HDGING MHODOLOGI IN QUIY-LINKD LIF INURANC Aleander Melnkov Unversy of Alera dmonon e-mal: melnkov@ualera.ca. Formulaon of he Prolem and Inroducory Remarks. he conracs we are gong o sudy have wo yes of uncerany: Uncerany n he framework of gven fnancal marke marke rsk Moraly of nsured. hese yes of uncerany are weakly correlaed and n our seng we shall model hem elong wo dfferen roaly saces: Ω and ~ ~ F P ~ F P Ω.
Comnng oh models we ge he roduc sace ~ ~ Ω Ω ~ F F P P. Fnancal Marke s reresened y a ar of asses non-rsky B and rsky ha are denfed y her rces B and Ω F P. as sochasc rocesses on Assumng for smlcy B we descre a flraon F F {... } as he avalale nformaon aou rces a me. o gve a full descron of he marke we should nroduce a varey of admssle oeraons wh he asc asses B and. Defne a -dmensonal sochasc rocess β γ π usually redcale adaed o F as a orfolo or a sraegy wh he value caal X π β B γ.
If π X follows he equaon dx hen we call π self-fnancng. π β db γ d Only orfolos wh non-negave caal are admssle n he framework of he marke. Any fnancal conrac s denfed wh s oenal laly ayoff H eercsed a he end of a conrac erod [ ]. In fac H s a F -measurale random varale. Any such nonnegave random varale H s called a conngen clam. In fnancal economcs a conngen clam s called a dervave secury of he uroean ye. Remark: A larger class of dervave secures he Amercan ye s no consdered here. he man rolem s o fnd he curren rce of a gven conngen clam durng a conrac erod [ ] n order o manage he rsk n he framework of he conrac. 3
An arorae soluon s found y hedgng he conngen clams: o consruc an admssle sraegy π so ha π X s close enough o H n some roalsc sense. Hedgng gves us a ossly o deermne he curren rce π X C a me C of he hedgng orfolo π. In arcular as he curren caal π reresens he nal rce of he clam. C X 4
yes of hedgng. Perfec hedgng: { } P X. Mean-varance hedgng: π H. π X H s mnmal. 3. ffcen hedgng: { } π l H s mnmal X where l s a loss funcon. In arcular for quanle hedgng we have l I { } π X H P and s mamal. 5
How o calculae he rce C? o answer hs queson we use he marngale characerzaon of arrage and comleeness of a fnancal marke n erms of esence and unqueness of rsk-neural or marngale measures gven y he frs and he second fundamenal heorems n fnancal mahemacs. o we reduce he mehod of rsk-neural valuaon of he conngen clam H o fndng he rce C H F C as where s an eecaon w.r. o a sngle marngale measure P comlee marke or C ncomlee marke. su H P F 6
Lfe nsurance conracs ased on rsky asses he source of randomness n fnancal conracs s he evoluon of sock rces. he source of randomness n lfe nsurance s he moraly of clens. Denoe a random varale reresenng he fuure lfeme of a clen of age. s gven on ~ ~ F P ~ Ω. An nsurance comany can ssue a med conrac for he erod [ ] where he ayoff funcon s a funcon of sock rces... and. hs ye of conrac s called equy-lnked lfe nsurance conrac. We wll consder only ure endowmen conracs wh he followng srucure of ayoffs: H H I >. { } 7
Remarks: radonal lfe nsurance deals wh he deermnsc ayoff he conrac s equal o { } H K cons. he rce of K where P ~ > s a survval roaly of he nsured of age. Pure endowmen wh a fed guaranee: { K} I { } H ma > where s a rsky asse K s a guaranee. 3 Pure endowmen wh a flele guaranee: where { } I { } H ma > s he rsky asse also a rsky asse guaranee. s he flele may e Assume ha n follows geomerc Brownan Moon Black-choles model: e µ W where W s a Wener rocess µ s a rae of reurn on sock and s he volaly. 8
A Bref Hsory We noe he aers y Brennan and chwarz 976979 Boyle and chwarz 977 whch recognzed a close connecon of equy-lnked lfe nsurance wh he oon rcng heory Black choles and Meron 973. hey found ha he ayoff from equy-lnked lfe nsurance conrac a eraon s dencal o he ayoff from a uroean call oon lus some guaraneed amoun. Hence he aearance of Black-choles formula n such rcng s que naural n he consrucon of he nal rce and he value of he orfolo: d U [ Φ d KΦ d ] K Φ d KΦ d ln ± K [ ] ± Φ π e y dy. 9
Furher develomens: Delaen 986 Bacnello and Oru 993 Aase and Persson 994. he moran se was done y Moeller 998 who aled he mean-varance hedgng echnque. He oaned he followng resul: U l [ Φ d KΦ] l d and γ l N F gve he nal rce and omal n he meanvarance sense hedgng sraegy for he conrac of ure endowmen wh a guaranee K for a grou of clens of age where l s he sze of he grou of he nsured wh he remanng lfe mes... l N l ~ I { } P > F sasfes he Black-choles fundamenal equaon r.
