ISABEL MARIA FERRAZ CORDEIRO ABSTRACT

Transcription

1 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM* BY ISABEL MARIA FERRAZ CORDEIRO ABSTRACT Coreiro (2002a) has presene a muliple sae moel for Income Proecion (formerly known as Permanen Healh Insurance) which enables us o analyse claims by cause of isabiliy. In ha paper average claim uraions coniione on recovery have been calculae. Since, when an Income Proecion claim is repore, he insurance company never knows wheher i will en in recovery, in eah or in expiry, o analyse only average claim uraions coniione on recovery coul be misleaing, specially for he people responsible for he claims conrol process. The main purpose of his paper is o calculae average claim uraions coniione on eah an average claim uraions no coniione on any paricular moe of claim erminaion. We calculae hese claim uraions for ifferen eferre perios, causes of isabiliy an ages a he beginning of sickness an we analyse he resuls obaine. KEYWORDS Income Proecion; Permanen Healh Insurance; Muliple Sae Moels; Analysis by Cause of Disabiliy; Average uraion of a claim; Average uraion of a claim coniione on eah. 1. PRESENTATION OF THE PROBLEM Income Proecion (IP for breviy), formerly known as Permanen Healh Insurance (see Coninuous Moraliy Invesigaion Commiee (2001) for he reasons of he re-naming), is a class of long-erm an non-cancellable sickness insurance which provies cover agains he risk of loss of income ue o isabiliy. In general erms, an IP policy eniles he policyholer o an income uring perios of isabiliy longer han he eferre perio specifie in he policy. Benefis only sar o be pai afer he en of he eferre perio. * This research was suppore by FCT Funaçao para a Ciência e Tecnologia, Porugal, uner program POCTI. Asin Bullein 37(1), oi: /AST by Asin Bullein. All righs reserve.

2 134 I.M. FERRAZ CORDEIRO There are several ypes of IP policy sol by UK insurance companies. However, in his paper we are only inerese in iniviual convenional policies wih level benefis. This ype of policy eniles he policyholer o a regular level income uring perios of isabiliy longer han he eferre perio. In exchange for hese benefis he policyholer has o pay a regular level premium hroughou he erm of he policy: from he ime when he effecs i, a which he is require o be healhy, o his 60h or 65h birhay (usually, he age of his reiremen). In general, he premiums are waive whenever he policyholer is claiming. We will assume hroughou his paper ha a policy expires when he policyholer reaches age 65 or ies, whichever occurs firs. We will also assume ha each policy has one of he following eferre perios: 1 week, 4 weeks, 13 weeks or 26 weeks (for breviy, hroughou he paper we will refer o hese eferre perios as D1, D4, D13 an D26, respecively). Coreiro (2002a) has presene a new muliple sae moel for IP which enables us o analyse claims by cause of isabiliy. This moel, which is a generalizaion of he moel for IP propose in Coninuous Moraliy Invesigaion Commiee (1991), is very useful in he unerwriing an claims conrol sages of IP business since i allows he calculaion of quaniies such as he average uraion of a claim an claim incepion raes by cause of isabiliy. In paricular, as i is argue in Coreiro (2002a), ables showing he average uraion of a claim for he ifferen classes of causes of isabiliy, for he ifferen eferre perios an for ifferen ages a he beginning of sickness can help he unerwriers in heir ecisions concerning proposals for new enries an he persons responsible for he claims conrol process in keeping a igher conrol over he claims which are being pai an, herefore, in reucing claim recovery ime. In ha paper, for illusraive purposes, examples of he ables menione above were shown. However, all he average claim uraions in hese ables were coniione on recovery, i.e. hey were all uraions of claims which ene in recovery. I coul be argue ha, since for almos all he causes of isabiliy he vas majoriy of claims ens in recovery, as we will see in Secion 2, he imporan average claim uraions for insurance companies woul be hose coniione on recovery. However, o analyse only hese average claim uraions coul be misleaing, specially for he people responsible for he claims conrol process, since when a claim is repore, he insurance company never knows wheher i will en in recovery, in eah or in expiry (i.e. he sickness will no en before he corresponing policy expires). On he oher han, insurance companies have o ake ino consieraion ha, in general, as we will see in Secion 2, he percenage of claims which en in eah an in expiry increases wih he lengh of he eferre perio (becoming imporan for he longer eferre perios) an, for cause of isabiliy malignan neoplasms, here is a grea number of claims which en in eah. In view of he poins mae above, we hink i is imporan ha insurance companies shoul be able o analyse, no only average claim uraions coniione

