Embedding the Natural Hedging of Mortality/Longevity Risks into Product Design

Transcription

1 Embing h Nurl Hging of Morliy/Longviy Rik ino Prouc Dign Bcky Fngwn Hung Jon Chng-Hin Ti 1 Dprmn of Rik Mngmn n Inurnc Rik n Inurnc Rrch Cnr Collg of Commrc, Nionl Chngchi Univriy Tipi, Tiwn ABSTRACT How o mng morliy/longviy rik i nil o h long-rm olvncy of lif inurnc compni. Th lirur propo o hg h prouc ubjc o h longviy rik (uch nnuii) by uing h prouc ubjc o h morliy rik (.g., whol lif inurnc) ol by n inurr. Such nurl hging i inuiiv bu my b ifficul o implmn u o h rigi l mrk n incniv iu. W propo o mb nurl hging ino prouc ign o h h hging my occur wihin prouc. Th ky i o off h impc of morliy on h iming of h h in urn rmin h prn vlu of h h bnfi by clvrly choo h growh r of h h bnfi. how much o py whilδ woul rflc h im vlu of pymn. W provi horicl rivion, grphicl illurion, n numricl nly o illur h i of mbing nurl hging ino prouc ign. Kywor: longviy rik; morliy rik; nurl hging 1 Corrponing uhor: ci@nccu.u.w; l & f: Th uhor i grful o h Miniry of Scinc n Tchnology (projc numbr: H MY3) n h Rik n Inurnc Rrch Cnr for hir finncil uppor. 1

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3 1. Inroucion Th rik mrging from h chng of morliy r i criicl problm o h inurnc provir. Whn quick pr of ly congiou i cu gr l of culy h morliy r woul incr n h morliy rik ri up. In gnrl iming, h morliy r i cring grully u o progriv micl chnology nowy. Th mk longviy rik ri up. Boh morliy rik n longviy rik r morliy r rik cu by h ynmic chng of morliy r. Whhr i i morliy rik or longviy rik, h miigion of morliy r rik h lo of icuion in lirur. Th longviy rik cch mor nion owing o i prin rn. A h cing living bcom norml c o vryon in h worl, i i volving ino ymic rik. On h ohr hn, h ynmic prn of morliy rik i focuing on un morliy r hock. Tho rik r no only rik o h prouc provir bu lo coly problm o conomic ociy. Th chniqu of rik miigion in boh rik i n imporn iu o rik mngmn. Coniring how o miig morliy r rik, mny ui r wih h longviy rik. Trnfrring longviy rik rnlly wih h finncil vhicl of cpil mrk i h rli uggion in lirur o olv h problm. Blk n Burrow (2001) propo oluion ccoringly hrough cpil mrk by iuing urvivor bon o miig h longviy rik po o living bnfi provir. Ohr prior ui lo provi miigion oluion hrough cpil mrk, incluing urvivor bon (Dnui, Dvolr, n Gorniu, 2007), urvivor wp (.g. Dow n Blk, 2006; Dow, Blk, Cirn, n Dwon, 2006), morliy wp (Lin n Co, 2007), morliy curiizion (.g. Dow, 2003; Lin n Co, 2005; Cirn, Blk, n Dow, 2006; Blk, Cirn, n Dow, 2006; Co, Lin, n Wng, 2006; Blk, Cirn, n Dow, 2006; Blk, Cirn, Dow, n McMinn, 2006). Th ui ugg miiging longviy rik by rnfrring h rik o h invor of cpil mrk. Trnfrring h rik rnlly my b poibl wy bu my no iminih h primry rik roo in h policy. Th mho r lo involv wih uncriny in mrk nvironmn n rncion co o h provir. 3

4 Nurl hging rgy i n lrniv. In of rnfrring rik oui h iniiiv rik bring compny, om ui k p o llvi rik by wy of innr miigion wihin h compny. Nurl hging rgy mk rik miigion cu in h inurnc compny inrnlly. A whol lif inurnc prouc i po o morliy rik n n nnuiy prouc i po o longviy rik. Compni cn k vng o iminih h hr of morliy r rik hrough lling lif inurnc prouc n nnuiy prouc imulnouly. Th min rm of h nurl hging rgy in ui i o opimiz h proporion of prouc porfolio h hlp mimiz h hging ffc. Co n Lin (2007) cr prouc porfolio wih lif inurnc prouc n n nnuiy prouc h how h inc of nurl hging ffc bwn lif inurnc prouc n nnuiy prouc. Wng l. (2010) oon propo n immunizion mol o chiv n opiml lif inurnc o nnuiy rio in miiging longviy rik. Thy vlu h hging ffc o h rrv of prouc porfolio n uiliz mching of urion n convii o gnr h opiml rio of lif inurnc o nnuiy prouc by h impc of morliy r chng. In mo rcn ui Ti n Chung (2013) n, Lin n Ti (2013) op h urion/conviy mching o h pric of lif inurnc n nnuiy prouc n n h pplicion in rmining h wigh of wo or hr prouc in prouc porfolio h cn obin h mimum ffc of miigion. Bi morliy urion/conviy chniqu in rmining opiml nurl hging rgy, Ti, Wng, n Tzng (2010) lo k h vrinc of prouc porfolio ino conirion n gnr nrrowr qunil of lo iribuion compr o immunizion mol (Wng, Hung, n Hong, 2013). Furhrmor, Wng, Hung, n Hong (2013) mploy porfolio of zro coupon bon, lif inurnc polici, n nnuiy polici o conruc fibl objciv funcion h mk poibl o cch h mipricing ffc n h vrinc ffc of h nir compny porfolio. Thy lo r nurl hging rgy prciclly. W fin wy of nurl hging rgy wihin policy h w cn immuniz/miig h morliy r rik hrough lic prouc ign. W r bl 4

