THE IPACT OF THE ECONDARY ARKET ON LIFE INURER URRENDER PROFIT Nadine Gazer, Gudrun Hoermann, Hao chmeiser Insiue of Insurance Economics, Universiy of. Gallen (wizerland), Email: nadine.gazer@unisg.ch, hao.schmeiser@unisg.ch, gudrun.hoermann@unisg.ch, el.: +41 71 243 412, fax: +41 71 243 44 JEL Classificaion: G1, G22 ABTRACT Life insurers ofen claim ha he life selemen indusry reduces heir surrender profis and leads o an adverse shif in heir porfolio of insured risks, i.e., bad risks remain in he porfolio insead of surrendering. In his paper, we aim o quanify he effec of alered surrender behavior subjec o he healh saus of an insured in a porfolio of life insurance conracs on he surrender profis of primary insurers. Our model includes moraliy heerogeneiy by applying a sochasic fraily facor o a moraliy able. In he course of our invesigaion, we addiionally analyze he impac of he premium paymen mehod by comparing resuls for annual and single premium paymens. 1. INTRODUCTION In he life selemen marke, life insurance policies of senior ciizens wih belowaverage life expecancy are raded. 1 Wih purchases of abou $6.1 billion in face value in 26, he U.. life selemen indusry is of considerable volume. 2 However, he benefis and derimens of a secondary marke for life insurance are conroversial. 3 In general, primary insurers have hisorically profied from lapse or surrender of 1 2 3 ee, e.g., Dohery and inger (22, p. 4). ee Conning & Company (27). As he populaion ages, he poenial of his secondary marke generally increases. ee Bhaacharya e al. (24, p. 643), Dohery and inger (22, p. 4), Giacolone (21, p. 6), and aple Life Financial (27, p. 3). From an insured s perspecive, see Deloie Consuling LLP and he Universiy of Connecicu (25), as well as he corresponding discussion in inger and allard (25). From an insurer s perspecive, see Jenkins (26).
2 policies, 4 especially by insureds wih impaired healh. 5 Adverse selecion agains insurance companies due o secondary marke aciviy may lead o a decline of hose profis, which is paricularly rue for lapse-suppored producs, i.e., policies ha are priced based on persisency assumpions. This may resul in he need o charge higher premiums or could decrease he safey level of life insurance companies. 6 Even hough his is a very opical issue in pracice, no quaniaive analyses have been conduced before. The aim of our paper is o fill his gap and invesigae he impac of alered surrender behavior on an insurer s surrender profi. We provide a model framework o quanify he effecs of reduced surrender raes subjec o he healh saus of insureds in a moraliy heerogeneous universal life insurance porfolio. To dae, he secondary marke for life insurance has no received much aenion in he academic lieraure. Giacolone (21) provides a shor overview, describing he developmen of he life selemen indusry, limiaions on he marke, and sources of compeiion. Bhaacharya e al. (24) empirically analyze he impac of sae regulaion on he viaical selemen marke by esimaing welfare losses. 7 The benefis of a secondary marke for policyholders and life insurance carriers are examined in Dohery and inger (22). These auhors discuss he effecs of modified surrender behavior due o secondary marke aciviy bu heir aim is no o perform quaniaive analyses in his respec from an insurer s perspecive (see also Dohery and inger, 23). Dohery and inger (23) sae ha more han 2% of all policyholders above age 65 could consider selling heir policy o he secondary marke as an aracive alernaive o lapse or surrender. os of he lieraure dealing wih he surrender of life insurance conracs concerns iself wih valuaion of he surrender opion, e.g., Albizzai and Geman (1994), Bacinello (21, 23a, 23b, 25), Grosen and Jørgensen (1997, 2), Jensen e al. (21), effensen (22), and Tanskanen and Lukkarinen (23). In addiion, Bacinello (25) reveals differences in surrender opion value beween policies wih single or annual premium paymens. In Oureville (199), he emergency fund hypohesis is examined, which claims ha surrender values serve as an emergency fund for policyholders in imes of personal financial illiquidiy. The hypohesis implies ha he surrender decision is no primarily riggered by he developmen of ineres raes. Tsai e 4 5 6 7 When a policy lapses due o insufficien premium paymens, he conrac is erminaed wihou payou o he policyholder. This undersanding of policy lapse is in conras o exercise of he surrender opion, where he cash surrender value of he policy is paid ou. ee Dohery and inger (22, pp. 15 16). ee Dohery and inger (22, p. 6). In he viaical selemen marke, policies of insureds wih a considerably reduced life expecancy of less han wo years are raded.
