Premium indexing in lifelong healh insurance W. Vercruysse 1, J. Dhaene 1, M. Denui 2, E. Piacco 3, K. Anonio 4 1 KU Leuven, Belgium 2 U.C.L., Louvain-la-Neuve, Belgium 3 Universià di Triese, Triese, Ialy 4 Universiy of Amserdam, The Neherlands, and KU Leuven, Belgium Absrac For lifelong healh insurance covers, medical in aion 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 appropriae premium indexing mechanisms, based on a given medical in aion index. Firs, we consider a general relaion beween bene, premium and reserve increases, which can be used on a yearly basis o resore he acuarial equivalence ha is broken due o observed medical in aion 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 in aion more sable over ime. Finally, we consider an aggregae premium indexing mechanism for a porfolio of new enrans, where he relaive premium increase above observed in aion is independen of age-a-enry, inroducing inergeneraional solidariy. Key words: medical expense insurance, lifelong conrac, medical in- aion index, reserve. This work is performed as par of he AG Insurance Chair on Healh Insurance a KU Leuven. 1
1 Inroducion We consider healh insurance conracs, more speci cally medical expense reimbursemen policies (or forfeiure daily allowance policies) o ered 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 bene s 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 in aion, insurers ofen do no ake ino accoun or are no able o fully accoun for his medical in aion 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 prede ned medical in aion index. This pracice is used in several EU member counries (for insance, in Germany and in Belgium). This approach e cienly couneracs he sysemaic risk induced by medical in aion 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 indusry-wide loss daa. Besides public agencies, also privae consuling rms develop indicaors for medical insurance. See, e.g., Da Silva (2007), Devolder and Yerna (2008) and Ranjee e al. (2011). 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 in aion, no only he increase in medical coss above he in aion aken ino acoun 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 in aion, 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 a premium indexing mechanism. This mehod is illusraed wih numerical examples in Secion 3. The nal secion brie y concludes. 2
2 Indexing for medical in aion 2.1 Bene 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 non-ransferabiliy of he reserves has a premium-reducing e ec. 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! (in case of a lifelong cover, he policy is assumed o cease a age!). The superscrip "(0)" is used o denoe quaniies deermined a policy issue. The average annual claim amoun a age x+j, j = 0; 1; 2; : : : ;! x 1, is denoed as c (0) x+j. Noice ha j refers o he ime passed since policy issue and ha he he superscrip "(0)" indicaes ha he bene s c (0) x+j are deermined a ime 0. Henceforh, we assume ha he annual claim amouns are subjec o in aion, 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 don 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 e ec of medical in aion. 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, Z k kp x+ = exp x++s + x++s ds ; 0 = 1 kq [d] x+ 1 kq [w] x+ ; where x++k is he insananeous deah rae a age x + + k, while x++k is he insananeous lapse rae a he same age. The noaions k q [d] x+ and 3
are used o denoe he absolue raes of decremen (also called he independen probabiliies of exiing), i.e. kq [w] x+ kq [d] x+ = 1 exp Z k 0 x++s ds and k q [w] x+ = 1 exp Z k 0 x++s ds : Assuming ha he bene s are paid a he beginning of he year (a convenien and conservaive, ye unrealisic assumpion), he expeced presen value of all fuure bene s, evaluaed a policy issue, is denoed by B (0) x = 1X k=0 c (0) x+k v(0; k) kp x ; where v(s; ); s ; is he discoun facor over he period (s; ). Noice ha he sum in his expression is a nie sum, as k p x = 0 if k! x. Assuming ha level premiums of amoun (0) x are paid yearly in advance as long as he policy is in force, he acuarial