5 o opimiz h rik miigion in imply h bnfi procion prouc n hv no o buil up wigh porfolio of lif inurnc n nnuiy. Evry inurnc clim conin l wo ky fcor of whn o py n how much o py h vn wrin in policy bu normlly popl ovrlook how much o py cn b vribl rl o rik pour. W my h h bnfi funcion of fc moun n vry long wih im in h conn of prouc ign. W mp o uiliz boh h iming of bnfi n h moun of bnfi o b vribl in rmining h rik of h prouc. If w ign h prouc in which h rik ruling from whn o py i mor/l hn h pc n h from how much o py i l/mor hn h pc ccoringly, w my immuniz h rik wihin h prouc. Th pc vlu i rmin by h prn vlu of h policy ch flow n h ch flow r ci by h fuur lifim iribuion b on morliy umpion. Th im vlu i h brig bwn whn o py n how much o py. Th rik rooing in whn o py conroll by h fcor, h forc of inr r, cn b rprn h im vlu of h bnfi. If h h bnfi i pi rlir hn pc iming h compny fc lo of h im vlu. In hi c, w my nvig h h moun of bnfi i pi l in rpon o h lo of im vlu. Thu, w inrouc nw fcor, h forc of moun, in rmining h moun of h bnfi n i i lo ngg o h rik rooing in how much o py. W hu r bl o cr h prouc h whn h bnfi i pi rlir/lr hn pc iming i moun of bnfi i pi l/mor ccoringly. W implmn our rgi by mn of rik in prouc ign illur prouc in hi ricl, in which w prn h opiml rgy of prfc miigion n h conry rgy in iminihing rik. Our rgi hrough rik immuniz prouc fcili h mrking n mngmn n lo mk morliy rik miigion fibl o crry on. Following our rgi in prouc ign, h inurnc prouc h lo mor poibiliy in ngging o finncil inrumn wihou coniring h morliy r rik. Our fining cn lo provi furhr rrch or r-ign of prviou ui h ignor h inc of morliy r 5

6 rik in n inurnc prouc vluion or rik mngmn conirion. A fr w know, hr r no uch pproch in h fil of h nurl hging lirur y. Th rminr of hi ricl i orgniz follow. In h cion Thoricl Dvlopmn, w uc n lbor our horicl vlopmn for nurl hging rgy uing h Lplc rnform h w gnr pcifi lif inurnc. In h n cion Numricl Anlyi w u n iing mol wih U.S. morliy princ n monr how o implmn our propo rgi o h nurl hging ffc wih numricl mpl in h cion. W n our nlyi o h c of nnuiy prouc in h n cion, n hn w conclu in h l cion. 2. Thoricl Dvlopmn 2.1 I Scrching W fir nlyz riionl whol lif inurnc prouc in h frmwork of Lplc rnform 2. Th n ingl prmium for h whol lif inurnc wih 1- uni fc moun pr by h noion of Bowr l. (1997) cn lo b pr in Lplc rnform form : b( ) ( ) ( ) 0 p b f L, (1) whr δ i h forc of inr, b() i h h bnfi im pr 1- uni fc moun, 3 f() i h probbiliy niy funcion of h fuur lifim rnom vribl S g : p+, in which μ+ i h forc of morliy g of + n p no h probbiliy of pron g who will urviv yr. No h b()=1 for riionl whol lif inurnc, which mn h h h bnfi i fi h fc moun of h policy. In uch c, quion (1) impli h h n ingl prmium i funcion of h forc of inr δ wih rpc o f(). Shoul morliy ri unpcly, h collc prmium h inufficin im o ccumul o h h bnfi; on h ohr hn, h policyholr whol py 2 Th Lplc rnform L f ( ) f ( ) 0 i n ingrl rnform wily u in mhmic wih mny pplicion in phyic n nginring. I urn convoluion ino muliplicion, h lr bing ir o olv bcu of i lgbric form. 3 6

7 oo much if morliy improv mor hn pc. Th prouc mk h inurr n inur ubjc o h morliy rik. Cn w miig uch rik hrough prouc ign? On ign i o mk h h bnfi n incring funcion of h im o h h ccumul vlu of h prmium cn mch h h bnfi no mr how morliy vri. Th Lplc rnform form h ligh on poibl oluion: chnging h prmr of (.) p f 0 L. (2) Th prouc impli by quion (2) i n incring whol lif inurnc policy. I h bnfi incr coninuouly h nnul r of. Th prmr γ conrol how much o py whilδ woul rflc h im vlu of pymn. Appropri choic on (,δ), uch δ, cn mk h prn vlu of h h bnfi inniiv o h iming of h h in urn i ffc by morliy. Such ign cn hu miig h morliy rik. 2.2 Forml Dvlopmn W lbor h bov i by mining h pc rrv of h pproprily clibr whol lif inurnc prouc o whhr i cn b immuniz from h morliy rik by ilf. Whn nw policy of h clibr prouc i ol now (i.., im 0) o cuomr g wih fc moun 1 (wihou lo of gnrliy), h pc rrv im of h policy i in quion (3): 0 F 0 p. (3) Rfrring o h rivion in Bowr l (1997, chpr 4), w my compo ino wo pr: ( S ) S E 0 F ( S ) S ( S ) S E F0 S h Pr S h E F 0 S h Pr S h, (4) whr S h q S h Pr n Pr h h p. Sinc h coniionl p..f. of S givn S h i 7

8 f p 0 h f S h F h q h 0 lwhr, p ( S ) S ( ) 0 S h 0 F0 q h E h F (5) n ( S ) S h ( S h) ( S h) E F0 S h E F0 ( S h) 0 h h ( S ) S h h E F 0 ( S h) 0 h. (6) Subiuing quion (5) n (6) ino (4) yil p h p h q. ( ) h q 0 h h h h (7) Muliplying boh i of quion (7) by -1, ing by h, w obin h h 1 1 p h h h h h. h h ( ) p 0 h, n hn iviing (8) Th limi of h wo im on h righ-hn i of quion (8) r follow: 1 lim h0 h p n h ( ) ( ) 0 p 0 p 0 1 p ( ) 0 lim h 1 p h 0 0 Th limi of quion (8) i hu h p ( ) p 0 0 ( ), (9). (10) 8