F wh he oundary condon F ma{ K}. F Remark: Our man focus s on calculang he remum of sze l we ge U for a sngle conrac. nce for a grou Also for every fed U l l U. we can reea he logcal ses for he nerval [ ] and fnd he corresondng remums as a roduc. l N U
.Condoned Conngen Clams n a semmarngale seng. Assume ha he rsky asse s a semmarngale F on F F P Ω. A self-fnancng admssle sraegy π β γ π has a caal X X γ udu. π M P s a se of marngale measures of hs marke. Frsly we assume ha M P { P }. Consder a nonnegave conngen clam H whch s F -measurale random varale wh H <. Accordng o he rsk-neural valuaon mehodology H H C H H F C H γ u d C H H β γ H π s a relcang orfolo erfec hedge. H u
Consder a nonnegave random varale τ defned ~ Ω ~ F P ~ and deermne on anoher roaly sace a condoned conngen clam H τ H I{ τ > } ~ ~ Ω Ω ~ F F P P. on he roduc sace o rce H τ we consder he value ~ ~ C τ H τ H I > C H { τ } < C H as a ound for he nal rce of H τ and defne he se of successful hedgng π π π H A X π { ω : X X }. If H π π hen P A C H π. We should consder a resrced se of sraeges wh he nal π caal X X C H. hus we canno rovde he aove equaly. 3
o rce a conngen clam H τ we should consder he followng ereme rolem: fnd an admssle sraegy π such ha π P X C τ H ma P A X π π π π H under he resrcon X X C τ < C. hs s eacly he rolem of quanle hedgng Follmer and Leuker 999. Accordng o hs heory he omal sraegy β γ π s a erfec hedge for he clam H H I where he se A F has a mamal P-roaly wh HI A X. hs se s called a mamal successful hedgng se. Remarks: In case of an ncomlee marke he ound for he nal caal s gven y C τ su H C H P M P A A. 4
5 3. Prcng of conracs wh a fed guaranee. Consder Black-choles model for he rsky asse W µ e or dw d d µ wh he marngale measure W dp dp Z P e : µ µ and W W µ he new Wener rocess wh resec o P. A ure endowmen conrac wh a fed guaranee s denfed wh he followng conngen clam: { } { } { } { }. ma I K I K I K I H H > > > > he nal rce U for hs conrac can e calculaed as
6. Φ Φ d K d K K K U C We can consder he quany K K U C as he ound of he nal caal avalale for a call oon. Alyng he mehodology of quanle hedgng we ge A I K K or he followng convenen form for furher acuaral analyss of he conrac:. K I K A he mamal successful hedgng se A has a secal srucure: { }. > > K a K a Z A µ
7 here are wo cases when he equaon K a µ has one or wo soluons. Case µ s reduced o { } { } W c A and. Φ Φ Φ Φ K d K d I K A Fnally. Φ Φ Φ Φ d K d K A consan or c can e found from gven roaly of successful hedgng: an nsurance comany can e agreed wh a ceran rsk level ε such ha A P ε. Due o he srucure of A we can fnd
