Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Transcription

1 Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical Sciences, Universiy of Copenhagen, Denmark 2 Edlund A/S, Denmark Sepember 27, 2013 Absrac We calculae reserves regarding expeced policy holder behavior. The behavior is modeled o occur incidenally similarly o insurance risk. The focus is on muli-sae modeling of insurance risk, e.g. in a disabiliy model, and of behavioral risk, e.g. in a premium paymen - free policy - surrender model. We discuss valuaion echniques in he cases where he behavior is modeled o occur independenly of insurance risk and where we ake explicily ino accoun ha e.g. disabled do no hold behavioral opions, respecively. Ordinary differenial equaions make i easier o work wih dependence beween insurance risk and behavior risk. We analyze he effecs of he underlying behavioral assumpions in wo conracs. For a new conrac, i.e. low echnical ineres rae relaive o he marke ineres rae, we obain he lowes reserve by working wih he correc model wihou inaccurae shorcu assumpions. For an old conrac, i.e. high echnical ineres rae relaive o he marke ineres rae, he picure is more blurred, depending on assumpions on reacivaion recovery and he roue of he shorcu. 1 Inroducion We characerize reserves under finie-sae Markov chain modeling of policy holder behavior and illusrae numerically he effecs on values from modeling policy holder behavior in various ways. By policy holder behavior we hink, in paricular, of policy holder inervenion like ranscripion o free policy and surrender. The reserves are characerized by ordinary differenial equaions and heir more or less explici soluions, depending on he behavior model and he underlying risk model. These soluions are paricularly racable if one assumes independence beween insurance risk and behavior, alhough such independence is ofen ruled ou by conrac design: Disabiliy Work suppored by he Danish Advanced Technology Foundaion Højeknologifonden

2 and life annuians are ypically no allowed o exercise such behavior opions. In he illusraions we calculae values for sandard conracs in order o analyze he effecs of aking ino accoun policy holder behavior in various ways. In paricular, we sudy he consequences of assuming independence beween insurance risk and behavior. Curren developmens in insurance accouning and solvency rules ake an explici approach o policy holder behavior. For calculaing reserves i is o an increasing exen required o ake ino accoun policy holder behavior. Policy holder behavior should be hough of as acions aken by he policy holder ha influence eiher he risk in he processes ha drive he paymen sreams of an insurance conrac or he paymens hemselves. In his paper we pay special aenion o he surrender opion, i.e. he opion o erminae he conrac in exchange for a lump sum paymen, and he free policy opion, i.e. he opion o sop paying he premium agains a reducion of benefis. Among oher opions held by he policy holder may be he annuiizaion opion in case he defaul coverage is a pension sum ha can hen, on basis of echnical assumpions abou ineres and moraliy, be convered o an annuiy. Alhough his disincion is no necessarily a clean cu in pracice, annuiizaion is an opion ha can be exercised upon reiremen and can herefore be hough of as a European ype opion. The surrender and free policy opions can, in general, be exercised a any poin in ime and can herefore be hough of as American ype opions. Anoher opion ha is someimes menioned explicily is he opion o raise he premiums. Typically such an opion is provided in connecion wih an occupaional pension scheme, where e.g. premiums are calculaed as a percenage of he salary. There exiss a range of approaches o modeling of behavior risk. One exreme posiion o ake is o assume ha he policy holder exercises his opions based on an economically opimal sraegy, i.e. in order o maximize he value of he paymen sream from he conrac. This approach is aken in Seffensen 2002 in order o characerize values in erms of so-called variaional inequaliies known in a financial conex from American opion pricing. Compared o a sandard American opion i is, of course, a delicae feaure of he conrac ha, possibly, boh a free policy and a surrender opion exis. Furhermore, if he free policy opion is exercised he conrac does no vanish bu coninues under differen erms and possibly including, sill, a surrender opion. This is all deal wih by Seffensen The same exreme American opion approach is aken by e.g. Grosen and Jørgensen 2000 and Bacinello A primiive approximaion of he value obained from his approach is o reserve, a any poin in ime, he larger of he value based on no exercise and he surrender value. This is wha has been called a now or never -reserve since i corresponds o opimizing over wo inervenion sraegies corresponding o exercising now or never. Due o is racabiliy, his is ofen seen as a firs approach o ake inervenion opions ino accoun in pracical accouning rules. Clearly, his approximaion underesimaes he rue value since he opimal sraegy may be o exercise somewhere beween now or never. Anoher exreme approach o ake is o assume ha he inervenion opions are exercised compleely incidenally. Then inervenion risk can be reaed formalisically as diversifiable insurance risk, alhough hey are differen conceps and he reamen considerably complicaes, in general, he saes of he world ha have o be aken ino accoun. This approach is aken e.g. by Buchard and Møller 2013 and Buchard e al A modern approach in accouning and solvency is o base reserves on expeced policy holder behavior raher han raional policy holder behavior. This approach is aken e.g. in preliminary 2