9 . (11) Th bov rivion mn h. (12) Rrrnging quion (12), w g. (13) No h i lwy poiiv inc 0. Thn w hv h following bounry coniion of h pc rrv: 1. whn μ++δ γ> 0 n 0, 2. whn μ++δ γ< 0 n 0, 3. whn μ++δ γ> 0n 0, 4. whn μ++δ γ< 0n 0,,,, n. Th quion of h bounry cn b wrin :. (14) A Figur 1 n 2 iply, Equion (14) rprn hyprbol cnr ( δ+γ, γ ) on h (μ+, ) pln. Th wo ympo of h hyprbol r givn by μ+ = δ+γ n = γ. Th hyprbol wih poiiv (δ γ) ly in h 2n n 4h qurn coorin wih rpc o h cnr whil h hyprbol wih ngiv (δ γ) ly in h 1 n 3r qurn hown by Figur 1 n Figur 2 rpcivly. Th hyprbol in Figur 1 hibi poiiv lop whil h in Figur 2 hv ngiv lop. 9

10 Coniion 2 Coniion 1 Figur 1 Th bounry of h pc rrv wih poiiv (δ γ) on h (μ+, ) pln No: W r wr h ngiv μ + i no ronbl bu rin hm o how compl hyprbol. Coniion 3 Coniion 4 Figur 2 Th bounry rrv wih ngiv (δ γ) on (μ+, ) pln In Figur 1, h curv on h own-righ of h cnr mch h criri of h 10

11 coniion 1. Auming h μ+ i incring funcion 4 of g, wih cririon 0 of coniion 1, w my uc h h i incring long wih μ+, n wih cririon μ++δ γ> 0, h lowr boun of pc rrv in coniion 1 houl b on h own-righ in rpc o h cnr of hyprbol. Th up-lf curv in Figur 1 mch h criri of h coniion 2. Wih criri μ++δ γ< 0 n 0, no h h coniion 2 hown on h own-lf pr of h hyprbol o no i in rliy inc h forc of morliy μ+ houl no b ngiv. Following h m ucion, w inify h hyprbol wih ngiv (δ γ) in Figur 2, of which h up-righ curv rprn h uppr boun of h pc rrv in h coniion 3 n h own-lf curv rprn h lowr boun of h pc rrv in h coniion 4. Th fur of bounry rrv in coniion 3 n 4 i cring long wih μ+. Th pc rrv my b mor/l hn or qul o h lowr/uppr boun of pc rrv in prcic. Whn nrrowing own h poibiliy of (δ γ) in h four coniion w fin ou h bounry rrv i qul o h pc rrv if (δ γ) = 0. Whn (δ γ) i qul o 0, h lowr/uppr boun of h pc rrv in Figur 1 i fol oghr wih h uppr/lowr boun of h in Figur 2. Th fur of h pcil c wih (δ γ) = 0 urn ino horizonl lin hown in Figur 3 which i cly on of h ympo of h hyprbol = γ. 4 11

12 Figur 3 Th bounry rrv wih (δ γ) = 0 on h (μ+, ) pln Furhr Anly on h (+, ) pln Morliy r rik i bcu w vlu rik ccoring o g. If inurnc compni pric lif inurnc prouc ccoring o rl ging iuion, h compni my no hv morliy r rik. In fc, nihr cn w obrv ging of iniviul ircly, nor cn w pric h morliy rik by ch iniviul ging iuion. Wh w cn obrv in nur i iniviul g n morliy r wih rpc o g iiclly. Thn, h forc of morliy i qunifi o pr ging wih rpc o g in mhoology. Sinc w pric inurnc prouc b on g, no ging, morliy r rik mrg ccoringly. Thu, cquiring h informion on (+, ) pln i imporn in rmining h morliy r rik of h pcifi prouc long wih h chng of morliy r. N, w inn o monr h rlionhip of pc rrv n g by h fur of h bounry rrv coorin on h (+, ) pln. Wih h informion on h (μ+, ) pln in l cion, w cn rnform h 12

13 fur of h bounry rrv wih rpc o h forc of morliy ino h fur of h wih rpc o g by uming h h forc of morliy μ+ i monoonic incring funcion of g +. W hn cn illur fur of h bounry rrv ccoring o g + on h (+, ) pln n plor h rlionhip of pc rrv n g. In c of h poiiv (δ γ), w rnform h fur of h bounry rrv on h (μ+, ) pln in Figur 1 ino h fur on h (+, ) pln hown in Figur 4. Alo, in c of h ngiv (δ γ), w rnform h fur of h bounry rrv in Figur 2 ino h fur on h (+, ) pln hown in Figur 5. Th ngiv vlu of μ+ on h (μ+, ) pln cnno hibi on h (+, ) pln for ch g i of poiiv forc of morliy. Coniion 1: Uppr boun Figur 4 Th bounry rrv wih poiiv (δ γ) on (+, ) pln No h h illurion i uing h Mkhm mol , which i ci from Mlnikov n Romniuk (2006) n h originl i b on h morliy r from 1959 o 1999 in Amricn (Pollr, 1973). 13

