Optimal life-contingent insurance under bid-ask spreads

Transcription

1 Opimal life-coningen insurance under bid-ask spreads 20 May 207 Geoffrey Kingson Deparmen of Economics, Macquarie Universiy 4 Easern Road, orh Ryde, Sydney, SW 209, usralia E: geoff.kingson@mq.edu.au T: F: bsrac re-examine opimal insurance in a Meron-ype seing when life insurance and/or life annuiies carry loads. Previous invesigaions of his quesion incorporae a parameer corresponding o he bid price of life insurance, bu overlook incorporaing a parameer corresponding o he ask price. posiive-bid-ask spread can induces a non-paricipaion inerval in he marke for life-coningen insurance. Valuaions of human capial for lifeinsurance decisions should apply hree differen discoun raes o prospecive wages, depending on wheher he wages are receivable when long, no invesed or shor in life insurance. Under logarihmic uiliy, annuiy demand is ypically more price-sensiive han life-insurance demand. Keywords: Life insurance, life annuiies, Meron porfolio model, insurance loads, human capial, discoun raes. JEL Classificaion: C5, C6, D9, G, G22

2 . nroducion re-examine opimal insurance when life insurance and/or life annuiies carry loads. Richard (975) and Pliska and Ye (2007) are he wo classic conribuions o his line of research. They exend he Meron porfolio model of an invesor s life cycle by exploiing he insigh ha a long posiion in life insurance is effecively a shor posiion in life annuiies, and vice versa. Using a racable insananeous-erm specificaion of insurance conracs, hey specify a single parameer o model he fac ha life-coningen insurance conracs ordinarily carry posiive loads. This parameer is essenially he bid price for life insurance. Bu models of markes wih bid-ask spreads also need o specify an ask price. Omiing a separae ask price has ushered in a concepual glich whereby assuming posiive loads on life insurance enails assuming life annuiies wih money s-worh raios in excess of 00 per cen. Tha is, loads on life annuiies have implicily been assumed negaive. Bid-ask spreads in he marke for life-coningen insurance have effecively been held a zero. nroducing a second parameer recifies his glich and leads o a number of new resuls. Formal noaion sharpens he poin. Le () 0denoe he force of moraliy, and le () () denoe he premium-insurance raio (Pliska and Ye 2007). 2 Life-insurance premiums paid a he rae p() secures for he upcoming insan a sum insured equal o The money s-worh raio for convenional life annuiies is he expeced presen value of annuiy income expressed as a fracion of is cos. s counerpar for insananeous-erm annuiies is annuiy income expressed as a fracion of he expeced paymen from he annuian s esae o he insurance company during he upcoming insan. The money s-worh meric exends readily o life insurance. The raio is ordinarily less han one, owing o loads and commissions, induced in par by he adverse-selecion problem, which explains he pauciy in he real world of insananeous-erm life annuiies. 2 The noaion in Richard (975) corresponding o () is (). 2

3 p()/ (). Life insurance carries a posiive load whenever he above inequaliy is sric. So far so good. However, Richard (975), Pliska and Ye (2007) and numerous oher conribuions make his premium-insurance raio do double duy for he invesor s annuiy phase as well as her life-insurance phase. They assume ha annuiy income a he rae p() enails a sum p()/ () payable by he invesor s esae o he annuiy provider if she dies during he upcoming insan. Bu if he inequaliy () () is sric, hen his annuiy is cheaper han an acuarially-fair one, rendering i unviable for he annuiy provider. For his reason inroduce a second premium-insurance raio 0 ( ) ( ) for life annuiies. f he invesor dies a ime, her esae pays he provider a sum p()/ (). Life-coningen insurance loads say nonnegaive if and when he invesor goes long in life annuiies. The bid-ask spread () () can be posiive hroughou he invesor s life. This second premium-insurance raio ges elescoped forward ino he demand for life insurance, which is forward-looking. ccordingly, opimal life-insurance rules here are more complicaed han ones in he previous lieraure. posiive bid-ask spread can induce an inerval during which he invesor sis on he sidelines of he marke for life-coningen insurance. The inuiion for his non-paricipaion inerval is simply ha self-insurance can be a beer deal when bid-ask spreads are posiive, a leas for a ime. By conras, he previous lieraure implies ha opimal paricipaion in he marke for life-coningen insurance is coninuous (i.e. lifelong) even when marke for life insurance is acuarially unfair. 3 3 See e.g. Richard (975, Eq. (43)), Pliska and Ye (2007, Eq. (33)) and Figures 5 and 6 of Pliska and Ye (2007). 3

