Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages 365-384 (Devoed o aricles on Compu. Sci., Info. Sci., Financial Manag. & Biol. Sci. PREMIUM INDEXING IN LIFELONG HEALTH INSURANCE W. Vercruysse, J. Dhaene, M. Denui 2, E. Piacco 3 and K. Anonio,4 KU Leuven Belgium e-mail: ward.vercruysse@kuleuven.be an.dhaene@kuleuven.be karien.anonio@kuleuven.be 2 U.C.L. Louvain-la-Neuve, Belgium e-mail: michel.denui@uclouvain.be 3 Universià di Triese Triese, Ialy e-mail: ermanno.piacco@econ.unis.i 4 Universiy of Amserdam The Neherlands Absrac For lifelong healh insurance covers, medical inflaion no incorporaed in he level premiums deermined a policy issue requires an appropriae increase of hese premiums and/or he corresponding reserves during he erm of he conrac. In his paper, we invesigae Received: June, 203; Acceped: July 3, 203 200 Mahemaics Subec Classificaion: 9B30, 97M30. Keywords and phrases: medical expense insurance, lifelong conrac, medical inflaion index, reserve.
366 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio appropriae premium indexing mechanisms, based on a given medical inflaion index. Firs, we consider a general relaion beween benefi, premium and reserve increases, which can be used on a yearly basis o resore he acuarial equivalence ha is broken due o observed medical inflaion over he pas year. Nex, we consider an individual premium indexing mechanism, depending on he age a policy issue, which makes he relaive premium increases above he observed medical inflaion more sable over ime. Finally, we consider an aggregae premium indexing mechanism for a porfolio of new enrans, where he relaive premium increase above he observed inflaion is independen of age-a-enry, inroducing inergeneraional solidariy.. Inroducion We consider healh insurance conracs, more specifically medical expense reimbursemen policies (or forfeiure daily allowance policies offered as erm or lifelong insurance covers wih level premiums. As is he case in life insurance, level premiums lead o asse accumulaion in a reserve. In general, he benefis ha will be paid over he years for a erm or lifelong healh insurance porfolio will be impaced by a number of unpredicable facors, such as changes in prices for medical goods and services and demographic evoluions of he insured populaion. Given he long-erm naure of healh insurance conracs and he impossibiliy o predic or hedge agains medical inflaion, insurers ofen do no ake ino accoun or are no able o fully accoun for his medical inflaion in he seing of he premium level a policy issue. Insead, during he erm of he conrac, hey adap he premium amouns a regular imes (e.g., yearly, based on some predefined medical inflaion index. This pracice is used in several EU member counries (for insance, in Germany and in Belgium. This approach efficienly couneracs he sysemaic risk induced by medical inflaion impacing all he policies of he porfolio in he same direcion. The reference medical index may be based on a represenaive baske of medical goods and services of which he price is followed over ime, or on
Premium Indexing in Lifelong Healh Insurance 367 indusry-wide loss daa. Besides public agencies, also privae consuling firms develop indicaors for medical insurance. See, e.g., Da Silva [, Devolder and Yerna [2 and Ramee e al. [4. In his paper, we do no discuss he consrucion of he index bu, given a cerain medical index, we propose several premium indexing mechanisms aimed a mainaining fairness beween policyholders and insurer. The medical index considered in his paper is assumed o accoun for all sources of inflaion, no only he increase in medical coss above he inflaion aken ino accoun by he usual consumer price index. Imporan o noice is ha no only fuure premiums need o be increased o ake ino accoun he medical inflaion, bu also he reserve may need o be adaped in order o resore he acuarial equivalence ha mus exis beween he liabiliies of boh parners of he insurance conrac. The remainder of his paper is organized as follows: Secion 2 discusses several premium indexing mechanisms. These mechanisms are illusraed wih numerical examples in Secion 3. The final secion briefly concludes. 