14 Coniion 3: Uppr boun Coniion 4: Lowr boun Figur 5 Th bounry rrv wih ngiv (δ γ) on (+, ) pln No h h illurion i uing h Mkhm mol , which i ci from Mlnikov n Romniuk (2006) n h originl i b on h morliy r from 1959 o 1999 in Amricn (Pollr, 1973). W monr h four coniion of quion (13) in corrponing fur on h (+,) pln. Th fur of h bounry rrv in Figur 4 how h g wih poiiv vlu of μ+ which i h c in coniion 1. Th c in coniion 2 o no how ou on h (+,) pln for h vlu of μ+ i ngiv. In Figur 5, h fur of bounry rrv on h up-righ pr i h c in coniion 3 h follow h cririon of μ++δ γ>0; whil h ohr on h own-lf pr i h c in coniion 4 h follow h cririon of μ++δ γ<0 on h (+, ) pln. Th fur of h bounry rrv in Figur 4 ppr fr incring hn h in 1 qurn of Figur 1 long wih h horizonl i. A h horizonl i of μ+ i chng ino h of h g +, h horizonl i i r-cl vnly h murmn of g on h (+, ) pln. Th lop Figur 4 i mor hn h in figur 1 inc of h lrly g i much highr hn h of h young g. 14 of lrly g in n In Figur 4, h bounry rrv of pcifi prouc wih poiiv (δ γ),

15 ppr h h rik rooing in whn o py i mor hn h rooing in how much o py. A for our h bnfi prouc, h rik rooing in whn o py rprn morliy rik bcu whn h rliz morliy r ri up, i mn h mor hn pc popl of cohor r pi rlir hn pc iming. Th inurnc compni g lo by h im vlu of r h bnfi clim u o incr of morliy r. Th rik rooing in how much o py in our ign i h w h incrmn of h bnfi ch im h i only provi o h urvivor h im. Th rik rooing in how much o py i form of longviy rik. Th morliy rik pour i mor hn longviy pour in Figur 4. In gnrl, mo lif inurnc prouc r group in hi yp of fur in Figur 4 h morliy rik i mor hn h longviy rik. Th fur in Figur 5 i mingly bu no c hyprbol on h (+, ) pln. Th g roun 33 yr ol i criicl g h h bounry rrv i unboun. Th bounry rrv in h c of ngiv (δ γ) i cring long wih g mor hn 33 yr ol. Whn (δ γ) i ngiv, w my y h h rik rooing in how much o py in our ign i mor hn h rooing in whn o py. Th i o y h longviy rik of h c i mor hn morliy rik following h imilr lborion in h c of poiiv (δ γ) bov. Thi yp of inurnc prouc i rr. Bu i vr horly ppr in Tiwn inurnc mrk. Th compni ign n incring whol lif inurnc wih γ > δ. For innc, h h bnfi i b on fc moun compoun by 4% nnully n nnul inr r i 2.5%. A for h pcil c on h (μ+, ) pln in Figur 3, w rnform h horizonl i o g + n iply h c of (δ γ)=0 on (+, ) pln hown in Figur 6. Th uppr boun i ovrlpp wih h lowr boun of h pc rrv. Th fur of h bounry rrv i horizonl lin h i h m in Figur 3. Th iffrnc of fur bwn Figur 3 n Figur 6 i h murmn of horizonl i h cnno lr h hp of horizonl lin. Th 15

16 funcion of h pcil horizonl lin = γ on (+, ) pln i h m h on h (μ+, ) pln inc h c i irrlvn o h vribl of horizonl i. In hi pcil c, h fur i no ju h bounry rvr bu lo h pc rrv. W hn com ou n only oluion h fulfill h criri of h four coniion of quion (13) rprn h rlionhip bwn pc rrv n g. Uppr boun=lowr boun Figur 6 Th bounry rrv wih δ γ = 0 on (+, ) pln No h h illurion i uing h Mkhm mol , which i ci from Mlnikov n Romniuk (2006) n h originl i b on h morliy r from 1959 o 1999 in Amricn (Pollr, 1973). Th pc rrv i horizonl lin on (+, ) pln rvl h h pc rrv i irrlvn o g. W my mk u of hi propry on rik miigion wihin prouc. Whn w ign uch pcifi prouc wih (δ γ)=0, h rik rooing in whn o py i cly qul o h rik rooing in how much o py. Th morliy rik i miig olly by h longviy rik for ch g in h c. Furhrmor, h pc rrv i lvl o ll g n o h forc of morliy of ch g. Th pc rrv of h policy will no b ffc by h 16

17 chng of morliy r n will b qul o h rliz rrv. A h rliz rrv rmin h m h pc rrv wih h chng of morliy r, h rik i immuniz wihin h policy. 3. Numricl illurion Th bic umpion r up in Tbl 1. W um h h fc moun i US$100,000 for h pcifi whol lif inurnc n h prmium i pi by ingl prmium. Aum h h compny only ll h pcifi whol lif inurnc prouc wih givn vlu of δ. Th nurl hging rgy pn on h policy ign of h forc of moun γ. γ cn b ny givn vlu in ronbl rik miigion rgy. Tbl 1 Bic umpion for h nw form of h whol lif inurnc prouc Ag of inur 25, 45 Gnr Ml Fc moun 100,000 Th iniil vlu of forc of inr r (δ) 4% Dh bnfi 100,000 compoun by γ() Bnfi prio Whol lif Mho of pying prmium Singl prmium 3.1 Th opiml rgy wih γ = δ W fir invig h c wih h vlu of γ qul o h vlu of δ. Our pricing morliy r i b on h Mkhm mol (Mlnikov n Romniuk, 2006). W k h 5 h policy yr n mpl o min h morliy r rik which i h iffrnc of h rliz rrv n h pc rrv. Th rliz rrv i vlu by 20% up hock or 20% own hock of h morliy r. Kping h ing of δ n γ o ify h quion (δ γ) = 0 in ll im, w cn offr hr iffrn yp of h ign prouc h following mpl. 17

