A Way of Hegig Moraliy ae iss i Life Isurace Prouc Develome Chagi Kim Absrac Forecasig moraliy imrovemes i he fuure is imora a ecessary for isurace busiess. A ieresig observaio is ha moraliy raes for a few ages are imrovig recely a ha here may eis moraliy rae riss. If he life able cosruce from a moraliy moel reics lower moraliy raes ha hose acually eeriece by he life isurace olicy holers he he comay will face losses from he sales of life isurace coracs. As a hegig sraegy, he isurace comay may romoe he sale of olices such as auiies or ure eowmes o offse he losses from he life isurace sales. We rese a way of hegig moraliy rae riss i eveloig eowme olicies usig he hege raios of ure eowme o offse he losses from erm life isurace. We also show a hegig sraegy uer a sochasic force of moraliy moel usig he resuls from Malliavi Calculus. Keywors: Moraliy ae iss, Force of Moraliy, Hege aios, Sochasic Moraliy rae Moels, Malliavi Calculus Dr. Chagi Kim is Lecurer a Acuarial Suies, Faculy of Busiess, The Uiversiy of New Souh Wales, Syey NSW 252 Ausralia. Tel: +6 2 9385 2647, Fa: +6 2 9385 883, mail: c.im@usw.eu.au
A Way of Hegig Moraliy ae iss i Life Isurace Prouc Develome Absrac Forecasig moraliy imrovemes i he fuure is imora a ecessary for isurace busiess. A ieresig observaio is ha moraliy raes for a few ages are imrovig recely a ha here may eis moraliy rae riss. If he life able cosruce from a moraliy moel reics lower moraliy raes ha hose acually eeriece by he life isurace olicy holers he he comay will face losses from he sales of life isurace coracs. As a hegig sraegy, he isurace comay may romoe he sale of olices such as auiies or ure eowmes o offse he losses from he life isurace sales. We rese a way of hegig moraliy rae riss i eveloig eowme olicies usig he hege raios of ure eowme o offse he losses from erm life isurace. We also show a hegig sraegy uer a sochasic force of moraliy moel usig he resuls from Malliavi Calculus. Keywors: Moraliy ae iss, Force of Moraliy, Hege aios, Sochasic Moraliy rae Moels, Malliavi Calculus 2
Iroucio Forecasig moraliy imrovemes i he fuure is imora a ecessary i life isurace rouc evelomes. A ieresig observaio is ha moraliy raes for a few ages are imrovig recely. We may refer o a few aers such as Friela (998), Guerma a Vaerhoof (998), Tuljaurar a Boe (998), Tuljaurar (998), a Goss e al. (998), which iscuss he res i moraliy chages a forecasig. For more rece ivesigaios o he res of moraliy imroveme, we refer o he wo reors The P-2 Moraliy Tables from he Sociey of Acuaries (2) a eor of he Sociey of Acuaries Moraliy Imroveme Survey Subcommiee from he Sociey of Acuaries (23). The Sociey of Acuaries couce a suy of uisure esio la moraliy i resose o he eireme Proecio Ac of 994 a resee he P-2 Moraliy Tables. Table. Aualize Moraliy Imroveme for Male Male Age Suy Daa 99-994 Social Securiy 99-994 2-24.3% 25-29 -.7%.99% 3-34 4.83% -.58% 35-39 2.5% -.4% 4-44 -.87% -2.85% 45-49 2.%.6% 5-54 3.63%.47% Feeral Civil Service 988-996 55-59 4.48%.83%.3% 6-64 2.45%.26%.72% 65-69.5%.96%.93% 7-74.75%.6%.22% 75-79.%.8%.59% 8-84.32%.47%.43% 85-89.8% -.49%.78% 9-94 -.8% -.82%.4% 3