3 al. (22) simulae he disribuion for policy reserves in a pool of policies being considered for early surrender. Their analysis is based on an esimaed empirical relaion beween surrender raes and ineres rae. Kim (25) describes surrender raes, using various explanaory variables based on differen surrender rae models, and finds appropriae modeling assumpions for four policy ypes. In his paper, we quaniaively examine he effecs of modified surrender behavior as implicaed by he secondary marke from an insurer s perspecive for he firs ime. oraliy heerogeneiy in he insurance porfolio is aken ino accoun by employing a coninuously disribued fraily facor o a deerminisic moraliy able. The surrender daes are generaed based on consan annual surrender raes. The join moraliy and surrender disribuion is implemened using a double-decremen model as presened in anders (1968). In a simulaion analysis, we quanify surrender profis for a porfolio of universal life policies using presen values. In he base case, consan surrender raes and a surrender charge induce a posiive surrender profi for he insurance company. In his seing, a decrease in surrender raes implies a reducion of surrender profis. However, we find ha his effec is considerably enhanced when aking ino accoun adverse selecion. In his case, only good risks surrender, whereas insureds wih reduced life expecancy choose he secondary marke alernaive and hus remain in he pool of insureds. Our resuls show ha his behavior no only reduces surrender profis, bu can even lead o a loss. One main finding is ha he premium paymen mehod single or annual has a subsanial impac on surrender profis reducion. In paricular, in he case of he more common annual premiums, surrender profis decline much more compared o he single premium case. The remainder of he paper is srucured as follows. In ecion 2, we presen our model framework including he life insurance conrac, moraliy heerogeneiy, and he double-decremen model. Numerical analyses and policy implicaions are discussed in ecion 3. ecion 4 summarizes he main findings. 2. THE ODEL FRAEWORK The model of moraliy heerogeneiy oraliy heerogeneiy is considered by means of a sochasic fraily facor 8 applied o a given deerminisic moraliy able. The one-year individual probabiliy of deah of a 8 ee Jones (1998, p. 81) and Vaupel e al. (1979, p. 44).
4 + person age x is hus given by he produc of he individual fraily facor d and he annual probabiliy of deah q x from he moraliy able: dq x, dq x < 1 qx ( d) = 1, x= min { x {,, ω} : d q 1 } wih x {,, ω}, x, oherwise where ω is he limiing age of he moraliy able. For d < 1, we le qω ( d): = 1. The superscrip sands for moraliy probabiliies. The parameer d specifies an insured s sae of healh. When < d < 1, he individual has an above-average life expecancy. The case of d = 1 corresponds o an insured wih normal healh, and when d > 1, he person is impaired. 9 For a given fraily facor d, he random variable K ( x, d ) denoes he individual remaining curae lifeime. Is disribuion funcion kqx ( d ) a a poin k resuls in ( ) ( ) k 1, 1 1 ( 1 ( )) + q ( d) = F ( ) ( ) k = P K x d k = p d = q d ( ), k x K x, d k x x l l= where p ( ) k x d is he individual k -year survival probabiliy. The fraily facor d is a realizaion of a random variable D. 1 For is disribuion F D, we follow he assumpions in Hoermann and Ruß (28): We le F D be a coninuous, righ-skewed disribuion on + wih an expeced value of 1, such ha he moraliy able describes an individual wih normal healh. As probabiliies of deah approaching zero are no realisic, he probabiliy densiy funcion f D is fla a zero, wih f D ( ) =. The disribuion of he sochasic fraily facor D represens he disribuion of differen saes of healh and hus of differen life expecancies in a porfolio. Ne presen value and premiums of he life insurance conrac We consider a porfolio of lifelong universal life insurance conracs purchased by insureds who are all he same age x a incepion. In case of deah, each policy pays a s fixed face amoun Y. Policyholders pay eiher a single premium B or consan annual a D premiums B. From he insurer s perspecive, he ne presen value ( NPV ) of one average policy in he pool can be calculaed by he difference of expeced premium paymens (paid a incepion or a he beginning of each year unil he sochasic year of K x, D ) and he expeced benefi paymen (paid a he end of year deah ( ) 9 1 ee Hoermann and Ruß (28, pp. 5 6). ee Jones (1998, pp. 8 83), Piacco (23, p. 14), and Vaupel e al. (1979, p. 44).
5 (, ) K x D ). The consan ineres rae is denoed by i. Hence, he ne presen value in he case of annual premiums resuls o 11 K ( x, D) a NPV = B Ε i Y ( ) + Ε ( + i ) = K ( x, D) + 1 ( 1 ) ( 1 ). (1) Given he disribuion of he fraily facor D, we calibrae he annual premium B a such ha he NPV of he policy is zero, i.e., B a ( K ( x, D) + 1) (( 1 i) ) Y Ε + = K ( x, D) Ε + = ( 1 i). (2) In he case of a single premium paymen B s, Equaion (2) simplifies o ( K ( x ) ) (( 1 ) ), D 1 s B Y i + = Ε +. (3) Hence, deah of he insured before reaching he average life expecancy based on he fraily disribuion D causes a negaive ne presen value for he insurer; an insured who survives longer han average generaes a posiive ne presen value. Policy reserves and surrender value of he life insurance conrac In general, policyholders have he righ o surrender heir life insurance policy. If his righ is exercised, a predeermined (cash) surrender value is paid ou ha depends on he policy reserve V (cash value) a he surrender dae. In our model, surrender may ake place only a he beginning of he year. As done in Tsai e al. (22), he surrender payou is given by =.8 +.2 V, = 1,, T, (4) T where a T = ω x he maximum aainable age is reached. We use his formula, as i accouns for common characerisics of he surrender value. I is, e.g., always higher 11 As, e.g., in Tsai e al. (22, p. 436), dividends, expenses, loadings, axes, and new business are no aken ino accoun.