18 Prouc Dign 1: Th prouc i yp of riionl incring whol lif inurnc. (). Wih δ = conn numbr L γ = δ = conn = 4% hrough h whol policy yr. Th h bnfi i compoun γ coninuouly n i i inic h following quion F = F0 p(γ), whr F0 i h fc moun, i h policy yr. Th oucom i hown in h Tbl 2. Tbl 2 Th Libiliy h En of h 5 h Policy Yr of Illur Inurnc Prouc for Diffrn Morliy B γ(=4%) = δ(=4%) (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) g Bi 20% Up Shock 20% Down Shock Rrv Chng Rrv Chng , , ,140 0% 0% , , ,140 0% 0%. A w cn in Tbl 2, h rrv rmin unchng fr h hock of morliy r. Th rul i conin wih h c of (δ γ) = 0 in our horicl nlyi. Thu, h morliy r rik of h pcifi prouc i non inc h rik pour o no incr or cr cu by h 20% hock h 5 h policy yr. Th pcifi whol lif inurnc prouc ppr no morliy rik or longviy rik in ll g. Th prouc ign 1 i riionl incring whol lif inurnc. If h inr r kp h m h forc of moun γ, h prouc fulfill h criri δ= γ n hr i no morliy r rik ll. Whil in rliy, h inr r i no fi long wih h whol policy yr, h prouc my b po o rik. (b). δ obin by CIR mol wih conn long rm mn To cpur h rik in rliy h bginning of h policy, w blih n 18

19 inr r mol of on fcor CIR 5 mol n morliy r mol of LC 6 (L & Crr, 1992) mol o cpur h ynmic of inr r n morliy in h whol policy yr. Th γ rmin conn in h prouc ign. All numricl rul r obin wih 10,000 imulion. Th prouc ign 1 of ynmic inr r (δ) wih conn long rm mn h i qul o γ. W iply h rul of γ = 6.5% n h ohr comprion pnl of γ = 0% hown in h Tbl 3. Tbl 3 Th Libiliy h En of h 5h Policy Yr of Illur Inurnc Prouc wih CIR Mol n LC Mol for Diffrn Morliy B Pnl A: γ (=6.5%) = Long Trm Mn of Inr R Mol (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) Morliy R Morliy R g Bi 20% Up Shock 20% Down Shock Rrv Chng Rrv Chng , , , % 0.12% , , , % 0.14% Pnl B: γ (= 0%) Long Trm Mn of Inr R Mol (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) Morliy R Morliy R g Bi 20% Up Shock 20% Down Shock Rrv Chng Rrv Chng 25 6,378 7,133 5, % % 45 17,509 19,195 15, % % Th rul in Tbl 3 h how miigion of rik in Pnl A i br hn ho in Pnl B. Evn hough h h prouc ign 1 in Pnl A i no cly kping γ() = δ() poin o poin, i ill br off h rul wih γ()=0. Prouc Dign 2: Th yp of prouc i n inr r vribl whol lif inurnc. In orr o kp h criri of γ() = δ() uring h m im prio, =1, 5 r( ) = κ(θ r()) + σ r( )W( ), wih κ = 0.25, θ = 0.065, σ = 0.07 ( Liu (2013), Chn l. (1992)) 6 log m(, ) = + b k + ε, Th in k i mol by rnom wlk wih rif rm: k = k 1 + c +. Th i US morliy r from 1961 o 2010 obin on Th prmr im for LC mol r: c = , σ =

20 2 c. W ci h γ() immily fr w obin nw δ() in h mrk ch im. W l γ() clo o δ() poibl. Th δ() i picwi coninuouly long wih im n o i h γ().w um δ(1) = δ, γ(1) = δ(1) for h fir policy yr. From h con policy yr on, w clr nw inr r h bginning of ch policy yr h gnr nw forc of inr r δ(). L γ() = γ = δ(). W obin h inicion of h h bnfi hi kin of policy i F = F0 p(σγ). A h inr r i vribl ch policy yr, w um h inr r cnrio for h 2 n o 5 h policy yr. Th vlu of h forc of inr r for h fir fiv policy yr r hown in Tbl 4. Tbl 4 Th Scnrio of h lu of h Forc of Inr R for h Fir Fiv Policy Yr. Policy Yr Scnrio A δ=4% δ(2)=4.25% δ(3)=4.50% δ(4)=4.50% δ(5)=4.75% Scnrio B δ=4% δ(2)=3.75% δ(3)=3.75% δ(4)=3.50% δ(5)=3% Th oucom of prouc ign 2 i hown in Tbl 5. W cn h h vlu of rrv r no chng by h hock of morliy r in Tbl 5. Th morliy rik n longviy rik inic in column (4) n column (5), rpcivly, r hown no rik by h chng of morliy r bcu h ol morliy r rik i immuniz wihin h policy. Tbl 5 Th Libiliy h En of h 5 h Policy Yr of Prouc Dign 2 for Diffrn Morliy B γ() = δ() (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) Scnrio g Bi 20% Up Shock 20% Down Shock Rrv Chng Rrv Chng A B , , ,608 0% 0% , , ,608 0% 0% , , ,722 0% 0% , , ,722 0% 0% 20

21 Th prouc inclu lvl bnfi whol inurnc n n r inurnc bnfi (i.. ivin or incrmn of h bnfi) h pn on h clrion of inr r ch policy yr. Th inr r vribl lif inurnc in h Uni S i on of h kin in hi prouc group. Our prouc ign i b on h inr r vribl lif inurnc n kping h cririon γ()=δ(). Th iffrnc from h h inr r vribl inurnc prouc provi ivin o policyholr, h prouc 2 provi h incrmn of h bnfi. Ech incrmn of h bnfi in our prouc houl b h m h ivin of inr r vribl inurnc prouc gnr by h clr inr r. A long h prouc m h cririon γ()=δ(), whhr h r bnfi i cll ihr h incrmn of h bnfi in our ign or h ivin in h conn of inr r vribl lif inurnc, i o no mr h chivmn of rik miigion. In our prouc ign hr, w k h r h bnfi incr by ch clr inr r n provi no ivin. A o fulfill h criri, h ign of inr r vribl lif inurnc houl k h bnfi n ivin ino ccoun on mn of rik. Sinc h inr r i clr in rl o mrk inr r, h yp of prouc clin h hr of inr r rik compring o h prouc wih fi inr r. Prouc Dign 3: Th inr r vribl incring whol lif inurnc. Th i of hi prouc i imilr o Prouc ign 2 h h ciion of γ() i oon fr h nw δ() w cn g in h mrk. L γ() = γ0 +Δγ= δ(), uring h m im prio, =1, 2 c. S h iniil vlu of h forc of moun γ0 = 2% < δ = 4% im 0. L δ(1) = δ, γ(1) = γ0 +Δγ1 = δ(1) in h fir policy yr. From h 2 n policy yr on, clr nw inr r in ch following policy yr h gnr nw forc of inr r δ(). A γ() = γ0 + Δγ = δ(), h h bnfi i F = F0 p(γ0 + ΣΔγ) = F0 p(γ0) + F0 p(σδγ). Th prouc i combinion of prouc 1 n prouc 2. Th prouc conin 21