Source : The P-2 Moraliy Tables. Sociey of Acuaries. Table 2. Aualize Moraliy Imroveme for Female Female Age Suy Daa 99-994 Social Securiy 99-994 2-24.2% 25-29 3.88% -.59% 3-34 -5.6% -.24% 35-39 -7.5% -2.9% 4-44 -.66% -.42% 45-49 -4.6%.56% 5-54 -5.72%.94% Feeral Civil Service 988-996 55-59 5.27%.9%.92% 6-64 -3.23%.49%.% 65-69.38% -.7%.44% 7-74 -.%.6%.7% 75-79 -.93% -.3%.% 8-84 -.24% -.3%.64% 85-89 -.25% -.49%.3% 9-94.5% -.47%.8% Source : The P-2 Moraliy Tables. Sociey of Acuaries. The ables were eveloe from eeriece o moraliy for uisure esio las. The raes of 992 base year were rojece o year 2 base o a review of hree ses of aa, Social Securiy aa, feeral reiree aa, a he suy aa. To beer observe he res of moraliy rae chages, hey calculae leas-squares regressio lies hrough he logarihms of he raw moraliy raes by year for each quiqueial age grou for each geer for each aa se. The bes-fi log-liear moraliy imroveme res were calculae usig he sloes of hese regressio lies. For each regressio lie, he bes-fi log-liear moraliy imroveme re equals oe mius he ailog of he sloe. Table reses he aualize bes-fi log-liear moraliy imroveme res for male by age a aa source. Table 2 shows he aualize bes-fi log-liear moraliy imroveme res for female by age a aa source. These ables comare rece moraliy imroveme from he aa collece for he suy o emloyees a healhy 4
auias combie (99-994), from Social Securiy aa (99-994), a from Feeral Civil Service aa (988-996). From Table a Table 2, we ca observe ha moraliy imroveme res for male from age 55 hrough age 8 for Social Securiy a Feeral Civil Service are all remarably osiive. Bu he res for females a all ages ehibi mie resuls icluig may egaive a isigifica res. Base o hese observaios hey rojece aual moraliy imroveme raes o P 2 moraliy able oly for male. I is o easy o ifer a secific relaioshi bewee aaie age a moraliy imroveme from he observaios. We o o ow ay reasos of he rece moraliy imroveme. We cao cojecure a quaiaive moraliy imroveme for he fuure eiher. We jus mae a observaio ha moraliy imroveme has bee eeriece i rece years. A we have oice he followig res o moraliy imroveme. () The re of moraliy imroveme is aare for male moraliy raes raher ha female raes. secially moraliy imroveme has bee higher for males ha for females. (2) There is o clear iicaio of moraliy imroveme a aaie ages uer 45 a aaie ages above 85. I is a ieresig observaio ha moraliy imroveme has bee eeriece esecially for he mile-age males for he las ecaes. (3) Almos.% of aual moraliy imroveme has bee realize for males ages 55 8 for Social Securiy a Feeral Civil Service aa. (4) For females, almos.5% -.% of aual moraliy imroveme has bee realize ages 45 64 for Social Securiy aa, a ages 55 84 for Feeral Civil Service aa. (5) For he suy aa, female moraliy has ee o ecrease. 2 Sesiiviy Tes a Hege aio From he revious observaio, we oice ha moraliy imroveme is a re i rece ecae esecially for he mile-age male a shoul be ivesigae by acuaries for he moraliy rae ris maageme. We efie he moraliy rae riss. Defiiio The moraliy rae ris is he ris ha he acual claims associae o eah are more freque ha he aiciae resulig i ueece losses. I his secio, we assume ha he force of moraliy may icrease for secific ages ueecely a ha here may eis moraliy rae riss. If he life able cosruce from a force of moraliy moel reics lower moraliy raes ha acually eeriece by he life isurace olicy holers he he comay will face losses i he fuure. We cosier he chages i moraliy raes, calle he moraliy rae shocs, a wa o chec he remium iffereces whe here eis moraliy rae shocs. For illusraio urose, we use Gomerz s moel, µ ( ) µ () c. (2.) The survival fucio, s (), is calculae from his force of moraliy fucio, s () e s ( µ () c ) s e[ mc ( )], (2.2) 5