2. Indexing for Medical Inflaion 2.. Benefi srucure We consider healh insurance conracs wih non-ransferable reserves (ha is, he reserve is no paid ou o he insured when he lapses he conrac, as i is ypically he case on he Belgian marke. I is obvious ha he nonransferabiliy of he reserves has a premium-reducing effec. Hereafer, ime measures he senioriy of he policy (i.e., he ime elapsed since policy issue. Policyholder s age a policy issue is denoed by x, so ha age a ime is x +. We denoe he ulimae age by ω (in case of a lifelong cover, he policy is assumed o cease a age ω. The superscrip (0 is used o denoe quaniies esimaed a policy issue. The average annual claim amoun a age x +, = 0,, 2,..., ω x, is denoed as ( 0 c x+. Noice ha refers o he ime passed since policy issue
368 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio and ha he superscrip (0 indicaes ha he benefis a ime 0. ( 0 c x+ are deermined Henceforh, we assume ha he annual claim amouns are subec o inflaion, whereas he oher elemens of he echnical basis (ineres rae, moraliy rae and lapse rae are in line wih he realiy ha unfolds over ime (which implies ha hese elemens do no have o be indexed over ime in order o mainain acuarial equilibrium. This simplifying assumpion is no realisic bu allows us o isolae and invesigae he effec of medical inflaion. 2.2. Level premiums The non-exi probabiliy k p x + is he probabiliy ha a policy in force a age x + is sill in force k years laer, ha is, k px+ k = exp ( μ + λ x+ + s x+ + s dx 0 = ( [ ( q [, d w k q x + k x + where μ x + + k is he insananeous deah rae a age x + + k, while λ x + + k is he insananeous lapse rae a he same age. The noaions [ d k q x+ and [ w k q x+, where d refers o deah and w o wihdrawal, are used o denoe he absolue raes of decremen (also called he independen probabiliies of exiing, i.e., [ d k q x + k = exp μ x+ + sds and 0 [ w k q x + = exp λ x+ + 0 k s ds. Assuming ha he benefis are paid a he beginning of he year (a convenien and conservaive, ye unrealisic assumpion, he expeced presen value of all fuure benefis, evaluaed a policy issue, is denoed by ( 0 ( 0 B x = cx+ k v 0, k k = 0 ( k p x,
Premium Indexing in Lifelong Healh Insurance 369 where v ( s,, s, is he discoun facor over he period ( s,. Noice ha he sum in his expression is a finie sum, as k p x = 0 if k ω x. Assuming ha level premiums of amoun ( 0 π x are paid yearly in advance as long as he policy is in force, he acuarial equivalence principle gives rise o ( 0 ( 0 B π x x =, where a = ( 0, a x v k x k = 0 Noe ha he premium calculaion is based on he expeced coss ( x0 k c + evaluaed a ime 0, wihou allowance for fuure inflaion. An alernaive, no sudied in he presen paper, consiss in compuing ( 0 π x from expeced ( 0 coss impaced by an assumed scenario for fuure medical inflaion. c x + k The framework described in his paper can be adaped o ake ino accoun such a scenario. 2.3. Indexing a differen imes 2.3.. Indexing a ime = Henceforh, he superscrip (, =, 2,..., is used o indicae ha he calculaions include medical inflaion from policy issue o ime. According o he equivalence principle, he level premium ( 0 π x is deermined such ha ( 0 he iniial reserve V 0 is equal o 0: ( 0 ( 0 ( 0 V 0 = B x πx a x = 0. ( The benefis paid in year ( 0, are denoed by ( 0 c x. As menioned before, we assume ha he observed moraliy, lapse and ineres raes follow he echnical basis assumpions. We denoe he available reserve per policy ( 0 in force a ime by V. This reserve is given by k p x.