and herefore P A Φ µ Φ µ ε. Illusrave Numercal amle Le s f a rsk level ε. and secfy oher arameers of he model and he conrac: µ.8;.3; ; K ; 3 5 years. We can fnd ha.9395; 3.9486; 5.9556. Usng Lfe ales Bowers e al 997 we can reconsruc he arorae age of he nsured: 78; 6; 53 years. Black-choles rces : 8.4; 6.876;.849. Quanle rces C ε : for ε. are 7.57; 6.3;.849 5-7% lower; for ε. 3 are 6.653; 4.54;.33-8% lower. 8
Remark. Consder cumulave clams l K where l s he numer of nsureds a me from he grou of sze l. he ermnal caal of a quanle hedge π π ε of he rsk level ε sasfes o π P X K ε. he mamal se of successful hedgng s nvaran w.r.o mullcaon y a osve consan δ. Hence π π ε reresens a quanle hedge for he clam δ K wh he nal rce δc ε. ake δ n α l where n α s defned from he equaly P nα l α. Parameer α characerzes he level of moraly rsk and hs roaly s calculaed wh he hel of Bnomal Dsruon wh arameer. Indeendence l and he marke mles 9
P l X π π α l K P X K P nα l l ε α ε α. n o wh he hel of sraegy rce C n π π ε and he nal α ε α Cε one can hedge he gven l cumulave clam wh he roaly ε α. For rsk levels ε. 3 α. 3 5 and l we have: n 8993 94 and C ε α 5.9;3.498;8.83. α herefore under he rsk level a 5% he nal conrac rce can e reduced y 8-8%. Case µ > leads o wo consans c < c or <. he srucure of a successful se wll e as follows A { } { } W W U > and we can do he same as n he revous case.
4. Conracs wh flele guaranees. he conrac under consderaon here has he srucure { } I { } ma H > where follows he equaon d µ d dw. he frs asse s suosed o e more rsky han he second one. Hence we assume ha > and lays he role of he flele guaranee. he marke modelled n hs case can e denfed wh a Black-choles model for he asse ecause can e eressed as a ower funcon of. Hence we can use a sandard marngale measure P wh he densy
. e W dp dp Z µ µ We can reduce he rolem o he equaly. A I Assume ha. and << < he srucure of he successful hedgng se A s { } Y Y a Z a Z > >. We reresen as cons Y Z α and fnd he characersc equaon defnng A as a cons α where. α herefore we can relace he equaon aove y
3 a cons and fnd s unque soluon. > c o we arrve o he equaly { } ~ ~ ~ ~ ~ ~ c c I c Y Φ Φ Φ Φ where. ln e ~ µ µ ± ± Boh he numeraor and he denomnaor are varans of Margrae s formula. We can fnd c from he gven level of rsk ε :. ln ε µ ε ε ε a c a c P Y A P a Φ.