3 formulaions of boh Solvency II and IFRS. This draws aenion owards he laer of he wo exremes. Then, however, he expecaion o policy holder behavior is explicily required o ake ino accoun also e.g. he economic environmen and/or wheher he opion is beneficial. This appears o be one sep back owards he firs exreme wihou really going ha far. Such inermediary modeling is an ineresing objec of sudies wih a lo of challenges concerning he saisical maerial available, economic inuiion and mahemaical racabiliy of he sudied objecs depending on he driving facors. Simple ideas are o le he inervenion inensiy depend on ineres raes, as was done by De Giovanni 2010, or a relaion beween he inervenion value and some noion of he marke reserve. These ideas address he quesions abou regarding he economic environmen and wheher he opion is beneficial. There exiss a large amoun of empirical lieraure discussing explanaory variables. These range from macro variables like ineres raes, e.g. sudied by Kuo e. al. 2003, o micro variables like for insance policy holder age. We refer o Eling and Kiesenbauer 2013 and references herein for a comprehensive lieraure overview. In his paper we ake he exreme approach o assume ha he inervenion opions are exercised compleely incidenally. This does no mean ha we do no believe ha working wih ineres rae or reserve dependen inervenion inensiies are ineresing, imporan, challenging, or relevan. We are jus focusing on somehing else. Also, we focus on somehing raher differen from Buchard and Møller 2013 who mainly concenrae on represenaion and calculaion of cash flows in one of he special cases of our sudy, and Buchard e al who in a more heoreical framework deal wih duraion dependence in he risk model. We are ineresed in discussing he dependence beween insurance risk and behavioral risk ha arises essenially from he produc design. We do his in a finie sae Markov chain framework. Tha allows us o characerize condiional expeced values by ordinary differenial equaions and represenaions of soluions. Their srucures make i clear in wha sense one can choose beween a complicaed, differenial equaion based approach and he wrong soluion. Or said in a differen way: Keep i simple or keep i righ! To make he good choice here, i is of course relevan o qualify his one-liner. How simple is simple? And if simple means wrong, hen how wrong is wrong? These quesions are discussed from a heoreical poin of view hroughou he firs par of he paper in Secions 2-5 and addressed numerically in he second par in Secion 6. In a horough analysis where several aspecs are aken ino accoun, including varying over he value of he inervenion opions, he answer is no surprisingly: I depends! This paper illuminaes on wha i depends. A conclusion is ha i really does maer for odays enry values, in general, wheher one akes he simple or he righ approach. This makes a case for our advanced mehods. All numerical resuls are obained by Aculus R Calculaion Plaform 2 Risk and Behavior Models In his secion we presen he idea of considering a combined model for risk and behavior as being decomposed ino wo separae models for risk and behavior, respecively, ha are or are no probabilisically dependen of each oher. We hink of a risk sae model Z risk and a behavior sae model Z behavior and consider he sae model Z risk, Z behavior. Given he whole process hisory of Z behavior, Z risk is assumed o be a finie-sae Markov chain aking values in Z risk. Thus, condiional on Z behavior, here exis ransiion inensiies µ jk 3

4 for j, k Z risk and 0, such ha for all k Z risk, µzrisk sk s ds is condiional on Z 0 behavior a compensaor for he couning process couning he number of jumps ino risk sae k. The ransiion inensiies may be independen of Z behavior and in ha case Z risk is, even uncondiional on Z behavior, a finie-sae Markov chain. A canonical muli-sae example of a risk model is he disabiliy model illusraed in Figure 1. We have labeled he saes {acive, disabled, dead} by he leers {a, i, d}. This risk model is a key example below and in he numerical illusraions in paricular. a acive µ ad µ ai µ ia i disabled d dead µ id Figure 1: Disabiliy risk model Given he whole process hisory of Z risk, Z behavior is assumed o be a finie-sae Markov chain aking values in Z behavior. Thus, condiional on Z risk, here exis ransiion inensiies ν jk for j, k Z behavior and 0, such ha for all k Z behavior, 0 ν Z behavior sk s ds is condiional on Z risk a compensaor for he couning process couning he number of jumps ino behavior sae k. The ransiion inensiies may be independen of Z risk and in ha case Z behavior is, even uncondiional on Z risk, a finie-sae Markov chain. A canonical muli-sae example of a behavior model is he free policy/surrender model illusraed in Figure 2. We have labeled he saes {premium paymen, free policy, surrender} by he leers {p, f, s}. This behavior model is a key example below and in he numerical illusraions in paricular. p premium paymen ν ps ν pf ν fp ν fs f free policy s surrender Figure 2: Behavior model The repeiions in he wo paragraphs above are no made o bore he reader bu o emphasize he up-fron symmery in he wo separae models. As can be seen above, we refer consequenly o specificaions and saes in he risk model by superscrips and o specificaions and saes in he behavior model by subscrips. When Z risk given Z behavior and Z behavior given Z risk are Markov models, he combined model Z = Z risk, Z behavior is a Markov model. Thus, here exis risk ransiion inensiies µ jk l for j, k Z risk, l Z behavior and 0, such ha for all k Z risk, sk 0 µzrisk Z behavior s s ds is a compensaor for he couning process couning he number of jumps ino risk sae k. Similarly, here exis risk ransiion inensiies νjk l for j, k Z behavior, l Z risk and 0, such ha for all k Z behavior, µzrisk s 0 Z behavior sk s ds is a compensaor for he couning process couning he number of jumps ino behavior sae k. We inroduce he noaion p jk lm, s for he ransiion probabiliy ha he risk model goes from 4