370 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio ( 0 [ ( 0 ( 0 V π c [ v( 0, p. = x x x Taking ino accoun he equivalence relaion (, one can ransform his ( 0 rerospecive expression for V ino he following prospecive expression: ( 0 ( 0 ( 0 V = B + πx x+, x a where and ( 0 ( 0 B x+ = cx+ + k k px+ v, + k k = 0 ( a x+ = v(, + k k px+. k = 0 0 0 x x x ( 0 ( 0 x+ πx a x+, Hence, he available reserve a ime, i.e., [ π ( c ( [ v( 0, p, is equal o he required reserve a ime, i.e., assumpions concerning he echnical basis are me. B provided all Medical inflaion is aken ino accoun ex-pos as i emerges over ime by adaping he premium amoun from year o year according o he procedure described hereafer. Le be he medical inflaion observed during he firs year. Due o his observed medical inflaion, a ime, he expeced ( 0 presen value of he fuure benefis B has o be replaced by x+ ( ( ( 0 B x+ = + Bx+. Noe ha we assumed ha he yearly expeced coss a all ages are impaced equally by he medical inflaion, i.e., he ideniy c ( = ( 0 cx ( + + is assumed o hold for all. An alernaive, no sudied in he presen paper, is ha medical inflaion depends on age. I is a raher x +
Premium Indexing in Lifelong Healh Insurance 37 sraighforward exercise o adap he ex-pos premium indexing mechanism ha we presen hereafer o he siuaion wih age-dependen medical inflaion. Due o he observed medical inflaion, we find ha ( 0 ( ( 0 ( 0 V +, Bx+ πx a x+ which means ha he acuarial equivalence is broken, i.e., he available reserve is differen from he required reserve. To resore he acuarial equivalence, he insurer has o adap he premiums and/or reserve for his conrac. Suppose ha he level premium ( 0 π x is from ime on replaced by ( π x, ( 0 while he available reserve V a ( ime is changed ino V. The proporional increases of he premium and he reserve are denoed by [ P and [ V, respecively, ha is, ( ( [ P ( 0 π x = + πx and Following Piacco [3, [ P and equivalence is resored a ime, i.e., such ha or, equivalenly, ( [ V ( ( [ V ( 0 V = + V. [ V are chosen such ha he acuarial V ( 0 ( ( 0 P 0 + = + Bx+ + πx a x+ ( ( ( V = B + πx x+. x a ( [ (, This means ha he available reserve a ime, i.e., ( required reserve a ime, i.e., ( B x+ πx a x+. ( V, is equal o he From ime on, he original level premiums ( 0 π x ha were deermined a policy issue, are replaced by new level premiums ( π x. Noice ha he premium increases [ P ( 0 π x are financed by he policyholder, while he
372 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio reserve increase [ V ( 0 V is financed by he insurer. In pracice, he insurer may finance he reserve increase, parially or fully, from echnical gains on ineres, moraliy and lapses. 2.3.2. Indexing a ime = 2, 3,... Le us now suppose ha we are a ime, = 2, 3,... Reevaluaions up o ime have led o ( ( ( [ 0 B c x+ + k = cx+ + k + h, k = 0,,..., h= ( ( B x + = cx+ + k k px+ v, + k k = 0 ( ( [ ( P 0 πx = + h πx. h= (, A each ime, 2,...,, he available reserve and he premium have been rese such ha available and required reserves are equal. In paricular, a ime, he available reserve such ha ( V and he premium ( π x have been rese ( ( ( V = B. x+ πx a x+ (2 The reserve available a ime for a person aged x a policy issue, aking ino accoun all informaion unil ime, is hen given by ( ( [ ( ( V V + π c [ v(, p. = x x x+ Taking ino accoun (2, he following prospecive expression can be derived for he available reserve: ( ( ( V = B x + πx a x+.
Premium Indexing in Lifelong Healh Insurance 373 Le be he medical inflaion observed during he year (,. ( Therefore, a ime, we have o replace B x + by ( ( ( B x + = + B x +. The acuarial equivalence is again broken, in he sense ha he available reserve is no equal o he required reserve: ( ( ( ( V + B x + πx a x+. In order o resore he acuarial equivalence, he premium and reserve are adaped o ( ( [ P ( π = + π x x and ( V = ( + [ V ( V such ha he available reserve and he required reserve are equal: V ( B ( P x + x x+ ( + [ V = ( + [ B ( + [ π ( a, (3 or, equivalenly, ( ( ( V = B x + πx a x+. The acuarial equivalence may be resored by an infinie number of pairs ( [ V [ P,. When [ V = 0, he benefi increase is compleely paid by he policyholder. On he oher hand, choosing [ P = 0 means ha he benefi increase is compleely financed by he insurer. 2.4. Relaionships beween, [ V and [ P The benefi inflaion is equal o a weighed arihmeic average of [ V and [ P, wih weighs ha sum up o, ha is, ( ( V [ V x a x [ P ( π = + + (. B x B + x + (4
374 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio This relaionship beween, [ V and acuarial equivalence condiion (3. [ P follows immediaely from he The equilibrium resoring procedure, expressed by (3 or equivalenly by (4, applied on a conrac per conrac basis, is an acuarial sound sysem (provided he assumpions we made are me. Noice however ha before he procedure can be applied in pracice, a choice has o be made abou how he addiional cos arising from he unanicipaed inflaion is shared beween he policyholder and he insurer. A simple and ransparen rule, unambiguously described in he policy condiions, is appropriae here. Taking ino accoun ha we assumed ha, apar from he inflaion, all assumpions made in he echnical basis are me, i may be reasonable o se [ V = 0, implying ha he insured finances he increased fuure benefis. The premium increase [ P can hen be deermined on a yearly basis from he equilibrium condiion (3. A problem wih he procedure explained above is ha he premium increases [ P may flucuae heavily from year o year. Therefore, we propose anoher procedure. In paricular, le us assume ha he policy sipulaes ha he yearly premium increase [ P is given by [ P ( = + α, =, 2,... (5 for some fixed value of α. Suppose, e.g., ha α = 0.5, hen a medical inflaion of 4% will lead o a premium increase of 6%. The exra increase α over he benefi inflaion can be inerpreed in erms of he policyholder s conribuion o he reevaluaion of he reserve. As a resul, he relaive exra premium increase above he increase of he medical index becomes more sable. Taking ino accoun (4 and (5, we find he following resuls for he case where he premium increase is se equal o he benefi increase: 0 [ P [ V α = = =.
