UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert


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1 UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse 2 D943 Nuremberg (Germany) Tel.: +49/9/ Gudrun Hoermann Conac (has changed since iniial submission): Leopoldsrasse 6 D882 Munich (Germany) Augus 29
2 UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES ABSTRACT Universal life policies are he mos popular insurance conrac design in he Unied Saes. They have eiher a level deah benefi paying a fixed face amoun, or an increasing deah benefi, which addiionally o a fixed benefi pays he available cash value, and boh ypes include he opion o swich from one o he oher. In his paper, we are ineresed in he fac ha unlike a swich from level o increasing a swich from increasing o level deah benefi requires neiher fees nor addiional evidence of insurabiliy. To assess he impac of he deah benefi swich opion, we develop a model framework of increasing universal life policies embedding he opion. Consideraion of heerogeneiy in respec of moraliy via a sochasic fraily facor allows an invesigaion of adverse exercise behavior. In a comprehensive simulaion analysis, we quanify he ne presen value of he opion from he insurer s perspecive using riskneural valuaion under sochasic ineres raes assuming empirical exercise probabiliies. Based on our resuls, we provide policy recommendaions for life insurers. Keywords: Increasing deah benefi; Deah benefi swich opion; Heerogeneiy in respec of moraliy
3 2. INTRODUCTION Firs inroduced in 979, universal life is now he mos imporan individual life insurance conrac ype in he Unied Saes. Lifelong universal life policies offer flexibiliy wih respec o frequency and amoun of premium paymens and wo deah benefi opions o choose from (see Cherin and Huchins (987, p. 69) and D Arcy and Lee (987, p. 453)). The level deah benefi pays a consan specified face amoun; he increasing deah benefi pays he available cash value (or policy reserve) in addiion o a fixed face value. Eiher ype of conrac ypically embeds he opion o swich from level o increasing or vice versa (he deah benefi swich opion). A swich from level o increasing benefis requires new evidence of insurabiliy and, possibly, an exra fee since he deah benefi immediaely increases by he curren amoun of cash value a he ime he opion is exercised. In conras, when swiching from increasing o level benefis, he deah benefi is fixed a he curren value. Thus, in he laer case, he swich does no affec he ne amoun a risk, i.e., he difference beween deah benefi and cash value, a he swich exercise ime and so here are usually no special requiremens or fees involved in making his ype of swich (see Smih and Hayhoe (25, p. 2); see also, e.g., However, developmen of he ne amoun a risk afer he swich depends on premium paymen behavior. Thus, here is some quesion as o wheher insurers should be concerned abou deah benefi swiches under oherwise unchanged acuarial assumpions. In he presen paper, we examine he deah benefi swich opion in a pool of increasing universal life policies wih he goal of enhancing undersanding of his feaure. To accomplish his, we develop a model framework for increasing universal life conracs wih deah benefi swich opion ha incorporaes heerogeneiy in respec of moraliy, swich probabiliies, and sochasic ineres raes. Based on his model, we evaluae he opion under differen premium paymen assumpions afer swiching and for various exercise scenarios. By considering
4 3 resuls for adverse exercise behavior depending on an insured s healh saus, we derive policy implicaions and, in paricular, analyze wheher a requiremen of charges or evidence of insurabiliy would be advisable. The lieraure abou universal life insurance is mainly concerned wih he reurn on universal life policies (e.g., Belh, 982; Cherin and Huchins, 987; Chung and Skipper, 987; D Arcy and Lee, 987). Carson (996) finds deerminans for universal life cash values, and Carson and Forser (2) examine policy yields of whole and universal life conracs. Coss of universal and erm life insurance are compared in Corbe and Nelson (992). Carson (996), Cherin and Huchins (987), and Chung and Skipper (987) empirically sudy he reurn characerisics of increasing universal life policies. However, o dae here have been no aemps o develop a model of universal life conracs wih increasing deah benefi, much less any sudy of he deah benefi swich opion. The same is rue regarding premium paymen opions in universal life policies. Mos sudies are resriced o he paidup opion (i.e., sopping premium paymens) in paricipaing life insurance conracs (Kling, Russ, and Schmeiser, 26; Linnemann, 23, 24; Seffensen, 22). In addiion o he paidup opion, Gazer and Schmeiser (27) inegrae he resumpion opion (i.e., resumpion of premium paymens afer having made he conrac paidup) in heir framework for paricipaing policies. To he bes of our knowledge, increasing universal life policies and he deah benefi swich opion have no ye been sudied. We provide an acuarial model framework of a universal life conrac wih increasing deah benefi and incorporae he deah benefi swich opion. Since universal life policies are lifelong conracs ha pay a deah benefi, we accoun for moraliy risk as a cenral risk facor. Moraliy varies among insureds, and hus heerogeneiy in respec o moraliy is modeled by a sochasic fraily facor on a given deerminisic moraliy able. The concep of fraily was originally defined by Vaupel e al. (979) in erms of he coninuous force of moraliy. In his paper, we use he erm "fraily
5 4 facor" in he discree conex in order o express he facor s sochasiciy, as well as respecive disribuional characerisics. An examinaion of adverse exercise behavior wih respec o an insured s healh saus is of imporance, as exercise of he deah benefi swich opion does no require evidence of insurabiliy. The new level deah benefi conrac is hus based on unchanged acuarial assumpions. Afer swiching, premium paymens are adjused since he former increasing policy premium is no longer adequae for he new level policy. Due o he full flexibiliy in premium paymens for universal life conracs, he modificaion is no prescribed by he insurer; he only resricion is prevenion of policy lapse. An evaluaion of he swich opion hus necessarily involves assumpions abou modified premium paymen behavior afer swich. I is his combinaion of opions he deah benefi swich opion and premium paymen opions ha can have subsanial negaive effecs for he insurer. We consider wo viable premium paymen scenarios, one wih consan premiums and one wih flexible paymens. To gain deailed insigh ino he deah benefi swich opion of increasing universal life policies, we conduc a comprehensive invesigaion for differen swich probabiliies. In a simulaion analysis, we quanify he ne presen value of he opion using riskneural valuaion under sochasic ineres raes based on he Vasicek model. We hen sudy he effec of adverse exercise behavior by assuming differen swich probabiliies depending on an insured s healh saus and on he ime since policy incepion (and hus on he amoun of policy cash value). This procedure allows an invesigaion of he necessiy of requiring evidence of insurabiliy. Finally, we conduc a sensiiviy analysis wih respec o he fraily facor disribuion. A universal life policy lapses if he cash value is insufficien o pay policy coss (see Carson, 996, p. 675). In his case, he conrac is erminaed wihou payou o he policyholder. During a onemonh grace period, cachup premium paymens can be made o avoid policy lapse. Afer ha period, reinsaemen of he policy requires new evidence of insurabiliy as well as paymen of all ousanding premiums (see Trieschmann, Hoy, and Sommer, 25, p. 34). This undersanding of policy lapse is in conras o exercise of he surrender opion, when he cash surrender value of he policy is paid ou.