6 han 8% of he policy reserve, and he surrender charge decreases wih ime. 12 According o U.. law and as se forh in Bacinello (23a), he cash surrender value of a life insurance policy mus be less han he ne single premium needed o fund fuure benefis. 13 Policy reserves V are calculaed based on he moraliy able according o he following formula: 14 ( )( 1 ) V + B + i Yq 1 1 x+ 1 V =, = 1,..., T 1 q x+ 1, (5) given V =. In he case of annual premium paymens, B = B a for all. For a single premium, B = B s and B = for = 1,,T. In year, he policy reserves V 1 and he premium are assumed o be compounded wih he consan ineres rae i. In case of survival, from his value, he cos of insurance given by he produc of he deah benefi Y and he probabiliy of deah in year is deduced, and he new reserve is hus given by V. Following he usual pracice, we do no consider he surrender opion when deermining he policy reserves as is done in Bacinello (23b). 15 To avoid policy lapses, in our model, premiums and reserves mus be calculaed based on he same acuarial assumpions, i.e. he same ineres rae i and he same moraliy able; oherwise, reserves could become negaive. 16 Therefore, given he premiums calculaed according o Equaions (2) and (3), which depend on he fraily facor disribuion, we need o adjus he moraliy able ha is used for calculaing he policy reserves in Equaion (5). By using a consan muliplier m, his leads o ( ),,..., ( ) q m = m q = T m, x+ x+ wih q ( )( m) = 1. We calibrae m such ha he premium calculaed under x+ T m consideraion of he sochasic fraily facor equals he expeced benefis calculaed based on he deerminisically shifed moraliy able. Thus, in he case of he single premium (Equaion (3)), m is adjused such ha 12 13 14 15 16 Beyond ha, by subsiuing T by a fixed number τ wih 1 τ < T, he formula allows o consider a resriced surrender charge period. ee Bacinello (23a, p. 466). ee Bacinello (21), Bowers e al. (1997), and Linnemann (24). ee Bacinello (23b, p. 3). A (universal) life policy lapses if he cash value is insufficien o pay policy coss, see Carson (1996, p. 675).
7! T( m) ( ( ) ) (( 1 ) ) x ( ) x+ ( )( 1 ) s K x, D 1 ( 1) B = Y Ε + i + = Y p m q m + i + (6) = (analogously for annual premium paymens). The deah and survival probabiliies in Equaion (5) are hen replaced, leading o V = ( V 1+ B 1)( 1+ i) Yqx+ 1( m) 1 q ( m) x+ 1, (7) for premiums calculaed according o Equaions (2) and (3), respecively. Based on he policy reserves given by Equaion (7), he corresponding surrender value can be compued by Equaion (4). The double-decremen model The difficuly wih double-decremen models lays in idenifying he cause of erminaion, since he dependence srucure beween surrender and deah disribuion canno be observed (one can only observe he minimum of he wo causes). In his analysis, we employ he model developed in anders (1968). We denoe he one-year surrender ( ) rae by qx+ ( d) for =,,T(d) wih q ( )( d) = as he corresponding probabiliy x+ T d of deah q ( )( d) = 1 for a given fraily facor d. The ime unil decremen K x+ T d ( x, d ) from eiher deah or surrender has he disribuion funcion k 1 ( ) F k q d q d q d K. ( ) ( ) ( ) = 1 1 ( ) (, ) k x = x d x+ l x+ l l= The parameer d sill represens a realizaion of he sochasic fraily facor D. For a generaed random number from he uniform disribuion ( u U(,1) ), he conrac is erminaed in year κ if κqx ( d) u < κ+ 1qx ( d). ince he one-year decremen probabiliy consiss of he one-year probabiliy of deah and he one-year probabiliy of sur- q d = q d + q d ), and can be decomposed o render ( ( ) ( ) ( ) x+ κ x+ κ x+ κ ( ) = ( ) + ( ) ( ) = κqx ( d) + κ px ( d) ( qx+ κ ( d) + qx+ κ ( d) ) = q ( d) + p ( d) q ( d) + p ( d) q ( d) q d q d p d q d κ+ 1 x κ x κ x x+ κ κ x κ x x+ κ κ x x+ κ he inerval of he year of erminaion q ( d x ), q 1 x ( d κ κ+ )) can be spli ino wo pars o deermine he cause of erminaion, namely,
8 ( ), ( ) ( ) + ( )) κqx d κqx d + κ px d qx κ d and + ( ) ( ) ( ), 1 ( )) + + κqx d κ px d qx κ d κ qx d. If he uniformly disribued random number occurs in he firs inerval, i.e., u < q d + p d q d, deah occurred; oherwise, he erminaion is due o ( ) ( ) ( ) κ x κ x x+ κ surrender. 17 In he case of annual paymens, he ne presen value surrender is hus given by NPV of he policy including K ( x, D) a NPV = B Ε i ( ) + Ε( L + i ) = K ( x, D) + 1 ( 1 ) ( ) ( 1 ), (8) K x, D + 1 where he benefi L paid o he policyholder a he ime of erminaion depends on he cause of erminaion. In case of surrender, L = ; in case of deah, L = Y. ince premiums are calculaed such ha NPV = (see Equaion (1)), NPV is he insurance company s surrender profi. urrender profis are generaed by way of he surrender charge. In our model, for a zero surrender charge (i.e., reserves V are fully paid ou) NPV =. The same is rue for zero surrender probabiliies, where NPV = NPV =. Thus, lowering posiive surrender probabiliies, ceeris paribus, reduces surrender profis. 