5 j o k and he behavior model goes from l o m over, s. In he case where he wo sub-models for Z risk and Z behavior are independen, i.e. he ransiion inensiies µ do no depend on Z behavior and he ransiion inensiies ν do no depend on Z risk, we can simplify his probabiliy ino a produc of probabiliies wih respec o each sub-model, i.e. p jk lm, s = pjk, s p lm, s. In case of independence we specify here he ransiion probabiliies in he wo models exemplified above. If µ ai and µ ia are boh posiive, we have no closed-form expressions for he probabiliies p aa, p ii, p ai, p ia, p ad, p id. Bu in he case of no reacivaion, i.e. µ ia = 0, we do: p aa, s = e µ ai τ+µ ad τdτ ; p ii, s = e p ai, s = µid τdτ ; p aa, τ µ ai τ p ii τ, s dτ; p ia, s = 0. The probabiliies p ad and p id are calculaed residually by summing condiional probabiliies o 1. Correspondingly, if µ pf and µ fp are boh posiive, we have no closed-form expressions for he probabiliies p pp, p ff, p pf, p fp, p ps, p fs. Bu in he case of no premium resumpion, i.e. µ fp = 0, we do: p pp, s = e ν pf τ+ν psτdτ ; p ff, s = e p pf, s = ν fsτdτ ; p pp, τ ν pf τ p ff τ, s dτ; p fp, s = 0. The probabiliies p ps and p fs are calculaed residually by summing condiional probabiliies o 1. A specific model for behavior is a model for he demand from policy holders. A probabilisic model for demand means ha here is a endency in a porfolio ha policy holders hold cerain ypes of conracs. There are many moivaions for hinking of he wo processes Z risk and Z behavior as being dependen. Two classical feaures in risk rading represen each one direcion of influence beween he wo sub-models. Adverse selecion, on one hand, means ha policy holders wih cerain risks end o demand cerain conracs. We can reflec his in our model by leing he ransiion inensiies in he behavior model be more or less explicily dependen on he risk process. Moral hazard, on he oher hand, means ha policy holders wih cerain behavior/demand end o cause cerain levels of risks. We can reflec his in our model by leing he ransiion inensiies in he risk model be more or less explicily dependen on he behavior model. Thus, causal effecs beween he models have direcions and each direcion may have a given economic inerpreaion. Bu a he end of he day, we observe a combined process where i may be difficul/impossible o deec he direcion of causal effecs from he experienced dependence. The canonical behavior model illusraed in Figure 2 above is also such a model for demand of cerain ypes of paymen profiles. Again here may be effecs of adverse selecion, i.e. policy holders in differen risk saes end o exercise heir behavioral opions, free policy and surrender, differenly. Or here may be effecs of moral hazard, i.e. policy holders in he premium paymen or free policy saes, which essenially means ha hey hold differen insurance conracs, have differen moraliy/disabiliy raes. Below we pay full aenion o a simple effec in he policy design ha conracualizes he dependence beween he risk and behavior models. I is common pracice ha e.g. only policy holders in he risk sae acive are allowed o ranscribe ino a free 5

6 policy or surrender. A sandard conracual formulaion is ha such exercise opions fall away when he conrac goes from a premium paymen conrac o a benefi receip conrac, eiher by ransiion of sae or by ransiion of ime. We assume hroughou ha he risk of policy holders aking up heir premium paymen afer having been ranscribed o free policy is zero, i.e. ν fp = 0. This is ofen a harmless assumpion since such a conrac is ypically handled as a new conrac and should herefore no be aken ino accoun. Wih such a dependence coming exclusively from he behavioral opions in he conrac, we have illusraed he wo-dimensional model in Figure 3. ν a ps a, p µ ad p µ ai p µ ia p a, s ν a pf d, p ν a fs a, f µ ad f µ ai f µ ia f d, f µ id p µ id f i, p i, f Figure 3: Combined model We conclude his secion wih a general remark regarding regime shif models. The Markov srucure of he model makes i relaively easy o allow for underlying regime shifs. Regime shifs underlying he risk model could be moivaed by an impac from he sae of he economy on disabiliy and recovery raes. This is obained by generalizing he model illusraed in Figure 1 by subdivision of he acive and disabiliy saes corresponding o he differen saes of he economy. I could be even more relevan o model an impac from he sae of he economy on free policy and surrender raes. This is done by a corresponding exension of he model illusraed in Figure 2. Noe ha if he sae of he economy influences ransiion raes in boh he risk model and he behavior model, we have inroduced a dependence beween he wo models, even in he case where here is no dependence via he conrac design. Alhough his consiues an ineresing source of dependence beween he underlying models, we will no pursue he idea furher here. 3 Values and Cash Flows in Risk and Behavior Models In his secion we describe he conracual paymens and presen formulas for calculaion of heir condiional expeced presen values. We assume a general risk model in combinaion wih he canonical behavior model illusraed in Figure 2 wih he premium resumpion rae se o zero, i.e. ν fp = 0. We ake as saring poin a conrac ha, in he firs place, specifies is paymens in he risk model, condiional on he behavior model being in he premium paymen sae p. We assume ha 6