6 5 Resuls show ha he value of he deah benefi swich opion is srongly dependen on premium paymen behavior afer exercise and on he healh saus of an exercising insured. From our findings, we derive policy implicaions and provide recommendaions for insurers, which can be applied depending on specific moraliy and behavioral experience in an insurance porfolio. The remainder of he paper is srucured as follows. Secion 2 presens he model framework, including he model of a universal life policy wih increasing deah benefi and he model of he deah benefi swich opion. In Secion 3, he valuaion approach is presened and Secion 4 conains numerical resuls. Policy implicaions for insurers are discussed in Secion 5; a summary is found in Secion THE MODEL FRAMEWORK The universal life conrac wih increasing deah benefi We consider a lifelong universal life insurance conrac wih increasing deah benefi. The policy is issued a ime = for an insured of age x { x ω },, a incepion, where min x min is he minimum enry age admied. The conrac maures a ime T = ω x +, where ω is he limiing age of a moraliy able, i.e., he oneyear probabiliy of dying a age ω, q ω, is equal o. In wha follows, deah or survival probabiliies based on he moraliy able will be denoed wih a prime (`) mark. The oneyear able probabiliy of deah a age x+ is hus given by q, =,, T. x+ In case of deah during policy year (beween ime and ), he deah benefi is paid in arrears a he end of he year, i.e., a ime {,, T} of he sum of a fixed face value Y and he cash value V a ime : Y = Y + V, =,, T.. The increasing deah benefi consiss
7 6 To focus on he pure effec of he deah benefi swich opion in increasing universal life policies, our model framework does no accoun for charges or surrenders. According o a sandard acuarial valuaion (see, e.g., Bowers e al. (997) and Linnemann (24)), for annual premium paymens B, =,, T paid a he beginning of each year in which he insured is alive, he cash value is given by he following recursive formula: q V = V + B + i q Y, =,, T, () x+ x+ where V =. We assume ha a consan annual ineres rae i is credied o cash value and premium. Each policy year, his amoun is reduced by he cos of insurance, i.e., he produc of deah benefi and able probabiliy of deah. Calculaions are hence based on he acuarial assumpions of a consan annual ineres rae i and probabiliies of deah according o he moraliy able. Wih Y = Y + V, he recursion formula for policy reserves in Equaion () reduces o V = V + B + i q Y, =,, T. (2) x+ Defining he savings premium a ime as ( S ) ( ) B = V + i V and he cos of insurance (risk premium) a he same ime as ( R) x+ B = q Y + i, i urns ou from Equaion (2) ha ( S ) ( R) B = B + B. From he definiion of he savings premium, we obain he following expression for he cash value: h= ( S ) ( ) V = B + i h h. Given ha ( S ) ( R) B = B B, we can also wrie ( R) h + h V = B B + i = B q Y + i + i h h h x h h= h= h h h= h= h = B + i Y q + i. x+ h (3)
8 7 Since universal life conracs allow for flexible premium paymens, we need o make cerain assumpions in his regard. Universal life policies are usually paid by means of consan periodic premiums. These consan paymens aim o reflec he general savings paern of life insurance policies, where savings are accumulaed during he earlier years of he conrac erm when he coss of insurance are low in order o finance he higher coss of insurance laer in life. We, herefore, base our analysis on consan annual premium paymens B = B, =,, T. Given he premium B, he cash value should be posiive unil mauriy o avoid policy lapse (see Carson (996, p. 675)). The minimum (consan annual) premium o fulfill his condiion is he amoun for which he cash value a mauriy equals. Thus, we solve V T = for B (see Equaion (3)), which is equal o solving he equivalence principle, and obain B = Y T x+ h h= T h= ( i) q + ( + i) h h. (4) In general, he ne amoun a risk R for a universal life policy a ime {,, T} is given as he difference beween he deah benefi Y and he cash value V : R = Y V, =,, T. (5) In he case of an increasing deah benefi, he deah benefi a ime is he sum of he fixed face value and he curren cash value, and, hus, he ne amoun a risk for an increasing policy is consan and equals he face amoun Y hroughou he conrac erm. Based on he above assumpions, Figure illusraes he premiums, cash value, deah benefi, and ne amoun a risk of a universal life policy wih increasing deah benefi from incepion o mauriy.