3 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 135 on recovery, bu also average claim uraions no coniione on any paricular moe of claim erminaion an coniione on eah. The purposes of his paper are o calculae he laer wo average claim uraions for ifferen eferre perios, classes of causes of isabiliy an ages a he beginning of sickness an analyse hese claim uraions. For hose purposes, we use he approximaions o he ransiion inensiies efine for he moel menione above obaine in Coreiro (2002b). The iea ha insurance companies shoul analyse average uraions no coniione on any paricular moe of claim erminaion is alreay implici in he morbiiy ables for valuaion which he Unie Saes isabiliy income inusry has use for years. In fac, he erminaion raes given in he well known 85 CIDA ables o no iffereniae erminaions beween recoveries an eahs (see Commiee of he Sociey of Acuaries o Recommen New Disabiliy Tables for Valuaion (1985)). The same will happen in he new ables for valuaion which are being creae for he Unie Saes inusry (see Iniviual Disabiliy Experience Commiee (2005)). However, we shoul make wo remarks on his maer. Firsly, boh he suies which lea o he ables menione above are mae in a ifferen perspecive, when compare o his paper. Their main purpose is o consruc ables for calculaing reserves boh for healhy an sick policyholers (in he laer case, claim reserves). Our objecive is o calculae average claim uraions by cause of isabiliy in orer o help unerwriers an, specially, he people responsible for he claims conrol process. On he oher han, in boh ables menione above only recoveries an eahs (no expiries) are consiere erminaions. Furhermore, in he suies associae wih hese ables he impac of expiries on claim coss is no invesigae. In his paper we emonsrae ha insurance companies shoul no analyse only average claim uraions coniione on recovery by showing ha eahs an expiries conribue o make claims no coniione on any moe of erminaion, in general, much longer han hose which en in recovery. We also show ha, in a significan number of cases (i.e. combinaions of eferre perio, class of causes of isabiliy an age a he beginning of sickness), expiries have a srong impac on average uraions no coniione on any moe of claim erminaion. This paper is relae irecly o he works presene in Coreiro (2002a, 2002b) an in Coninuous Moraliy Invesigaion Commiee (1991) (for breviy we will refer o his repor as CMIC (1991) hroughou he paper). Oher imporan references on he subjec muliple sae moels applie o life an isabiliy insurance are: Haberman an Piacco (1999), Hoem (1969, 1972, 1976, 1988), Waers (1984, 1990) an Wolhuis (1994). In Secion 2, we escribe briefly he moel propose in Coreiro (2002a), we presen he approximaions o some of he ransiion inensiies efine for his moel an, finally, we give formulae for he average uraion of a claim coniione on eah an he average uraion of a claim no coniione on any paricular moe of claim erminaion. In Secion 3, we presen ables wih

4 136 I.M. FERRAZ CORDEIRO claim uraions coniione on eah an claim uraions no coniione on any moe of claim erminaion. We also analyse hese resuls. 2. FORMULAE FOR AVERAGE CLAIM DURATIONS Firsly, le us escribe briefly he moel menione in Secion 1, which is going o be he basis for our calculaions. This moel has (n + 2) saes (n > 1): healhy (enoe by H), ea (enoe by D), sick wih a sickness from class 1 (enoe by S 1 ), sick wih a sickness from class 2 (enoe by S 2 ),..., sick wih a sickness from class n (enoe by S n ). Each sae S i represens a ifferen class of causes of isabiliy. These n saes, consiere ogeher, group all possible causes of isabiliy. The imporan quaniies for his moel are he ransiion inensiies: s(i) x (associae wih he ransiions from H o S i ), r(i) x,z (associae wih he ransiions from S i o H), n(i) x,z (associae wih he ransiion from S i o D) (i =1, 2,...,n) an m x (associae wih he ransiion from H o D). The acion of hese ransiion inensiies governs he movemens of a policyholer beween he (n + 2) saes. The ransiion inensiies s(i) x (for a fixe i) an m x, which can be esignae as sickness inensiy for class i an moraliy of he healhy inensiy, respecively, epen only on x, he policyholer s aaine age. r(i) x,z an n(i) x,z (for a fixe i), which can be esignae as recovery inensiy an moraliy of he sick inensiy for class i, respecively, epen on x an on z, he uraion of he policyholer s curren sickness. Obviously, he s(i) x, r(i) x,z an n(i) x,z epen no only on x, or on x an z, bu also on he class of causes of isabiliy i hey are associae wih. Noe ha in he moel we are consiering here are no ransiions beween he saes S 1, S 2,..., S n. Once he policyholer is in sae S i, he can ener sae S j (i! j; i, j =1,2,...,n) only afer ransferring back o sae H. In oher wors, in his moel i is assume ha once he policyholer is suffering from a given coniion, he can become sick wih a ifferen coniion only afer recovering from he firs one. A firs sigh, we coul hink ha i woul be more realisic o efine for he moel ransiions beween he saes S 1, S 2,..., S n, bu his is no he case. In fac, i is no uncommon for someone who is sick wih a given illness o sar suffering also from anoher coniion or o recover from he original illness an wihin a very shor perio of ime o become sick again bu wih a ifferen illness. However, i is common pracice for he insurance companies, even in he cases where a sickness sops an anoher sars wihin a very shor perio of ime, o coninue wih he paymens of benefis o he policyholer uner he original claim an no o consier ha he original claim has ene an a new one has sare. Thus, no o allow hese ransiions in he moel, no only is convenien, since i makes he moel less complicae in mahemaical erms, bu also i is more consisen wih he IP claims aminisraion sysem an, ulimaely, wih he aa available concerning IP claims.

