Life Settlement: Securitization in the Insurance Market

Transcription

1 Lfe Settlement: Securtzaton n the Insurance Market Abstract: We nvestgate the mpact of securtzaton of the lfe nsurance contract on the prmary nsurance market. We analyze an economc model n whch polcyholders have dfferent lqudty rsks n an overlappng generaton settng, and polcyholders and the monopolstc nsurer ncur lqudty costs. We fnd that securtzaton may lead to the ncrease n nsurance demand and the decrease n fnancng costs. he nsurer wll have a hgher proft when the decrease n fnancng costs or the ncrease n demand s suffcently large. We also fnd that securtzaton enhances the effcency of the nsurance market, whch s n contrast wth the theoretcal studes on settlement. Keywords: Securtzaton, lfe nsurance, settlement market, overlappng generaton, welfare JEL Classfcaton: D6, G, G3 1

2 Lfe Settlement: Securtzaton n the Insurance Market I. Introducton Securtzaton can be understood as a process of convertng llqud assets or a group of varous assets nto securtes (lqud assets). Promnent examples nclude mortgage backed securtes (MBS), asset backed securtes (ABS), and collateralzed debt oblgatons (CDs). Compared wth these securtzatons, a relatvely less focus has been placed on the securtzaton of lfe nsurance. In general, nsurance polces are llqud because they are not tradable. Whle securtzaton s prohbted for most nsurance polces, lfe nsurance s an excepton. Under some condtons, lfe nsurance polcyholders are permtted to sell ther polces to thrd partes who are typcally nsttutonal nvestors, such as penson funds, endowments, and prvate equty funds. 1 hs trade s often called lfe settlement (or lfe settlement securtzaton). Settlement opens up the secondary market for lfe nsurance wheren polces become traded as securtes. he sze of the secondary market of lfe nsurance n the US s known to be 1 bllon dollars n 7 and 16 bllon dollars n 13 (Connng Research & Consultng, 8; IB Lfe, 13). he settlement n 13 totaled 4 bllon dollars (IB Lfe, 13). Securtzaton of lfe nsurance polcy s potentally attractve to both polcyholders and nvestors. Wthout securtzaton, polcyholders have to surrender ther nsurance polces to meet cash needs. he cash value obtaned from surrender s called the surrender value. Snce the surrender value s usually extremely low or zero, polcyholders are not lkely to receve suffcent cash. Consequently, polcyholders have an ncentve to sell ther polces to nvestors as long as the sellng prce s hgher than the surrender value. Investors can enjoy the beneft of dversfcaton of rsks by addng the securtes nto ther portfolos, as the mortalty rsk s not correlated wth the captal market. Insurers may also have hgher proft from ncreased demand. hs study theoretcally nvestgates the effect of lfe settlement on effcency. Specfcally, we consder the case n whch polcyholders face heterogeneous lqudty rsks n the monopolstc nsurance market. Consderng monopolstc market s meanngful n the followng aspects. Whle a large number of nsurers compete n nsurance markets, t s also well known that a few large nsurers enjoy market powers, due to reputaton and fnancal strength among others. herefore, studyng monopoly can provde nsghts on the effects of market structure on settlement. Second, studyng monopoly allows to analyze the strategc reacton of the nsurer to settlement, whch can mprove our understandng about the relaton between the nsurance market and the settlement market. Lqudty rsk refers to the possblty that polcyholders need cash n urgency, such as unexpected medcal expenses and lablty. o meet the lqudty need, polcyholders may receve the cash surrender value by surrenderng ther polces f no settlement s allowed. When settlement s allowed, polcyholders may opt to sell ther polces f the settlement prce s hgher than the surrender value. hs mples that settlement provdes polcyholders wth an addtonal alternatve, whch s benefcal to polcyholders. However, the possblty of settlement may also alter the behavor of the nsurer. In prncple, the nteracton between settlement and the nsurer may mprove or deterorate welfare. 1 wo types of settlement are allowed: vatcal settlement and lfe settlement. he former s avalable for the termnally-ll patents who have less than two years of lfe expectancy and the latter s avalable for those aged above 65.

3 As an extenson of Hong and Seog (18), we consder the case n whch the nsurer cannot offer the menu of contracts to dscrmnate polcyholders based on ther lqudty needs. ur analyss utlzes a "hybrd" of two exstng approaches lke Hong and Seog: economcs and actuaral approaches. In the economcs approach utlzed n Daly, Hendel and Lzzer (8) and Fang and Kung (11), death benefts are endogenously determned dependng on mortalty rsk. n the other hand, death benefts are usually fxed n the actuaral approach utlzed n Gatzert, Hoermann, and Schmeser (8) and Zhu and Bauer (13). Note that ndvduals choose to settle (surrender) or to hold the nsurance contract based on the surrender ncentve and mortalty rsk. Both approaches have strength and weakness. Whle the economcs approach provdes the nsghts based on the ratonal and nteractve behavor n equlbrum, the equlbrum outcome of full dscrmnaton over lqudty rsk s hardly observed n realty. herefore, the actuaral approach seems to better reflect the contract desgn n practce. However, t tends to overlook the strategc reacton between the nsurer and polcyholders. Please refer to the lterature revew secton for detaled explanaton. Based on these observatons, we compromse two approaches: Whle we consder the strategc nteracton between the nsurer and polcyholders as n the economcs approach, we assume that nsurance contracts do not dscrmnate across lqudty rsks as n the actuaral approach. We also assume that death beneft s the same for each mortalty rsk. hs approach enhances the tractablty n analyss of the strategc nteractons between the nsurer and polcyholders. ur study dstngushes tself from the exstng studes as follows. Frst, we consder lqudty costs of the nsurer as well as polcyholders whle most exstng studes do not consder these costs. Lqudty costs are ncurred f cash s not readly avalable when needed. Polcyholders have to ncur costs when they are unable to meet the cash requrements. he cost can be nterpreted as the hgher nterest rates when they have to make an urgent cash loan. In addton, the nsurer has to ncur costs when t has to meet the surrender request by polcyholders. he costs are ncurred because, n order to meet the surrender request, the nsurer may have to ncur hgh external fnancng costs, or to forego proftable nvestment opportuntes and nvest n lqud assets. 3 hs study nvestgates the nteracton between securtzaton and lqudty costs. Second, we consder an overlappng generaton model to examne the effect of the ntroducton of the settlement market. As the ntroducton of settlement affects the new contracts wthout alterng the exstng contracts, an overlappng generaton model allows us to examne dverse effects on dfferent contracts. hrd, polcyholders are heterogeneous n ther surrender rsks (represented by lqudty rsks n ths study). 4 Note that a majorty of exstng studes focus on homogeneous surrender nly a few contracts wth menu are offered n the real world. hus, the assumpton s acceptable as long as the number of contracts s too small to fully dscrmnate over lqudty rsks, whch s the case because lqudty rsk s contnuous. 3 Accordng to Damodaran (15), the cost of captal of the lfe nsurance ndustry s 5.76%, whch s relatvely hgher than that of the fnancal servces ndustry of.85%. hus, t s realstc that the lqudty cost of the settlement nvestor s lower than that of the nsurer. In ths paper, we frst suppose that there s no cost n the settlement market whereas the lqudty cost s ncurred for the surrender to the nsurer. ( 4 Whle surrender occurs due to lqudty needs n ths study, surrender s also modeled n the context of the dsappearng bequest motve n lteratures (Daly, Hendel, and Lzzer, 8; 3