where m µ () / log c. For illusraio urose, we assume ha he arameer c i he force of moraliy fucio is icrease by %, i.e. c is chage o c*.. Uer his assumio, we will cosruc he chage life able, calculae he remiums of -year erm isurace a -year ure eowme, chec he gais or losses of remiums, a comue he hege raios. The remiums of -year erm isurace A are icreasig afer he moraliy : rae shoc, a he remiums of -year ure eowme A are ecreasig afer he : moraliy rae shoc. Table 3 shows he chages of remiums. To fi he hege raio, le us eoe he remium of -year erm isurace afer ~ moraliy rae shoc by A, a eoe he remium of -year ure eowme afer : ~ moraliy rae shoc by A. The remium loss amou from -year erm isurace is : ~ Loss A A >, : : a he remium gai from -year ure eowme is ~ Gai A A >. : To offse he losses from he -year erm isurace sales, we have o sell -year ure eowme a he same ime, ha is i is beer for isurace comaies o sell - year eowme isurace olicies ha o sell oly -year erm isurace olicies. The hey ca hege he moraliy rae ris from he icreasig moraliy rae shoc. The hege raio,, is efie o be he face amou of -year ure eowme o be sol o offse he losses from he -year erm isurace of face amou ayable a he e of he year whe () ies. The hege raio is such ha ~ A : A : ( : A : ~ A ), a we have ~ ~ ( A A )/( A A : : : ). (2.3) : Table 4 shows he gais a losses from moraliy rae shoc a he hege raio. We assume µ ()., c.4987, a aual ieres rae i 5%. Age() : Table 3. Chage of Premiums wih Moraliy ae Shoc Chage of Premiums Before Shoc Term Life Pure owme Age() A A : : Afer Shoc Term Life ~ A : Pure owme ~ A 35.283.59598 35.3239.58722 : Thorough ou his aer, we ry o follow he geeral rules for symbols of acuarial fucios. See he Aei 4 i Bowers e al (997). 6
4.3299.58684 4.59.5778 45.4967.5738 45.83.5479 5.7438.553 5.246.58 55.47.52366 55.8995.4589 6.624.4895 6.2892.38544 Table 4. Premium Gais a Losses from Moraliy ae Shoc Age() Losses from Term Life Gais from Pure owme (Hege aio) 35.56.876.2548 4.82.56.2845 45.364.2527.225 5.522.42.2864 55.7948.6475.22749 6.988.965.2425 3 Hegig Sraegy for Moraliy ae iss i Develoig owme Policies We mae a assumio ha he moraliy rae shoc follows a aricular moveme a he arameer c i he force of moraliy fucio is icrease by %. This is jus for illusraio urose a o realisic. We cao reic he eac amou of chage i he moraliy rae shoc, eve hough we ry o observe he res of moraliy imroveme as recisely as ossible. We geeralize his assumio. I his secio, we o o assume he amou of moraliy rae shoc. We jus assume ha here eis moraliy rae shocs. For a give life able, he remium of - year erm life isurace wih face amou ayable a he e of he year whe () ies is A : + + v, (3.) l where v /( + i), i aual effecive ieres rae, + is he eece umber of eahs bewee age + a ++, a l is he umber of olicy holers a age. ε Le us assume ha he force of moraliy µ + a age + is icrease o µ +, by ε (, ) >, The he survival robabiliy µ + ε µ + + will be chage o µ + ε (, ),. (3.2) ε, ε e + ( µ + s ε (, s)) s e µ + ss e ε (, s) s e ε (, s) s <. (3.3) 7