equivalence principle gives rise o (0) x = B(0) x where a x = a x 1X v(0; k) k p x : Noe ha he premium calculaion is based on he expeced coss c (0) x+k evaluaed a ime 0, wihou allowance for fuure in aion. An alernaive, no sudied in he presen paper, consiss in compuing (0) x from expeced coss impaced by an assumed scenario for fuure medical in aion. The c (0) x+k framework described in his paper can be adaped o ake ino accoun such a scenario. k=0 2.3 Indexing a ime = 1 Henceforh, he superscrip "()", = 1; 2; : : :, is used o indicae ha he calculaions include medical in aion from policy issue o ime. According o he equivalence principle, he level premium (0) x is deermined such ha he iniial reserve V (0) 0 is equal o 0: V (0) 0 = B (0) x (0) x a x = 0: (1) 4
The bene s paid in year (0; 1) are denoed by c (0) 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 in force a ime 1 by V (0) 1. This reserve is given by V (0) 1 = (0) x c (0) x [v(0; 1) 1 p x ] Taking ino accoun he equivalence relaion (1), one can ransform his rerospecive expression for V (0) 1 ino he following prospecive expression: where and B (0) V (0) 1 = B (0) x+1 (0) x a x+1 ; x+1 = 1 P a x+1 = c (0) k=0 1 : x+1+kkp x+1 v(1; 1 + k) 1X v(1; 1 + k) k p x+1 : k=0 Hence, he available reserve a ime 1, i.e. [ (0) x c (0) x ] [v(0; 1) 1 p x ] 1, is equal o he required reserve a ime 1, i.e. B (0) x+1 (0) x a x+1, provided all assumpions concerning he echnical basis are me. Medical in aion 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 j [B] 1 be he medical in aion observed during he rs year. Due o his observed medical in aion, a ime 1 he expeced presen value of he fuure bene s B (0) x+1 has o be replaced by B (1) x+1 = (1 + j [B] 1 )B (0) x+1: Noe ha we assumed ha he yearly expeced coss a all ages are impaced equally by he medical in aion, i.e. he ideniy c (1) x+ x+ is = (1 + j [B] 1 )c (0) assumed o hold for all. An alernaive, no sudied in he presen paper, is ha medical in aion depends on age. I is a raher sraighforward exercise o adap he ex-pos premium indexing mechanism ha we presen hereafer o he siuaion wih age-dependen medical in aion. Due o he observed medical in aion, we nd ha V (0) 1 6= (1 + j [B] 1 )B (0) x+1 (0) x a x+1 ; 5
which means ha he acuarial equivalence is broken, i.e. he available reserve is di eren 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) x is from ime 1 on replaced by (1) x, while he available reserve V (0) 1 a ime 1 is changed ino V (1) 1. The proporional increases of he premium and he reserve are denoed by j [P ] 1 and j [V ] 1, respecively, ha is, (1) x = (1 + j [P ] 1 ) (0) x and V (1) 1 = (1 + j [V ] 1 )V (0) 1 : Following Piacco (1999), j [P ] 1 and j [V ] 1 are chosen such ha he acuarial equivalence is resored a ime 1, i.e. such ha or, equivalenly, (1 + j [V ] 1 )V (0) 1 = (1 + j [B] 1 )B (0) x+1 (1 + j [P ] 1 ) (0) x a x+1 ; V (1) 1 = B (1) x+1 (1) x a x+1 : This means ha he available reserve a ime 1, i.e. V (1) 1 is equal o he required reserve a ime 1, i.e. B (1) x+1 (1) x a x+1. From ime 1 on, he original level premiums (0) x ha were deermined a policy issue, are replaced by new level premiums (1) x. Noice ha he premium increases j [P ] 1 (0) x are nanced by he policyholder, while he reserve increase j [V ] 1 V (0) 1 is nanced by he insurer. In pracice, he insurer may nance he reserve increase, parially or fully, from echnical gains on ineres, moraliy and lapses. 2.4 Indexing a ime = 2; 3; : : : Le us now suppose ha we are a ime, = 2; 3; : : :. Reevaluaions up o ime 1 have lead o ( 1) c x++k = c (0) ( 1) B x+ = 1 P k=0 ( 1) x = Q 1 h=1 x++k h=1 ( 1) c Q 1 (1 + j [B] ); k = 0; 1; : : : ; x++kkp x+ v(; + k); (1 + j [P ] h h )(0) x : 6