3. NUERICAL ANALYE We use he U.. 21 Commissioners andard Ordinary (CO) male ulimae composie 18 moraliy able wih limiing age ω = 12 as he basis for our numerical analyses. According o he NAIC andard Nonforfeiure Law for Life Insurance, his able may be used o calculae cash surrender values. 19 We consider a pool of policyholders aged 45 a incepion of he conrac. For he fraily facor, we le D follow a generalized gamma disribuion, D Γ( α, βγ, ), wih shape parameer α = 2, scale parameer β =.25, and shifed by γ =.5, as used in Hoermann and Ruß 17 18 19 ee also Glasserman (24, p. 57). Composie means ha no disincion is made beween smokers and nonsmokers. ee inger and allard (25, p. 13).
9 (28). 2 For his parameerizaion, abou 4% of fraily facors lie beween 1. and 3.5, which for a 65-year-old male leads o life expecancies beween abou 9 (for d = 3.5) o 17 (for d = 1) years. For a 75-year-old male, life expecancies lie approximaely beween 5 (for d = 3.5) and 11 (for d = 1. ) years. 21 The ineres rae is i = 3%, he deah benefi Y = $1,, and he consan surrender rae in he porfolio is se o qx+ ( d) 4%, d, =,, T. The laer is a raher conservaive assumpion, as, e.g., he average surrender rae for all individual U.. life insurance policies in 26 was 6.6% according o he Life Insurers Fac Book 26. According o A.. Bes (27), he lapse raio of oal U.. life, healh, and fraernal insurance was 5.9% in 26. urrender occurs independen of he ineres rae, meaning ha no opimal exercise behavior is assumed. This assumpion is suppored by Tsai e al. (22, p. 439), who sae ha, hisorically, no dependence beween acual surrender behavior and ineres rae has been observed. Numerical resuls are derived using one Carlo simulaion wih 1, sample pahs (corresponds o a porfolio of 1, policies). To sample from moraliy and surrender raes, we use anders s (1968) mehod, as deailed in he previous secion. Base case: urrender leads o a posiive ne presen value for he insurer In he base case, as described above, we firs calibrae he premium such ha he ne presen value wihou surrender ( NPV ) is zero (see Equaions (2) and (3)). This implies a single premium of B s = $38,126 and an annual premium of B a = $1,795. econd, he muliplier m is calculaed in order o deermine he policy reserves. In boh cases, i is given by m =.9518 (Equaion (6)). The surrender profis for single and annual premium paymens are se ou in Table 1. 2 21 The gamma disribuion is a common choice for fraily models (see Olivieri, 26, pp. 29 3). Furher analysis revealed ha he resuls are no very sensiive o changes in disribuional assumpions of D. The chosen figures are realisic values for policies raded in he secondary marke. ee Dohery and inger (22, p. 4), Giacolone (21, p. 2), odu (25, p. 1), and OA Record (25, p. 3).
1 Table 1: Base case wih 45-year-old male policyholder a incepion premiums and surrender profis NPV (resuls for one average conrac) ingle premium Annual premium B s, B a $38,126 $1,795 NPV, qx+ ( d) % d, $ $ NPV, q + ( d) 4% d, $4,17 $1,36 x As a surrender charge is applied o he cash value, he surrender profis are posiive for qx+ ( d) 4%. Table 1 shows ha surrender profis are much higher for he single premium paymen ($4,17) han for annual premiums ($1,36). The reason for his oucome is illusraed in Figure 1. Par a) shows he number of deahs wihou surrender ( qx+ ( d) % ) a each age, saring a 45 (firs year of he conrac) o he limiing age 12, for he 1, policies. Given he premiums in Table 1, he ne presen value of one conrac is zero on average. When inroducing surrender raes as a second ype of decremen in he porfolio, he curve showing number of deahs changes, as laid ou in Par b) of Figure 1. The graph shows ha he number of surrenders a he beginning of he policy duraion is subsanially higher compared o he number of deahs, which is due o he consan surrender rae of qx+ ( d) 4% d, and very low annual probabiliies of deah a early ages. Deah probabiliies increase wih age, which is why he curve of he number of deahs increases unil he age of 8. The absolue number of deahs decreases afer age 8 since he number of insureds in he porfolio has been subsanially reduced due o previous surrenders and deahs. The oal number of decremens due o deah and surrender over all ages sums up o 1,. Par c) of Figure 1 displays deerminisic ne presen values in he case of single and annual premiums. NPV (D) is he difference beween premium paymens and he deah benefi if an insured dies a age x+, =,,T and NPV () is he corresponding ne presen value in case of surrender, boh discouned o policy incepion. NPV () represens he surrender profi, which depends on he surrender charge and he policy reserves (see Equaion (4)).