9 8 Figure : Premiums, cash value, deah benefi, and ne amoun a risk of universal life policy wih increasing deah benefi Premium Cash Value Deah Benefi Ne Am a Risk Y Y Time Time Time Time For consan annual premium paymens hroughou he policy erm calculaed according o Equaion (4) he cash value firs increases and hen decreases over ime unil i becomes zero a mauriy. The decrease in cash value near mauriy is due o high coss of insurance a higher ages ha exceed he ineres earnings of he cash value and premiums (see Equaion (2)). The deah benefi is given by he sum of he fixed face value Y and he cash value and hus develops analogously o he laer. Hence, he erm increasing deah benefi is employed irrespecive of he fac ha he deah benefi may also decrease if he cash value does. Chung and Skipper (987) accoun for his poin and use he more precise erm nonlevel deah benefi. In insurance pracice, however, he erm increasing is common. I suggess ha he policy in conras o a policy wih a level deah benefi includes a dynamic componen ha increases deah benefi coverage in he course of accumulaing cash value. Since he cash value mus be posiive o keep he policy in force, he increasing deah benefi is always a leas as high as a consan level deah benefi for he same face amoun. The ne amoun a risk is equal o he fixed face value Y from policy incepion o mauriy. The deah benefi swich opion Increasing universal life policies ypically give he policyholder he righ o swich he deah benefi from increasing o level wihou charges or addiional evidence of insurabiliy. When exercising he deah benefi swich opion a ime {,, T }, he deah benefi is swiched o level and fixed a he curren value Y = Y + V. In our model, he opion may be exercised only once and a discree exercise imes, namely, a he beginning of each policy
10 9 year. Exercise of he opion a ime can also be inerpreed as erminaing he increasing deah benefi conrac and, based on oherwise unchanged acuarial assumpions, purchasing a new conrac wih level deah benefi Y. The deah benefi a ime given exercise a ime is denoed by ( ) Y, =,, Y = Y, = +,, T. (6) When he deah benefi swich opion is exercised in he accumulaion phase of he cash value, he swich hals furher increase of he deah benefi by fixing i a he aained level Y. Compared o he case wihou swich, fuure deah benefi amouns are hus lower unil he increasing deah benefi falls below he fixed level again. A swich a or afer he peak of he deah benefi curve implies a higher level deah benefi unil mauriy han under increasing policy condiions. However, in boh cases, a he poin in ime when he swich opion is exercised (and only a his poin), he ne amoun a risk remains unchanged. This is in conras o a swich from level o increasing, which immediaely increases he deah benefi, and hus he ne amoun a risk, by he curren amoun of cash value. Therefore, a swich from increasing o level does no require any charges or evidence of insurabiliy. However, fuure developmen of ne amoun a risk depends on fuure premiums. Hence, when evaluaing he deah benefi swich opion, i is crucial o ake ino accoun possible changes in premium paymen behavior afer exercise of he opion. When swiching before he peak of he cash value curve, previously calculaed premiums for he increasing deah benefi conrac (see Equaion (4)) are oo high for he new level policy. A swich near (some policy years before), a, or afer he peak resuls in higher premiums due o fixing a higher deah benefi han in he increasing case. Thus, i is no possible o simply analyze he deah benefi swich opion alone: we need o make assumpions abou he premium paymen behavior afer swich, which leads o a combined examinaion of he deah benefi swich opion and premium paymen opions. Again, wih universal life policies, policyholders are
11 free o choose he frequency and amoun of premium paymens as long as he cash value says posiive. In he following, we resric our analysis o wo premium paymen scenarios ha can be regarded as general cases from he insurer s perspecive as hey consiue minimum premium paymen schedules where premiums are jus high enough o avoid policy lapse. In paricular, hey represen he minimum consan annual premium paymens and minimum flexible annual premium paymens ha will ensure a posiive cash value hroughou he conrac erm. Any oher consan annual or flexible paymens keeping he conrac in force unil mauriy need o exceed hese premium amouns. In he firs, level premium, scenario, consan annual level premiums B paid afer he swich are calculaed based on he equivalence principle, aking ino accoun he presen cash value V a he exercise dae as an addiional single paymen. This can be inerpreed as erminaing he former increasing deah benefi conrac and saring a new level deah benefi conrac wih an iniial premium paymen in he amoun of he curren cash value. For universal life policies, insurers chiefly use consan annual level premiums o projec policy values (cash value, cash surrender value, deah benefi) ha imply a zero cash value a mauriy (socalled policy illusraions). Afer opion exercise, updaed policy illusraions are usually provided. Annual premium noices are ofen based on he premium values conained in hese projecions. Alhough holders of universal life policies are no forced o pay he saed premium amoun, hey likely do so, unless a cerain even makes hem depar from he prescribed premium schedule. Since a swich from an increasing o a level deah benefi does no require addiional evidence of insurabiliy, moraliy and ineres rae assumpions remain he same. The equivalence principle requires he presen value of fuure premium paymens o equal he presen value of fuure benefis (see, e.g., Bowers e al. (997) and Linnemann (24)) i.e.,
12 T T ( ) ( + ) x+ ( + ) + = x+ x+ + ( + ) = = B p i V Y p q i. If he iniial single premium V exceeds he presen value of fuure benefis of he new level policy, he annual premium is se o zero. Solving for B hus yields B ( ) T ( + ) Y p x q + x+ + ( + i) V = = max, T. p x+ ( + i) = For simplificaion purposes, we do no include he scenario where, if he available cash value exceeds he presen value of fuure benefis, he deah benefi amoun of a universal life policy migh as well be increased in order o mainain a fair conrac according o he employed echnical basis. However, his assumpion would be favorable from he policyholder s perspecive and would increase negaive effecs of he swich opion value for he insurer, hus implying ha he obained swich opion value in he presen analysis represens a lower bound o he acual opion value (which can already be subsanial). As regards he policyholder perspecive, we assume ha he decision o swich may someimes be made for oher han financially raional reasons, and ha, despie disadvanages in he premium amoun, doing so can sill be beneficial for he policyholder, despie he fixed deah benefi. Premium paymens a ime for a policy swiched a ime are denoed by B ( ) B, =,, =. (7) ( ) B, =,, T The new deah benefi Y and new premium paymens B mus be aken ino accoun when calculaing cash value V and ne amoun a risk R afer exercise of he swich opion, analogously o Equaion () and Equaion (5), respecively. Figure 2 shows premium paymens, cash value, deah benefi, and ne amoun a risk based on he level premium scenario. In Par a), he swich occurs before he peak of he cash