5 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 137 A full escripion an all he imporan mahemaical an saisical aspecs of he moel menione above can be foun in Coreiro (2002a, 2002b). In orer o obain approximaions o he ransiion inensiies menione above (excluing m x ), Coreiro (2002b) has use he grauaions of he ransiion inensiies efine for he moel propose in CMIC (1991), alreay menione in Secion 1, an a se of IP aa from UK insurance companies prouce by he Coninuous Moraliy Invesigaion Bureau of he Insiue of Acuaries an he Faculy of Acuaries: he Cause of Disabiliy Experience, Iniviual Sanar Experience, We shoul noe ha he moel presene in CMIC (1991) has only hree saes: healhy, sick (which groups all possible causes of isabiliy) an ea, an he ransiion inensiies efine for he moel are s x, r x,z, n x,z an m x (he sickness inensiy, he recovery inensiy, he moraliy of he sick inensiy an he moraliy of he healhy inensiy, respecively). Grauaions of hese inensiies have been obaine also in CMIC (1991). We shoul also noe ha in he se of aa menione above he claims are classifie ino he 18 sickness caegories which are presene in Table 1. Taking ino consieraion his fac, Coreiro (2002b) has assume ha he number of classes of causes of isabiliy in our moel is n = 18 an obaine approximaions o r(i) x,z, n(i) x,z an s(i) x for each of he 4 eferre perios we consier an each of hese 18 sickness caegories. TABLE 1 SICKNESS CATEGORIES INTO WHICH ARE CLASSIFIED THE DATA USED IN CORDEIRO (2002b) AND ESTIMATES OF THE CORRESPONDING k i. Sickness Caegory k3 i 1 Oher Infecive Malignan Neoplasms Benign Neoplasms Enocrine an Meabolic Menal Illness Nervous Disease Hear/Circulaing Sysem Ischaemic Hear Disease Cerebro Vascular Disease Acue Respiraory Bronchiis Respiraory Digesive Genio-Urinary Arhriis/Sponyliis Oher Musculoskeleal 1 16 R.T.A. Injuries Oher Injuries All Ohers 1

6 138 I.M. FERRAZ CORDEIRO For a given eferre perio, he approximaions o he ransiion inensiies have been obaine assuming ha r(i) x,z = k i r x,z i = 1,...,18 n(i) x,z = w i n x,z i = 1,...,18 s(i) x = exp{a i + b i x}s x i = 1,...,18 (1) where k i, w i, a i an b i (i = 1,...,18) are unknown parameers which can be esimae from he aa. As far as he recovery an moraliy of he sick inensiies are concerne, his means i is assume ha, for a given eferre perio, alhough r(i) x,z an n(i) x,z have he same shapes as r x,z an n x,z, respecively, hey can have ifferen levels. In Table 1, for each of he 18 sickness caegories we consier, we presen he esimae level of he corresponing r(i) x,z (ue o anoher assumpion mae by Coreiro (2002b), his level is he same for he 4 eferre perios we consier). The approximaion propose for each n(i) x,z is he grauaion of n x,z obaine in CMIC (1991) (which is he same for he 4 eferre perios we consier), excep for sickness caegory 2 (malignan neoplasms), in which case he esimae level of n(2) x,z is much higher (w62 =13.55, for he 4 eferre perios we consier), as we woul expec. Since we will no nee o use he approximaions o he s(i) x an m x in his paper, we o no presen here more informaion abou hem. A full escripion of he esimaion processes which lea o he approximaions o he ransiion inensiies efine for our moel can be foun in Coreiro (2002b). I is imporan o noe ha, since he approximaions o he r(i) x,z an n(i) x,z we use in his paper have been obaine wih aa from , i is very likely ha he average claim uraions we presen in Secion 3 are no upae. In fac, we have o consier ha changes over ime in ifferen facors, such as meical echnology an claims managemen echniques, imply changes in he ransiion inensiies. The ieal siuaion woul be o work wih grauaions of he inensiies base on more recen aa bu, unforunaely, hese grauaions are no available. The fac ha he ransiion inensiies have been changing wih ime is confirme by he invesigaions carrie ou in successive Coninuous Moraliy Invesigaion repors (CMIC (1996), CMIC (2000), CMIC (2001) an CMIC (2005)). These invesigaions ry o ienify rens in IP claims aa for successive quarennia by using a mehoology base on a comparison of acual versus expece recoveries an eahs (analyse separaely). This mehoology was propose iniially in CMIC (1996). Unforunaely, he IP aa analyse in he menione invesigaions are no by cause of isabiliy an he aa afer 1990 refer o a wier IP experience (i.e. he experience o which our aa refer is a subse of he experience analyse in hose invesigaions). More recenly, an analysis of IP aa by cause of isabiliy covering he perio has become available (see Income Proecion Commiee (2006)). One