5. Numercal amle. Consder he fnancal ndces he Russell RU-I and he Dow Jones Indusral Average DJIA as rsky asses and. We esmae µ and µ for hese ndces emrcally usng daly oservaons of rces from Augus 997 o July 3 3: µ µ.48.47.3.89. he nal rces of hese ndces are 44. and 894.4. herefore we use 894.4 44. as he value of he frs asse o make nal values of oh asses he same 894. 4. Ulzng he formulae for c and wh 3 5 and ε..5. 5 we ge he 4
corresondng survval roales for he conrac wh flele guaranee ale. We also do he same for he conrac wh fed guaranee K. ale. Usng Lfe ales Bowers e al 997 we defne he corresondng age of he nsured for hese conracs ales 3 and 4. Whenever he rsk ha he comany wll fal o hedge successfully ncreases he recommended ages rse as well. hs means ha he nsurance comany should comensae y choosng safer older clens. We also oserved ha wh longer conrac maures he comany can wden s audence o younger clens. In oh cases fed and flele guaranees quanle rces for eamle ε. 5 are reduced y 7-8% and 9-%. If he comned α ε. 5 rsk s 5% he corresondng rce reducon wll e - 8%. 5
urvval Proales ale ε. ε. 5 ε. 5.9447.8774.78 3.95.89.84 5.9549.8989.874.965.98.8378 ale ε. ε. 5 ε. 5.9733.936.8585 3.97.947.85 5.976.966.8553.973.933.8679 6
Age of he Insured ale 3 ε. ε. 5 ε. 5 78 87 94 3 6 7 79 5 53 63 7 4 5 58 ale 4 ε. ε. 5 ε. 5 68 8 88 3 55 67 76 5 48 59 68 36 47 56 7
6. Furher Develomens. We resen here how our aroach o he rcng of equy-lnked lfe nsurance conracs can e eended o oher yes of effcen hedgng and oher models of fnancal marke. 6.. ffcen hedgng wh ower loss funcon. l >. Model wh zero neres rae: d µ d dw. he omal sraegy π for a gven c.c. H s defned π π from l H X l H nf X π 8
Where he nf s aken over all self-fnancng sraeges wh nonnegave values sasfyng he udge resrcon X π X < H. For he med conrac wh H I H > { } he ound X H ma. { } he effcen hedge π for hs rolem ess and concdes wh a erfec hedge for a modfed c.c. H wh he srucure H H H H a Z { Z > a H } { Z > a } H for > H I for < < H I for where Z s he densy of he unque marngale measure P consan H X. a s deermned from 9
We reduce he rolem o effcen hedgng of he and fnd he followng key relaon oon >. We gve he analyss of hs equaly. For eamle n µ he case where < < and consderaons lead us o he formula: such ~ ~ ~ C Φ C ~ ~ ~ Φ Φ Φ where consan C s he unque soluon of he characersc equaon wh y α cons y y µ µ α. 3
d 6.. Jum-Dffuson Model. µ d dw ν dπ > ν <. Here W s a Wener rocess and Π s a Posson rocess wh nensy λ >. In he framework of hs model we can fnd he unque marngale measure P wh he densy Z e α W α λ λ Π λ ln λ where α µ ν µ ν ν ν λ. µ µ ν ν We can consder he same rcng rolems for he gven model. For eamle quanle mehodology leads o he key relaon 3
I A where A s he mamal se of successful hedgng. for In case α we have he followng eresson for n erms of Margrae s formula averaged y Posson dsruon: where n [ ~ ] ~ ~ ~ ~ ~ Φ C Φ C n n [ ~ ] ~ ~ ~ ~ ~ Φ Φ n n n n n n n n n n n ~ n n e λ λ n! n ν eν λ n and C s he unque soluon of he equaon α y cons y y. 3
6.3. Quanle hedgng n wo-facors model generaed y correlaed Wener rocesses. d µ d dw W W ρ. > > cov hs marke s comlee and he unque marngale measure P has a densy Z ϕ e ϕw ρµ µ ϕ ϕ ρϕ ϕ ϕ µ. ρ ρ ρµ Boh asses and are marngales w.r. o P. long hs measure we reroduce he same quanle echnque for rcng as n econ 4. { } ma I H > { } 33
Frsly we fnd he nal rce of hs conrac as U reducng he rolem o quanle hedgng of. econdly he quanle hedge π s a erfec hedge for he modfed c.c. I A where A s he mamal se of successful hedgng. for hese consderaons agan lead o he key relaon I A. he se A can e reresened as A Y ϕ ϕ wh α. α cons Y 34