13 2 value curve; in Par b), he swich occurs a his peak. Figure 2: Level premium scenario premiums, cash value, deah benefi, and ne amoun a risk of universal life policy wih increasing deah benefi swiched o level a ime a) Swich before peak of cash value curve Premium Cash Value Deah Benefi Ne Am a Risk Y Y Time Time Time Time b) Swich a peak of cash value curve Premium Cash Value Deah Benefi Ne Am a Risk Y Y Time Time Time Time Afer he swich, he conrac, in principle, works like a radiional whole life insurance conrac wih consan premiums. If he swich occurs a ime before he cash value curve peaks (see Figure 2, Par a)), premiums drop o consan annual level premiums. These reduced paymens resul in slower growh of he cash value. A swich a he peak of he cash value curve (see Figure 2, Par b)) implies ha higher level premiums are necessary, wih a consequen increase of he cash value. The increasing deah benefi is fixed a he swich exercise ime. As he cash value increases afer swich, he ne amoun a risk lies below he consan amoun Y in he nonswich case. In he second premium paymen scenario ( risk premium ), premium paymens are sopped immediaely afer swich a ime and no resumed unil he cash value is exhaused. When swiching from an increasing o a level deah benefi, he deah benefi amoun is frozen a he swich exercise ime, offering he policyholder he opporuniy o mainain he aained deah benefi level by deferring premium paymens unil depleion of he cash value. From hen on, he risk premium is paid in only such an amoun ha he cash value remains zero
14 3 unil mauriy. The premium arrangemen is hus based on naural premiums, as hey are relaed o he amoun of benefi. We refer o his seing as a risk premium scenario o emphasize ha once he cash value is exhaused, he policyholder mus pay he full risk premium o keep he conrac in force. This scenario is ypically exercised in he secondary marke for life insurance, where he policies of insureds wih reduced life expecancy are raded. Life selemen companies aim o opimize premium paymens in he sense of he risk premium scenario by paying only he minimum premium necessary o keep a policy in force, speculaing on early deahs of he insureds in heir porfolio (see, e.g., The above assumpions imply he following formula for premium paymens, which is derived from he recursive developmen of he cash value in Equaion (), where V + is se o zero: B ( ) B, =,, = ( ). (8) max{, q x+ Y + ( + i) V }, =,, T If he cash value V a ime exceeds he discouned risk premium for year (i.e., ( ) q Y + i ), no premium paymen is necessary. Once x+ + premium for he firs ime, he remainder of V V is less han he required risk is exhaused and he ousanding difference is covered by he premium paymen. Afer he zero level of V has been reached, i is susained by premiums equaling exacly he amoun of he discouned annual cos of insurance ( ) q Y + i. Again, we illusrae he course of premiums, cash value, deah x+ + benefi, and ne amoun a risk in Figure 3.
15 4 Figure 3: Risk premium scenario premiums, cash value, deah benefi, and ne amoun a risk of universal life policy wih increasing deah benefi swiched o level a ime a) Swich before peak of cash value curve Premium Cash Value Deah Benefi Ne Am a Risk Y Y Time Time Time Time b) Swich a peak of cash value curve Premium Cash Value Deah Benefi Ne Am a Risk Y Y Time Time Time Time 3. CONTRACT VALUATION The previous secion makes apparen ha he effecs of he deah benefi swich opion depend on swich exercise ime and premium paymen behavior afer swich. When evaluaing he opion, however, moraliy as a hird (random) componen also needs o be considered. Since he opion value is deermined by he combinaion of premium paymen mehod, swich exercise ime, and ime of deah, we accoun for adverse opion exercise behavior. Tha is, we consider moraliy heerogeneous insureds whose exercise behavior depends on heir healh saus or moraliy expecaion. This enables a comprehensive examinaion of he opion and an invesigaion of wheher fees or evidence of insurabiliy are recommended. Opion valuaion can be conduced in differen ways depending on policyholder exercise behavior. Generally, wo approaches can be disinguished. Firs, under financially raional exercise, policyholders aemp o idenify an opimal exercise sraegy ha maximizes he opion value. This is implemened by solving an opimal sopping problem. In our seing, deermining an opimal exercise sraegy is highly ambiious because of he complex ineracion beween moraliy and financial facors as well as furher opions embedded in a
16 5 universal life conrac. In paricular, he swich exercise decision, iner alia, depends on decisions regarding frequency and amoun of premium paymens before and afer swich, lapse opion exercise, he insured s healh saus, and he ineres rae. Therefore, ackling his problem is an exensive underaking and requires assumpions regarding many decision variables. In addiion, even hough an ever greaer number of policyholders may be aking advanage of increased ransparency in he insurance marke and are hus making more raional exercise decisions, empirically observed exercise behavior can sill vary from his assumpion. Hence, he opion value under raional exercise is likely o overesimae he value acually generaed in insurance porfolios. From he opion value based on an opimal exercise sraegy, i is herefore difficul o derive policy recommendaions for insurers. For hese reasons, we focus on he second valuaion approach and inegrae exercise probabiliies ino our model. The invesigaion is conduced from an insurer s perspecive for a pool of policyholders who do no necessarily exercise heir opions in a raional way. Insead, exercise decisions are exogenously made for financial or oher, possibly personal, reasons. In he curren marke siuaion, our model allows an assessmen of he risk associaed wih he swich opion in a porfolio of insureds as well as he derivaion of policy implicaions, as is done in Secion 4 and 5, respecively. Thus, in he following, opion value or ne presen value of he opion refers o he value of he deah benefi swich opion calculaed using his approach. As daa regarding empirical swich exercise behavior are no available, we conduc our analysis by sudying comprehensive exercise scenarios. An insurer can employ he model using is own swich exercise experience o deermine he impac of he swich opion in a porfolio. However, cauion is needed when implemening his approach as using exercise probabiliy esimaes from hisorical daa is no enirely wihou problems because deviaions beween acual and esimaed probabiliies can represen a risk for he insurer.