7 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 139 of he purposes of his work is also o analyse rens in he aa. However, neiher hese aa are classifie ino exacly he same sickness caegories as our aa, nor a saisically robus analysis of hese aa (e.g. an analysis base on he mehoology propose in CMIC (1996)) has been carrie ou (only a simple saisical aa analysis has been presene). Furhermore, he aa afer 1990 also refer o he wier IP experience menione above. Alhough i is no possible o use he informaion presene in Income Proecion Commiee (2006) in orer o upae he approximaions o our ransiion inensiies, his informaion suggess i is very likely ha many of he conclusions in Secion 3 which resul from comparisons beween sickness caegories are sill vali oay. This is cerainly rue for many of he conclusions concerning claims which en in recovery presene in Coreiro (2002a), where he same approximaions are use (e.g. he sickness caegories which have he shorer an he longer claim uraions are broaly he same; i is sill rue ha, in general, he average uraion of a claim increases as he eferre perio becomes longer an ha he n(2) x,z ominae he corresponing inensiies for all he oher sickness caegories, making he caegory malignan neoplasms he main responsible for eahs). Table 2 shows a summary of par of he se of IP aa menione above. A full accoun of he characerisics an limiaions of hese aa can be foun in Coreiro (2002b). TABLE 2 NUMBER OF CLAIMS WHICH ENDED IN RECOVERY, IN DEATH AND IN EXPIRY. Sick. Ca. Number of claims which Number of claims which Number of claims which ene in Recovery ene in Deah ene in Expiry D1 D4 D13 D26 D1 D4 D13 D26 D1 D4 D13 D All S.C As we can see from Table 2, he percenage of claims which ene in eah an in expiry increases wih he lengh of he eferre perio, being 47.1% for eferre perio D26. We can also see ha, for sickness caegory 2 (malignan neoplasms), he number of claims which ene in eah is significan for all he eferre perios consiere (for D13, i represens half of he claims which ene in eah for all causes of isabiliy). Furhermore, if we analyse he se of aa menione above, we can fin a significan number of cases where, alhough he number of claims which ene in eah or in expiry is small, he average uraion of hese claims is exremely long (hese aa are no shown). There are even a few cases where he number of claims which ene in expiry is significan an he corresponing average uraion is quie long.

8 140 I.M. FERRAZ CORDEIRO Le us give some examples of such cases. For sickness caegory 8 (ischaemic hear isease) an eferre perios D1 an D26, here are 22 an 17 claims which ene in expiry, respecively, an he corresponing average uraions are an weeks, respecively. For sickness caegory 5 (menal illness) an D13, he number of claims which ene in eah is 4 an he respecive average uraion is weeks. Also for sickness caegory 5 an D26, he number of claims which ene in expiry is 12 an he corresponing average uraion is 413 weeks. Consiering all he facs menione above, as we have sae in Secion 1, insurance companies selling IP policies (specially he people responsible for he claims conrol process) can only benefi from analysing, no only average claim uraions coniione on recovery, bu also average claim uraions no coniione on any moe of claim erminaion an coniione on eah. We shoul noe ha, for a given age a he beginning of sickness, he uraion of a claim which ens in expiry is obvious. Coreiro (2002a) has presene he formula for he average uraion of a claim coniione on recovery for a given class of causes of isabiliy. If we enoe by T r iy he uraion of a claim in class i, which ens in recovery, for a claiman age y a he beginning of he corresponing sickness, he average uraion of his claim (in years) is given by r iy EaT k # 65 - y y p r^ih y+, = y p r^ih # y y +, i = 1,2,,n (2) where is he eferre perio expresse in years an SS p i i wih z = 0, being y + z, z p SS i y SS i i y + z, z = exp - r + n 0 y + z + s, z + s y + z + s, z + s i is he noaion for p ) # a ^ih ^ih k s3 (3) he probabiliy of a policyholer remaining sick unil a leas age (y + z + ), given ha he is sick a age (y + z) wih a sickness from class i an wih uraion of sickness z. p y + z, z is a basic probabiliy for our moel an he erivaion of is formula can be foun in Coreiro (2002a). Wih wo simple changes of variable an some rivial manipulaion, i can be shown ha formula (2) is equal o: r iy EaT k # 65 - y- 0 = 65 - y- # 0 y +, p r^ih y +, y + +, + p r^ih y + +, + i = 1,2,,n (4) In orer o obain he formula for he average uraion of a claim coniione on eah, we only have o make small changes in formula (4). In fac, if we