7 he conrac pays ne benefis o he policy holder a rae b j as long as he policy holder is in sae j and a lump sum ne benefi b jk upon a ransiion from sae j o sae k. By ne benefi we mean ha premium paymens are aken ino accoun by a negaive sign. We can now formalize he expeced paymen rae a ime s given ha he policy holder is in risk sae k a ime s as c k s = b k s + l:l k µ kl p s b kl s We assume ha he conrac specifies ha upon surrender from risk sae k a ime, all fuure paymens are canceled and a surrender sum G k is paid ou in reurn. We assume ha he conrac specifies ha if he policy is ranscribed ino a free policy while he policy holder is in risk sae h a ime, he fuure paymens are changed in he following way. The negaive elemens of b j and b jk, i.e. premiums, are se o zero whereas posiive elemens of b j and b jk, denoed by b j+ and b jk+, are muliplied by a so-called free policy facor ha is, exclusively, depending on and h and ha we denoe by f h. Firs we inroduce he expeced paymen rae of posiive paymens before muliplicaion by f as c k+ s = b k+ s + µ kl f s b kl+ s. l:l k We can also wrie he expeced paymen rae a ime s given ha he policy holder is in risk sae k a ime s and jumped ino he behavior sae free policy a ime while being in risk sae h as f h c k+ s = f h b k+ s + s b kl+ s. One can imagine a series of alernaive recalculaions of paymens upon ransiion ino he free policy sae. E.g. he policy holder may wan his risk coverages like erm insurance and disabiliy annuiies o eiher fall away or o be fully kep upon ranscripion, and hen he saving coverages like deferred life annuiies are changed residually. We elaborae briefly on such alernaives in Secion 5 below. Bu we develop he valuaion formulas under he assumpion ha all fuure benefis are changed proporionally. Tha even goes for he surrender paymen in he following sense. If he policy was ranscribed ino a free policy while he policy holder was in risk sae h a ime, and he policy is surrendered a ime u > while he policy holder is in risk sae j, he policy pays ou a surrender sum f h G j+ u. I is imporan o noe he following. Since he process Z is Markovian he inensiy of making a jump a ime depends on he posiion of Z only. However, his does no mean ha he paymen rae a ime only depends on Z. Since he expeced paymen rae a ime s, f h c k+ s, depends on and h hrough he free policy facor, we have inroduced a specific duraion dependence in he paymen process which is no presen in he probabiliy model. l:l k µ kl f 3.1 Given he free policy sae In his subsecion we presen a differenial equaion characerizing he reserve defined as he expeced presen value of fuure paymens a ime > τ given ha he policy jumped o he free policy sae while he policy holder was in risk sae h a ime τ. We also specify is soluion. Here and hroughou we refer o Seffensen 2000 for all differenial equaions and heir soluions. 7

8 Denoing by V j f τh he reserve if he policy holder is in risk sae j a ime and became a free policy while being in risk sae h a ime τ, we can characerize his reserve by he differenial equaion d d V j f τh = rv j f τh f h τ b j+ k:k j ν j fs f h τ G j+ V j f τh. µ jk f f h τ b jk+ + Vf k τh V j f τh The soluion can be wrien as V j f τh = f h τ e s r k p jk ff, s c k+ s + ν k fs s G k+ s ds, where p jk ff, s is he probabiliy ha he policy holder moves from j o k in he risk model while saying in sae f in he behavior model. This inerpreaion of p jk ff, s relies on he assumpion ha ν fp = 0, such ha p jk ff, s = pjk ff, s. Alhough we speak of his as he soluion, i is no in closed form, since hese ransiion probabiliies, in general, do no exis in closed form. In he inegral soluion we can see ha he reserve consiss of paymens during sojourn in he free policy sae he c k+ s erms and paymens paid upon surrender he G k+ s erms. In he special case where he behavior and he risk models are independen, hen we have he simple form p jk ff, s = p ff, s p jk, s such ha V j f τh = f h τ e r p ff, s k p jk, s c k+ s + ν fs s G k+ s ds. This formula is paricularly convenien since i can be buil around he original expeced cash flow raes p jk, s c k+ s and he raes p jk, s ν fs s G k+ s. Thus, when making use of k k he inegral soluion, one may, for compuaional convenience, be inclined o assume probabilisic independence. This is no correc, bu he numerical consequences may or may no be negligible. We elaborae on his in Secion 6 below. When working wih he differenial equaions, on he oher hand, using he correc model does no inroduce any addiional complexiy and hence, in his case, here is no excuse for no performing he righ calculaions. 3.2 Given he premium paymen sae In his subsecion we presen a differenial equaion characerizing he reserve given ha he policy is in he premium paymen behavior sae and in risk sae j a ime. We also specify is soluion. Denoing his reserve by V j, acily skipping he subscrip p on all reserves below, we can characerize his reserve by he differenial equaion d d V j = rv j b j k:k j ν j pf V j f j V j µ jk p b jk + V k V j 1 ν j ps G j V j. 8

9 Noe ha he reserve V j f j is obained by solving he differenial equaion for V j f τj for fixed τ and subsequenly replacing τ by. The soluion can be wrien as V j = + e s r k e s r k p jk pp, s c k s + G k s ν k ps s ds 2 W jk, s c k+ s + ν k fs s G k+ s ds where p jk pp, s is he probabiliy ha he policy holder moves from j o k in he risk model while saying in sae p in he behavior model. This inerpreaion of p jk pp, s relies on he assumpion ha ν fp = 0, such ha p jk pp, s = p jk, s. Furhermore, W jk, s = pp h p jh pp, τ ν h pf τ p hk ff τ, s f h τ dτ. In he inegral soluion we can see ha he reserve consiss of paymens during sojourn in he premium paymen sae he c k s erms, paymens due upon surrender from he premium paymen sae he G k s erms, paymens during sojourn in he free policy sae he c k+ s erms, and paymens upon surrender from he free policy sae he G k+ s erms. The raio W jk, s /p jk pf, s is he expeced free policy raio given ha he policy holder jumps in risk saes from j o k and in behavior saes from p o f over, s. In he special case where he behavior and he risk models are independen, hen we have he simple form recall he simple forms for p pp, p ff, and p pf from Secion 2 p jk pp, s = p pp, s p jk, s, p jk ff, s = p ff, s p jk, s, p jk pf, s = p pf, s p jk, s, such ha V j = W jk, s = + e r p pp, s k e s r k p jk, s c k s + ν ps s G k s ds W jk, s c k+ s + ν fs s G k+ s ds, p pp, τ ν pf τ p ff τ, s h p jh, τ p hk τ, s f h τ dτ. This formula appears convenien since he firs line can be buil around he original expeced cash flow raes p jk, s c k s and he raes p jk, s ν ps s G k s. Thus, when making k k use of he inegral soluion, one may, for compuaional convenience, again be inclined o assume probabilisic independence. However, his shorcu is no as appealing as i seems. In spie of he probabilisic independence, he second line is sill an involved quaniy. In order o really benefi from original expeced cash flow raes, we could even furher assume ha f h τ does no depend on h. Then W jk, s = p jk, s W, s 9