17 6 Heerogeneiy in respec of moraliy To examine adverse opion exercise behavior and hus he impac of an insured s healh saus, we evaluae he deah benefi swich opion by aking ino accoun heerogeneiy in respec of moraliy. As in Hoermann and Russ (28), we inegrae heerogeneiy in respec of moraliy by use of a fraily model (see, e.g., Jones (998, pp. 8 83), Piacco (24, p. 4), and Vaupel, Manon, and Sallard (979, p. 44)). Since he dae of he benefi paymen and hus also he amoun of premiums paid ino he conrac depend on insureds moraliy, he value of he opion o swich from one deah benefi scheme o anoher will also so depend. The inroducion of a fraily facor and, in paricular, is sochasiciy will allow a more deailed analysis of he deah benefi swich opion wih respec o he policyholder s individual moraliy level and exercise behavior. The oneyear individual probabiliy of deah for a person age x is obained by muliplying an individual fraily facor d wih he probabiliies of deah q x of a deerminisic moraliy able: d q x, d q x < qx =, x = min xɶ {,, ω} : d q xɶ for x {,, ω} and qω : = for d <., oherwise If he resuling produc is greaer han or equal o for any ages xɶ, he individual probabiliy of deah is se equal o for he younges of hose ages; for all oher ages xɶ, i is se o. The random variable K ( x ) describes he remaining curae lifeime of an individual age x. Is disribuion funcion k q x a a poin k N is given by K x k k x k x ( x+ h ) F k = P K x k = q = p = q, where k h= p x is he individual k year survival probabiliy of an xyearold and P denoes he objecive (realworld) probabiliy measure. The disribuion of he remaining curae lifeime
18 7 hus depends on he individual fraily facor. A person wih a fraily facor d less han indicaes ha he insured is impaired wih a reduced expeced remaining lifeime. The parameer d can be inerpreed as a realizaion of a sochasic fraily facor D. The disribuion FD of D specifies he porion of individuals whose moraliy is lower or higher han a cerain percenage of able moraliy. I is characerized as follows (see, e.g., Ainslie (2, p. 44), Bu and Haberman (22, p. 5), Hougaard (984, pp. 75, 79), and Piacco (24, p. 5)). We assume a coninuous fraily disribuion such ha i can represen fine differences beween remaining life expecancies. I is only defined for posiive values of d and for d =, i equals zero. The disribuion is righskewed, i.e., high values of d corresponding o high moraliies can occur. Is expeced value is equal o, such ha he deerminisic moraliy able describes an individual wih average life expecancy. For our analyses, we use a disribuion ha employs as a suiable choice of parameers for he characerisics lised above, and ha is a common choice for fraily models: a gamma disribuion (see, e.g., Bu and Haberman (22, pp. 8 9), Hougaard (984, p. 76), Jones (998, p. 82), Olivieri (26, pp. 29 3), and Piacco (24, p. 7), all of which refer o Vaupel e al. (979, pp )). Vaupel e al. (979) iniially chose he gamma disribuion as i is one of he besknown nonnegaive disribuions, is convenien o work wih, and is very flexible. Alhough some advanageous properies of he gamma fraily disribuion are los when applied o a deerminisic moraliy able insead of a coninuous moraliy law, i remains a reasonable assumpion. Since moraliy probabiliies near zero are unrealisic, he disribuion is shifed by a posiive value of γ, resuling in a generalized gamma disribuion, Γ ( α, β, γ ). For is probabiliy densiy funcion, we employ he following formula: α Γ β f(,, ) ( d ) = ( d γ ) e, for d γ, γ R, α, β >. α β γ α Γ α β d γ
19 8 Swich probabiliies Le denoe he ime of swich and le s( ) be he swich probabiliy ha depends on he ime since policy incepion. As here is a prescribed developmen of he cash value in our seing, dependence on ime can be inerpreed as he swich probabiliy depending on he amoun of cash value in he policy. The disribuion of a a poin in ime k N is given by k h = ( ) = ( ν ) F k P k s h s. h= ν = Moreover, he swich probabiliy can ake differen values depending on an insured s healh saus measured by he fraily facor d a realizaion of D F, i.e., s(, d ). However, in D he following we omi he index d o simplify he noaion. Shorrae process For he shorrae process, we follow Briys and de Varenne (994), Hansen and Milersen (22), and Jørgensen (26) and use he Vasicek model (Vasicek, 977), which is a Gaussian OrnseinUhlenbeck process. Under he riskneural measure Q, he shorrae process r( ) evolves as Q dr = κ θ r d + σ dw Q where W ( ),, T is a sandard Brownian moion on a probabiliy space ( Ω,, Q) F, and (F ), T is he filraion generaed by he Brownian moion. The ineres rae volailiy σ is deerminisic, he mean reversion level is denoed by θ, and he parameer κ deermines he speed of mean reversion. P r (, ) denoes he price of a zerocoupon bond a ime paying $ a mauriy, where = r. Since he zerocoupon bond price in he Vasicek model has an affine erm srucure, he expecaion can be represened by
20 r( u) du e 2 P κ, e σ σ σ κ = Ε = exp θ r θ ( e ) Q κ. (9) 2κ 2κ 4κ Hence, once all inpu parameers have been defined, he enire erm srucure can be deermined as a funcion of he curren shor rae r. Ne presen value of he deah benefi swich opion Based on he above moraliy assumpion and he shorrae process, he ne presen value (NPV) of an increasing deah benefi policy can be calculaed as he expeced discouned premium paymens less he expeced discouned deah benefi, using riskneural valuaion given a complee, perfec, and fricionless financial marke (see, e.g., Björk (24)). In he analysis, we assume independence beween shorrae and moraliy dynamics. Furhermore, he marke is assumed o be riskneural wih respec o moraliy risk, such ha he objecive (realworld) probabiliy measure P coincides wih he riskneural probabiliy measure Q (see, e.g., Bacinello (23, p. 468) and Dahl (24, p. 24)). From he insurer s perspecive, pooling effecs are achieved for a sufficien number of policyholders since, a he porfolio level, only expeced values and hus he moraliy disribuion in he pool are of relevance in evaluaing he conrac. According o our assumpions on he fraily disribuion, he expeced value of he fraily facor is equal o, implying ha, on average, moraliy in he pool is described by he deerminisic moraliy able. For a policy wih increasing deah benefi hroughou is erm, he ne presen value condiional on D = d under he riskneural measure Q hus resuls in K( x) K( x) + r( u) du r( u) du Q K( x) + Q NPV d = Ε B e Ε Y e = T T + r( u) du r( u) du B { } e Q Y K x + { K( x) } e = Q = Ε Ε = = T T = B p P, Y p q P, +. x + x x+ = = ()