A each ime 1; 2; : : : ; 1, he available reserve and he premium have been rese such ha available and required reserve are equal. In paricular, a ime ( 1) ( 1) 1, he available reserve V 1 and he premium x have been rese such ha ( 1) ( 1) ( 1) V 1 = B x+ 1 x a x+ 1 : (2) The reserve available a ime for a person aged x a policy issue, aking ino accoun all informaion unil ime 1, is hen given by h i ( 1) ( 1) ( 1) ( 1) V = V 1 + [v( 1; ) 1 p x+ 1 ] 1 : x c x Taking ino accoun (2), he following prospecive expression can be derived for he available reserve: ( 1) V = B ( 1) ( 1) x+ x a x+ : Le j [B] be he medical in aion observed during he year ( 1; ). Therefore, a ime we have o replace B ( 1) by x+ B () x+ = (1 + j [B] )B ( 1) x+ : The acuarial equivalence is again broken, in he sense ha he available reserve is no equal o he required reserve: V ( 1) 6= (1 + j [B] )B ( 1) ( 1) x+ x a x+ : In order o resore he acuarial equivalence, he premium and reserve are adaped o () x = (1 + j [P ] ) ( 1) x and V () = (1 + j [V ] )V ( 1) ; such ha he available reserve and he required reserve are equal: (1 + j [V ] )V or, equivalenly, ( 1) = (1 + j [B] ( 1) )B x+ (1 + j [P ] ( 1) ) x a x+ ; (3) V () = B () x+ () x a x+ : The acuarial equivalence may be resored by an in nie number of pairs j [V ] ; j [P ]. When j [V ] = 0, he bene increase is compleely paid by he policyholder. On he oher hand, choosing j [P ] increase is compleely nanced by he insurer. 7 = 0 means ha he bene
2.5 Relaionships beween j [B], j [V ] and j [P ] The bene in aion j [B] is equal o a weighed arihmeic average of j [V ], wih weighs ha sum up o 1, ha is, j [P ]! = V ( 1) ( 1) B j [B] x+ j [V ] + ( 1) x a x+ ( 1) B x+ This relaionship beween j [B], j [V ] and j [P ] acuarial equivalence condiion (3).! and j [P ] : (4) follows immedialy 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 in aion is shared beween he policyholder and he insurer. A simple and ransparan rule, unambigously described in he policy condiions, is appropriae here. Taking ino accoun ha we assumed ha, apar from he in aion, all assumpions made in he echnical basis are me, i may be reasonable o se j [V ] = 0, implying ha he insured nances he increased fuure bene s. The premium increase j [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 j [P ] may ucuae heavily from year o year. Therefore, we propose a more sable procedure. In paricular, le us assume ha he policy sipulaes ha he yearly premium increase j [P ] is given by j [P ] = (1 + ) j [B] ; = 1; 2; : : : (5) for some xed value of. Suppose e.g. ha = 0:5, hen a medical in aion of 4% will lead o a premium increase of 6%. The exra increase j [B] over he bene in aion j [B] can be inerpreed in erms of he policyholder s conribuion o he reevaluaion of he reserve. Taking ino accoun (4) and (5), we nd he following resuls for he case where he premium increase is se equal o he bene increase: = 0 ) j [P ] = j [V ] = j [B] : 8
Hence, in case he proporional premium increase is chosen equal o he proporional bene increase, we nd ha he reserve has o be increased by he same proporion in order o resore he acuarial equivalence. Also, > 0 ) j [P ] < 0 ) j [P ] > j [B] and j [V ] < j [B] ; < j [B] and j [V ] > j [B] : This means ha if he proporional premium increase is se larger (respecively smaller) han he proporional bene increase, hen he required proporional increase of he reserve is lower (respecively higher) han he bene- increase. Taking ino accoun our assumpion ha here are no echnical gains, a sricly posiive value of will be appropriae. From equaion (4) i follows ha he relaive required reserve increase j [V ] is a decreasing funcion of = j [P ] j [B] =j [B]. 2.6 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 over ime more sable.the value of in (5) could be xed in he policy. Alernaively, i could be deermined on a regular basis (e.g. every couple of years) according o a well-speci ed procedure, or i could be provided by he regulaor on a regular basis. The choice of a fair value of is crucial. If is oo low, he insurer will have o nance he fuure increases of he reserves himself. On he oher hand, if 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. 2.6.1 Opimal for a given age a policy issue To deermine an appropriae value for he facor 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 AP V x () = 1X j [V ] ( 1) V =1 p x v(0; ): (6) 9