10 wih W, s = such ha he second line becomes e r W, s k p pp, τ ν pf τ p ff τ, s f τ dτ, 3 p jk, s c k+ s + ν fs s G k+ s ds. Finally, we have now reached he mos elegan, bu incorrec, expression since all elemens are buil around original cash flow raes p jk, s c k s and p jk, s c k+ s and he raes k k p jk, s ν ps s G k s and p jk, s ν fs s G k+ s. k k 4 Imporan special cases In his secion we specialize he main resuls from Secion 3 o wo paricularly imporan special cases. We consider he canonical risk model illusraed in Figure 1, he disabiliy model, and he survival model respecively. In boh cases we presen relevan differenial equaions and heir more or less explici soluions depending on wha we assume abou he underlying model or conrac. Paricular aenion is paid o he various simplifying assumpions ha one can make in order o ease calculaions. We concenrae on he valuaion of policies ha are in he behavior sae premium paymen since his is where he main calculaion challenges arise. 4.1 The disabiliy model Firs we consider he disabiliy model. We assume ha all paymens are zero in he sae dead. We label he differen saes according o Figure 1. Then he reserve corresponding o he policy holder being premium paying and acive a ime can be characerized by a special case of 1 ha becomes d d V a = rv a b a µ ai b ai + V i V a µ ad b ad V a ν a pf V a f a V a ν a ps G a V a. Here, he second line conains he risk premia relaed o he sae ransiions in he risk model whereas he hird line conains risk premia relaed o he sae ransiions in he behavior model. The general soluion represened in 2 becomes n V a = e r p aa pp, s c a s + νps a s G a s +p ai pp, s c i s + νps i s G i s ds + e r W aa, s c a+ s + νfs a s Ga+ s +W ai, s c i+ s + νfs i s ds, Gi+ s 10

11 where W aa, s = W ai, s = p aa pp, τ νpf a τ paa +p ai pp, τ ν i pf p aa pp, τ νpf a τ pai +p ai pp, τ ν i pf ff τ, s f a τ τ pia ff τ, s f i τ ff τ, s f a τ τ pii ff τ, s f i τ We now make he realisic assumpion ha he conrac specifies ha behavioral evens only ake place as long as he policy holder is acive. This means ha νpf i = νi ps = 0 such ha W aa and W ai reduce o W aa, s = W ai, s = p aa pp, τ ν a pf τ p aa ff τ, s f a τ dτ, p aa pp, τ ν a pf τ p ai ff τ, s f a τ dτ. These formulas are, of course, no explici due o he allowance for posiive reacivaion rae. This makes he probabiliies impossible o calculae explicily. However, if we furher do no allow for reacivaion, we ge simpler expressions for he probabiliies, e.g. p aa pp, s = p aa p aa ff, s = p aa pp, s = e ff, s = e µai p +µad p +νa pf +νa ps, µai f +µad f +νa fs. This has simplifying consequences for calculaion of p ai ff, s, W aa, s, and W ai, s. Insead of assuming ha he conrac allows for behavioral evens from he acive sae only, we now make he opposie assumpion and say ha he risk and behavior models are independen. We know from he previous secion ha his may help making he calculaions much simpler. We ge he expression for he reserve, where V a = W aa, s = W ai, s = + e r p pp, s e r dτ, dτ. p aa, s c a s + ν ps s G a s +p ai, s c i s + ν ps s G i s W aa, s c a+ s + ν fs s G a+ s +W ai, s c i+ s + ν fs s G i+ s p pp, τ ν pf τ p ff τ, s p pp, τ ν pf τ p ff τ, s p aa, τ p aa τ, s f a τ +p ai, τ p ia τ, s f i τ p aa, τ p ai τ, s f a τ +p ai, τ p ii τ, s f i τ There are sill several difficulies wih his represenaion. Even hough some elemens relae o condiional expeced cash flows from he original conrac, we noe ha we canno calculae he ransiion probabiliies explicily as long as we allow for reacivaion. Furhermore, he expeced cash flows from he free policy sae are sill quie complicaed and do no relae o original cash flows in an easy manner. Referring o he resuls in he previous secion, we propose he addiional assumpion ha f i = f a. Then W aa, s = p aa, s W, s, W ai, s = p ai, s W, s, ds, ds dτ, dτ. 11