Premium Indexing in Lifelong Healh Insurance 375 Hence, in case he proporional premium increase is chosen equal o he proporional benefi increase, we find ha he reserve has o be increased by he same proporion in order o resore he acuarial equivalence. Also, [ P α > 0 > and [ P α < 0 < and [ V <, [ V >. This means ha if he proporional premium increase is se larger (respecively, smaller han he proporional benefi increase, hen he required proporional increase of he reserve is lower (respecively higher han he benefi increase. Taking ino accoun our assumpion ha here are no echnical gains, a sricly posiive value of α will be appropriae. From equaion (4, i follows ha he relaive required reserve increase [ V is a P B decreasing funcion of α = ( [ [. 2.5. A sable premium indexing mechanism The advanage of a premium indexing mechanism of he form (5 is ha i makes he relaive increase of he premium above he observed medical inflaion over ime more sable. The value of α in (5 could be fixed in he insurance conrac or in he legal framework and be deermined on a regular basis (e.g., every couple of years according o a well-specified procedure. The choice of a fair value of α is crucial. If α is oo low, he insurer will have o finance he fuure increases of he reserves himself. On he oher hand, if α is oo high, he policyholder will consider he insurance conrac as an unfair deal, and evenually no buy he conrac. Hereafer, we presen some possible ways o deermine he facor α. 2.5.. Opimal α for a given age a policy issue To deermine an appropriae value for he facor α on a single policy corresponding o he age a policy issue x, we propose o calculae he acuarial presen value of all fuure required reserve increases as
376 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio = [ V ( APV ( α = V p v( 0,. (6 x Thus, APV x( α expresses he acuarial value of he fuure reserve increases for his conrac. Under he appropriae assumpions, i can be inerpreed as he exra capial o be ineced by he insurer in order o fund all fuure required reserve increases. In case of a negaive value, he APV x( α expresses he acuarial value of he exra capial paid by he insured above he necessary exra capial he had o pay o resore he acuarial equivalence in case [ V = 0. A posiive value of APV ( α poins o an acuarial loss while a negaive x APV x( α is an acuarial gain on his conrac for he insurer. Taking ino accoun ha we assumed ha here emerge no echnical gains on ineres, moraliy and lapse raes, he conrac can be considered as fair for boh paries if APV ( α = 0. The opimal α for a given age a policy issue, which x will be denoed by α x, is hen deermined by seing he expeced presen value of all fuure required reserve increases equal o 0, i.e., he equaion APV ( α = 0. x x α x is he roo of Of course, he deerminaion of he opimal α a ime 0 requires he knowledge of [ V, =, 2,..., which correspond o he fuure medical inflaion, =, 2,..., unknown a policy issue. Thus, deermining α x according o he principle explained above requires an assumpion for he fuure medical inflaion. The numerical illusraions performed in Secion 3 show ha α x is quie robus o moderae deparures from he cenral inflaion scenario. Therefore, he sysem can accommodae uncerainy abou he fuure pah of inflaion, o some exen.