11 Figure 1: Base case number of decremens in simulaion; NPV for surrender and deah a deerminisic daes a) Number of decremens due o deah wihou surrender for qx+ ( d) % d, Resuls for porfolio: number of decremens (deah only) Resuls for one conrac (on average) Number of decremens 4. 3.5 3. 2.5 2. 1.5 1. 5 Deah Premium NPV ingle $ Annual $ 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age qx+ d d b) Number of decremens due o deah and surrender for ( ) 4%, Resuls for porfolio: number of decremens Resuls for one conrac (on average) Number of decremens 4' 3'5 3' 2'5 2' 1'5 1' Deah urrender Premium NPV ingle $4,17 Annual $1,36 5 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age c) Deerminisic ne presen values a ime for single and annual premium paymens Number of decremens ingle premium NPV () NPV (D) 4' 2' 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115-2' -4' -6' -8' -1' Age s ( ) ( )( ) ( NPV = B m 1+ i + 1) s ( ) ( ) ( NPV D = B Y 1+ i + 1) Number of decremens Annual premium NPV () NPV (D) 4' 2' 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115-2' -4' -6' -8' -1' Age a k ( ) ( 1 ) ( )( 1 ) ( NPV = B + i m + i + 1) k = a k ( ) ( 1 ) ( 1 ) ( NPV D = B + i Y + i + 1) k =
12 To calculae he ne presen value of an average conrac in he porfolio (Equaion (8)) for sochasic imes of surrender and deah, he deerminisic values NPV (D) and NPV () are weighed wih he number of decremens due o deah and surrender, respecively, a he ime of he decremen given in Par b) of Figure 1. As shown in Par c) of ha figure, NPV (D) is much more negaive in case of annual premium paymens during he firs 3 years of he conrac han in he case of a single premium. When inroducing he possibiliy of surrender, a subsanial porion of negaive ne presen value realizaions are replaced by posiive surrender profis. During he early years of he conrac, NPV () is higher for a single premium paymen han in he annual premium case, which given he same deah and surrender raes leads o he much higher surrender profi of $4,17 compared o $1,36 for he annual premium. The impac of he secondary marke on surrender profis urrender behavior depends on he insured s healh saus. 22 Individuals wih aboveaverage healh are generally considered more likely o surrender; however, he paern is no as clear-cu for hose wih impaired healh. On he one hand, heir ill healh makes i less likely ha hey will surrender bu, on he oher hand, he same ill healh may make hem more in need of money and hus more likely o surrender. The second effec is said o be sronger, bu boh will have adverse effecs on he insurer. To assess he impac of adverse selecion on surrender profis, we specifically focus on a change in he surrender behavior of impaired individuals as implicaed by he secondary marke. We firs assume ha surrender raes are se o zero for all policyholders wih a reduced life expecancy, i.e., wih a fraily facor d greaer han some barrier d*, and ha surrender raes remain a 4% for all oher policyholders over all ages (i.e., for all, qx+ ( d) % if d > d* and qx+ ( d) 4%, else). This means ha, generally, he average surrender rae in he porfolio decreases. We compare he resuls in his secondary marke scenario wih he surrender profis in he base case given in Table 1. Figure 2 displays resuls for d* = 1 and d* = 1.25 (Par a) and b), respecively), i.e., impaired individuals wih below-average life expecancy do no surrender heir policies. The graphs in Figure 2 show he number of decremens due o deah and surrender ( Deah (d*) ; urrender (d*) ) for he case of alered surrender raes. The oucomes show ha he secondary marke scenario leads o much fewer surrenders compared o he base case (see Par b) in Figure 1). Thus, as we are only considering wo causes of 22 ee Dohery and inger (22, pp. 16, 21, 23, pp. 63 73).