4 noher novely is ha valuaions of human capial for making decisions abou life insurance should apply up o hree differen discoun raes o prospecive wages. Richard (975), Pliska and Ye (2007) and Ye (2008) argue ha human capial should be valued by means of he single discoun rae r () (), where r () is he shor-erm ineres rae. argue insead ha he required discoun raes are r () (), r () and r () () depending on wheher he wages are receivable when he invesor is long, no invesed or shor in life insurance.. This complicaes he so-called replacemen mehod of needs analysis whereby life-cover requiremens are assessed primarily by reference o proecing a household from premaure loss of human capial. My penulimae secion specializes he uiliy funcion from consan relaive risk aversion o logarihmic uiliy. This gives rise o wo simple demand funcions, one for life insurance and he oher for life annuiies. They are no only linear in wealh, as in he more general case of consan relaive risk aversion (an implicaion of CRR poined ou by Richard 975), bu in relaive prices as well. 4 They sugges ha he demand for life annuiies will ypically be more price-sensiive han he demand for life insurance, even hough he coefficien on price is he produc of he force of moraliy and financial wealh in boh cases. The reason for his comparaive price sensiiviy of annuiies is ha in laer life he force of moraliy ypically rises faser han financial wealh declines. So a given across-he-board deparure from acuarial fairness by life-coningen insurance insrumens will ypically induce a bigger fall in he demand for life annuiies han life insurance. Moreover, a long nonparicipaion inerval is likely o increase his dispariy, as he produc of he force of moraliy 4 These relaive prices are he money s-worh raios of insananeous-erm life annuiies and life insurance see Secion 7. 4

5 and financial wealh is likely o grow during ha inerval. These observaions help o explain he annuiy puzzle whereby annuiy markes end o be shallow Model Following Richard (975), Pliska and Ye (2007) among ohers, assume a fixed planning horizon T, which can be inerpreed as he maximum possible span over which an invesor could conceivably consume and earn a wage. 6 n invesor alive a ime has a random lifespan, 0, wih probabiliy densiy funcion f (). The invesor s wage is a general deerminisic process y () 0, 0 T. To highligh he implicaions of inroducing an ask price for life insurance, absrac from risky asses, in line wih Pliska and Ye (2007), bu deparing from Richard (975), Ye (2008) and many ohers. The invesor s wealh consrain is dx () rxd () () cd () pd ( ) y( d ) () where he new variables are wealh X (), consumpion c (), and wage y(). niial wealh is x. The relevan inerval of ime for () is min[, T]. The invesor s legacy when she dies 0 a ime = is 5 See Peijnenburg e al. (206) for a Meron-syle analysis of he annuiy puzzle. These auhors find ha if life annuiies are acuarially fair, hen he puzzle persiss under a variey of circumsances ha have previously been regarded as lessening he puzzle. 6 Pliska and Ye argue persuasively ha heir reamen of he erminal dae improves on ha of Richard (975). 5

6 p() Z () X () () X() p() X() according as wheher p() is posiive, zero or negaive, (2) () i.e., according as wheher she is long, no invesed or shor in life insurance. Concerning preferences, all secions here bu he penulimae one assume CRR, following par of Richard (975) and mos of Pliska and Ye (2007). Maximum expeced uiliy over he se of admissible consumpion and insurance policies is T C () Z () XT ( ) V( x) sup E0, x[ e de { T} e { T} (3) cp, 0 where he new variables are he expecaions operaor E, uiliy-funcion curvaure parameer <, and rae of ime preference > 0. Following he bulk of he previous lieraure, solve for opimal policies by dynamic programming. By a sandard argumen, he associaed remainder-of-uiliy funcion can be resaed as one wih a non-random erminal ime T: T u Zu ( ) u Cu ( ) XT ( ) J(, xc ;, p) [ f(,) ue Fue (,) ] du FTe (,), (4) where f ( u, ) is he probabiliy densiy for deah a ime u condiional upon surviving a ime s, and F( u, ) is he probabiliy a ime u for survival a ime u condiional upon survival a ime s. These probabiliies are relaed o () by 6