9 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 141 enoe by T m iy he uraion of a claim in class i, which ens in eah, for a claiman age y a he beginning of he corresponing sickness, he average uraion of his claim (in years) is given by EaT m iy k # 65 - y- 0 = 65 - y- # 0 y +, p n^ih y +, y + +, + p n^ih y + +, + i = 1,2,,n (5) Since when a claim is repore, i is no known in avance wheher i will en in recovery, in eah or in expiry, i woul be useful for he insurance company o have available ables wih average claim uraions no coniione on any paricular moe of claim erminaion (i.e. average uraions of claims which en in recovery or in eah or in expiry while he corresponing policies are in force). The formula for he average uraion of a claim in class i, no coniione on any moe of claim erminaion, for a claiman age y a he beginning of he corresponing sickness, is he following: # 65 - y- i y 0 y +, ET _ i = p ar^ih + n^ih k y + +, + y + +, + y +, + ^65 - y - h p i = 1, 2,..., n 65 - y- (6) where T iy enoes he uraion of such a claim. The firs erm on he righ han sie of (6) concerns he case where he claim ens in recovery or in eah whereas he secon erm concerns he case where he claim ens in expiry (i.e. he corresponing sickness ens in recovery or in eah afer he policy expires a age 65). T iy has a mixe isribuion wih posiive ensiy from 0 o (65 y ) an a probabiliy mass a (65 y ) (he uraion of he claim when i ens in expiry). 3. ANALYSIS OF THE AVERAGE CLAIM DURATIONS In his secion we use he approximaions o he r(i) x,z an n(i) x,z propose in Coreiro (2002b) o calculae he average uraion of a claim coniione on eah an he average uraion of a claim no coniione on any moe of claim erminaion for he 4 eferre perios we consier, he 18 sickness caegories presene in Table 1 an ifferen ages a he beginning of sickness (20, 25, 30, 35, 40, 45, 50, 55 an 60). The inegrals in formulae (5) an (6) have been calculae numerically, using he repeae rapezoial rule wih a sep size equal o 1/156 of a year, an we assume here are exacly 52 weeks in a year. We shoul noe ha, alhough here, ue o limiaions of space, we show only he resuls for eferre perios D1 an D26 an ages a he beginning of sickness 20, 30, 40, 50 an 60, our conclusions refer o all he resuls obaine. We shoul also noe ha, as i can be noice, he average claim uraions

10 142 I.M. FERRAZ CORDEIRO TABLE 3 AVERAGE DURATION OF A CLAIM WHICH ENDS IN DEATH (IN WEEKS). DEFERRED PERIODS D1 AND D26. Sick. Ca. D1 D26 Age a he Beginning of Sickness Age a he Beginning of Sickness have he same values for some pairs of sickness caegories (1 an 3; 4 an 8; 15 an 18). The reason for his is he fac ha he sickness caegories in each pair have he same approximaions o r(i) x,z an n(i) x,z (see Secion 2). The average claim uraions coniione on eah (in weeks) are shown in Table 3. The mos prominen feaure in his able is ha, in general, he average claim uraions are very long. Furhermore, here is a significan number of cases where he claim uraions are exremely long (see, for example, he claim uraions for sickness caegories 4 (enocrine an meabolic), 5 (menal illness), 8 (ischaemic hear isease) an 14 (arhriis/sponyliis), he younger ages a he beginning of sickness an all he eferre perios we consier). If we compare hese uraions wih he corresponing average claim uraions coniione on recovery, which appear in Table 4 (he values in his able are exrace from Coreiro (2002a, Tables 3 o 6)), we can see ha, for a given combinaion of eferre perio, sickness caegory an age a he beginning of sickness, he average uraion of a claim coniione on eah is always greaer han he average uraion of a claim coniione on recovery. We were expecing his feaure. In fac, if we consier he moel escribe in Secion 2 an