21 2 When policyholders have he opion o swich he deah benefi scheme, he sochasic swich exercise dae is included in he ne presen value calculaion. For he ne presen value of he policy wih swich opion for D = d, we hence obain T T + ( ) Q ( ) r u du Q r u du = Ε { } K x Ε + { K( x) = }. () NPV d B e Y e = = The deah benefi Y is given by Equaion (6) and premiums + B are given by Equaions (7) and (8) for he level premium and risk premium scenario, respecively. Equaion () conains hree sources of randomness, namely, he remaining lifeime K(x), he ime of swich, and sochasic ineres raes. The equaion furher illusraes ha K x, i.e., he opion can be exercised only as long as he insured is alive. Since we le he swich rae depend on an insured`s healh saus and hus on he fraily facor d, swich probabiliies and probabiliies of deah are dependen. Again, assuming independence beween he sochasic fraily facor and ineres raes, Equaion () can be rewrien as NPV ( d ) ( ( ) ( ) { } ) ( + { } ) Q ( ) ( B { } k ) P( k ) P(, K x ) T T Q Q K x K x = = = = Ε B P, Ε Y P, + T T = Ε = = = k = ( ) ( Y + { } k ) P( k ) P(, K x = ) T T Q Ε = = + = k = T T ( k ) = B P( K ( x) = k ) P( = k ) P(, ) = k = T T ( k ) Y + P( K ( x) = = k ) P( = k ) P(, + ) = k = T T T T ( k ) ( k ) = B P( = k ) P( K ( x) ) P(, ) Y + P ( = k ) P ( K ( x) = ) P(, + ) = k = = k = T T k T T k ( k ) ( k ) = B s ( k ) ( s ( h) ) pxp (, ) Y + s ( k ) ( s ( h) ) pxqx+ P (, + ). = k = h= = k = h= Thus, he expeced value of Equaions () and () is obained by
22 NPV = E Q ( NPV ( D) ) 2 (2) and ( ) ( E ) NPV = Q NPV D, (3) respecively. In he conex of heerogeneiy in respec of moraliy implied by a gamma disribued fraily facor, closedform soluions are generally no feasible for he above ne presen values. To assess he value of he deah benefi swich opion, we subrac he ne presen value of he increasing policy wihou swich in Equaion (2) from Equaion (3) and denoe he value by Op NPV. Hence, ( ) Op NPV NPV NPV =. (4) 4. NUMERICAL ANALYSIS This secion presens resuls from a simulaion sudy so as o quanify he impac of he deah benefi swich opion. Firs, we consider he increasing universal life conrac. Nex, we inegrae he swich opion and illusrae effecs for deerminisic swich exercise imes and imes of deah. We hen derive ne presen values of he opion from he insurer s perspecive for differen swich probabiliies depending on he healh saus of insureds and for some specific exercise scenarios. In addiion, a sensiiviy analysis wih respec o he parameerizaion of he fraily disribuion is provided. Inpu parameers We examine a universal life insurance conrac wih increasing deah benefi wih a policy face value of Y = $, for a male insured aged x = 45 years a incepion. The acuarial minimum ineres rae is se a i = 3.5%. The minimum guaraneed ineres rae for universal
23 22 life producs is usually around 4%. For newer producs, i is ofen 3% (see, e.g., To be conservaive, numerical analyses are based on he U.S. 98 Commissioners Sandard Ordinary (CSO) male ulimae composie moraliy able wih a limiing age ω = 99. Composie means ha smokers and nonsmokers are no disinguished. An older moraliy able wih low limiing age like he 98 CSO able is conservaive regarding deah risk in he sense ha i ends o oversae probabiliies of deah. In conras, modern life ables accoun for moraliy improvemen and usually have a limiing age of 2. For he generalized gamma disribuion of he fraily facor, we employ he parameerizaion used in Hoermann and Russ (28), given by D Γ ( 2.;.25;.5 ). The parameer values lead o a fraily disribuion ha fulfils he requiremens laid ou in Secion 3. A shif by γ =.5 means ha individual probabiliies of deah can be a mos half he size of he moraliy able probabiliies bu no less han ha. We laer vary disribuional assumpions o examine he sensiiviy of swich opion values o parameerizaion changes. For he sochasic ineres rae, we use he inpu parameers given in Hansen and Milersen (22) wih speed of mean reversion κ =.3723, mean reversion level θ = 3.7%, ineres rae volailiy σ =.2258, and r() = 3.7%. As is common in he life insurance business, he ineres rae credied o he accoun value (here, 3.5%) is slighly below he ineres earned by he insurance company in he long erm (his difference is larger in European counries, e.g., in Germany he minimum guaraneed ineres rae is currenly 2.25%). Numerical resuls are derived using Mone Carlo simulaion wih 5, sample pahs (see Glasserman, 24). In all simulaion runs, we use he same se of random numbers o ensure comparabiliy of resuls. Value of he universal life conrac wih increasing deah benefi The consan annual premium for he increasing policy calculaed according o Equaion (4) is given by B = $5,937. The riskneural ne presen value from he insurer s perspecive
24 23 resuls in NPV = $2,866, as deermined by Equaion (2) under consideraion of sochasic ineres raes and he sochasic fraily facor. More precisely, in a Mone Carlo simulaion, 5, fraily facors are generaed ha imply 5, individual moraliy disribuions. Based on hese probabiliies of deah, he NPV can be deermined. I would be zero when using D, i.e., solely he moraliy able, as well as he calculaion ineres rae i insead of sochasic ineres raes. Value of he deah benefi swich opion by swich exercise ime and ime of deah The opion o swich from an increasing o a level deah benefi can be exercised only once during he policy erm and if done, mus be done a he beginning of a year unil he year of deah. Afer swich, premiums are adjused. In he following, we evaluae he opion and compare resuls for he wo previously described premium scenarios o idenify he effec of fuure premium paymens on he swich opion value. In he level premium case, consan annual premiums are paid afer swich, which are calculaed based on he equivalence principle, aking he curren cash value a he ime of swich as a single premium. In he risk premium case, premium paymens are sopped a he exercise dae and no resumed unil he cash value is exhaused. From hen on, he minimum risk premium is paid ha will keep he cash value a zero and hus avoid policy lapse. The is given by he difference beween he NPV he NPV of he conrac wihou swich (see Equaion (4)). Op NPV of he deah benefi swich opion of he increasing policy wih swich opion and To provide a firs impression of he impac of he deah benefi swich opion, we calculae riskneural values for differen deerminisic imes of swich exercise and imes of deah. For deerminisic swich dae and dae of deah K(x), Equaion () simplifies o K ( x) ( ) ( ) ( ) K ( x) ( ) NPV = B P, Y P, K x + + =. Noe ha in order o examine he effec of he swich opion in a porfolio, hese deerminisic