7 f( s) s f ( s, ) ( s)exp{ ( udu ) } F () F( s) s F( s, ) exp{ ( u) du}. F () (5) n he following hree secions and an appendix solve his model piecewise. 3. nnuiizaion inerval Consider he annuiizaion inerval T, where is he ime when he invesor firs goes long in life annuiies. Define c () () p (, xcp ;, ) [( rx () y () c pv ) x (, x) e e ( x ). (6) () Equaion (6) differs from corresponding expressions in he previous lieraure only by using () raher han η() o price he payou from he invesor s posiion p() in life-coningen insurance. Thus, he soluion o he pos-annuiizaion problem can be obained simply by subsiuing () for η() in a number of he expressions derived by Pliska and Ye. n paricular, he associaed HJB equaion and erminal condiion are ( c, p) V (, x) () V (, x) sup (, x; c, p) 0 V e X( T) ( T, x).. (7) The firs-order condiions are * * * c(, xc ;, p) 0 Vx (, x) e ( c) (8) 7

8 and * * * () p p(, xc ;, p) 0 Vx (, x) e ( x ). (9) () () From (8) and (9), * c () ( ) (0) V e x and x * p () () ( () () ). () V e x Equaions (7) o () give x x V () V [(() r ()) x y()] V e K ()( V ) 0 e V ( T, x) x (2) where K () (). (3) () Following cosmeic changes o expressions obained by Pliska and Ye, he soluion o (2) is seen o be e a () V (, x) ( x b ()) (4) where 8

9 T b ( ) y( s)exp{ {[ r( v) ( v)] dv} ds s, (5) T T s ( ) [exp{ ( ) } exp{ ( ) } ( ) ], (6) a H v dv H v dv K s ds and () H () (() r ()). (7) From (0), (), and (4)-(7), and again following rivial modificaions o expressions obained by Pliska and Ye (or Richard for ha maer), he opimal consumpion and annuiy demands are c () ( x b ()) (8) [ a ( )] and p () () Z () x ( ) ( x b ()). (9) () () [ a ( )] From (9) he life-annuiy rule is ] x ( () () b () p () (){[( ) ) } (20) () () [ a ( )] [ a ( )] where p () is (he negaive of) annuiy income. Equaion (20) differs from is counerpars (43) in Richard (975) and (33) in Pliska and Ye (2007) by incorporaing he ask price of life insurance raher han he bid price. can be simplified wih he naural counerpar here of he expression D() in Lemma 3 of Pliska and Ye, namely 9

10 () D () ( ). (2) () [ a ( )] n conras o he expression D() in Pliska and Ye, however, D ( ) is no necessarily less han one. Equaions (20) and (2) help deermine he greaes lower bound of he annuiizaion inerval. The earlies ime when p ( ) = 0 enails x D ( ) b ( ). D ( ) (22) Remaining human capial a any poin wihin he annuiizaion inerval is defined by (5). The implied discoun rae for ha inerval is r () (). ccordingly, wages receivable during ha inerval are discouned more lighly here han in he previous lieraure, unless annuiies are acuarially fair. 4. on-paricipaion inerval Consider nex he non-paricipaion inerval, where is he ime when he invesor firs chooses no o paricipae in he marke for life insurance. The equaions characerizing he value funcion V and an envelope condiion: wihin ha inerval consis of he following parial differenial equaion 0

11 () [() ()] x ( x ) 0, V V r x y V e V a ( ) V (, x) e ( xb ( )), (23) where s (24) b ( ) y( s)exp{ r( v) dv} ds b ( )[ { r( s)exp{ r( v) dv} ds], s s ( ) ( )[exp{ ( ) } exp{ ( ) } ], a a H v dv H v dv ds (25) and () r() H (). (26) Equaions (23)-(26) sugges a candidae soluion for he value funcion wihin he nonparicipaion inerval: e a () V (, x) ( x b ( )). (27) Opimal consumpion is c () ( x b () ) (28) [ a ( )] and he beques process reduces o Z () x. (29)

12 Equaion (25) shows ha human capial wihin he non-paricipaion inerval is given by he presen value of remaining wages receivable during ha inerval, plus human capial a he ouse of he annuiizaion inerval brough back o a presen value. Wages are discouned by r(), i.e., more lighly han in he annuiizaion inerval. 5. Life insurance inerval Consider finally he life-insurance inerval 0. The equaions describing i again consis of a parial differenial equaion and an envelope condiion: x x V () V [(() r ()) x y()] V e K ()( V ) 0, a ( ) V (, x) e ( x b () ), (30) where b () y()exp{ s [( r v) ( v)] dv} ds s b ( )[ [() r s ()]exp{ s [() r v ()] v dv} ds], s (3) s ( ) ( )[exp{ ( ) } exp{ ( ) } () s ], (32) a a H v dv H v dv K ds () (() r ()) H (), (33) 2