11 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 143 TABLE 4 AVERAGE DURATION OF A CLAIM WHICH ENDS IN RECOVERY (IN WEEKS). DEFERRED PERIODS D1 AND D26. Sick. Ca. D1 D26 Age a he Beginning of Sickness Age a he Beginning of Sickness ha, in general, for a given sickness caegory i, he grauae values of r(i) x,z are much higher han hose of n(i) x,z (see Coreiro (2002a, 2002b) an CMIC (1991)), i is easy o see ha, on average, a policyholer remains more ime in he sae S i before he ies from a sickness in caegory i han before he recovers from a sickness in he same caegory. The fac ha, in general, he claims which en in eah are much longer han he claims which en in recovery confirms ha unerwriers an he people responsible for he claims conrol process shoul analyse no only claim uraions coniione on recovery bu also claim uraions coniione on eah. This conclusion is relevan specially when consiering claims in sickness caegory 2 (malignan neoplasms), where here is a significan number of eahs, an claims in caegories where he average claim uraions coniione on eah are exremely long. Anoher prominen feaure in Table 3 is ha, in general, for a given combinaion of sickness caegory an age a he beginning of sickness, he average uraion of a claim increases as he eferre perio becomes longer (his oes no happen only when we compare he claim uraions for D4, sickness caegory 2 an ages a he beginning of sickness higher han 40 wih he corresponing ones

12 144 I.M. FERRAZ CORDEIRO for D1). We also observe his feaure in he average claim uraions coniione on recovery (see Table 4 an Coreiro (2002a)). The explanaion for his feaure is he same as for he claim uraions coniione on recovery: he average uraion of a claim ens o be longer for longer eferre perios because in he calculaion of he average uraion, for a given eferre perio, we only consier sicknesses wih uraions longer han (see formula (5)). We can also see from Table 3 ha, for mos of he combinaions of eferre perio an sickness caegory, he average uraion of a claim coniione on eah increases wih age a he beginning of sickness up o some age an hen ecreases. For all he eferre perios we consier, here are a few sickness caegories for which he average uraion of a claim always ecreases wih y (he number of such cases ens o increase as he eferre perio becomes longer). We also observe hese feaures in he average claim uraions coniione on recovery (see Table 4 an Coreiro (2002a)). However, in Table 3: in general, he claim uraions sar o ecrease a earlier ages; for D1, we observe a greaer number of cases where he average uraion is a ecreasing funcion of y; an, unlike in he average uraions coniione on recovery, we o no see any case where he average uraion always increases wih y. While having an average uraion which increases up o some age an hen ecreases can be consiere as a non-inuiive feaure for claims which en in recovery (see Coreiro (2002a)), his is no he case for claims which en in eah. In fac, we woul expec he average uraion of a claim coniione on eah o ecrease wih y, a leas afer some age. The explanaions for he average claim uraions which ecrease afer some age (or which always ecrease wih y) are he same as hose given for he claim SS uraions coniione on recovery: he behaviour of he probabiliy p i i y +, when y increases, which epens on he levels an shapes of r(i) x,z an n(i) x,z, an he policy expiraion effec (i.e. claims have o be shorer han (65 y )) (see Coreiro (2002a)). As far as he former explanaion is concerne, noe ha he average claim uraions which always ecrease wih y are hose for he sickness caegories wih he r(i) x,z having he lowes levels: sickness caegories 2, 4, 5, 8, 9 an 14 (malignan neoplasms, enocrine an meabolic, menal illness, ischaemic hear isease, cerebro-vascular isease an arhriis/sponilyis, respecively; see Table 1). A reason ha can be given for claim uraions coniione on eah saring o ecrease a earlier ages han hose coniione on recovery (an also for no having uraions coniione on eah which always increase wih age) is ha he longer he claim uraions, he sronger is he policy expiraion effec. As we have seen above, in general, he claim uraions coniione on eah are very long. The average claim uraions no coniione on any moe of claim erminaion (in weeks) are shown in Table 5. We can see from Tables 3, 4 an 5 ha, for a given combinaion of eferre perio, sickness caegory an age a he beginning of sickness, he average uraion of a claim no coniione on any moe of claim erminaion is