The remium of erm isurace afer moraliy rae shoc ε (, ) is A ~ ~ v + + : l > + + v A. (3.4) : l The amou of loss from erm isurace afer he moraliy rae shoc is ~ Loss A A >. (3.5) : As a hegig sraegy, we cosier -year ure eowme o offse he losses from -year erm isurace. The e sigle remium of -year ure eowme issue o () before moraliy rae shoc is A : v. (3.6) ε Afer he force of moraliy µ + is icrease o µ +, by ε (, ) >, he e sigle ~ remium of -year ure eowme will be ecrease o A :. The amou of gai from -year ure eowme afer moraliy rae shoc is Gai : A : ~ A : ~ >. (3.7) We wa o offse he losses from -year erm isurace wih he gais from - year ure eowme. The hege raio,, is he raio of -year ure eowme o offse he losses from -year erm isurace i -year eowme olicies. Noe ha he raio bewee erm isurace a ure eowme is i raiioal eowmes. Here we wa o evelo ew i of eowmes, calle moifie eowmes, ha ay whe he isure () ies i years or ay whe he isure survives a age +. Defiiio 2 The hege raio for age is he raio of -year ure eowme o offse he losses from -year erm isurace i a moifie eowme, i.e. is he umber such ha Loss A ~ : A : ( ~ ) Gai. (3.8) We wa o calculae for each age. From he efiiio of, we have A ~ + ~ : A + :. (3.9) For coveiece, le us eoe he liabiliy of a moifie eowme before moraliy rae shoc o be L a he liabiliy afer moraliy rae shoc o be L ~, L A + :, (3.) a L ~ A ~ + ~ :. (3.) We wa o fi he hege raio such ha L A + : L ~ A ~ + ~ :. (3.2) Noe ha he remium of raiioal -year eowme is A A + : : 8
a :, where a : is he remium of -year emorary auiy-ue. The liabiliy L of a moifie eowme before he moraliy rae shoc is L A + : A : + A + ( : ) a : + ( ) v + ( ) v. (3.3) By he same way, he liabiliy L ~ afer he moraliy rae shoc is L ~ v ~ + ( ) v ~. (3.4) The ifferece L bewee he liabiliies L ~ a L is L L ~ L ~ v ( ) + ( ) v ( ~ ). (3.5) We wa o fi such ha he ifferece bewee he liabiliies, L, is as small as ossible, L L ~ L ~ v ( ) + ( ) v ( ~ ). Le us aalyze he ifferece L L ~ L bewee he liabiliies. L L ~ L ~ v ( ) + ( ) v ~ ) ( ~ ( ) ( ~ ) v + ( ) v. Defie he fucio ~ f (). (3.6) Noe ha f () ()/. The ifferece bewee he liabiliies is L v f ( ) + ( ) v f ( ). (3.7) If we assume ha he fucio f () is wice iffereiable he, by Taylor s formula wih iegral remaier, he fucio f () ca be eresse as f () f () + f () + ( w) f ( w) w 9
f () + ( w) f ( w) w. Now he ifferece bewee he liabiliies is L v f () [ v ( ) w f ( w) w] + ( ) v f () + ( ) v ( w) f ( w) w where a f () [ v + ( ) v ] [ v ( ) w f ( w) w] + ( ) v ( w) f ( w) w I + II, I f () [ v + ( ) v ], II [ v ( ) w f ( w) w] + ( ) v ( w) f ( w) w. Now le us fi such ha he ifferece L bewee he liabiliies is as small as ossible, L I + II. Oe sraegy is o fi such ha he firs erm I of L equals, I f () [ v + ( ) a he seco erm II of L becomes early, v ], (3.8) II [ v ( ) w f ( w) w] + ( ) v ( w) f ( w) w. From he equaio (3.8) we obai where v + ( ) v (3.9) Ia ) a ] + ( : : ) (3.2) Ia ) ) : (3.2) ( ( ) ( ) : ( Ia ) : Now we have he followig heorem. Ia, (3.22) v.