22 fi incrmn h bnfi h mk hi pr of prouc look lik riionl incring whol lif n vrin incrmn h bnfi h mk hi pr of prouc m lik n inr r vribl lif inurnc. Th vrin incrmn h bnfi i lik prouc ign 2 rmin by h clrion of inr r. W um h inr r cnrio for h 2 n o 5 h policy yr. Th vlu of h forc of inr r for h fir fiv policy yr r hown in Tbl 6. Tbl 6 Th Scnrio of h lu of h Forc of Inr R for h Fir Fiv Policy Yr Policy Yr Scnrio C δ=4% δ(2)=4% δ(3)=4.50% δ(4)=4.50% δ(5)=4.50% Δγ1=2% Δγ2=2.25% Δγ3=2.50% Δγ4=2.50% Δγ5=2.75% Scnrio D δ=4% δ(2)=3.50% δ(3)=3.50% δ(4)=3.50% δ(5)=3.25% Δγ1=2% Δγ2=1.75% Δγ3=1.75% Δγ4=1.50% Δγ5=1% Th oucom of prouc 3 i hown in Tbl 7. W cn lo h h vlu of rrv r no chng by h hock of morliy r in Tbl 7. Th morliy rik n longviy rik inic in column (4) n column (5), rpcivly, r hown no rik by h chng of morliy r bcu h ol morliy r rik i immuniz wihin h policy. Tbl 7 Th Libiliy h En of h 5 h Policy Yr of Prouc 3 for Diffrn Morliy B γ() = γ0 + Δγ = δ() (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) Scnrio g Bi 20% Up Shock 20% Down Shock Rrv Chng Rrv Chng C D , , ,986 0% 0% , , ,986 0% 0% , , ,423 0% 0% , , ,423 0% 0% 22

23 3.2. Th conry rgy wih 0<γ<δ 7 Whn h rgy i wih γ<δ, our objciv i o iminih h morliy r rik, no o immuniz h rik. A w lbor in hi ricl, h rik rooing in whn o py of h bnfi lif inurnc i morliy rik long wih h chng of morliy r, n h rik rooing in how much o py of h bnfi lif inurnc i longviy rik. W k h bnfi whol lif inurnc n mpl. Th oucom of h chng of h morliy r wih 20% hock i in Tbl 8. Tbl 8 Th Libiliy h En of h 5 h Policy Yr of Illur Inurnc Prouc for Diffrn Morliy B Pnl A: γ (=2%) <δ (=4%) (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) g Bi 20% Up Shock 20%Down Shock Rrv Chng Rrv Chng 25 45,368 47,287 43, % % 45 63,758 66,020 61, % % Pnl B: γ (=0%) <δ (=4%) 25 18,612 20,153 16, % % 45 35,216 37,608 32, % % Th prouc in Pnl A wih δ, 4% n γ, 2% i compr o h in Pnl B wih δ, 4% n γ, 0%. Th prouc in Pnl A i po o boh h morliy rik n h longviy rik n h longviy rik i l hn h morliy rik. Th prouc in Pnl B i po o morliy rik only. Whn h morliy r i chng by 20% hock, h pc rrv of boh prouc i chng. Wih incr of rrv inic in column 4 wih rpc o 20% up hock of morliy r, prouc in Pnl A i po o rik of 3%~4% mor hn h in bi morliy r. An h prouc in Pnl B i po o h of 6~8% mor hn h in bi morliy r by h chng of morliy r. Th conry rgy i o iminih h rik wihin h policy by cring longviy rik in lif inurnc prouc o lowr h pour of morliy rik n h rgy hlp u o ign uch prouc upon h mn of rik pour. 7 Wih prouc ign of γ>δ, h lif inurnc prouc bcom po o h longviy rik only. Th i no norml in pur h bnfi lif inurnc prouc. In h norml iuion, inurnc compni my no ign uch prouc o confu hmlv in inificion of h longviy rik ruling from pur h bnfi lif inurnc. 23

24 4. Annuiy W my pply h m chniqu wih h forc of moun γ in cion 2 for whol lif nnuiy prouc of 1 uni fc moun pr nnum pybl coninuouly whil () urviv. Sinc h nnuiy prouc i pying bnfi long h inur urviv, i i po o longviy rik of whn o op pying. Th rm i h m whn o py of lif inurnc prouc i h im vlu rmin by δ. W r ing h forc of moun γ in prouc rucur o cr morliy rik of how much o py o miig h longviy rik. 4.1 Thoricl vlopmn on nnuiy prouc Thu, w h funcion of urvivor bnfi for h nnuiy prouc, b"()= -γ, pr 1- uni fc moun im, whr γ >0. Wih S rprning h fuur lifim of (), h prn vlu of h nnuiy pymn m up unil h i ( ) S S u u 1 0 Y u. (14) Th n ingl prmium of h pcifi nnuiy prouc i no Y p ( ) 1 1 p p 0. (15) Ingrion by pr, h quion (15) cn lo b pr 1 1 r r Y p r 0 P P. (16) Whn h nnuiy prouc i ol o cuomr g wih 1-uni fc moun, policy yr ping by, h pc rrv h n of policy yr for hi nnuiy policy cn b pr quion (17) or quion (18) 24

25 25 P r r r 0 1, (17) or P r r 0. (18) Hving quion (17) iffrni by n pplying quion (10), w obin r r r M r r r r P (19) P P P r r r r r r r r r r M whr Rrrnging h quion (19) yil. (20) Th four coniion of h nnuiy prouc for quion (19) r follow: 1. Whn μ++δ+γ> 0 n 0, hn. 2. Whn μ++δ+γ< 0 n 0, hn. 3. Whn μ++δ+γ> 0 n 0, hn. 4. Whn μ++δ+γ< 0 n 0, hn.