13 decremen, a much higher number of policies is erminaed by deah han by surrender. In he single premium case (Par a) of Figure 2), he original surrender profi of he base case is considerably reduced from $4,17 o $1,781, which means a reducion of 56.6%. In he annual premium paymen seing, he ne presen value even becomes negaive, implying a reducion of more han 12%. This effec is explained by he negaive selecion of insured risks and he adverse ineracion of surrender and deah probabiliies. Due o he highly negaive ne presen value of he deah benefi NPV (D) during he early years of he conrac (see Par c) of Figure 1), he annual premium paymen case is considerably more affeced. Figure 2: econdary marke scenario number of decremens due o surrender and deah wih qx+ ( d) % if d > d*, qx+ ( d) 4%, else, for =,, T a) d * = 1: urrender rae q + ( d) % if d > 1; q + ( d) 4%, else; =,, T x x Number of decremens Resuls for porfolio: number of decremens Deah (d*) urrender (d*) 4' 3'5 3' 2'5 2' 1'5 1' Ne presen value and reducion wih respec o base case (average conrac) Premium NPV ( d *) Reducion ingle $1,781-56.6% Annual $-313-123.% 5 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age b) * 1.25 d = : urrender rae qx+ ( d) % if d > 1.25; qx ( d) 4% +, else; =,, T Number of decremens Resuls for porfolio: number of decremens Deah (d*) urrender (d*) 4' 3'5 3' 2'5 2' 1'5 1' Ne presen value and reducion wih respec o base case (average conrac) Premium NPV ( d *) Reducion ingle $2,786-32.2% Annual $236-82.7% 5 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age The effecs are reduced if we consider only insureds wih d > 1.25 (Par b) in Figure 2). In his case, surrender profis are sill subsanially diminished by 32.2% (single premium paymen) and 82.7% (annual premium paymen). Overall, he resuls emphasize he impac of he premium paymen mehod, since ne presen values are much
14 more affeced in he case of annual premiums, which, i should be noed, is by far he mos common mehod of paymen. To idenify he impac of adverse selecion, i.e., of seing q ( d) % x+ for impaired individuals ( d > d* ) only, we consider a modified surrender rae in he porfolio aking d* = 1.25 as an example. For his barrier value, in he simulaion, 19,911 insureds ou of 1, have a fraily facor d > d* and hus do no surrender heir policy. The remaining individuals surrender a he usual rae of qx+ 4%. The new average surrender rae in he whole porfolio of insureds (independen of healh saus) is obained by ( ) ( ) s qx+ d = 19,911 % + 8,89 4% /1, 3.2% d,. The surrender profis for his average surrender rae are se ou in Table 2. Table 2: urrender profis (resuls for one conrac on average) q + d d NPV for average surrender rae ( ) 3.2%, x Premium NPV Reducion wih respec o base case ingle $3,632-11.6% Annual $1,296-4.7% Compared o he remendous reducion of surrender profis in Figure 2, Par b) 32.2% for a single premium he decline is reduced o 11.6% when he decrease in surrender raes is disribued over he enire porfolio insead of seing qx+ ( d) % for impaired individuals ( d > d* ) only. This effec is even greaer for annual premium paymens: surrender profis are reduced by 4.7% insead of 82.7%. In conras o Figure 2, Par b), reducing he overall surrender rae leads o a sronger decline of ne presen value in he case of a single premium han for annual paymens. We nex modify he underlying assumpion ha all impaired individuals wih a fraily facor d > d* do no surrender during he whole policy duraion. In fac, i is predominanly policyholders older han 65 who make up he arge group for he life selemen marke. Hence, we now assume ha impaired individuals have an average surrender rae of qx+ ( d) 4% unil age 64, afer which q x+ ( d) %, x+ 65. As before, all * oher policyholders coninue o surrender a qx+ ( d) 4%, d d, =,, T. The resuling decremen curves and surrender profis NPV are illusraed in Figure 3.
15 Figure 3: econdary marke scenario number of decremens due o surrender and deah wih qx+ ( d) % if d > d* saring a age 65, qx+ ( d) 4%, else a) d * = 1: urrender rae q ( d) % x+ ; q ( d) 4% + if d > 1 and 65 x +, else x Number of decremens Resuls for porfolio: number of decremens Deah (d*) urrender (d*) 4' 3'5 3' 2'5 2' 1'5 1' Ne presen value and reducion wih respec o base case (average conrac) Premium NPV ( d *) Reducion ingle $3,761-8.4% Annual $1,25-24.6% 5 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age b) * 1.25 d = : urrender rae q ( d) % + if d > 1.25 and x 65 x + ; q ( d) 4% x +, else Number of decremens Resuls for porfolio: number of decremens Deah (d*) urrender (d*) 4' 3'5 3' 2'5 2' 1'5 1' 5 45 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age Ne presen value and reducion wih respec o base case (average conrac) Premium NPV ( d *) Reducion ingle $3,928-4.4% Annual $1,172-13.8% Compared o Figure 2, he reducion of surrender profis shown in Figure 3 is considerably less, bu he general rend is very similar. In paricular, a change of surrender raes has a much sronger effec on he ne presen value for he annual premium paymens scenario han in he single premium case. Furhermore, surrender profis are sill reduced by 8.4% (single) and 24.6% (annual) in Par a) of Figure 3. An addiional cushioning effec occurs when aking ino consideraion ha only a cerain percenage of insureds wih d > d* have a zero surrender probabiliy afer age 65. Realisically, only a porion of insureds wih reduced life expecancy will sell heir policy o he secondary marke. This furher reduces he effec wih respec o losses in he surrender profi. However, he key resuls and cenral effecs remain he same.