4 rsks except Hong and Seog (18). In contrast, we assume homogeneous mortalty rsks to focus on the surrender behavor. he analyss of ths study provdes the followng fndngs. Frst, settlement may help to ncrease nsurance demand by makng nsurance more attractve to polcyholders. hs mples the possblty that settlement may ncrease consumer welfare. Second, t s possble that the nsurer prefers to transfer the surrender functon from the nsurance market to the fnancal market, whch enhances effcency and consequently, the proft of the nsurer may ncrease. hs s because the nsurer may reduce the lqudty costs f the settlement market s more effcent as assumed here. hs fndng s n lne wth those of Cowley and Cummns (5) who suggest that securtzaton provdes a mechansm through whch nsurers can mprove captal effcency. hs fndng s novel because most exstng studes (for example, Daly, Hendel, and Lzzer, 8; Fang and Kung, 1; Zhu and Bauer, 13; and Hong and Seog, 18) consder settlement a compettor, not a complement, to the nsurer. II. Lterature Revew he effects of securtzaton have been studed n lterature. Black, Garbade, and Slber (1981) ndcate that securtzaton lowers the mortgage rate of the Federal Housng Authorty (FHA). her emprcal results show that the mortgage rate of the FHA decreased as much as 14 bass ponts durng Kolar, Fraser, and Anar (1998) fnd that there exsted a long-term nverse relatonshp between the prme mortgage rate and the growth of securtzaton durng In other studes such as Chloewck (1985), Hendershott, and Shllng (1989), Rothberg, Nothaft, and Gabrel and (1989), Passmore, Sparks, and Ingpen (), and Naranjo and oevs (), authors fnd that securtzaton lowered the nterest rate offered to consumers and ncreased the demand for loan. Whle a majorty of studes focused on the mortgage market, probably due to ts sze of securtzaton, there exst studes that are focused on other debt-related securtzaton. For example, Ivashna and Sun (11) and Nadauld and Wesbach (1) nvestgate the securtzaton of corporate bank loan. Smlar to the studes focusng on the mortgage market, ther fndngs suggest that securtzaton lowers the loan rate and attracts more borrowers. Securtzaton may also mtgate undernvestment problem and lowers agency costs (James, 1988) Although the aforementoned studes ndcate the postve sde of securtzaton, the negatve sde of securtzaton also exsts. Gorton and Souleles (7) fnd that banks wth low credt ratng tend to securtze more than banks wth hgh credt ratng. hs mples that securtzaton s postvely related wth rsk. Berndt and Gupta (8) and Keys, Benjamn J., et al. (1) argue that securtzaton leads to weak underwrtng standards. Man, Rehman and Suf (9) also fnd that securtzaton ncreased the supply of rsker mortgage, whch s consdered one of the culprts of the 8 fnancal crss. An ncreasng number of studes have focused on the effects of securtzaton of lfe nsurance. Doherty and Snger (3) argue that lfe settlement ncreases consumer welfare, thereby reducng the monopsony power of the nsurer. he nsurer has a monopsony power n surrender because the polcyholder can sell her polcy only to the nsurer. he ntroducton of settlement ncreases the competton between nsurers and nvestors; thus, consumer welfare can be mproved. Gatzert, Hoermann, and Schmeser (8) utlze an actuaral smulaton Fang and Kung, 1; and Seog and Hong, 1). 4

5 model to show that settlement lowers the nsurer s proft, mplyng that consumer welfare can be mproved. Zhu and Bauer (13) fnd that the realzed return of settlement nvestors s markedly low compared to the expected return because of the nformaton asymmetry problem regardng mortalty rsk. n the other hand, a strand of economc research ndcates that settlement lowers consumer welfare except Hong and Seog (18). Daly, Hendel and Lzzer (8) and Fang and Kung (1) argue that the ntroducton of settlement lowers consumer welfare when the nsurance market s compettve. he ntroducton of settlement prevents nsurers from provdng effcent nsurance, resultng n lower consumer and socal welfare. From the study of the monopolstc nsurance market, Seog and Hong (1) fnd that settlement does not change consumer welfare, but lowers socal welfare due to the decrease n the monopolstc proft. n the contrary, Hong and Seog (18) show the condton to enhance consumer welfare when settlement s allowed. he remander of ths study s organzed as follows. Secton II provdes the model descrpton n detal. Secton III nvestgates the basc model wthout a lfe settlement market. Secton IV examnes the model wth a settlement market. Secton V provdes an llustratve example. Secton VI concludes. III. Model descrpton We consder a monopolstc nsurer n an overlappng generaton model. Wthout loss of generalty, we consder two generatons: old () and young (). Each generaton s denoted by G, where =,. me s denoted by t =, 1,, 3. We assume that settlement may be ntroduced at t = 1. In each generaton, we can denote the probablty of lqudty rsk of polcyholder j as q j. Gven that the nsurer s prohbted from dscrmnatng polcyholders aganst ther lqudty rsks, an equal premum should be charged to the polcyholder n each generaton. hs mples that all polcyholders n G purchase nsurance wth premum Qj = Q and surrender value Sj = S. 5 Let us frst consder the old generaton. At t =, polcyholder n G may choose to buy nsurance payng premum Q. At t = 1, death beneft D s pad to the (dependents of) polcyholder n G when she des. he death beneft s gven exogenously. All polcyholders have the same mortalty probablty p 1 at t = 1. 6 Polcyholder j who survves at t = 1 may need cash V wth probablty q qj = n [, 1]. We assume that polcyholders are homogeneous except for ther lqudty probabltes q. herefore, the populaton dstrbuton of G s dentfed wth that of q whose densty 5 hat s, the nsurer does not offer a separatng nsurance contract to polcyholders wth dfferent lqudty rsks snce the lqudty needs are not verfable, and consequently unnsurable. 6 When polcyholders have dfferent mortalty rsks, our approach can be appled nto each mortalty rsk group. o smplfy analyss, polcyholders n the same generaton are assumed to be homogeneous except for ther lqudty rsks. 5