26 Th quion of bounry rrv i whn i qul o zro. Wih poiiv vlu of -γ, rrrnging h quion of bounry rrv (20) ino, (21) w obin h only poibl fur of hyprbol wih cnr ( δ γ, 0) on h (μ+, ) pln. Th wo ympo of h hyprbol r givn by μ+ = δ γ n = 0. Th fur wih poiiv vlu of -γ ly in h 1 n 3 r qurn wih rpc o h cnr, hown in Figur 7. Coniion 1 Coniion 2 Figur 7 Th bounry rrv wih poiiv γ on h (μ+, ) pln Sinc h lop of h fur in Figur 7 i ngiv n if w um h μ+ i n incring funcion of g w cn g h rquir cririon in coniion 1 n 2. Th poiiv i ngiv rquir cririon in 26

27 coniion 3 n 4 i impoibl iing in our pcifi nnuiy prouc unl μ+ i cring funcion of g, bu hi i gin nur of ging. W hn only focu on h c whn i ngiv in coniion 1 n 2. N, w rnform h fur on h (μ+, ) pln ino h on h (+, ) pln by proviing μ+ givn funcion of g hown in Figur 8. Th fur in Figur 8 i h c in coniion 1 wih h cririon of μ++δ + γ>0 on h (+, ) pln. Thi fur iply h uppr boun of h pc rrv in coniion 1. For h m ron in cion 2, h c of coniion 2 i no hown on h (μ+, ) pln. Coniion 1: Uppr boun Figur 8 Th bounry rrv wih poiiv γ on (+, ) pln No h h illurion i uing h Mkhm mol , which i ci from Mlnikov n Romniuk (2006) n h originl i b on h morliy r from 1959 o 1999 in Amricn (Pollr, 1973). Unlik h oucom in lif inurnc, h nnuiy prouc wih poiiv γ cnno 27

28 yil fur of horizonl lin h i confirmion in rmining if h opiml miigion rgy i. Evn hough w cn only g h conry rgy for h nnuiy prouc in uch prouc rucur, h cr ffc of morliy rik in h nnuiy prouc cn ill miig pr of h longviy rik wihin h policy. Thu, givn poiiv γ w cn u h conry rgy, rik iminihing, o pply on h nnuiy prouc o rpon h mn of prouc rik. 4.2 Numricl Illurion W um h h fc moun i US$10,000 for h nnuiy prouc n h prmium i pi by ingl prmium. Aum h h compny only ll h pcifi nnuiy prouc wih givn vlu of δ. Th rgy i o h vlu of γ. γ cn b ny givn poiiv vlu o lowr h ol rik of h prouc. Th bic umpion of nnuiy prouc hown in h Tbl 9 Tbl 9 Th Bic Informion of h Illur Annuiy Prouc Ag of inur 25, 45 Gnr Ml Fc moun 10,000 Th iniil vlu of forc of inr r (δ) 4% Annum pybl 10,000 compoun by -γ() Bnfi prio Whol lif Mho of pying prmium Singl prmium Unr h rgy of h nnuiy prouc wih h forc of moun, w cn iminih h longviy rik of h nnuiy prouc wihin h policy whn morliy r i chng. W k whol lif nnuiy n illur c. In Tbl 10, w how h h longviy rik of h nnuiy prouc i l hn h wih h forc of moun. 28

29 Tbl 10 Th Libiliy h En of h 5 h Policy Yr of Illur Annuiy Prouc for Diffrn Morliy B A: γ (=4%) <δ (=4%) (1) (2) (3) (4)=[(2)-(1)]/(1) (5)=[(3)-(1)]/(1) g Bi 20% Up Shock 20%Down Shock Rrv Chng Rrv Chng 25 90,472 89,608 91, % 1.014% 45 79,509 77,469 81, % 2.830% B: γ (=0%) <δ (=4%) , , , % 2.286% , , , % 4.703% Th prouc A wih δ, 4% n γ, 4% i compr o prouc B wih δ, 4% n γ, 0%. Th prouc A i po o boh h morliy rik n h longviy rik n h morliy rik i l hn h longviy rik. Th prouc B i po o longviy rik only. Whn h morliy r i chng by 20% hock, h pc rrv of boh prouc i chng. Wih incr of rrv inic in column 5 wih rpc o 20% own hock of morliy r, prouc A i po o rik of 1%~3% mor hn h in bi morliy r. An h prouc B i po o h of 2~5% mor hn h in bi morliy r by h chng of morliy r. W how h h conry rgy cn b lbor o iminih h rik wihin h policy by cring morliy rik in h nnuiy prouc o lowr h pour of longviy rik n h rgy hlp u o ign uch prouc upon h mn of rik pour. 5. Concluion W icovr h nurl hging rgy hrough prouc ign i n imporn p h w cn immuniz/miig h morliy r rik wihin policy. Th ky 29