25 24 values have o be weighed wih respecive probabiliies of swich and survival. Resuls are displayed in Figure 4 for level premiums (Par a)) and risk premiums (Par b)). Figure 4: Ne presen value (NPV Op ) of swich opion by ime of swich exercise and ime of deah for 45yearold insureds a) Level premium $2' $ $2' NPV Op $4' 5 4 Time of deah Swich exercise ime $6' $8' b) Risk premium $' $8' $6' $4' $2' $ $2' $4' Time of deah Swich exercise ime
26 25 If policyholders pay level premiums afer swich as shown in Par a), he swich opion value falls below zero for insureds, deceasing afer abou 4 policy years (age 85). This is because insureds wih high life expecancy save so o speak on premiums over ime when choosing o exercise he swich opion in combinaion wih level premium paymens. Wihou swich, in conras, hey are likely o survive unil he ime when he deah benefi decreases again oward mauriy (approaching Y). Hence, hey do no benefi from he increasing deah benefi feaure anyway. Opion values are lower he earlier he year of swich and he laer deah occurs. In he level premium case, negaive values are hus generaed by insureds wih high life expecancy, especially when exercising he swich opion in early policy years. In he risk premium scenario, opion values can also become negaive from he insurer s perspecive. This is he case if deah occurs early afer swich, such ha premiums for he remaining lifeime are covered by he available cash value and no high risk premiums become due. In conras, opion values are exremely high if deah occurs lae and risk premiums are paid afer he cash value is exhaused. Value of he deah benefi swich opion by swich probabiliy To obain he Op NPV of he deah benefi swich opion, individual swich probabiliies, as well as individual probabiliies of deah, need o be aken ino accoun. Resuls for differen consan swich probabiliies beween s = % and s = % are displayed in Figure 5 for risk and level premium paymens. Figure 5 shows ha he wo premium paymen scenarios have very differen oucomes. In he level premium case, he ne presen value from he insurer s perspecive is negaive for all swich probabiliies; however, i remains posiive in he risk premium scenario. In he laer case, he Op NPV a mos reduces o $5 as he swich probabiliy approaches %, implying early swich. Hence, for risk premiums, high ne presen values (see Figure 4 Par
27 26 b)) cause he Op NPV of he swich opion o always remain posiive in he example even hough he probabiliy of occurrence of such exreme evens (e.g., survival unil = 45, i.e., age 9) is very low. This implies swich profis for he insurer if swich probabiliies are consan in he porfolio of insureds. However, if insureds erminaed conracs (policy lapse) insead of paying high risk premiums afer depleion of he cash value, as discussed in Secion 2, he Op NPV urns negaive, looks similar o he level premium curve. Figure 5: Ne presen value ( 45yearold insureds NPV Op ) of swich opion for differen swich probabiliies for $2' "risk premium" "level premium" $' NPV Op $ $' 2.5% 5% 7.5% % 2% 3% 4% 5% 6% 7% 8% 9% % $2' $3' Swich probabiliy For swich probabiliies higher han or equal o 2%, he level premium case leads o negaive ne presen values o abou $2, in he calibraion employed. This is due o considerably negaive values for early exercise imes, which are weighed more heavily for high swich probabiliies (see Figure 4, Par a)). Thus, depending on he premium paymen mehod, he swich opion can have negaive effecs on an insurer s porfolio even if swich probabiliies are assumed o be consan over ime, an assumpion ha we will relax in he following analysis. Value of he deah benefi swich opion by swich probabiliy and healh saus Since he value of he deah benefi swich opion is srongly dependen on an insured s life expecancy, as demonsraed in Figure 4, we nex examine he effec of adverse exercise
28 27 behavior wih respec o healh saus on he opion s riskneural value Op NPV. This is done by calculaing he opion value for swich probabiliies ha vary depending on an insured s individual moraliy. We disinguish persons wih a fraily facor greaer han or equal o one (d, average or belowaverage life expecancy) and persons wih a fraily facor d < (aboveaverage life expecancy). Resuls are displayed in Figure 6 for level premium (Par a)) and risk premium (Par b)). The level premium graph in Par a) of Figure 6 reveals srong discrepancies in he Op NPV if he opion exercise behavior depends on an insured s healh saus. In his case, from he insurer s perspecive, riskneural values remain posiive only if persons wih aboveaverage life expecancy have very low swich probabiliies and hus end o swich if a all lae in he conrac erm. The value of he deah benefi swich opion becomes negaive if hey exercise he opion wih higher probabiliy. This effec is more pronounced he lower he swich probabiliies are for insureds wih belowaverage life expecancy, wih he Op NPV reaching negaive values up o abou $3,5. This is in line wih resuls in Figure 4 Par a), where negaive values are generaed for early swich imes and lae imes of deah. In he risk premium scenario, shown in Par b) of Figure 6, differences depending on he healh saus are less disinc, bu sill visible. In paricular, he risk premium scenario generaes negaive values for he insurer only if persons wih belowaverage life expecancy exercise he opion and swich probabiliies are zero for insureds wih aboveaverage life expecancy. This observaion is in line wih he reasoning ha individuals wih impaired healh are likely no o pay high risk premiums afer depleion of he cash value due o expecaions of early deah. If insureds survive unil cash value exhausion, increasing deah probabiliies imply high risk premiums and hus lead o posiive ne presen values from he insurer s perspecive. Alogeher, srong adverse effecs can be observed.