6 functon and dstrbuton functon are denoted by f( q ) and Fq ( ), respectvely. We assume that the densty functon s dfferentable. When the polcyholder surrenders her polcy, she receves cash surrender value S. At t =, the mortalty rsk for polcyholders n G s p. he dscount factors n each perod are all the same and denoted as. We frst assume that the settlement market s perfect and compettve and that the nvestors (settlement provders) are rsk neutral for techncal smplcty. 7 Followng the assumpton, the polcyholder can sell her polcy at prce pd, when settlement s allowed at t = 1. It s assumed that the contract s not renegotable once purchased. hus, the ntroducton of settlement does not affect the exstng contracts. 8 he structure of the young generaton s bascally the same except that t enters the market at t = 1. A polcyholder n G who purchases an nsurance wll pay a premum Q at t = 1. he death beneft, D are exogenously gven, and cash surrender value s S. Lqudty rsk q s dstrbuted on [,1] and the populaton dstrbuton s dentfed wth the dstrbuton of q. he mortalty rsks for t = and t = 3 are denoted as p 1 and p, respectvely. If settlement s allowed at t = 1, then the settlement prce s denoted by pd. Note that the nsurer may reflect the ntroducton of settlement on the contracts wth the young generaton. he populaton dstrbuton of the young generaton s Fq ( ) whch has densty functon f( q ). Generally, the surrender value s lower than the present value of death beneft n realty. hus, we assume that S p D. We also assume that V p D. he tme lne s shown n Fgure 1. Fgure 1. me lne of model A monopolstc rent s denoted by R, whch represents the markup n premum. he rent comes from the polcyholder n G who holds the nsurance. he polcyholder n G can enjoy consumer surplus of M from keepng nsurance, whch wll be lost f she surrenders. If the polcyholder wth cash requrement s unable to receve suffcent cash from surrender or settlement, she needs to fnance the gap ncurrng unt costs k where k 1, whch wll be called the lqudty costs of the polcyholder. n the other hand, the nsurer may have to ncur lqudty costs to meet the surrender request by polcyholders. he costs can be nterpreted as the expected losses due to the urgent sales of assets to meet the cash need. he unt lqudty cost s denoted by c, where c 1, so the total lqudty cost of the 7 he assumpton of rsk neutralty s based on the observaton that nvestors can have welldversfed portfolos. Snce the mortalty rsk s no correlated wth the captal market returns, nvestors may requre the rsk premum for settlement close to zero accordng to the fnancal prcng models lke the CAPM. 8 In realty, t s prohbted to change the contents of nsurance contracts such as nsurance premum and death beneft already sold because of the deteroraton of the busness envronment. 6

7 nsurer for surrender s cs. We assume that k c as well as that companes can have a better access to the fnancal market than ndvduals. IV. he benchmark model: no settlement market Suppose that there s no settlement market. Gven that the nsurer cannot dscrmnate across dfferent lqudty rsks, t should offer the same contract to polcyholders. For generaton, the monopolstc nsurer needs to offer one nsurance contract, ( Q, S ), to maxmze ts proft. hat s, the contract ( Q, S) s decded to extract consumer welfare from the polcyholder n each generaton. he nsured decde whether to purchase nsurance or not gven nsurance contract. o understand the contract offer of the nsurer, let us frst suppose that full dscrmnaton s possble, n contrast to our assumpton. Let us denote that the rent from polcyholder wth lqudty rsk q s denoted as R. In ths case, the premum for lqudty rsk q can be decomposed as follows. (1) j j Q = p ( D + R ) + (1 p ) q S + p (1 p )(1 q )( D + R ), =, j 1 j 1 j j 1 j j he premum Q j s the sum of the expected payment when polcyholders de or choose to surrender and rent. hus, n the expresson (1), the frst and thrd terms of the RHS are the pure premum plus rent for each tme. he second term s the expected payment of cash surrender value. Now, the gross nsurance beneft of polcyholders j wth lqudty rsk purchasng nsurance, GB( q j ), can be denoted as: GB( qj ) = p 1( D + M ) + (1 p 1) qj Sj k max( V Sj,) + p p 1 qj D + M (1 )(1 )( ) j qj from he frst and the thrd terms of the RHS are the expected nsurance beneft plus consumer surplus for each tme. he second term represents the expected cash surrender value receved net of lqudty costs. he net nsurance beneft of the polcyholder s defned as the dfference between gross nsurance beneft and nsurance premum. NB( q ) = GB( q ) Q j j j = p ( M R ) + (1 p ) q k max( V S,) + p (1 p )(1 q )( M R ) 1 j 1 j j 1 j j If dscrmnaton s allowed, the nsurer wll offer a contract ( Q, S ) to each polcyholder j n G such that NB( q ) =, fully extractng consumer surplus from each polcyholder. j Now let us consder our case n whch dscrmnaton s not allowed. In ths case, the nsurer needs to offer one contract, ( Q, S ), to all polcyholders. Gven ( Q, S ), the gross j j 7

8 nsurance beneft of polcyholder j should be wrtten as GB q = p ( D + M ) + (1 p ) q S k max( V S,) ( ) j 1 1 j + p p 1 qj D + M (1 )(1 )( ) he polcyholder wll purchase nsurance only f her net nsurance beneft s nonnegatve. hus, gven contract ( Q, S ), we can fnd a polcyholder wth zero net nsurance beneft who wll be called the target (lqudty) rsk. Let us denote Followng (1), we can express premum Q as: q for the target lqudty rsk. Q = p ( D + R ) + (1 p ) q S + p (1 p )(1 q )( D + R ) () where R s the rent from the target rsk. 9 Now, the net nsurance beneft of the target polcyholder s: NB( q ) = GB( q ) Q = p ( M R ) + (1 p ) q k max( V S,) + p (1 p )(1 q )( M R ) = (3) From ths expresson, rent s determned as follows: (1 p 1) q k max( V S,) =, p 1 + p (1 p 1)(1 q ) = M, S V R M S V (4) For polcyholder j, the net nsurance beneft s: NB( q ) = GB( q ) Q j j = p ( M R ) + (1 p )[ q S k max( V S,) q S ] 1 1 j + p (1 p 1)[(1 qj )( D + M ) (1 q )( D + R )] (5) Polcyholders who have postve net nsurance benefts would buy nsurance. he followng lemma shows that a lower lqudty rsk s related wth a hgher nsurance beneft. Lemma 1. In each generaton, polcyholders wth lqudty rsks are less than or equal to the target lqudty rsk purchase nsurance. Proof. See the appendx. // 9 Note that R denotes the rent only from the target polcyholders. Rent vares across polcyholders because premum s fxed. 8