Theorem The hege raio of -year ure eowme for age o offse he losses from he sale of ui of -year erm isurace (wih face amou ) is aroimaely + ( Ia ) :. (3.23) emar The hege raio i heorem is ieee of he amou of moraliy rae shoc ε (, ). A i is greaer ha for ay age, his may o be rue for some age a we will solve his roblem usig a sochasic moraliy rae moel a Malliavi calculus laer. We ca ierre heorem usig he sesiiviy of he liabiliies L wih resec o he chage of moraliy raes wih he followig assumio. For he survival robabiliy, e µ + ss, we assume ha he sesiiviy of wih resec o he chage of moraliy raes is aroimaely µ e µ + ss. (3.24) Theorem 2 Wih he assumio above, he hege raio of -year ure eowme for age o offse he losses from -year erm isurace is he umber which saisfies L. (3.25) µ Proof From (3.3) we have L A + : v + ( ) v. We have he followig equivaleces L µ v µ + ( ) v µ v ( ) v ( ) ( Ia ) : + ( Ia ) :. emar 2 If he force of moraliy µ µ, a cosa, he we have
From (3.3) we have L A + : + ss e µ e µ. (3.26) v + ( ) v. Now we cosier L as a fucio of boh moraliy rae µ a ieres rae i. The we have he followig equivaleces o he chage of he liabiliies L wih resec o he chage of moraliy rae µ, L (3.27) µ v + ( ) v µ v µ + ( )v µ v ( ) v ( ) ( Ia ) : + Ia ( ) : ecall ha he ifferece L bewee he liabiliies is L I + II, where a If we se I f () [ v + (. ) v ], II [ v ( ) w f ( w) w] + ( ) v ( w) f ( w) w. such ha he he firs erm I of L equals, a I f () [ + Ia ( ) : v + ( L II., ) v ], 2
So he hegig sraegy wih + ( Ia ) : oes o guaraee a erfec hegig. There sill remai moraliy rae riss sice L II may o equal o. Defiiio 3 The resiual ris hegig sraegy wih + ( Ia ) : for age is he amou L II whe we have he (i.e. I ). Now le us aalyze he resiual ris L II [ v ( ) w f ( w) w] + ( ) v ( w) f ( w) w. (3.28) [ v ( f ( ) f ())] + ( ) v ( f ( ) f ()) [ v ( f ( ) f ())] + v f ( ) v f ( ) + f () + f () ( Ia) ( f ( ) f : v + ( ) : Ia + ()) ( Ia) ( f ( ) f : ( Ia) ( f ( ) f : v f ( ) + ( f ( ) Ia) : [ ( f ( ) Ia) v f ( ) : ]. (3.29) Sice we cao eermie he fucio f () eacly, (3.29) oes o give us he resiual ris. Bu we have aoher way o esimae if we assume ha is a measure of he ifferece bewee a o-erfec hegig sraegy a a erfec hegig sraegy. If we se u he liabiliy L as follows L A + : + a :, (3.3) he L, i.e. a erfec hegig sraegy. For he erfec hegig sraegy, he liabiliy L ~ afer he moraliy rae shoc is L ~ L + L A + : + a :. For he o-erfec hegig sraegy, he liabiliy L ~ afer he moraliy rae shoc is L ~ L + L ()) ()) 3
If we se he he resiual ris is where L + I + II A + : + +. + a :, + a : ( ) ( Ia ) : ) ( : + a : + a : Ia + a : [ a : Da ), ( : ( Da ) : Now we have he followig heorem. ( ) : Ia ] ( ) v., by (3.23) Theorem 3 Le us assume ha is a measure of he ifferece bewee he oerfec hegig sraegy wih he hege raio + ( Ia ) : a he erfec hegig sraegy (3.3). We have ha he resiual ris of he o-erfec hegig sraegy is Da ). (3.3) ( : emar 3 Le us fi a erfec hege raio, Comarig L A + : a we have L erfec A + : erfec + a : erfec,. + a :, of -year ure eowme. 4