30 poin i h w houl k h rik rooing in whn o py n h rik rooing in how much o py ino conirion in rik miigion chniqu. W inrouc fcor γ, h forc of moun, rik fcor rooing in how much o py in h prouc ign. W uiliz h γ o ign h bnfi procion lif inurnc h h morliy r rik i immuniz. W uc h opiml nurl hging rgy wihin policy wih h criri of ing on γ o l γ = δ. Whn h γ i qul o δ, h pcifi prouc ppr no rik wih h chng of morliy r. Th morliy r rik i immuniz in uch kin of prouc wih h bnfi procion in ll g. Whn h γ i no qul o δ, h rgy i u o iminih h rik rhr hn o immuniz h rik. If δ > γ, h h bnfi procion prouc i po o morliy rik rhr hn longviy rik wih h chng of morliy r n h lif inurnc prouc wih ing of γ i l po o morliy rik hn h on wihou ing of γ. In h c of h h bnfi lif inurnc prouc, h γ fcor i cring wy o miig h morliy rik of lif inurnc. An in h c of nnuiy prouc, h γ fcor i u o miig h longviy rik of h nnuiy prouc. B on our horicl nlyi, h nnuiy prouc cn only uc conry rgy o iminih rik wihin h policy by cring morliy rik fcor γ. Following h opiml rgy in h prouc ign, h inurnc prouc h lo mor poibiliy in ngging o finncil inrumn wihou coniring h morliy r rik. Our fining cn lo provi furhr rrch or r-ign of prviou ui h ignor h inc of morliy r rik in n inurnc prouc vluion or rik mngmn conirion. 30

31 Rfrnc Bur, D., Borgr, M., Ru, J., On h Pricing of Longviy-Link Scurii. Inurnc: Mhmic n Economic 46, Blk, D., Burrow, W., Survivor Bon: Hlping o Hg Morliy Rik. Th Journl of Rik n Inurnc 68(2), Blk, D., Cirn, A.J.G., Dow, K., Living wih Morliy: Longviy Bon n Ohr Morliy-Link Scurii. Briih Acuril Journl 12(1), Blk, D., Cirn, A.J.G., Dow, K., McMinn,R., Longviy Bon: Finncil Enginring, luion, n Hging. Th Journl of Rik n Inurnc 73(4), Cirn, A.J.G., Robu Hging of Longviy Rik. Th Journl of Rik n Inurnc 80(3), Cirn, A.J.G., Blk, D., Dow, K., Pricing h: Frmwork for h vluion n curiizion of morliy rik. ASTIN Bullin 36, Cirn, A.J.G., Blk, D., Dow, K., 2006b.A Two Fcor Mol for Sochic Morliy wih Prmr Uncriny: Thory n Clibrion. Th Journl of Rik n Inurnc 73(4), Chn, K., Krolyi, A., Longff F. A., n Snr, A. B., An Empiricl Comprion of Alrniv Mol of h Shor-Trm Inr R. Journl of Finnc 47, Co, S. H., Lin, Y., Nurl Hging of Lif n Annuiy Morliy Rik. Norh Amricn Acuril Journl 11(3), Co, S. H., Lin, Y., Wng, S., Mulivri Eponnil Tiling n Pricing Implicion for Morliy Scuriizion. Th Journl Rik n inurnc 73(4), 31

32 Lin, T., Ti, C. C., On h Morliy/Longviy Rik Hging wih Morliy Immunizion. Inurnc: Mhmic n Economic 53(3), Lin, T., Tzng, L.Y., An Aiiv Sochic Mol of Morliy R: An Applicion o Longviy Rik in Rrv Evluion. Inurnc: Mhmic n Economic 46(2), Liu, Xioming, Annuiy Uncriny wih Sochic Morliy n Inr R. Norh Amricn Acuril Journl, 17(2), Lucino, E. Rgi, L., Efficin vru Infficin Hging Srgi in h Prnc of Finncil n Longviy (lu ) Rik. Inurnc: Mhmic n Economic 55, P McMinn, R., Brock, P., Blk, D., Longviy Rik n Cpil Mrk. Th Journl of Rik n Inurnc 73(4), Mlnikov, A., Romniuk, Y., Evluing h prformnc of Gomprz, Mkhm n L Crr Morliy Mol for Rik Mngmn wih Uni-Link Conrc. Inurnc: Mhmic n Economic 39(3), Pollr, J.H., 1973, Mhmicl Mol for h Growh of Humn Populion, Cmbrig Univriy Pr, Lonon, Gr Briin Ti, C.C., Chung, S., Acuril Applicion of h Linr Hzr Trnform in Morliy Immunizion. Inurnc: Mhmic n Economic 53(1), Wng, C., Hung, H.C., Hong, D., 2013.A Fibl Nurl Hging Srgy for Inurnc Compni. Inurnc: Mhmic n Economic 52(3), Wng, J.L., Hung, H.C., Yng, S.S., Ti, J.T., An Opiml Prouc Mi for Hging Longviy Rik in Lif Inurnc Compni: Th Immunizion Thory Approch. Th Journl of Rik n Inurnc 77,

33 Yng, S.S., Yu, J.C., Hung, H.C., 2010.Moling Longviy Rik Uing Principl Componn Approch: A Comprion wih Eiing Sochic Morliy Mol. Inurnc: Mhmic n Economic 46(1), Gzr, N., Wkr, H., Morliy Rik n i Effc on Shorfll n Rik Mngmn in Lif Inurnc. Th Journl of Rik n Inurnc 81(1), Ti; J. T., Wng, J. L., Tzng, L.Y., On h Opiml Prouc Mi in Lif Inurnc Compni Uing Coniionl lu Rik. Inurnc: Mhmic n Economic 46, Pl, R., On yr lu Rik for longviy n morliy. Inurnc: Mhmic n Economic 49, Shn, Y., Siu, T.K., Longviy bon pricing unr ochic inr r n morliy wih rgim-wiching. Inurnc: Mhmic n Economic 52, Hung, Y.L., Ti, J.T., Yng, S.S., Chng, H.W., Pric boun of morliy-link curiy in incompl inurnc mrk. Inurnc: Mhmic n Economic 55, Dow, K., Blk, D., Cirn, A.J.G., Dwon, P., Survivor wp. Journl of Rik n Inurnc 73,