9 he exstence of postve net nsurance beneft mples that the monopolstc nsurer fals to fully extract consumer surplus from polcyholders. Snce q s unquely determned by ( Q, S ), the nsurer's problem of selectng ( Q, S) s equvalent to that of selectng ( q, S ). Now, the nsurer needs to solve the followng proft maxmzaton problem: Max ( q, S ), q, S =, q q ( ) 1 ( ) (1 1)(1 ) ( ) (1 1) (1 ) ( ) = Q F q p D F q p + c S q f q dq p p D q f q dq q = [ p + p (1 p )(1 q )] R F( q ) + (1 p ) S [ q F( q ) (1 + c) q f ( q ) dq ] q p p 1 D q F q q f q dq (1 ) [ ( ) ( ) ] where Q = p ( D + R ) + (1 p ) q S + p (1 p )(1 q )( D + R ) (6) Solvng the problem wth (4), we obtan proposton 1. We denote the optmal target rsk and surrender value as q * and S *, respectvely. Proposton 1. Suppose that settlement s not allowed. he optmal nsurance contract s characterzed as follows. (1) he optmal surrender value S * s between V and pd. In partcular, f q* F( q*) c 1, then the surrender value s equal to p q * D. q f ( q ) dq () arget lqudty rsk q * s determned by: [ p + p (1 p )(1 q *)] M f ( q *) + (1 p ) S *[ F( q *) cq * f ( q *)] p (1 p 1)( M + D ) F( q*) = (7) Proof. See the appendx. // From (4) and proposton 1 (1), we have R = M. he nsurer can fully extract consumer surplus from the polcyholder wth target lqudty rsk by settng the surrender value at or above V. hs s because lqudty costs k can be saved. Moreover, the nsurer may set surrender value above V when lqudty cost of the nsurer c s suffcently low. In ths case, polcyholder wth target lqudty rsk reduces the costs k as well; thus, the rent R s stll equvalent to M. In addton, the optmal target lqudty rsk s determned where margnal revenue of lqudty rsk equals margnal cost of the rsk. Note that margnal revenue of lqudty rsk s the ncrease n premum and margnal cost of lqudty rsk s the ncrease n payment of 9

10 nsurance as target lqudty rsk ncreases. hs result s summarzed n (7). Let us defne consumer welfare for G ( CW ) as the sum of net nsurance beneft of polcyholders n G. hus, consumer welfare for G can be expressed as follows: 1 q* q* CW = NB( q ) f ( q ) dq = GB( q ) f ( q ) dq Q * F( q *) j j q * = (1 p 1)[ q * F( q*) q f ( q ) dq ][ p ( D + M ) S*] where V S * p D (8) For each generaton, let us defne socal welfare for proft and consumer welfare from SW = + CW G. hat s, socal welfare for G ( SW ) as the sum of the nsurer s G s: 11 q* q* = [ p + p (1 p )] M F( q *) p (1 p ) M q f ( q ) dq (1 p ) cs * q f ( q ) dq where V S* p D (9) Fnally, total consumer welfare s defned as the sum of consumer welfare n each generaton, whch s evaluated at t =. hat s, CW = CW + CW. Smlarly, we defne total socal welfare as follows: SW = SW + SW. V. he model wth a settlement market 1. he effect on polcyholders of the old generaton Now, suppose that settlement s allowed at tme 1 and that the settlement market s compettve. In addton, there s no transacton cost for the settlement, whch mples that the settlement prce s pd whch s equvalent to the present value of death beneft. Gven these assumptons, polcyholders of the old generaton wll choose to sell ther contracts, nstead of surrenderng them as long as S* p D. When S* = p D, polcyholders are ndfferent between settlement and surrender. As a te break rule, we assume that polcyholders stll choose to surrender when S* = p D, even f settlement s allowed. Recall that the nsurer s prohbted from renegotatng over the exstng contracts wth the old generaton. Nevertheless, the nsurer s proft and the polcyholder s net nsurance beneft are affected by the polcyholder s selecton of settlement. Let us frst examne the effects of 1 In ths research, dfferences n preferences followng the lqudty rsks ( q ) are mportant to explan varaton n ndvduals behavor. here s however no consensus n economc lteratures how to take nto account these dfferences to measure welfare. hus, we use the standard utltaran crteron such as Harsany (1955) and Sen (14) to defne consumer welfare as a sum of ndvdual net beneft. 11 We follow the approach of role (1988) to defne socal welfare. 1

11 settlement on the nsurer s proft. Proposton. When settlement s allowed, the followng changes n the nsurer's proft are observed compared wth the case wth no settlement. (1) When S* p D, the change n the nsurer's proft from the old generaton s expressed as follows. q * (1 p )[(1 + c) S * p D ] q f ( q ) dq (1) 1 p D herefore, the nsurer s proft ncreases f and only f S *. (1 + c) () When S* = p D, the nsurer s proft from the old generaton does not change. Proof. See the appendx. // Proposton shows that the effect on the nsurer s proft from the old generaton s not n one drecton. Settlement brngs two countervalng effects when S* pd. n the one hand, settlement ncreases the death beneft payment of the nsurer, whch lowers the proft ( pd). hs effect ndcates that monopsony power of the nsurer s weakened due to the settlement. n the other hand, settlement lowers the possblty of surrender, thus reducng the lqudty costs, whch ncreases the proft ( (1 + cs ) * ). he proft ncreases as the latter effect s greater than the former. When S* = p D, t s obvous that the proft s not affected by settlement because the surrender value s equal to the (dscounted) death beneft payment. 1 Proposton 3. When settlement s allowed, the followng changes n welfare are observed compared wth the case wth no settlement. (1) Consumer welfare for the old generaton ncreases. he ncrease n consumer welfare s: q * (1 p )( p D S *) q f ( q ) dq (11) 1 () Socal welfare for the old generaton ncreases. (3) Both consumer and socal welfares reman the same when S* = p D. Proof. See the appendx.// Proposton 3 states that both consumer and socal welfare for the old generaton ncrease. Polcyholders wll choose to settle over to surrender facng lqudty needs when the 1 hs result comes from the assumpton that polcyholders choose to surrender rather than settle when S* = p D. If some polcyholders choose to settle, then the proft may ncrease by savng lqudty costs. 11