Thus, AP V x () 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 injeced by he insurer in order o fund all fuure required reserve increases. A posiive value of AP V x () poins o an acuarial loss while a negaive AP V x () 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 AP V x () = 0. The opimal for a given age a policy issue, which will be denoed by x, is hen deermined by seing he expeced presen value of all fuure required reserve increases equal o 0, i.e. x is he roo of he equaion AP V x () = 0. Of course, he deerminaion of he opimal a ime 0 requires he knowledge of j [V ], = 1; 2; : : :, which correspond o he fuure medical in aion, = 1; 2; : : :, unknown a policy issue. Thus, deermining x according o j [B] he principle explained above requires an assumpion for he fuure medical in aion. 2.6.2 Opimal for a given porfolio of new enrans In general he opimal "exra premium increase facor " is dependen on he age a policy issue. Alhough from an acuarial poin of view i is possible o work wih an age-dependen x, consumers and regulaors may prefer a more sraighforward and simple approach, where he opimal is independen of he age a policy issue. Hereafer, we propose a possible way o deermine his age-independen opimal which will be denoed by. We rs de ne AP V () =! P 1 x=x 0 n x AP V x (); where x 0 is he younges age of enry and n x is he esimaed number of enrans a age x in his porfolio. Hence, AP V () expresses he acuarial value of he fuure reserve increases for his porfolio of new enrans. A posiive value of AP V () corresponds o an acuarial loss, while a negaive value of AP V () is an acuarial gain on his porfolio for he insurer. The opimal value of, which will be denoed by, is hen deermined as 10
he roo of he equaion AP V () = 0. Remark ha he use of an ageindependen opimal has he advanage (or disadvanage) ha i inroduces inergeneraional solidariy. Deermining according o he principle explained above again requires an assumpion for he fuure medical in aion. The numerical illusraions carried ou in he nex secion show ha several scenario s of fuure in aion lead o similar values of, indicaing ha he opimal is raher robus o he magniude of medical in aion. 3 Numerical illusraion 3.1 Technical basis In he numerical examples, he discoun facors correspond o a consan yearly ineres rae of 2%. The absolue rae of decremen due o deah q y [d] conforms o he rs Heligman-Pollard law, ha is, q [d] y 1 q [d] y = A (y+b)c + De E(ln y ln F )2 + GH y wih A = 0:00054, B = 0:017, C = 0:101, D = 0:00013, E = 10:72, F = 18:67, G = 1:464 10 5 and H = 1:11. Furhermore, we consider a lifelong cover and we x he ulimae age o! = 110. In line wih curren pracice on he Belgian marke, we assume ha he one-year absolue rae of decremen due o lapse q y [w] is equal o 0:1 0:002(y 20) a age y = 25; 26; : : : ; 70 and 0 oherwise. The lapse rae only depend on he aained age and no on he ime elapsed since policy issue. Figure 1 displays he one-year independen probabiliies q y [w] and 1 q y [d], as well as he non-exi probabiliies p y enering he compuaions. Based on healh insurance daa colleced by he Ialian Naional Insiue of Saisics (ISTAT) graduaed by he Ialian Associaion of Insurance Companies (ANIA), we choose he annual average claim amouns a age y and esimaed a ime 0, equal o c (0) y = 0:204476472 exp(0:038637y); y 20: 11
Figure 1: q y [w], 1 q y [d] and p y. 3.2 Iniial premium and reserves The level premium (0) x for an insured aged x a policy issue, x = 25; 26; : : : ; 70; is shown in Figure 2. The rajecory of he non-ransferable reserves for a policyholder aged 25 a policy issue, assuming ha no medical in aion is occurring during he erm of he conrac, is shown in Figure 3. Figure 2: Level premiums (0) x for di eren ages. 12