16 Impac of age a incepion on surrender profis We nex look a he impac of he insured s age a incepion of he conrac on surrender profis. In his secion, we consider a porfolio of older policyholders where he insureds iniial age is 55 insead of 45. As in he base case, we firs need o calibrae he premiums such ha he ne presen value NPV is zero. Equaion (1) is saisfied for B s = $48,915 and B a = $2,789. The corresponding muliplier for he policy reserves is given by m =.9486. The resuling surrender profis are summarized in Table 3. Table 3: Base case wih 55-year-old male policyholder a incepion premiums and surrender profis NPV (resuls for one conrac on average) ingle premium Annual premium B s, B a $48,915 $2,789 NPV, qx+ ( d) % d, $ $ NPV, q + ( d) 4% d, $4,522 $1,455 x Table 3 shows ha premiums and surrender profis for an average surrender rae of 4% are higher when he porfolio is comprised of 55-year-old policyholders han when i conains 45-year-olds (see Table 1). Figure 4 illusraes resuls ha are derived under he same scenario as was used in Figure 2 ( qx+ ( d) % if d > d*; qx+ ( d) 4% else; =,, T ). Wih a porfolio of 45-year-old insureds, he decline in profis is considerably sronger for he annual premium scenario han for he single paymen case. In he porfolio of 55-year-olds, he surrender profi wih respec o he corresponding base case is less reduced for he single premium, and more reduced for he annual premium paymen mehod compared o he porfolio of 45-year-olds in Figure 2. Overall, however, he difference beween he wo porfolios is no very grea due o he adjusmen in he amoun charged for premiums.
17 Figure 4: econdary marke scenario number of decremens due o surrender and deah in a porfolio wih 55-year-old insureds a conrac incepion wih qx+ ( d) % if d > d*, qx+ ( d) 4%, else, for =,, T a) d * = 1: urrender rae q + ( d) % if d > 1; q + ( d) 4%, else; =,, T x x Number of decremens Resuls for porfolio: number of decremens Deah (d*) urrender (d*) 4' 3'5 3' 2'5 2' 1'5 1' 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age Ne presen value and reducion wih respec o base case age 55 (average conrac) Premium NPV ( d *) Reducion ingle $2,62-54.4% Annual $-471-132.4% d = : urrender rae qx+ ( d) % if d > 1.25; qx ( d) 4% b) * 1.25 +, else; =,, T Number of decremens Resuls for porfolio: number of decremens Deah (d*) urrender (d*) 4' 3'5 3' 2'5 2' 1'5 1' 5 55 6 65 7 75 8 85 9 95 1 15 11 115 Age Ne presen value and reducion wih respec o base case age 55 (average conrac) Premium NPV ( d *) Reducion ingle $3,176-29.8% Annual $23-86.1% eleced addiional numerical resuls Over ime, he number of insureds in he porfolio decreases because of decremens due o deah or surrender. The former especially concerns impaired insureds wih reduced life expecancy. Thus, if we increase he year of age age 65 in previous analyses afer which all impaired insureds (wih d > d* ) change heir surrender behavior o qx+ ( d) %, he discussed effecs will be less disincive. For example, seing he age o 75, he decline of ne presen value compared o he base case is abou 2.7%, ha is, $1,323 for annual premium paymens (single: a.9% reducion, or $4,72); given an age of 65, he ne presen value was reduced abou 13.8% o $1,172 (single: 4.4% o $3,928; see Figure 3). imilar effecs occur when surrender raes are assumed o decrease over he policy duraion.
18 Furhermore, modificaions of he surrender payou have an effec on surrender profis. For example, lowering he surrender charge reduces profis, which also implies less disinc effecs of alered surrender behavior. The same is rue if he surrender charge is imposed during only he firs, e.g., 1 o 15 years. Afer his period, he surrender payou is equal o he policy reserves, which in our model leads o surrender profis of zero. When changing he ineres rae from i = 3% o i = 4%, lower premiums are obained when solving Equaions (2) and (3). The single premium B s goes from 38,126 o 28,651; annual premiums B a are $1,545 insead of $1,795. In he base scenario wih a consan surrender rae of 4%, an ineres rae of 4% leads o ne presen values of $2,966 and $1,22 for single and annual premiums, respecively. In he secondary marke scenario qx+ ( d) 4% for insureds wih d > d* = 1.25 saring a age 65 (see Figure 3) he corresponding ne presen value is $2,829 (single premium) and $881 (annual premium). Compared o he base scenario, his means a decline of 4.6% and 13.8%, respecively. These values approximaely coincide wih he 3% ineres case (see Figure 3). Policy implicaions Our analyses revealed ha reduced surrender raes by insureds wih impaired healh caused by secondary marke aciviy resul in a decline in profis for insurance companies. No only are he surrender profis reduced, bu here are negaive effecs from adverse selecion. In pracice, boh effecs are probably inensified due o he fac ha he life selemen marke, in order o minimize ransacion coss, is mainly ineresed in policies wih large face amouns (see OA Record, 25). In he fuure, life selemens will probably become an alernaive for an increasing number of policyholders, i.e., i will no only be he large policies held by seniors ha are raded, bu also hose held by younger aduls wih below-average life expecancy. To preserve heir surrender profis, U.. life insurers have looked for ways o compee wih he secondary marke. The simples answer would be o pay healh-dependen surrender values. Thus, a person surrendering his or her policy would receive he curren (ne presen) value from he insurance company, which should be close o or even higher han (because of less ransacion coss) he price in he secondary marke. However, according o Dohery and inger (22, p. 18), regulaory, acuarial, and adminisraive difficulies seem o ouweigh he benefis gained from offering more compeiive surrender values o impaired insureds.