12 settlement prce s hgher than the surrender value. Hence, consumer welfare s mproved as much as the dfference between the settlement prce and the surrender value. n the other hand, the ncrease n consumer welfare can be nterpreted as the opton value gven to polcyholders. Wthout the settlement market, polcyholders have no choce other than surrenderng ther polces. he settlement market provdes polcyholders wth an opton to settle at a hgher prce; thus, consumer welfare s mproved as much as the opton value. Furthermore, socal welfare for the old generaton s mproved because of the costs savng. Whle the ncrease n consumer welfare s offset by the change n the nsurer s proft, the lqudty costs savng by settlement ncreases socal welfare.. he effect on the polcyholders of the young generaton Unlke n the case of the old generaton, the nsurer can reflect the possblty of settlement on determnng the surrender value and premum for the polcyholders of the young generaton. In prncple, the nsurer may set the surrender value above or below the settlement prce. However, there s no ncentve to set the surrender value hgher than the settlement prce snce the lqudty costs should be ncurred followng the surrender. Gven that the surrender value s lower than or equal to the settlement prce, the polcyholders wll prefer settlement to surrender. 13 Note that the surrender value s ndetermnate, because no polcyholder wll surrender. In ths case, the premum s equal to the case where all polcyholders receve the death beneft. We use subscrpt s to denote settlement. Followng secton III, let us frst express the premum and nsurance beneft under full dscrmnaton for polcyholder l: Q = p ( D + R ) + p (1 p ) q D + p (1 p )(1 q )( D + R ) ls 1 ls 1 ls 1 ls ls = p ( D + R ) + p (1 p ) D + p (1 p )(1 q ) R 1 ls 1 1 ls ls GB( q ) = p ( D + M ) + p (1 p ) q D + p (1 p )(1 q )( D + M ) ls 1 1 ls 1 ls = p ( D + M ) + p (1 p ) D + p (1 p )(1 q ) M ls As noted earler, the nsurer cannot dscrmnate across polcyholders lqudty rsks. he nsurer has to offer the same premum Q to all polcyholders of the young generaton. s here exsts a target lqudty rsk q s wth whch the polcyholder s net nsurance beneft equals zero. nce premum s determned, rent from the target rsk, R s, wll be determned. Now, the premum and the gross nsurance beneft of the target rsk can be expressed as: Q = p ( D + R ) + p (1 p ) D + p (1 p )(1 q ) R (1) s 1 s 1 1 s s 13 It s assumed that polcyholders of the young generaton act correspondng to the nsurer s nterest, snce the nsurer can adjust the contract offer unlke that for old generaton. he nsurer can slghtly lower surrender value than settlement prce to nduce settlement rather than surrender to save lqudty cost. hus, polcyholders opt to settle ther polces when surrender value equals settlement prce, pd. 1

13 GB( q ) = p ( D + M ) + p (1 p ) D + p (1 p )(1 q ) M (13) s s By defnton, the net nsurance beneft of the polcyholder wth q s should be zero. ( NB q ) = GB( q ) Q s s s = p ( M R ) + p (1 p )(1 q )( M R ) = (14) 1 s 1 s s From (14), we obtan that rent R s s equvalent to consumer surplus M as well. Smlar to (4), the net nsurance beneft of polcyholder l s expressed as follows. NB( q ) = GB( q ) Q ls ls s = p p 1 qls M qs Rs (1 )[(1 ) (1 ) ] = p (1 1)( p qs qls ) M (15) From (15), we have qs qls smlar wth lemma 1 snce polcyholder wth non-negatve net beneft purchases nsurance. Now, the nsurer solves the followng problem. Max s ( qs ) q s = [ Q ( p + p (1 p )) D ] F( q ) s 1 1 s = [ p + p (1 q )(1 p )] F( q ) M 1 s 1 s where Q = p ( D + R ) + p (1 p ) D + p (1 p )(1 q ) R (16) s 1 s 1 1 s s By solvng the problem, we obtan the followng proposton. Proposton 4. When settlement s allowed, the optmal target lqudty rsk q s * s determned by: [ p + p (1 q *)(1 p )] f ( q *) p (1 p ) F( q *) = (17) 1 s 1 s 1 s Proof. See the appendx. // Analogous to proposton 1 (), the target lqudty rsk s determned where margnal revenue equals margnal cost of the nsurer. he followng proposton shows the effects of settlement on the nsurance contract. Proposton 5. he allowance for settlement has the followng effects on the contract. (1) he target lqudty rsk s hgher. () he premum ncreases f q *( p D S *) p ( qs * q *) M. therwse, the premum decreases. 13

14 Proof. See the appendx. // When settlement s allowed, polcyholders sell ther polces n the settlement market, nstead of surrenderng them. hus, settlement reduces lqudty costs, whch n turn the nsurer can sell more nsurance. As a result, the target rsk wll be ncreased. n the other hand, settlement ncreases the nsurer s payment. However, the nsurer can extract consumer surplus from more polcyholders as target lqudty rsk s hgher. herefore, the premum ncreases when the ncrease n payment s hgher than the ncrease n consumer surplus extracton. Proposton 5 confrms the fndngs of the fnance lterature that securtzaton lowers nterest rate and consequently attracts more borrowers. Proposton 6. When settlement s allowed, the nsurer s proft from the young generaton ncreases. Proof. See the appendx. // In contrast to the case of the old generaton, the proft from the young generaton s always ncreased when the settlement market s ntroduced. here are two sources for the proft ncrease. Frst, settlement saves the lqudty costs. Second, the nsurer can sell more nsurance as the target rsk gets hgher. Based on propostons 4, 5, and 6, the welfare for the young generaton can be expressed as: q s * CW = GB( q *) f ( q ) dq Q * F( q *) SW s s s s q * s = p (1 p 1) M[ qs * F( qs *) q ( ) ] f q dq (18) = + CW s s s q * s 1 1 s 1 = [ p + p (1 p )] M F( q *) p (1 p ) M q f ( q ) dq (19) Comparng (8) to (18) and (9) to (19), we have proposton 7. Proposton 7. When settlement s allowed, the followng changes n welfare are observed compared wth the case wth no settlement. (1) Consumer welfare for the young generaton ncreases f the followng condton holds: q * q * F( q *) q f ( q ) dq p M q F q q f q dq ( ) * qs * * ( *) ( ) s s p D + M S hs condton holds n case of S * = p D. () Socal welfare for the young generaton always ncreases. Proof. See the appendx. // V S * p D (), 14