Premium Indexing in Lifelong Healh Insurance 377 2.5.2. Opimal α for a given porfolio of new enrans In general, he opimal exra premium increase facor α is dependen on he age a policy issue. Alhough from an acuarial poin of view i is possible o work wih an age-dependen α x, consumers and regulaors may prefer a more sraighforward and simple approach, where he opimal α is independen of he age a policy issue. Hereafer, we propose a possible way o deermine his age-independen opimal α which will be denoed by α. We firs define ω x= x0 APV ( α = n x APV x ( α, where x 0 is he younges age of enry and n x is he number of enrans aged x in his porfolio. Hence, APV ( α expresses he acuarial value of he fuure reserve increases for his porfolio of new enrans. A posiive value of APV ( α corresponds o an acuarial loss, while a negaive value of APV ( α is an acuarial gain on his porfolio for he insurer. The opimal value of α, which will be denoed by α, is hen deermined as he roo of he equaion APV ( α = 0. Remark ha he use of an age-independen opimal α has he advanage (or disadvanage ha i inroduces inergeneraional solidariy. Deermining α according o he principle explained above again requires an assumpion for he fuure medical inflaion. The numerical illusraions carried ou in he nex secion show ha several scenarios of fuure inflaion lead o similar values of α, indicaing ha he opimal o he magniude of medical inflaion. 3.. Technical basis 3. Numerical Illusraion α is raher robus In he numerical examples, he discoun facors correspond o a consan
378 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio yearly ineres rae of 2%. The absolue rae of decremen due o deah conforms o he firs Heligman-Pollard law, ha is, [ d q y [ d q y [ d q y = C ( y+ B E( ln y ln F y A + De + GH 2 wih A = 0.00054, B = 0.07, C = 0.0, D = 0.0003, E = 0.72, F = 5 8.67, G =.464 0 and H =.. Furhermore, we consider a lifelong cover and we fix he ulimae age o ω = 0. In line wih curren pracice on he Belgian marke, we assume ha he one-year absolue rae of decremen due o lapse q [ w y is equal o 0. 0.002( y 20 a age y = 25, 26,..., 70 and 0 oherwise. The lapse rae only depends on he aained age and no on he ime elapsed since policy issue. This age srucure of lapse raes is reasonable because very few conracs are cancelled a higher ages as he reserves are non-ransferable. Figure displays he one-year independen probabiliies q [ w y and d ( q [ y as well as he non-exi probabiliies p y enering he compuaions. Figure. [ w q y, ( [ d q y and p y. Based on healh insurance daa colleced by he Ialian Naional Insiue of Saisics (ISTAT graduaed by he Ialian Associaion of Insurance
Premium Indexing in Lifelong Healh Insurance 379 Companies (ANIA, we choose he annual average claim amouns a age y and esimaed a ime 0, equal o ( 0 = 0.204476472 exp( 0.038637 y, y 20. c y 3.2. Iniial premium and reserves The level premium π ( x 0 for an insured aged x a policy issue, x = 25, 26,..., 70, is shown in Figure 2. The raecory of he non-ransferable reserves for a policyholder aged 25 a policy issue, assuming ha no medical inflaion is occurring during he erm of he conrac, is shown in Figure 3. Figure 2. Level premiums ( 0 π x for differen ages. ( 0 Figure 3. Reserves V for a person aged 25 a policy issue when = 0.
380 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio 3.3. Opimal α as a funcion of he age a enry Figure 4 displays he expeced presen value of all fuure reserve increases APV 25( α as a funcion of α for 3 differen scenario s of a consan inflaion over ime: = 2.5%, 4% and 6%, respecively, while [ P = ( + α for all. Obviously, for a given inflaion scenario, APV ( α is 25 a decreasing funcion of α: he higher α, he more he policyholder finances he benefi increases himself. Furher, for a given value of α, he funcion APV 25( α is an increasing funcion of he level of inflaion: a higher level of he inflaion leads o higher required reserve increases. For he scenario where = 2.5%, he opimal α 25 lies beween 0.6 and 0.7. Increasing he yearly medical inflaion o 4% or 6% leads o a seeper decreasing funcion APV ( α and decreases he value of he opimal value α. The opimal 25 α 25 urns ou o be a decreasing funcion of he assumed medical inflaion. The previous calculaions have been repeaed for all ages x a policy issue beween 20 and 70. The opimal values α x, for he hree scenarios of medical inflaion ( = 2.5%, = 4% and = 6%, are depiced in Figure 5. The opimal facor α x is a decreasing funcion of age x a policy issue. This is due o he shorer remaining period of he conrac and he fac ha he premium is an increasing funcion of age a policy issue. From Figure 5, i is also clear ha for older ages x, he benefi increase facor has a raher moderae effec on he opimal facor α x. The explanaion for his observaion lies again in he shorer remaining erm of he conrac. 25