13 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 145 TABLE 5 AVERAGE DURATION OF A CLAIM NOT CONDITIONED ON ANY MODE OF CLAIM TERMINATION (IN WEEKS). DEFERRED PERIODS D1 AND D26. Sick. Ca. D1 D26 Age a he Beginning of Sickness Age a he Beginning of Sickness always greaer han he average uraion of a claim which ens in recovery an, for some cases, i is even greaer han he average uraion of a claim which ens in eah. Furhermore, when we consier all he resuls obaine, he number of he laer cases seems o be an increasing funcion of simulaneously he age a he beginning of sickness, he inverse of he level of he approximaion o r(i) x,z for sickness caegory i (1/k3 i ) an he lengh of he eferre perio, becoming quie significan for eferre perios D13 (32 cases ou of 162) an D26 (55 cases). The fac ha, on average, a claim no coniione on any moe of claim erminaion is always longer han a claim which ens in recovery is no a all surprising. In fac, he claims no coniione on any moe of claim erminaion comprise also he claims which en in eah (which, on average, are longer han he claims which en in recovery) an he claims which en in expiry (which have he longes possible uraions). On he oher han, he oher feaures menione in he paragraph before las seem o sugges ha he claims which en in expiry are a phenomenon which can become imporan (in he sense ha, on average, i can make he claims

14 146 I.M. FERRAZ CORDEIRO even longer han hose which en in eah), specially for he cases where he age a he beginning of sickness is high, he sickness caegory has a r(i) x,z wih a low level an he eferre perio is long. In orer o hrow some ligh on he maer, we ecie o calculae y +, 65 - y- p (i.e. he probabiliy ha a claim in sickness caegory i, for a claiman age y a he beginning of sickness, ens in expiry) for all he eferre perios we consier an some sickness caegories an ages a he beginning of sickness. Par of hese resuls are shown in Table 6 (we o no show he resuls for eferre perios D4 an D13). TABLE 6 PROBABILITY OF A CLAIM ENDING IN EXPIRY. DEFERRED PERIODS D1 AND D26. Sickness Caegory D1 D26 Age a he Beginning of Sickness Age a he Beginning of Sickness , Afer analysing all he resuls obaine, we have conclue ha: for a given combinaion of sickness caegory an age a he beginning of sickness, he probabiliy increases as he eferre perio becomes longer; for a given combinaion of eferre perio an sickness caegory, he probabiliy increases as he age a he beginning of sickness becomes higher; an, in general, he probabiliy is higher for sickness caegories wih he r(i) x,z having lower levels (in his las analysis we o no consier sickness caegory 2 (malignan neoplasms) which has an approximaion o he n(2) x,z wih a much higher level han he ohers). If we hink carefully abou hese feaures, we can see ha, inuiively, all of hem were expece (recall ha approximaions o he r(i) x,z wih lower levels mean longer claims). We can also conclue ha, epening on he values of, i an y, 65 - y-py +, can be surprisingly high. See, for example, he values of S5S5 S15 S / 2p an /, 12 / p. The value of his probabiliy can be / /, 12 / significan even for early ages a he beginning of sickness an no very long S5S5 eferre perios. Consier, for example, / 4 p /, 14 / = In he ligh of he informaion given in he previous paragraphs, we can conclue ha, afer all, he behaviour of he claim uraions no coniione on any moe of claim erminaion for high ages a he beginning of sickness, sickness caegories wih approximaions o he r(i) x,z having low levels an long

15 SOME NOTES ON THE AVERAGE DURATION OF AN INCOME PROTECTION CLAIM 147 eferre perios was expece. The behaviour of he uraions of claims which en in expiry, menione above, conribues o make he claims in Table 5, in general, much longer han hose which en in recovery, specially for ages a he beginning of sickness above 40 (for he wors cases we have claims which are 8, 9 an 10 imes longer han hose which en in recovery). The oher feaures we can observe in Table 5 were alreay observe in claim uraions coniione on recovery an in claim uraions coniione on eah: in general, for a given combinaion of sickness caegory an age a he beginning of sickness, he average claim uraion increases as he eferre perio becomes longer an, for mos of he combinaions of eferre perio an sickness caegory, he average claim uraion increases up o some age a he beginning of sickness an hen ecreases. However, in his able: in general, he claim uraions sar o ecrease a much laer ages han hose coniione on recovery; we observe more cases where he average uraion is an increasing funcion of y han in he uraions coniione on recovery; an, unlike in he claims which en in recovery an in eah, he average uraion is a ecreasing funcion of y only for sickness caegory 2 an all he eferre perios we consier. I is obvious ha he main reason for hese ifferences is also he behaviour of he claim uraions coniione on expiry. The resuls in Table 5 show ha insurance companies shoul also analyse claim uraions no coniione on any moe of claim erminaion. In fac, consiering ha, when a claim is repore, he insurance company oes no know in avance is moe of erminaion, an he uraions in Table 5, in general, are much longer han hose coniione on recovery, o analyse only he laer average uraions coul be quie misleaing for he persons who are conrolling he uraions of he claims which are being pai. Anoher conclusion we can raw, i is ha claims which en in expiry are a phenomenon which shoul no be ignore by insurance companies. As we have seen above, hese are he claims wih he longes possible uraions an heir number becomes quie imporan for longer eferre perios, laer ages a he beginning of sickness an sickness caegories wih approximaions o he r(i) x,z having lower levels. ACKNOWLEDGEMENTS I am graeful o he associae eior responsible for his paper an hree anonymous referees for heir invaluable commens an suggesions. REFERENCES COMMITTEE OF THE SOA TO RECOMMEND NEW DISABILITY TABLES FOR VALUATION (1985) Repor of he Commiee o Recommen New Disabiliy Tables for Valuaion. Transacions of Sociey of Acuaries 37, CONTINUOUS MORTALITY INVESTIGATION COMMITTEE (1991) The Analysis of Permanen Healh Insurance Daa. Coninuous Moraliy Invesigaion Repors 12. The Insiue of Acuaries an he Faculy of Acuaries.

