isurace Premium S = 0.5 S = 0.50 S = 0.75 S = 0.90 Upper bouds 06.8% 6.5% 89.8% 380.5% %.8% 79.4% 353.0% % 0.3% 7.6% 39.6% % 8.5% 65.6% 3.3% % 3.6% 49.0% 59.% 0 0.4% 09.9% 35.9% 7.0% % 07.% 6.0% 84.9% % 05.3% 9.% 63.0% % 03.8% 3.8% 45.% % 0.7% 09.8% 3.% % 0.9% 07.0%.9% % 0.4% 05.0% 6.3% % 0.0% 03.5%.5% % 00.7% 0.5% 08.% % 00.5% 0.8% 05.8% The cotet of the Table show that upper bouds give by (0) probably produce quite tight bouds whe S is smaller tha half. O the other had, bouds for S = 0.75 ad S = 0.90 seem to be heavily overestimated. This stems from the guess that for give S the ratio g(s, ) should be a decreasig fuctio of. So the value g(s, ) should be smaller tha g(s, ). Thus, the value gs (, ) larger tha g(s, ) for each case preseted i the Table, seem to be a effect of overestimatio of the true value by the boud (0). This overestimatio is very small for S = 0.5, moderate for S = 0.50, quite substatial for S = 0.75 ad very high for S = Leavigdoubts cocerig the accuracy of boudig aside, we ca try to draw practical coclusios that stem from the preseted results. Let us assume for istace that we accept the approximatio of the Shapley value by the variace priciple if g(s, ) > 0%. Results preseted i Table allow to coclude that the approximatio is acceptable: i case of S = 5%: for ay (so that ay m 5% is acceptable); i case of S = 50%: for 0 (so that m.5% is acceptable); i case of S = 75%: for 300 (so that m 0.5% is acceptable); i case of S = 90%: for 4800 (so that m 0.09% is acceptable). 8. Coclusios ad extesios Results preseted i this article cofirm that the variace priciple applied to allocate the premium loadig that has bee set for the whole portfolio as proportioal to the stadard deviatio ca be justified as a approximatio to the Shapley value. This result is also relevat for oe reaso that has ot bee exploited i the paper. Despite the soud iterpretatio, the ekoomia 9 09

15 ojciech Otto Shapley value has a obvious deficiecy whe used for pricig. This is because it does ot remai uchaged uder aggregatio of risks the Shapley value of a subportfolio of risks does ot equal the sum of Shapley values of idividual risks from this subportfolio. Perhaps the best approximatio of the Shapley value that is free of this deficiecy is the variace priciple. The approximatio may appear to be poor whe the portfolio cotais both very large risks ad very small risks, ad whe the share of small risks i the variace of the whole portfolio is small. However, whe the portfolio cotais some very large risks (say, m larger tha oe percet), the the distributio of the aggregate amout of claims sigificatly departs from the ormal oe. Thus the stadard deviatio priciple is o loger adequate for settig the portfolio premium. It is more difficult to defed the preseted coclusio i the case whe the share of small risks i the variace of the whole portfolio is small, 0% for istace. The the maximum share of a large risk i variace m at which the approximatio is still acceptable is less tha 0.0%. However, the requiremets o the structure of the portfolio seem to be far too restrictive for two reasos: e have oly studied i more detail the case whe S = m, whereas i practice collectios of large risks are heterogeeous. It seems that for a give S ad m the heterogeeity reflected by the umber of large risks greater tha S/m will lead to more moderate ratio g of Shapley loadig to the variace loadig. Eve i the case S = m we have derived upper bouds for the ratio g that seem to overestimate the true values, especially i the case of a relatively large S. Both limitatios of aalysis preseted i this paper are due to the aim of exploitig as much as possible Theorem, ad to avoid goig too far beyod. Durig the EM Coferece some prelimiary results based o direct aalysis of the mixed game (mixed of atoms ad the ocea, as it is called i the laguage of the Game Theory) have bee preseted. More mature results have bee preseted few weeks later o the IME Coferece i Leuve. Literature Borch Karl H., 96, Applicatio of Game Theory to Some Problems i Automobile Isurace, ASTIN Bulleti Vol., No., pp. 08. Buhlma Has, 985, Premium Calculatio from Top Dow, ASTIN Bulleti Vol. 5, No.. Miœta Pawe³, ojciech Otto, 005. Premiums, Ivestmets ad Reisurace Chapter 0, pp , i Statistical Tools for Fiace ad Isurace, ed. Pavel Cizek, olfgag Haerdle ad Rafa³ ero, Spriger Verlag. Otto ojciech, 004, Ubezpieczeia maj¹tkowe Czêœæ I Teoria ryzyka, NT, arsaw, Polad. Shapley L.S., 953, A value for -perso games, Aals of Mathematical Studies, Priceto, pp ekoomia 9

16 The Top-Dow Approach to Calculatio of the Isurace Premium Abstract he the risk loadig for the whole portfolio is set proportioally to the stadard deviatio, the the problem of coheret pricig of idividual risks arises. Borch (96), proposed a solutio based o Shapley s value of the -perso game. However, the solutio is suited oly for small, rather reflectig the game played by few compaies that egotiate poolig their portfolios. Otto (004) proposed a ituitively appealig approximatio for the case of large that leads to allocatio of the risk loadig proportioaly to variaces. The paper is devoted to formally justify that the variace priciple ca be justified as a approximatio to the Shapley s solutio. ekoomia 9