15 he effect of settlement on consumer welfare s unclear even f target lqudty rsk s hgher. hs s because the beneft from polcyholders whose lqudty rsk s lower than the target lqudty rsk pror to the ntroducton of settlement decreases as much as the payment dfference, whle more polcyholders can enjoy consumer surplus from nsurance whch cannot be fully extracted by the nsurer after the settlement s allowed. hat s, the nsurer may also take a beneft from settlement. As a result, proposton 7 (1) states that consumer welfare can be mproved f demand s suffcently ncreased, dependng on the populaton dstrbuton and the optmal surrender value. However, socal welfare always ncreases. hs results from the costs reducton on the one hand and from the demand ncrease on the other hand. Gven that socal welfare for each generaton ncreases, except the case that S* = p D, total socal welfare always ncreases. However, the drecton of change n total consumer welfare s unclear because consumer welfare for the young generaton can decrease, whle that of the old generaton always ncreases except the case that S * = p D. hese observatons are summarzed as follows. Corollary 1. When settlement s allowed, the followng changes n the total welfare for both generatons are observed compared wth the case wth no settlement. (1) otal consumer welfare ncreases f consumer welfare for the young generaton ncreases, or f the ncrease n consumer welfare for the old generaton s greater than the decrease n consumer welfare for the young generaton. () otal socal welfare always ncreases. In general, the change n consumer welfare for the young generaton depends on relatve szes of D, M, and V. he dstrbuton of lqudty rsk also affects consumer welfare. otal consumer welfare can be lower f the decrease n consumer welfare for the young generaton s larger than the ncrease n that of the old generaton. VI. An example hs secton provdes an example n whch both consumer and socal welfare ncrease. Suppose that the populaton dstrbuton of each generaton follows a unform dstrbuton: q ~ U [,1]. ptmal surrender value s pd by proposton 1 (1), so the premum s: Q * = p ( D + M ) + p (1 p ) q * D + p (1 p )(1 q *)( D + M ) (1) By expressons (6) and (7), we have the target lqudty rsk q * and consumer welfare for the young generaton CW wthout the settlement market as follows. p 1M + p (1 p 1) M q * = p (1 p ) M + cp (1 p ) D 1 1 () 1 CW = p (1 p 1) q * M (3) 15

16 By (1), the premum after the settlement s: Q * = p ( D + M ) + p (1 p ) q * D + p (1 p )(1 q *)( D + M ) (4) s 1 1 s 1 s he ntroducton of the settlement market changes the target lqudty rsk and consumer welfare for the young generaton as follows: q s * = p 1 + p (1 p 1) p (1 p ) 1 (5) 1 CWs = p (1 p 1) qs * M (6) By comparng () and (5), we observe that the target lqudty rsk s hgher. We also observe that the premum decreases because q *( p D S *) p ( qs * q *) M. hese results confrm Proposton 5. In addton, the condton n Proposton 7 (1) s also satsfed as follows. q * * ( *) ( ) * qs * * ( *) ( ) qs * p ( D M ) S * p M s s + q F q q f q dq q p M p M = = = 1 q F q q f q dq (7) From these observatons, we show that consumer welfare for the young generaton ncreases as settlement s allowed. Snce consumer welfare for the old generaton remans unchanged n ths example when the settlement market s ntroduced, we have the followng result. Corollary. Under the assumpton of the unformly dstrbuted populaton, the allowance for settlement ncreases not only consumer welfare for the young generaton, but also total consumer welfare. Proof. See the appendx. // n the other hand, by (5) and (15), the profts from the young generaton wthout settlement market and wth settlement market are expressed as follows. 1 = [ p + p (1 p )(1 q *)] M q * c p (1 p ) D q * 1 [ p 1 + p (1 p 1 )] M = (8) p (1 p ) M + cp (1 p ) D = [ p + p (1 p )(1 q *)] M q * s 1 1 s s 16

17 [ p + p (1 p )] M = 4 p (1 p ) (9) he proft from young generaton ncreases as follows. cd[ p 1 + p (1 p 1)] M s = 4[ p (1 p ) M + cp (1 p ) D ] 1 1 (3) In the meanwhle, the proft from the old generaton remans unchanged n ths example because the optmal surrender value equals the settlement prce. As a result, we have the followng results. Corollary 3. Under the assumpton of the unformly dstrbuted populaton, the allowance for settlement ncreases not only socal welfare for the young generaton, but also total socal welfare. In the above example, optmal surrender value s equal to the settlement prce. n the other n 1 hand, let us suppose that the populaton dstrbuton f ( q ) = nq, on the support of [,1]. hen optmal surrender value can be any value n between V and pd n case of 1 q * n n c =. hs s because that q * F( q *) = q * = (1 + c) ( ) (1 ) * n q f q dq = + c q n + 1 by proposton 1 (1). If the surrender value satsfes the condton that n 1 n 1 n 1 q * + [ p D S *] p M [ q * + q * + ], consumer welfare ncreases wth s settlement by proposton 7 (1) snce + 1 n+ 1 q * * ( *) ( ) n 1 + * = qs * n+ 1 * ( *) ( ) qs * s s q F q q f q dq q q F q q f q dq. VII. Concluson here have been debates over the effects of the ntroducton of settlement on consumer welfare. Pros argue that settlement expands the consumer s rghts, thereby enhancng welfare. Settlement may also provde nvestors wth the opportuntes to dversfy ther portfolo. n the other hand, cons argue that nsurers may react to settlement by ncreasng premums, whch eventually deterorates consumer welfare. Economc theores further suggest that settlement may work as an addtonal constrant n maxmzng the consumer s utlty, whch wll lower consumer and socal welfare. We examne the effects of securtzaton of lfe nsurance, called settlement, on the prmary nsurance market. We propose an overlappng generaton model n whch polcyholders have dfferent lqudty rsks n the monopolstc nsurance market. We assume that lqudty costs are ncurred to both polcyholders and the nsurer. ur analyss provdes a case n whch the ntroducton of settlement ncreases socal welfare whle consumer welfare may ncrease or decrease. If the settlement market operates more effcently than the nsurance market n dealng wth lqudty rsk (as s the case n our analyss), settlement ncreases socal welfare. he effect on consumer welfare s ambguous because the nsurer may extract rents greater than the ncreases n polcyholders' utltes. We 17