19 As an answer o he viaical selemen marke, he concep of acceleraed deah benefis (ADBs) was developed by life insurance carriers in he early 199s. 23 An ADB rider on a policy provides he opporuniy of receiving beween 25 1% of he deah benefi in he case of dread disease, long-erm care, or erminal illness accompanied by a remaining life expecancy of (usually) less han 12 monhs. A furher aemp o successfully compee wih he life selemen marke involves expanding he ADB rider o cover chronic illnesses. 24 Furhermore, Dohery and inger (22, 23) sae ha life insurers are lobbying for regulaion of he life selemen indusry and are refusing o allow heir agens o deal wih life selemen firms, a siuaion ha is currenly changing. Life insurers also aemp o idenify so-called premium financed policies polices purchased for he sole purpose of selling hem o he secondary marke. 25 4. UARY In his paper, we sudy he impac of modified surrender raes on insurance company profi ha occurs due o he opporuniy of selling one s policy o he secondary marke. This kind of analysis has no been conduced before, even hough i is of grea ineres o insurers. By use of a sochasic fraily facor, we model a moraliy heerogeneous pool of life insurance conracs. In he analysis, we firs calibrae annual and single premiums such ha he acuarial ne presen value of an average conrac wihou consideraion of surrender is zero. Nex, surrender profis (generaed due o surrender charges) are calculaed by means of a double-decremen simulaion analysis for differen scenarios. In he base case, surrender raes are consan for he enire porfolio. The secondary marke scenario assumes an asymmeric surrender behavior, i.e., impaired insureds do no surrender (bu, e.g., sell heir policies o he life selemen indusry insead), while only good risks coninue o surrender. In general, surrender profis are reduced when he porfolio s surrender rae declines. However, our resuls showed ha his effec is inensified by he secondary marke scenario. We furher found ha he single premium paymen mehod resuls in considerably higher surrender profis and ha negaive effecs from asymmeric surrender behavior are less severe wih his ype of paymen scheme han hey are when annual paymens are made. Hence, in he case of he more common annual premiums, originally lower surrender profis experience a much sronger decline in he secondary 23 24 25 ee Dohery and inger (22, pp. 31 33) and Giacolone (21, p. 5). ee Dohery and inger (23, p. 77). ee Dunmore (26) and Giacolone (21).
2 marke scenario. This reducion has shown o be even higher in a porfolio comprised of insureds who are older a conrac incepion. If only impaired insureds above age 65 sop surrendering in he secondary marke scenario, he effecs are less disinc bu sill quie eviden. Effecs are furher reduced if only a porion of impaired insureds or decreasing surrender raes are aken ino accoun. In he long run, boh consumers and life insurance carriers will benefi from a compeiive secondary marke. On he one hand, increasing compeiion in he life selemen marke will allow consumers o obain higher prices for heir policies. On he oher hand, primary insurers may benefi if he secondary marke causes a sronger demand for life insurance. However, life insurers will need o abandon lapse-suppored pricing, which could also aid in reducing he volailiy of heir profis. REFERENCE Albizzai,.-O., and Geman, H. (1994): Ineres Rae Risk anagemen and Valuaion of he urrender Opion in Life Insurance Policies. Journal of Risk and Insurance, 61(4), 616 637. A.. Bes (27): 27 aisical udy: U.. Life 26 Ordinary Life Average Policy ize & Lapse Raios. Available a www3.ambes.com (download on 1/9/28). Bacinello, A. R. (21): Fair Pricing of Life Insurance Paricipaing Policies wih a inimum Ineres Rae Guaranee. ATIN Bullein, 31(2), 275 297. Bacinello, A. R. (23a): Fair Valuaion of a Guaraneed Life Insurance Paricipaing Conrac Embedding a urrender Opion. Journal of Risk and Insurance, 7(3), 461 487. Bacinello, A. R. (23b): Pricing Guaraneed Life Insurance Paricipaing Policies wih Annual Premiums and urrender Opion. Norh American Acuarial Journal, 7(3), 1 17. Bacinello, A. R. (25): Endogenous odel of urrender Condiions in Equiy-Linked Life Insurance. Insurance: ahemaics and Economics, 37(2), 27 296. Bowers, Jr., N., Gerber, H., Hickman, J., Jones, D., and Nesbi, C. (1997): Acuarial ahemaics. The ociey of Acuaries, chaumburg, Illinois. Carson, J.. (1996): Deerminans of Universal Life Insurance Cash Values. Journal of Risk and Insurance, 63(4), 673 681.
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