Figure 3: Reserves V (0) for a person aged 25 a policy issue when j [B] = 0. 3.3 Opimal as a funcion of he age a enry Figure 4 displays he expeced presen value of all fuure reserve increases AP V 25 () as a funcion of for 3 di eren scenario s of a consan in aion over ime: j [B] = 2:5%; 4% and 6%, respecively, while j [P ] = (1+)j [B] for all. Obviously, for a given in aion scenario, AP V 25 () is a decreasing funcion of : he higher, he more he policyholder nances he bene increases himself. Furher, for a given value of, he funcion AP V 25 () is an increasing funcion of he level of in aion: a higher level of he in aion leads o higher required reserve increases. For he scenario where j [B] = 2:5%, he opimal 25 lies beween 0:6 and 0:7. Increasing he yearly medical in aion o 4% or 6% leads o a seeper decreasing funcion AP V 25 () and decreases he value of he opimal value 25. The opimal 25 urns ou o be a decreasing funcion of he assumed medical in aion. The previous calculaions have been repeaed for all ages x a policy issue beween 20 and 70. The opimal values x, for he hree scenarios of medical in aion (j [B] = 2:5%; j [B] = 4% and j [B] = 6%), are depiced in Figure 5. The opimal facor 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 incrasing funcion of age a policy issue. From Figure 5, i is also clear ha for older ages x, he bene increase facor j [B] has a raher moderae e ec on he opimal facor x. The explanaion for his observaion lies again in he shorer remaining erm of he conrac. 13
Figure 4: AP V 25 () when j [P ] = (1 + )j [B]. Figure 5: The opimal facor x as a funcion of age a policy issue. 3.4 Opimal for a porfolio of new enrans 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. The acuarial presen value of he fuure reserve increases AP V () as a funcion of he facor is given in Figure 7 for hree scenarios of medical in aion (j [B] = 2:5%; j [B] = 4% and j [B] = 6%). We observe ha for a given 14
in aion scenario, AP V () is a decreasing funcion of, while for a given value of, he funcion AP V 25 () is an increasing funcion of he level of in aion. For j [B] = 2:5%, he opimal lies beween 0:4 and 0:5. Increasing he yearly medical in aion o 4% or 6% leads o a seeper decrease of he funcion AP V () and decreases he value of he opimal value. Despie his decreasing e ec, he high of he medical in aion seems o have only a moderae e ec on he opimal value. Figure 6: Disribuion of he age of new enrans. Figure 7: AP V () as a funcion of in case j [P ] = (1 + )j [B]. 15
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 (upredicable) medical in aion. We propose some premium indexing mechanisms which yearly resore he acuarial equivalence, aking ino accoun he observed medical in aion over he pas year. Firs, we discussed he general relaion ha has o hold beween yearly bene, premium and reserve increases in order o accoun for he unanicipaed in aion ha has occured. 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 in aion muliplied wih a facor (1 + ) for some > 0 which is chosen upfron. The proposed opimal value for 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 approch is supposed o make he yearly relaive premium increases above he observed medical in aion more sable. Finally, we proposed an aggregae approach which is applicable o a whole porfolio of new enrans, where an overall opimal is deermined. The laer approach leads o age-independen relaive premium increases above he medical in aion. 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 o ered 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, de ning appropriae moraliy indices. This approach, which is an e cien hedge for sysemaic longeviy risk, which is inheren in aging populaions, is also a opic for fuure research. Acknowledgemen 16
We would like o hank AG Insurance for he nancial suppor via he Chair on Healh Insurance a KU Leuven. References Da Silva, R. (2007). Indexaion of medical coss for Souh African medical Schemes. IAAHS Colloquium, Cape Town, May 2007. Devolder, P., Yerna, B.-L. (2008). Consrucion d une méhode spéci que d indexaion des conras privés d assurance maladie. Belgian Acuarial Bullein 8, 22-36. Piacco, E. (1999). Mulisae models for long-erm care insurance and relaed indexing problems. Applied Sochasic Models in Business and Indusry 15, 429-441. Ramjee, S., Kooverjee, A., Dreyer, K. (2011). The consrucion of a price index for Souh African medical scheme conribuions. Acuarial Sociey of Souh Africa s 2011 Convenion, Johannesburg, November 2011. 17