16 148 I.M. FERRAZ CORDEIRO CONTINUOUS MORTALITY INVESTIGATION COMMITTEE (1996) Recovery an Moraliy Raes of Those Claiming Uner PHI Policies, Iniviual an Group Coninuous Moraliy Invesigaion Repors 15. The Insiue of Acuaries an he Faculy of Acuaries. CONTINUOUS MORTALITY INVESTIGATION COMMITTEE (2000) Sickness Experience for Iniviual PHI Policies. Coninuous Moraliy Invesigaion Repors 18. The Insiue of Acuaries an he Faculy of Acuaries. CONTINUOUS MORTALITY INVESTIGATION COMMITTEE (2001) Sickness Experience for Iniviual Income Proecion Policies. Coninuous Moraliy Invesigaion Repors 20. The Insiue of Acuaries an he Faculy of Acuaries. CONTINUOUS MORTALITY INVESTIGATION COMMITTEE (2005) Sickness Experience for Iniviual Income Proecion Policies. Coninuous Moraliy Invesigaion Repors 22. The Insiue of Acuaries an he Faculy of Acuaries. CORDEIRO, I.M.F. (2002a) A Muliple Sae Moel for he Analysis of Permanen Healh Insurance Claims by Cause of Disabiliy. Insurance: Mahemaics an Economics 30(2), CORDEIRO, I.M.F. (2002b) Transiion Inensiies for a Moel for Permanen Healh Insurance. Asin Bullein 32(2), HABERMAN, S. an PITACCO, E. (1999) Acuarial Moels for Disabiliy Insurance. Chapman & Hall/CRC, Lonon. HOEM, J.M. (1969) Some Noes on he Qualifying Perio in Disabiliy Insurance II. Problems of Maximum Likelihoo Esimaion. Mieilungen er Vereinigung schweizerischer Versicherungsmahemaiker 69, HOEM, J.M. (1972) Inhomogeneous Semi-Markov Processes, Selec Acuarial Tables an Duraion Depenence in Demography. In Populaion Dynamics (e. T.N.E. Greville), Acaemic Press, New York. HOEM, J.M. (1976) The Saisical Theory of Demographic Raes A Review of Curren Developmens. Scaninavian Journal of Saisics 3, HOEM, J.M. (1988) The Versailiy of he Markov Chain as a Tool in he Mahemaics of Life Insurance. Transacions of he 23r Inernaional Congress of Acuaries R, INCOME PROTECTION COMMITTEE (2006) Analysis of Iniviual Income Proecion Experience by Cause of Disabiliy. Coninuous Moraliy Invesigaion Working Paper 23. The Insiue of Acuaries an he Faculy of Acuaries. INDIVIDUAL DISABILITY EXPERIENCE COMMITTEE (2005) Iniviual Disabiliy Experience Commiee Repor. Sociey of Acuaries. WATERS, H.R. (1984) An Approach o he Suy of Muliple Sae Moels. The Journal of he Insiue of Acuaries Par II: 111(448), WATERS, H.R. (1990) The Recursive Calculaion of he Momens of he Profi on a Sickness Insurance Policy. Insurance: Mahemaics an Economics 9, WOLTHUIS, H. (1994) Life Insurance Mahemaics (The Markovian Moel). CAIRE Eucaion Series, Vol. 2. ISABEL MARIA FERRAZ CORDEIRO Escola e Economia e Gesao Universiae o Minho Campus Universiário e Gualar Braga Porugal Tel: Fax: icoreiro@eeg.uminho.p