12 wih W defined as in 3, such ha V a = + e r p pp, s e r W, s p aa, s c a s + ν ps s G a s +p ai, s c i s + ν ps s G i s ds p aa, s c a+ s + ν fs s G a+ s +p ai, s c i+ s + ν fs s G i+ s Finally, he ingrediens relae o original cash flows. In hese cash flows ransiion probabiliies appear. How accessible hey are, depends on wheher or no we allow for reacivaion. If we do no, he probabiliies are explici and we have reached he simples represenaion of our reserve. 4.2 The Survival model Now we specialize o he survival model by skipping he disabiliy sae in he Subsecion 4.1. We skip he superscrip a in he reserve V since all quaniies are condiional on he policy holder being alive. Then he reserve corresponding o he policy holder being premium paying and alive a ime can be characerized by a special case of 1 ha becomes d d V = rv ba µ ad b ad V ν pf V a f a V ν ps G a V As above, he second line conains he risk premium relaed o he deah sae ransiion in he risk model whereas he hird line conains risk premia relaed o sae ransiions in he behavior model. The general soluion represened in 2 becomes where V = + W aa, s = e r p aa pp c a s + ν ps s G a s ds e r W aa, s c a+ s + ν fs s G a s ds, p aa pp, τ ν pf τ p aa ff τ, s f a τ dτ. If we assume independence beween he models we essenially assume ha he moraliy rae is no affeced by he sae of he behavior model and he ransiion inensiies in he behavior model are no affeced by he sae of he risk model. The laer assumpion is harmless since we have assumed ha here are no paymens in he deah sae. This means ha i does no affec he value o allow for ranscripion ino a free policy or surrendering among dead policy holders. If insead here were paymens in he deah sae i would make, of course, a difference wheher we allow for behavioral inervenion in hese paymens or no. Anyway, under he assumpion of independence we ge he simplificaions, ds. V = + e r p pp, s p aa, s c a s + ν ps s G a s ds e r W aa, s c a+ s + ν fs s G a+ s ds, 12

13 where W aa, s = = p aa, s p pp, τ ν pf τ p ff τ, s p aa, τ p aa τ, s f a τ dτ p pp, τ ν pf τ p ff τ, s f a τ dτ. Then we have reached an expression based on he original cash flows. In he survival model, he final simplificaion, f h = f, is no necessary. We sress, however, ha his is rue only because we have no paymens in he deah sae. 5 The free policy raio In he calculaions above we have assumed ha all fuure benefis are muliplied by he same facor f j upon ranscripion ino free policy from risk sae j a ime. Bu we have no discussed wha his f j should be and we have no discussed he siuaion where differen raios apply o differen fuure benefis. If f j applies o all fuure benefis, a naural idea is o le f be deermined by f j = V j V j+ where denoes valuaion of he conracual paymens corresponding o a echnical basis consising of r, µ, possibly differen from our valuaion basis, r, µ. To see why his idea is naural, consider he free policy sum a risk V j f j V j. Since V j f j = f j V j+, we have ha V j f j V j = f j V j+ V j. Now, a paricular version of our valuaion basis would of course be he echnical basis. In ha case, as f j = V j V j+, we ge V j f j V j = f j V j+ V j = 0. In ha sense we can say ha under he echnical basis he policy holder pays himself fully for he free policy risk. An imporan consequence of his approach is ha one can disregard he free policy opion for echnical valuaion purposes, e.g. for seing an equivalence premium. We can keep his as a consrain on he free policy raio, ha under he echnical basis he free policy sum a risk is zero. Then we can consider siuaions where differen reducion facors apply o differen benefis. If a group of benefis keep is fully kep during ranscripion while anoher group of benefis is deleed delee, a residual group is reduced by he facor f j. I is found by solving he equaion 0 = V j+keep + f j V j+ V j+keep V j+delee V j. 4 Hence f j = V j V j+keep V j+ V j+keep V j+delee 5 13

14 If here is a prioriized order in which he benefis are o be kep and he res deleed, one could sar filling he group keep as long as V j V j+keep. The firs benefi ha violaes his inequaliy should be broken up by he f j above wih he residual benefis lef as deleed. Two special cases of his are when eiher he group keep or he group delee is empy. If he group keep is empy and he group delee is no, hen f j = V j V j+ V j+delee for he residual benefis. Noe ha his may lead o f > 1. If he group delee is empy and he group keep is no, hen f j = V j V j+keep V j+ V j+keep, for he residual benefis. Noe ha his leads o f 1. However, in principle we may now end up wih f < 0. In all hese cases he free policy sum a risk is V j f j V j = V j+keep + f j V j+ V j+keep V j+delee V j. 6 Plugging f j defined in 5 ino 6 under he echnical basis gives ha he desired echnical free policy sum a risk is equal o zero, cf Numerical resuls and discussion We consider here he siuaion in Secion 4.1, i.e. where he risk chain is a disabiliy model wih saes acive a, disabled i and dead d cf. Figure 1 and he behavioral chain consiss of he saes premium paymen p, free policy f and surrender s cf. Figure 2. For he various seups modeling of dependen/independen chains, using he same/differen free policy raios in risk saes and including/disregarding reacivaion from disabiliy menioned in Secion 4.1, regarding he disabiliy model, we compue he reserve condiional on he policy holder being in he risk sae acive and he behavioral sae premium paymen. The purpose of his is o quanify he implicaions of using he various alernaives o he correc model, which is he dependen model where reacivaion is included. Here dependen corresponds o including policy holder opions only from he risk sae acive, cf. Secion 6.1 below. 6.1 Model parameers There are wo compuaional bases in play; he echnical someimes referred o as firs order basis and he marke someimes referred o as hird order basis. In oher words, we omi including a separae so-called second order basis which is someimes used o model bonus disribuion schemes. As usual, he paymens iniiaed by exercise of policy holder opions i.e. surrender and free policy are defined such ha he corresponding sums a risk under he echnical basis are zero cf. Secion 6.2. Hence we disregard hem here, see also he discussion in Secion 5. We sar off by defining he ransiion inensiies in he risk chain under he bases; hroughou his secion hey are independen of he behavioral chain implying e.g. ha he moraliy of a premium paying policy holder is he same as he moraliy of a free policy holder of he same age. 14