13 and () K (). (34) () Equaions (3)-(34) suggess a candidae soluion for he value funcion wihin he life insurance inerval: e a () V (, x) ( x b ()). (35) Opimal consumpion is x b () c () (36) [ a ( )] and he beques process is p () () x b () Z () x ( ). (37) () () [ a ( )] From (37) he life-insurance rule is x () () b () p () (){[( ) ] ( ) } (38) () () [ a ( )] [ a ( )] where p () is life-insurance premiums. Requiring ha annuiy loads be non-negaive feeds forward on o he demand for life insurance via boh he relevan a and b funcions. Lemma 3 of Pliska and Ye carries over o he life-insurance inerval here: 3

14 () D () ( ). (39) () [ a ( )] lso consisen wih previous conribuions a his poin, (3) shows ha wages receivable during he life-insurance inerval are discouned by r () (), i.e., by he ineres rae plus he bid price for life insurance. Equaions (38) and (39) help deermine he leas upper bound of he life insurance inerval. The earlies ime when p ( ) = 0 enails x D ( ) b ( ). D ( ) (40) 6. on-paricipaion and posiive bid-ask spreads se of condiions sufficien for a posiive bid-ask spread o induce non-paricipaion under CRR can be summarized as follows: Proposiion. Given posiive paricipaion inervals for boh life insurance and life annuiies, posiive financial wealh a he end of he life-insurance inerval, and a coefficien of relaive risk aversion less han or equal o one, a posiive bid-ask spread in he marke for lifeconingen insurance insrumens induces a posiive non-paricipaion inerval. 4

15 Proof. The proof is by conradicion. Suppose insead ha he non-paricipaion inerval is of zero lengh, i.e.. By coninuiy, ( ) ( ), x ( ) x ( ), a ( ) a( ), and b ( ) b ( ). Hence, recalling (20) and (38), p ( ) p ( ) ( ) ( ) ( ) ( ) ( ( )) ( ( ) ( )) ( ) ( ) ( ) x b x ( a ( )) (4) 0, conrary o a defining characerisic of boh p ( ) and p ( ), namely, zero values for boh funcions evaluaed a hese imes. coefficien of relaive risk aversion greaer han one, i.e. 0, implies ha he expression wihin square brackes has a negaive sign, leading o an ambiguous sign for he expression (4) as a whole. coefficien of relaive risk aversion less or equal o one evidenly reduces ambiguiy. The reason is ha low risk aversion encourages vigorous subsiuion away from life-coningen insurance by an invesor facing a posiive bid-ask spread. Special cases of HR uiliy oher han CRR, noably generalized logarihmic uiliy, enable high risk aversion o co-exis wih a vigorous subsiuion response. 7. Logarihmic uiliy 5

16 Logarihmic uiliy, i.e. 0, simplifies opimal insurance. The life-annuiy rule (20) reduces o ( ) b ( ) p () () ( ) x(), T, a () () a () (42) and he life-insurance rule (38) reduces o ( ) b ( ) p () () ( ) x(), 0, a () () a () (43) where ()/ () is he money s-worh raio for insananeous-erm life annuiies and ( ( )/ ( )) is he raio for insananeous erm life insurance. Under logarihmic uiliy, hen, he wo rules are linear in boh wealh and relaive prices. The coefficien on relaive prices, i.e. () x (), is qualiaively idenical across hem. However, i is likely ha he force of moraliy rises faser han financial wealh declines, implying ha he price sensiiviy of life annuiies is likely o be higher. The longer he non-paricipaion inerval, he greaer his difference in magniudes is likely o be. ccordingly, loads are likely o impac more on he demand for life annuiies han he demand for life insurance. 8. Concluding commens solved a simple version of he Meron porfolio model exended o insurance quesions, having firs recified a longsanding misspecificaion of he load on life annuiies. One new finding was ha valuaions of human capial for making decisions abou life insurance should apply 6

17 up o hree differen discoun raes o prospecive wages, depending on wheher he wages are receivable when he invesor is long, no invesed or shor in life insurance. There is ample scope for furher research. One possibiliy is richer specificaions of he invesor s opporuniy se. The wage income process could be modelled wih a realisic ageearnings profile, i.e., a hump shape followed by a posiive reiremen inerval, along wih labor supply flexibiliy, and labor-income risks linked o he sockmarke (Hugge e al. 206). Risky financial asses are readily incorporaed ino he kind of model invesigaed here (Richard 975, Milevsky 2008, Ye 2008). noher possibiliy is richer specificaions of preferences. Huang and Milevsky (2008) and Ye (2008) employ HR preferences. Huang and Milevsky make a sar on inegraing he Meron model wih radiional needs analysis of insurance requiremens. They incorporae a change in household preferences following he deah of he household s breadwinner. n such ways, ime-varying shif parameers in he household s uiliy funcion can capure changes wihin he household. relaed exension builds on he observaion ha luxury bequess end o characerize he preferences of affluen elderly households. posiive shif parameer in he household s beques-uiliy funcion capures his preference (Carroll 2002). n conjuncion wih annuiy loads, luxury bequess may help o explain he annuiy puzzle (Lockwood 202). 7 ppendix 7 For a conrary view, see Peijnenburg e al. (206). Luxury bequess also help explain he surprising appeie for risky porfolios on he par of affluen elderly invesors (Carroll 2002, Ding e al. 204). 7