18 also fnd that the welfare ncrease results partly from the ncreased sales of nsurance followng settlement. hs fndng s consstent wth that of exstng securtzaton studes, ndcatng that securtzaton lowers the loan rate, thereby ncreases the demand for loan. A novel fndng s that the nsurer may also beneft from settlement, because the savng n lqudty costs, along wth the ncrease n demand, helps ncrease the proft. hs fndng s nterestng snce settlement may be complementary, not substtutve, to the nsurer, whch s n contrast to the conventonal vews on the relaton between settlement and the nsurer. Another contrbuton of the analyss s that we separate the short term and long term effects of the ntroducton of the settlement market. he short term effect s addressed through the fact that the ntroducton of settlement does not change exstng contracts wth the old generaton. Consumer welfare for the old generaton always ncreases because settlement provdes a settlement opton to the old generaton. he long term effect s addressed through the fact that the nsurer can reflect the settlement on the new contracts wth the young generaton. As a result, consumer welfare for the young generaton may ncrease or decrease. In sum, total consumer welfare for both generatons may ncrease or decrease. However, socal welfare ncreases because lqudty costs are saved and more polcyholders purchase nsurance. References Berndt, Antje, and Anurag Gupta (9), Moral Hazard and Adverse Selecton n the rgnate-to-dstrbute model of Bank Credt, Journal of Monetary Economcs 56 (5): Black, Deborah G., Kenneth D. Garbade, and Wllam L. Slber (1981), he Impact of the GNMA Pass-through Program on FHA Mortgage Costs, he Journal of Fnance 36 (): Connng Research & Consultng (8), Lfe settlements: New Challenges to Growth, Avalable at Cowley, Alex, and J. Davd Cummns (5), Securtzaton of lfe nsurance assets and labltes, Journal of Rsk and Insurance 7 (): Chloewck, V. (1985), CMs ransform Mortgage Credt Markets, Mortgage bankng 46: Daly, Glenn, Igal Hendel and Alessandro Lzzer (8), Does the Secondary Lfe Insurance Market hreaten Dynamc Insurance?, Amercan Economc Revew 98 (): Damodaran Aswath (15), Cost of Captal by sector (US), Doherty, N. A. and Snger, H. J. (3), he Benefts of a Secondary Market for Lfe Insurance Polces, Real Property, Probate and rust Journal?: Fang, Hanmng and Edward Kung (1), How Does the Lfe Settlement Affect the Prmary Lfe Insurance Market?, Natonal Bureau of Economc Research. Gatzert, Nadne (1), he Secondary Market for Lfe Insurance n the Unted Kngdom, Germany, and the Unted States: Comparson and vervew, Rsk Management and Insurance Revew 13 (): Gatzert, Nadne, Gudrun Hoermann and Hato Schmeser (8), he Impact of the Secondary Market on Lfe Insurers' Surrender Profts, Workng Papers on Rsk Management and Insurance 54, Unversty of St. Gallen. 18

19 Gorton, Gary B., and Ncholas S. Souleles (7), Specal Purpose Vehcles and Securtzaton, he Rsks of Fnancal Insttutons, Unversty of Chcago Press: Harsany, John C. (1955), Cardnal welfare, ndvdualstc ethcs, and nterpersonal comparsons of utlty, Journal of poltcal economy 63 (4): Hendel, Igal and Alessandro Lzzer (3), he Role of Commtment n Dynamc Contracts: Evdence from Lfe Insurance, Quarterly Journal of Economcs 118 (1): Hendershott, Patrc H.and James D. Shllng (1989),he Impact of the Agences on Conventonal Fxed-rate Mortgage elds, he Journal of Real Estate Fnance and Economcs (): IB lfe (13), US-Lfe Settlement Market, Hong, Jmn & Seog, S. H. (18), Lfe nsurance settlement and the monopolstc nsurance market, Insurance: Mathematcs and Economcs, Ivashna, V., Sun, Z. (11), Insttutonal Demand Pressure and the Cost of Corporate Loans, Journal of Fnancal Economcs 99: 5-5. James C. (1988), he Use of Loan Sales and Standby Letters of Credt by Commercal Banks, Journal of Monetary Economcs, (3): Keys, Benjamn J., et al. (1), Dd Securtzaton Lead to Lax Screenng? Evdence from Subprme Loans, he Quarterly Journal of Economcs 15 (1): Kolar, James W., Donald R. Fraser, and Al Anar (1998), he Effects of Securtzaton on Mortgage Market elds: A Contegraton Analyss, Real Estate Economcs, 6 (4): Man, Atf Rehman, and Amr Suf (9), he Consequences of Mortgage Credt Expanson: Evdence from the US Mortgage Default Crss, he Quarterly Journal of Economcs 14 (4): Nadauld,., Wesbach, M. (1), Dd Securtzaton Affect the Cost of Corporate Debt?, Journal of Fnancal Economcs 15 (): Naranjo, Andy, and Alden oevs (),he Effects of Purchases of Mortgages and Securtzaton by Government Sponsored Enterprses on Mortgage eld Spreads and Volatlty, he Journal of Real Estate Fnance and Economcs 5 (-3): Pas, Amela, (8), Securtzaton and Rate Settng n the UK Mortgage Market, Internatonal Revew of Fnance 8 (1 ): Passmore, Wayne, Roger Sparks, and Jame Ingpen (),GSEs, Mortgage Rates, and the Long-run Effects of Mortgage Securtzaton, he Journal of Real Estate Fnance and Economcs 5 (-3): Rothberg, James P., Frank E. Nothaft, and Stuart A. Gabrel (1989), n the Determnants of eld Spreads between Mortgage Pass-through and reasury Securtes, he Journal of Real Estate Fnance and Economcs (4): Sen, Amartya Kumar (14), Collectve choce and socal welfare, North-Holland. Seog, S. Hun and Jmn Hong (1), A Study on the Introducton of Lfe Settlement Based n Monopoly Insurance Market, he Journal of Rsk Management 3 (): role, J., (1988), he theory of ndustral organzaton, MI press. Zhu, Nan, and Danel Bauer (13), Coherent Prcng of Lfe Settlements Under Asymmetrc Informaton, Journal of Rsk and Insurance 8 (3):