29 28 Figure 6: Ne presen value ( depending on healh saus for 45yearold insureds a) Level premium Op NPV ) of swich opion for differen swich probabiliies Swich probabiliy if d< (aboveaverage life expecancy) % 2.5% 5% 7.5% % 2% 3% 4% 5% 6% 7% 8% 9% % 9% 8% 7% 6% 5% 4% 3% 2% % 7.5% 5% 2.5% % $2' $'5 $' $5 $ $5 $' $'5 $2' $2'5 $3' $3'5 $4' NPV Op Swich probabiliy if d>= (average or belowaverage life expecancy) b) Risk premium $2' $'5 $' $5 NPV Op $ $5 $' % 2.5% 5% 7.5% % 2% 3% 4% Swich probabiliy if d< (aboveaverage life expecancy) 5% 6% 7% 8% 9% % 9% 8% 7% 6% 5% 4% 3% 2% % 7.5% 5% 2.5% % Swich probabiliy if d>= (average or belowaverage life expecancy)
30 29 Addiional exercise scenarios Based on he previous analyses, cerain subsanial adverse effecs can be seen for specific exercise scenarios ha depend on healh saus and swich opion exercise ime. If he swich opion is exercised around he ime he cash value reaches is peak, for insance, he level deah benefi for he remaining conrac erm is higher han in he case of he original increasing deah benefi as se ou in Secion 2. This is because wihou swich he increasing deah benefi would acually decrease in line wih he cash value, down o he fixed level Y a mauriy (see Figure ). Hence, when swiching, he new level deah benefi is in fac higher han he original deah benefi of he increasing conrac a cerain imes during he conrac erm. Such comparably higher deah benefi amouns can be obained wihou having o pay addiional fees or providing new evidence of insurabiliy. Thus, depending on he insured s healh saus, paricular exercise behavior can have a considerable influence on conrac value, which may have serious consequences when considering a pool of insureds. To furher emphasize he poenial risk of adverse effecs regarding he deah benefi swich opion, we sudy several alernaive exercise scenarios (see Table ). Table : Ne presen values ( depending on he healh saus for 45yearold insureds Level premium s=% a =4 (peak) Op NPV ) of swich opion for specific exercise scenarios s=% =25 o =4 s=% o s=%* =25 o =4 s=% =5 o =5 All d d < All d d < All d d < All d d < Risk premium Risk premium (lapse) Noes: d : insureds wih average or belowaverage life expecancy, d < : insureds wih aboveaverage life expecancy, *: linear increase.
31 3 Firs, for insureds wih aboveaverage life expecancy, he swich opion is valuable in combinaion wih he level premium scenario. If exercised around he peak of he cash value curve, a high deah benefi is mainained compared o he decreasing benefi ha occurs wihou swich. Even hough new level premiums are higher han he original premiums in his case, opion exercise may sill give rise o negaive values for he insurer, which can be observed in Table (firs column, Level premium ). The scenario s=% a =4 (peak) compares resuls when eiher all insureds, only insureds wih average or belowaverage ( d ), or only insureds wih aboveaverage life expecancy ( d < ) exercise he swich opion wih probabiliy a he peak of he cash value curve (i.e., a age 86). Second, one would suspec ha he swich opion is especially valuable for insureds wih belowaverage life expecancy in combinaion wih he risk premium scenario. When exercised around he peak of he cash value curve a =4 or age 86, persons wih reduced life expecancy preserve a high deah benefi wihou having he underlying moraliy able adjused. Furhermore, fuure premiums can mosly be financed from he available cash value. For insureds wih higherhanaverage life expecancy, on he oher hand, his exercise paern would imply high risk premium paymens as he policy approaches mauriy and hus swich profis for he insurer. These expecaions are confirmed by he numerical resuls in Table ( s=% a =4 (peak), Risk premium ). However, riskneural values are much less negaive in his case han hey are for adverse exercise by healhy insureds in he level premium case. For risk premium paymens, we addiionally consider a scenario in which policyholders le he policy lapse, e.g., due o financial disress, as soon as risk premium amouns exceed % of he new level deah benefi (firs column, Risk premium (lapse) in Table ). The value of % was chosen by inuiion; however, furher ess revealed ha resuls remain robus wih respec o changes in he percenage parameer. Compared o he risk premium scenario wihou lapse (second row of Table ), such behavior has a considerable negaive impac on
32 3 he ne presen value from he insurer s perspecive if only insureds wih aboveaverage life expecancy are concerned. In paricular, he Op NPV is almos cu in half due o he lack of high risk premium paymens. In conras, if impaired insureds le he policy lapse when high risk premiums become due, negaive effecs are alleviaed and he ne presen value is increased because soon expeced deah benefi paymens do no have o be made. We furher exend he analysis and consider opion exercise prior o cash value peak given a consan swich probabiliy of % from age 7 (=25) o age 86 (=4) (second column in Table ). Resuls show ha opion values are subsanially affeced. In paricular, hey are more negaive from he insurer s perspecive in he level premium case for insureds wih aboveaverage life expecancy. The laer ne presen value decreases even more if he swich probabiliy is linearly raised from s = % a age 7 o s = % a age 86 (hird column in Table ). There are wo effecs responsible for hese aggravaed resuls. Firs, if he ime of cash value peak is no he only possible ime o swich (bu insead ranges from beween age 7 and age 86), opion exercise, on average, occurs earlier. From Figure 4 Par a) we know ha he earlier he opion is exercised by insureds wih long remaining lifeime, he lower are he opion values. And second, observed effecs are sronger due o he larger number of insureds sill alive a age 7, compared o a age 86, and hus able o exercise he opion. We now urn o he case where he swich opion is exercised afer five o fifeen policy years, i.e., beween ages 5 and 6 (fourh column in Table ), given a consan swich probabiliy of %. A reason for swiching early during he erm of he policy could, e.g., be he wish o reduce premium paymens. In his scenario, resuls are even more pronounced han in he case of exercising around he cash value s peak. As discussed previously, i is paricularly in he level premium scenario ha adverse exercise behavior by insureds wih high life expecancy generaes negaive ne presen values in an insurance porfolio. These adverse exercise scenarios assume ha insureds are well informed abou heir individual moraliy, i.e., wheher hey have an above or belowaverage life expecancy. The
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