Premium Indexing in Lifelong Healh Insurance 38 Figure 4. APV ( α when 25 [ P B = ( + α [. Figure 5. The opimal facor 3.4. Opimal α for a porfolio of new enrans α x as a funcion of age a policy issue. Le us suppose ha he age of new enrans in a given year is disribued as shown in Figure 6. This disribuion is based on Belgian daa. The high number of new enrans a age 20 is due o he fac ha conracs for ages younger han 20 are yearly renewable and priced on a risk premium basis, while he level premium srucure wih indexaion as described above is only applied from age 20 onwards. As a resul, here are a huge number of policy issuances a age 20. The acuarial presen value of he fuure reserve increases APV ( α as a funcion of he facor α is given in Figure 7 for hree scenarios of medical
382 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio inflaion ( = 2.5%, = 4% and = 6%. We observe ha for a given inflaion scenario, APV ( α is a decreasing funcion of α, while for a given value of α, he funcion APV ( α is an increasing funcion of he level of inflaion. For = 2.5%, he opimal α lies beween 0.4 and 0.5. Increasing he yearly medical inflaion o 4% or 6% leads o a seeper decrease of he funcion APV ( α and decreases he value of he opimal value α. Despie his decreasing effec, he heigh of he medical inflaion seems o have only a moderae effec on he opimal value α. Figure 6. Disribuion of he age of new enrans. Figure 7. APV ( α as a funcion of α in case [ P ( = + α.
Premium Indexing in Lifelong Healh Insurance 383 4. Conclusion In his paper, we considered lifelong healh insurance conracs, wih level premiums ha are se up a policy issue, no aking ino accoun fuure (unpredicable medical inflaion. We propose several premium indexing mechanisms which yearly resore he acuarial equivalence, aking ino accoun he observed medical inflaion over he pas year. Firs, we discussed he general relaion ha has o hold beween yearly benefi, premium and reserve increases in order o accoun for he unanicipaed inflaion ha has occurred. This equaion can in principle be used as he basis for indexing he premiums on a policy per policy and year o year basis, implying ha he relaive premium increase is a funcion of age a policy and of he number of years ha he policy is in force. Nex, we invesigaed a framework where he premium amoun is supposed o be yearly impaced by he observed medical inflaion muliplied wih a facor ( + α for some α > 0 which is chosen upfron. The proposed opimal value for α for a given age x a policy issue is hen chosen such ha he acuarial value of all fuure required reserve increases of he conrac is equal o 0. This individual approach is supposed o make he yearly relaive premium increases above he observed medical inflaion more sable. Finally, we proposed an aggregae approach which is applicable o a whole porfolio of new enrans, where an overall opimal α is deermined. The laer approach leads o age-independen relaive premium increases above he medical inflaion. Hence, i inroduces inergeneraional solidariy in he considered porfolio. Throughou he paper, we have assumed ha reserves are no ransferable, which is in line wih producs currenly offered on he Belgian marke. Allowing for fully or parially ransferable reserves is a opic for fuure research. Noe ha he indexing mechanisms described in he presen paper may also apply o oher long-erm life and healh insurance producs. In life insurance for insance, adapaion o a changing moraliy paern can be performed in a similar way, defining appropriae moraliy indices. This
384 W. Vercruysse, J. Dhaene, M. Denui, E. Piacco and K. Anonio approach is an efficien hedge for sysemaic longeviy risk, which is inheren o aging populaions and also a opic for fuure research. Acknowledgemen We would like o hank AG Insurance for he financial suppor via he Chair on Healh Insurance a KU Leuven. References [ R. Da Silva, Indexaion of medical coss for Souh African medical Schemes. IAAHS Colloquium, Cape Town, May 2007. [2 P. Devolder and B.-L. Yerna, Consrucion d une méhode spécifique d indexaion des conras privés d assurance maladie, Belgian Acuarial Bullein 8 (2008, 22-36. [3 E. Piacco, Mulisae models for long-erm care insurance and relaed indexing problems, Applied Sochasic Models in Business and Indusry 5 (999, 429-44. [4 S. Ramee, A. Kooveree and K. Dreyer, The consrucion of a price index for Souh African medical scheme conribuions, Acuarial Sociey of Souh Africa s 20 Convenion, Johannesburg, 20.