15 From To Technical basis Marke basis acive dead µ ad age = age µ ad = µ ad acive disabled µ ai age = age µ ai = µ ai disabled dead µ id age = µ ad age µ id = µ id disabled acive µ ia age 0 µ ia age = e 0.06age or 0 Table 1: Transiion inensiies, risk chain We remark ha for he echnical basis we use he sandard inensiies for a female occurring in he Danish G82 risk able. We le he moraliy and disabiliy inensiies of he wo bases be he same; he ineres raes, on he oher hand, are differen cf. Secion 6.2 below. Furhermore, we consider boh he case when he marke reacivaion inensiy, µ ia, is non-zero and when i is zero as he laer assumpion generally simplifies he semi-closed formulae, cf. Secion 4.1. Finally, we inroduce he ransiion inensiies in he behavioral model. We consider wo siuaions. 1. The behavioral inensiies are dependen of he policy holder s curren sae in he risk chain in he sense ha hey are zero unless he policy holder is in he risk sae acive ; his corresponds o sopping policy holder opions in he risk saes disabled and dead. 2. The behavioral inensiies are independen of he risk chain; his corresponds o coninuing he policy holder opions in he risk saes disabled and dead since here are no paymens in he sae dead, he laer consequence is irrelevan. Transiion inensiies no menioned in he able are zero. From To Independen model Dependen model premium paymen free policy ν pf age = e 0.07age ν pf = 1 {acive} ν pf premium paymen surrender ν ps age = ν pf age ν ps = 1 {acive} ν ps free policy surrender ν fs age = ν ps age ν fs = 1 {acive} ν fs Table 2: Transiion inensiies, behavioral chain Hence, as is he general assumpion hroughou his paper, we do no model he opion of reenering he premium paying sae from he free policy sae, and furhermore he sae surrender is absorbing. There does no seem o exis a sandard parameerizaion for he behavioral ransiion inensiies; he idea behind heir forms used here is simply ha he inclinaion o exercise policy holder opions decreases wih age. This reflecs a cerain loyaly effec ha is usually seen among policy holders. 6.2 Conracs The surrender and free policy opions The surrender opion gives he policy holder he choice of abandoning he conrac in exchange for a lump sum paymen, G j, where j is he risk sae in which he policy holder resides a 15

16 ime. In wha follows, we le G j be he value of he conrac under he echnical basis, i.e. he echnical reserve, G j := V j in realiy he sum received is someimes reduced by some facor; we disregard ha in wha follows. Noe ha his choice of G j makes he sum a risk upon surrender equal o zero under he echnical basis. The free policy opion gives he policy holder he choice of sopping he premium paymen; he conrac is hen kep bu he benefis are scaled by a cerain facor, f j, where, again, j denoes he risk sae from which he ranscripion ook place. Before we urn o specifying he f, we noe ha he size of he echnical reserve in relaion o he marke reserve is wha deermines if he surrender opion increases or decreases he marke value of he conrac; in a siuaion where he echnical reserve is higher han he marke reserve, i is o be considered profiable for he policy holder o surrender and conversely when i is lower as menioned above, under he echnical basis he sum a risk is zero by definiion. In order o accoun for boh siuaions, we consider wo differen conracs as specified below. In all realisic scenarios he policy holder opions are only allowed from he risk sae acive. I is hen sandard o define f a = V a V a +, i.e. he quoien beween he echnical reserve and he echnical benefi reserve premium removed, all benefis kep; his causes he sum a risk upon ranscripion o free policy, under he echnical basis, o be zero, cf. Secion 5. Oher alernaions of paymen sreams upon premaure sopping of premium paymen can also be sudied numerically, cf. Secion 5, bu we focus on he proporional benefi reducion in his numerical example. When allowing policy holder opions also from he disabled sae, he mos naural choice appears o be f i = V i V i + 1 again, his causes he sum a risk, under he echnical basis, upon ranscripion o free policy o be zero. However, as is menioned in Secion 4.1 above, seing f i := f a yields even simpler closed-form soluions, and we herefore consider boh hese varians of f i. We refer o he siuaions as using Separae f and Same f, respecively Common We ouline he common feaures of he wo conracs considered. Conrac expiry age 65 Premium paymen of inensiy 20,000 USD p.a. Disabiliy annuiy of inensiy 100,000 USD p.a. Term insurance a 400,000 USD Pure endowmen a expiry corresponding o a life annuiy deermined a iniiaion ime of he conrac such ha i gives he conrac a echnical value i.e. V a of zero a he ime of iniiaion Furhermore, he marke ineres rae, r, is he forward rae equivalen o he yield curve as published by he Danish FSA a cf. appendix A New conrac We consider here he siuaion where a relaively young policy holder has jus signed he conrac, and where he echnical ineres rae r is low relaive o he curren marke ineres rae; his 16