18 This appendix solves he hree HJB equaions (2), (23) and (30). verifies he corresponding rial soluions by plugging hem ino he HJB equaions, hereby obaining ordinary differenial equaions of he Bernoulli variey. The sae variables used here are oal (raher han financial) wealh variables, as his approach urns ou o simpler.. nnuiizaion inerval Begin wih he oal differenial of oal wealh w ( ) x b ( ) : dw dx db ( rx y c p) d[ y ( r) b ] d [ rw c ( p b )] d ( rw c q ) d (.) where q p b is life annuiy purchases ne of moraliy-securiy income impued o human capial. Define he value funcion when oal wealh is he sae variable as J ( w, ). The associaed HJB equaion is * * * e * q b 0 J J ( rw c q ) J w ( c ) e ( w b ) (.2) * * * e * q J J ( rw c q ) J w ( c ) e ( w ) where * c and * q sand for opimal consumpion and adjused beques, i.e. c * ( ) (.3) J we and 8

19 q (4) * {[ ( )] w }. J we Plug hese expressions ino (.2) and hen rial he candidae soluion e a J ( w, ) ( w ) o ge he required ordinary differenial equaion: e J ( r ) w Jw Jw 0 J ( ) [( ) ]( ) d ln a ( ) ( r) ( ) K ( a ). d (5) For an explanaion of how o solve equaions of his ype, see e.g. Ye (2008)..2 on-paricipaion inerval The oal differenial of oal wealh w () x b () in his inerval is dw dx db ( rx y c) d ( y rb ( rw c) d, ) d (.6) and he HJB equaion associaed wih he value funcion J ( w, ) is e * * 0 J J ( rw c ) J w ( c ) (.7) where 9

20 c * ( ). J w e (.8) e a Plug (.8) ino (.7) and hen rial J ( w, ) ( w ) : rw J w J w 0 J J ( ) e ( ) d ln a ( ) r ( )( a ). d (9).3 Life insurance inerval The oal differenial of oal wealh w () x b () in his inerval is dw dx db ( rx y c p) d [ y ( r ) b ] d ( rw c q ) d (.0) where is life insurance purchases ne of moraliy-securiy income impued o q p b human capial. The HJB equaion associaed wih he value funcion J ( w, ) is e q 0 J J ( c ) e ( w ) * * * * ( rw c q ) J w (.) where c * ( ) (.2) J we 20

21 and q (.3) * {[ ( )] w }. J we e a Plug (.2) and (.3) ino (.) and hen rial J ( w, ) ( w ) : e J ( r) w Jw Jw 0 J ( ) [( ) ]( ) d ln a ( ) ( r) ( ) K ( a ). d (.4) References Carroll, C. D., Porfolios of he rich. n: Guiso, J., Haliassos, M., Jappelli, T. (Eds), Household Porfolios: Theory and Evidence. MT Press, Cambridge, M. Ding, J., Kingson, G., Purcal, S., 204. Dynamic asse allocaion when bequess are luxury goods. Journal of Economic Dynamics and Conrol 38, Huang, H., Milevsky, M., Porfolio choice and moraliy-coningen claims: The general HR case. Journal of Banking & Finance 32, Hugge, M., Kaplan, G., 206. How large is he sock componen of human capial? Review of Economic Dynamics Lockwood, L.M., 202. Beques moives and he annuiy puzzle. Review of Economic Dynamics 5,

22 Peijnenburg, K., ijman, T., Werker, B., 206.The annuiy puzzle remains a puzzle. Journal of Economic Dynamics and Conrol 70, Pliska, S.R., Ye, J., Opimal life insurance purchase and consumpion/invesmen under uncerain lifeime. Journal of Banking and Finance 3, Richard, S.F., 975. Opimal consumpion, porfolio and life insurance rules for an uncerain lived individual in a coninuous ime model. Journal of Financial Economics 2, Ye, J., Opimal life insurance, consumpion and porfolio: dynamic programming approach. merican Conrol Conference. 22