20 Appendx 1. Proof of Lemma 1. Polcyholders wth lqudty rsk q k satsfyng NB( qk ) NB( q ) wll purchase nsurance. hen, the followng relaton s satsfed by (5). NB( q ) NB( q ) = (1 p )( q q ) S k max( V S,) p (1 p )( q q )( D + M ) k 1 k 1 k (1 p )( q q ) S k max( V S,) p ( D + M ) (A.1) 1 k By assumpton that S p D, we have qk q n (A.1). As a result, polcyholders whose lqudty rsk s lower than the target lqudty rsk wll buy nsurance. //. Proof of proposton 1. (1) From the expresson (6), we have the followng frst order condton for S : d ds = (1 p )[(1 + k) q F( q ) (1 + c) q f ( q ) dq ], where S V q 1 q 1 = (1 p )[ q F( q ) (1 + c) q f ( q ) dq ], where S V (A.) In (A.), the term that ( ) q q ( ) f q dq q F q q q q F( q ) q f ( q ) dq = ( q q ) f ( q ) dq s postve because. hus, for optmal target lqudty * q * (1 + k) q * F( q *) (1 + c) q f ( q ) dq s always postve. hat s, optmal cash surrender value s equal to Meanwhle, f V n case of S V. q * q * F( q *) (1 + c) q f ( q ) dq, then the optmum s pd. he nsurer can choose any value between [ V, p D ] dependng on the nsurer s strategy when q * q * F( q *) = (1 + c) q f ( q ) dq. Lastly, f surrender value s equal to V. q * q * F( q *) (1 + c) q f ( q ) dq, then the () he frst order condton wth respect to q s as follows: q, d dq = p 1 + p p 1 q M f q + p S F q cq f q p (1 p 1)( M + D ) F( q ) = (A.3) 14 // [ (1 )(1 )] ( ) (1 ) [ ( ) ( )] 14 he second order condton s satsfed, that s,

21 3. Proof of proposton. We use subscrpt s to denote settlement hereafter. he payment of nsurer s always equal to the death beneft wth settlement. herefore, f S* p D, then the nsurer s proft from the old generaton s: = Q * F( q *) p D F( q *) p (1 p ) D F( q *) (A.4) s 1 1 As a result, by the expresson (6), the change n proft becomes: = s = (1 p )[(1 + c) S * p D ] q f ( q ) dq (A.5) q * 1 Hence, f (1 + c) S* p D, then the nsurer s proft may be ncreased. In the case of S * = p D, the proft does not change. // 4. Proof of proposton 3. (1) When the settlement s allowed, then consumer welfare for the old generaton s: q * s = 1( ) ( *) + (1 1) ( ) CW p M R F q p p D q f q dq (1 p ) S * q * F( q *) 1 q * + p p 1 q F q q f q dq D + M (1 )[ * ( *) ( ) ]( ) (A.6) = (1 p )[ p D S *] q * F( q *) 1 q * + (1 p )[ q * F( q *) q f ( q ) dq ] p M 1 By comparng (8) and (A.6), we obtan the change n consumer welfare as below: CW = CWs CW q * = (1 p )( p D S *) q f ( q ) dq (A.7) () he change n socal welfare s expressed as the sum of the change n consumer welfare and nsurer s proft from the old generaton. d = [ p 1 (1 1)(1 )] '( ) (1 1) ( + p p q M f q p p M f q ) dq + (1 p ) S [ f ( q ) cf ( q ) cq f '( q )] p (1 p )( M + D ) f ( q ) 1 1 1

22 q * SW = CW + = (1 p 1) cs * q f ( q ) dq when S * p D (A.8) n the other hand, socal welfare also does not change when S* = p D // 5. Proof of proposton 4. From (15), we have (16) whch s the frst order condton wth respect to q s *. 15 // 6. Proof of proposton 5. (1) For q * and q *, let us defne hq ( ) usng (7) and (17) as follows. s h( q) = [ p + p (1 p )(1 q)] M f ( q) + (1 p ) S *[ F( q) cqf ( q)] p (1 p 1)( M + D ) F( q), where V S * D (A.9) hen, hq ( *) = and hq ( s*). As a result, q * qs *. ( h( q *) = [ p + p (1 p )(1 q *)] M f ( q *) p (1 p ) M F( q *) s 1 1 s s 1 s (1 p )( p D S *) F( q *) (1 p ) cq * f ( q *) S * ) (A.1) 1 s 1 s s () From (13), we see that M = R = Rs. Comparng the expresson () and (1), we fnd that the premum ncreases when q *( p D + p M S *) qs * p M. If the optmal surrender value S * = p D, then the premum decreases. // 7. Proof of proposton 6. Comparng (6) and (16), the change n proft from the young generaton becomes: = [ p + p (1 p )(1 q *)] M F( q *) [ p + p (1 p )(1 q *)] M F( q *) 1 1 s s 1 1 q* q* + (1 p )( p D S *)[ q * F( q *) q f ( q ) dq ] + (1 p ) cs * q f ( q ) dq where V S * p D (A.11) We already know that q * q * by proposton 5. In addton, we obtan that s 15 he second order condton s satsfed, that s, d = [ p 1 (1 1)(1 )] '( ) (1 1) ( + p p q M f q p p M f q ) dq

23 d[ p 1 + p (1 p 1)(1 q)] M F( q), where q q s * by proposton 4. Hence, dq [ p 1 + p (1 p 1)(1 qs *)] M F( qs *) [ p 1 + p (1 p 1)(1 q *)] M F( q *), and the proft from the young generaton ncreases. // 8. Proof of proposton 7. (1) We obtan (A.1) by comparng (8) and (18) for the young generaton: q * s 1 s s CW = p (1 p ) M [ q * F( q *) q f ( q ) dq ] (1 p )( p ( D + M ) S *)[ q * F( q *) q f ( q ) dq ], q * 1 where V S * p D (A.1) We have (18) by rearrangng (A.1). Partcularly, f S * = p D, followng relaton s satsfed. q * * s q s s q * * s qs (1 p ) M [ q s * f ( q ) dq ( ) ] q* q f q dq q* (A.13) CW = (1 p ) M [ q * F( q *) q f ( q ) dq q * F( q *) + q f ( q ) dq ] () We know the change n socal welfare from (9) and (19). SW = CW + q * * s q = p p 1 M q * f q q dq + p 1 cs q f q dq (1 ) (1 ) ( ) (1 ) * ( ) ] + p 1 M [ F( qs *) F( q *)] (A.14) // 9. Proof of Corollary. Snce c 1, we know q * q * F( q *) (1 + c) q f ( q ) dq 1 q* (1 + c) q* (A.15) herefore, optmal surrender value s pd by proposton 1 (1). By (3) and (6), the change n consumer welfare for the young generaton s: 1 CW = (1 p 1 )[ S * q * + q * p ( D + M ) qs * p M ] 1 = p (1 p 1)[ qs * q * ] M ( qs * q *) (A.16) // 3