We agan nroduce he characersc equaon y α y cons and fnd ha for α hs equaon has he unque soluon C and for α > here are wo soluons C < C. We consder only he frs case and fnd ha { } Y C A Usng log-normaly of Y and Margrae s formula for where we oan. C Φ C Φ Φ Φ ± C ln C ± ρ. 35
36 Alyng hs o he rcng of remum U we arrve o he followng equaon [ ]. C C U Φ Φ Φ Φ Φ Φ Hedgng sraegy π wh he caal d d dx γ γ π can e calculaed n a smlar way. Usng he ndeendence of ncremens of Wener rocesses we calculae π X as he condonal eeced value F I A and ge [ ]. C C Φ Φ Φ Φ γ γ
7. Concludng Remarks.. ome smlar calculaons can e done for Black- choles model wh sochasc volaly as a reresenave model of ncomlee marke.. Furher develomens of hs ssue may nclude he effec of ransacon coss. 3. We consdered condoned conngen clams under assumon ha marke and condoned facor τ are ndeenden. hs s no necessary. One can sudy condoned conngen clams hnkng aou τ as a source of nsder nformaon: τ F a F. As a measure of omaly of he sraegy he crera of eeced uly can e chosen. 4. here ess a close connecon wh defaulale dervaves and cred rsks. 37
8. References AA K. and PRON. 994. Prcng of un-lnked nsurance olces. candnavan Acuaral Journal : 6-5. BACINLLO A.R. and ORU F. 993. Prcng of unlnked lfe nsurance wh endegeneous mnmum guaranees. Insurance: Mah. and conomcs :45-57. BOWR N.L. GRBR H.U. HICKMAN J.C. JON D.A. and NBI C.I. 997. Acuaral Mahemacs. ocey of Acuares chaumurg Illnos. BOYL P.P. and CHWARZ.. 977. qulrum rces of guaranees under equy-lnked conracs. Journal of Rsk and Insurance 44: 639-68. BOYL P.P. and HARDY M.R. 997. Reservng for maury guaranees: wo aroaches. Insurance: Mah. and conomcs : 3-7. BRNNAN M.J. and CHWARZ.. 976. he rcng of equy-lnked lfe nsurance olces wh an asse value guaranee. Journal of Fnancal conomcs 3: 95-3. BRNNAN M.J. and CHWARZ.. 979. Alernave nvesmen sraeges for he ssuers of equy-lnked lfe nsurance wh an asse value guaranee. Journal of Busness 5: 63-93. DLBAN F. 986. quy-lnked olces. Bullen Assocaon Royal Acuares Belges 8: 33-5. 38
FOLLMR H. and LUKR P.. ffcen hedgng: cos versus shor-fall rsk. Fnance ochas. 4: 7-46. FOLLMR H. and CHID A.. ochasc Fnance: An nroducon n dscree me. Berln N.Y.: Waler de Gruyer. HARDY M.R. 3. Invesmen guaranees: Modelng and rsk-managemen for equy-lnked lfe nsurance. J.Wley. MARGRAB W. 978. he value of an oon o echange one asse o anoher. J. of fnance 33: 77-86. MLNIKOV A. VOLKOV. and NCHAV M.. Mahemacs of Fnancal Olgaons. Amercan Mah. oc. MLNIKOV A. 3. Rsk analyss n Fnance and Insurance. Chaman&Hall/CRC. MLNIKOV A. 4. Quanle hedgng of equy-lnked lfe nsurance olces Doklady Mahemacs Proceedngs of Russan Acad.c. MLNIKOV A. 4. On effcen hedgng of equylnked lfe nsurance olces Doklady Mahemacs Proceedngs of Russan Acad.c.. MOLLR. 998. Rsk-mnmzng hedgng sraeges for un-lnked lfe-nsurance conracs. Asn Bullen 8: 7-47. MOLLR.. Hedgng equy-lnked lfe nsurance conracs. Norh Amercan Acuaral Journal 5: 79-95. 39
H quy-lnked lfe nsurance Pure endowmen nsurance H H I > { } Pure endowmen wh fed guaranee ma{ K} I { } Black-choles model Jum-dffuson model H > Pure endowmen wh flele guaranee { } I { } H ma > Prces and hedgng sraeges are gven n erms of Black-choles formula and s Posson averagng Illusrave one-facor model d µ d dw wo-facor models d Jum-dffuson model µ d dw ν dπ d Black-choles model µ d dw Prces and hedgng sraeges are gven n erms of Margrae s formula and s Posson averagng 4 Prces and hedgng sraeges are gven n erms of Margrae s formula