17 makes he echnical reserve higher han he marke reserve. More precisely we have he following addiional parameers. Conrac iniiaion age 30 Age 30 a he ime = 0 of calculaion r = 1% p.a. coninuously compounded Pure endowmen a expiry of 552,796 USD corresponding o he reserve of a life annuiy, a expiry, of USD p.a. compued under he marke value basis Old conrac We consider here he siuaion where an older policy holder signed he conrac 20 years earlier, and where he echnical ineres rae r is high relaive o he curren marke ineres rae; his makes he echnical reserve lower han he marke reserve. The raionale for his is ha when he conrac was signed, r was indeed low compared o he conemporary marke ineres rae. More precisely we have he following addiional parameers. Conrac iniiaion age 30 Age 50 a he ime = 0 of calculaion r = 5% p.a. coninuously compounded Pure endowmen a expiry of 1,597,593 USD corresponding o he reserve of a life annuiy, a expiry, of 110,023 USD p.a. compued under he marke value basis Remarks We commen on he significan difference in he pure endowmens of he wo conracs. The endowmen sum compued a iniiaion of he conrac should be viewed as wha is hen guaraneed. If he insurance company can obain a higher ineres han he echnical ineres rae, his sum is ypically increased via surplus bonus as ime goes by. Assuming, in he conex of he new conrac, ha he insurance company realizes an ineres rae of 5% p.a. and uses all surplus conribuions o immediaely and coninuously increase he guaraneed pure endowmen sum, i will be exacly he same as ha compued for he old conrac when he policy holder reaches he age of Numerical resuls We now display and discuss he numerical resuls obained in he wo conracual conexs. We menion firs ha model variaions occur in wo dimensions ; on he one hand regarding wheher or no we include reacivaion from disabiliy and on he oher hand wheher or no we model he risk and behavioral chains independenly. Finally, in he case when he chains are modeled independenly, we consider he wo siuaions when he free policy raios are he same, f i := f a, from he saes acive and disabled, and when hey are no hey are precisely defined in Subsecion above. 17

18 Figure 4 below illusraes he various compuaional seups ha we consider; for each box we have compued he corresponding reserve for boh conracs. The obained numerical resuls allow us o find variaions in he reserve by alernaing he model one sep a a ime. In he figure below he arrows illusrae a siuaion where we sar off by using a model wihou reacivaion and independen chains and hen addiionally regard dependence and reacivaion, one a a ime. Hence, we have a means of decomposing he change beween wo models; in he case below he pars being a consequence by regarding dependence and regarding reacivaion, respecively. We emphasize ha he op lef box, corresponding o he correc model, is always considered as he benchmark resul in comparisons. Figure 4: The included model variaions, and a possible pah beween wo of hem. Some inequaliies beween reserves compued under differen seups can be obained in general by heoreical consideraions. For example i is fairly obvious ha using, in he independen models, f i := f a 1 raher han f i 1 1 yields, fixing he remaining parameers, a smaller marke reserve. We show, however, by means of examples ha oher inequaliies are indeed dependen on he concree seup e.g. he reacivaion inensiy New conrac Figure 5 below displays wo reserves; he echnical reserve Technical, and he reserve in he model including reacivaion from disabiliy and he aforemenioned dependence beween he risk and behavioral Markov chains Dep., reac., he correc model as funcions of ime, condiional on being in he saes acive and acive; premium paymen, respecively. Recall ha a ime zero, he policy holder is 30 years old. As expeced, he magniude of he reserves are a differen levels, primarily due o he difference in ineres rae levels of he echnical and marke bases. 18

19 600, , , , ,000 Technical Dep., reac. 100, , Figure 5: The echnical reserve and he dependen reserve including reacivaion We now presen all compued reserves evaluaed a a few ime poins and furhermore plo he difference beween he marke reserves and he correc marke reserve Dep., reac. where we model reacivaion and dependence beween he chains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igure 6: Compued reserves in descending order in USD 19

20 16,000 14,000 12,000 10,000 8,000 6,000 4,000 Indep., no reac., separae f Dep., no reac. Indep., no reac., same f Indep., reac., separae f Indep., reac., same f 2, Figure 7: Differences compared o Dep., reac. More precisely, for a given reserve V Model, where Model is e.g. Dep., no reac., we have ploed he funcion V Model V Dep., reac.. We draw a few conclusions from he resuls above. 1. For his paricular seup an insurance company would in fac benefi from using he correc model; i yields he smalles reserve. 2. One canno conclude from he above ha using he model yielding he simples closed form expression for he reserve, Indep., no reac., same f, is a leas on he safe side of he correc reserve. Namely, from he numbers above i is clear ha leing he reacivaion inensiy approach zero, he reserve in Dep., reac converges o ha in Dep., no reac., which is larger han he one in Indep., no reac., same f. Hence he order relaion beween he wo reserves is in fac inensiy dependen. 3. I is by no means a surprise ha he larges marke reserve is Indep., no reac., separae f. We omi a formal argumen bu noe ha when assuming independen chains we have a surrender and free policy opion in he disabiliy sae. When using separae f, i.e. f i 1, exercising he free policy opion from he disabiliy sae gives he policy holder V i+, in oher words he value of his own conrac wih premium paymen sreams removed from all saes. The surrender opion, when exercised from he disabiliy sae, gives he policy holder an amoun equal o he value of his conrac compued under an ineres rae lower han he marke ineres rae and furhermore disregarding reacivaion and free policy namely, he 20