So a erfec hege raio of -year ure eowme o hege he moraliy rae riss i - year eowme2 is a erfec : + + s :. (3.32) Table 5 shows he hege raios a resiual riss whe we assume ha he force of moraliy follows Gomerz s moel. I he firs colum of Table 5, we assume ha he arameer c i he force of moraliy fucio is icrease by %. I he seco colum of Table 5, we calculae he hege raios, a i he hir colum of Table 5, we calculae he resiual riss. Age() Table 5. Hege aios a esiual iss (% icrease i c ) + Ia ( ) : 35.2548.8792.22639 4.2845.957.22564 45.225.2386.22452 5.2864.248.22283 55.22749.22867.223 6.2425.24689.2656 D ( a) : 4 Sochasic Force of Moraliy usig Browia Gomerz Moel Uil ow we assume ha here eis moraliy rae shocs a fi he hege raio which is ieee of he amou of moraliy rae shocs. Bu he hegig sraegy is o erfec a here remai resiual riss sice he aroimaio usig Taylor s formula may have error erms. To imrove he hegig sraegies we ca cosier sochasic moraliy rae moels a calculae he sesiiviies of liabiliies irecly usig Malliavi calculus. There are several families of eermiisic aalyical laws of moraliy such as De Moivre, Gomerz, Maeham, a Weibull. The Gomerz law of moraliy is give by µ () µ ()c, (4.) where µ () >, c >,. 2 Noe ha a erfec hegig sraegy is L A + : + a :. So a erfec are he ris measures oly whe L. We wa o fi geeralize hegig sraegies. 5
For he sochasic moraliy rae moels, we refer o Lee a Carer (993), Lee (2), Yag (2), Dahl (23), Biffis (25), Milevsy a Promislow (2), Woobury a Mao (997), a Yashi e al. (985). For illusraio urose, we cosier a sochasic law of moraliy, esecially he Browia Gomerz (BG) moel base o Milevsy a Promislow (2)3. The BG force of moraliy rocess is eece o grow eoeially, he variace is roorioal o he value of he hazar rae, a his rocess ever his zero. I his aer, we cosier a simle sochasic force of moraliy rocess, { µ (); }, as follows4, µ () µ ()e(σ Z()), (4.2) where σ >, µ () >, a he yamics of {Z(); } are escribe by he sochasic iffereial equaio, Z() b Z() + W(), (4.3) wih Z(), b, a W() is he saar Browia moio. Noe ha if b, he rocess µ () is he geomeric Browia moio. The rocess Z() ossesses mea reversio wih mea reversio coefficie b. We ca solve (4.3) for Z(), b( s) Z() e W ( s). (4.4) The mea value of Z() is [Z()], (4.5) a he variace of Z() is Var(Z()) [Z() 2 2 b( s) 2 e ] b { e } ( s). (4.6) 2b Noe ha Var(Z()) <, so he rocess has a smaller variace ha W(), a Var(Z()) coverges o as b goes o. The eece value of he sochasic force of moraliy µ () is 2 2 σ e b [ µ ()] µ () e. (4.7) 2 2b Noe ha [ µ ()] coverges o he Gomerz law of moraliy as b goes o, [ µ ()] µ () e 2, as b. (4.8) Le us cosier he yamics of { µ (); }. Usig Iô s lemma, we have µ () σ µ () Z() + 2 σ 2 µ () (4.9) 2 σ σ µ () {b Z() + W()}+ 2 σ 2 µ (), by (4.3), (4.) 3 The choice a jusificaio of sochasic moraliy rae moels may ee o he aa a isurace olicies. 4 Milevsy a Promislow (2) suggess he moel, µ (,) ζ e(ξ + σ Z()), for he force of moraliy. Here we assume ha age for illusraio urose. 6
σ µ () {b l ( ) σ µ () + W()}+ σ 2 µ (), by (4.2), 2 (4.) { σ 2 + b l µ () b l µ ()} µ () + σ µ () W(). 2 (4.2) We will use his formula o fi a hegig sraegy usig Malliavi calculus laer. 5 Malliavi Calculus We assume ha he yamics of a asse {X(); T}, which is a - Marov rocess, are escribe by he sochasic iffereial equaio, X() β (X()) + σ (X()) W(), (5.) where {W(); T} is a Browia moio wih values i. We cosier he rice of a coige claim efie by he followig form V() [ψ (X( ),, X( m )) X() ], (5.2) where ψ is he ay off fucio o he imes < m T. Now, we wa o calculae he rice of he ah eee coige claims a he sesiiviy of V() wih resec o he iiial coiio. We ee o comue a Moe Carlo simulaio of V() a a Moe Carlo esimaor V(+ε ) for a small ε, a esimae he sesiiviy of V() by he followig value V V ( + ε ) V ( ). (5.3) ε Oe way o calculae he sesiiviy of V() is o use he Malliavi calculus. We briefly irouce a few resulig formulas ee i his aer. For more eails o Malliavi calculus a is alicaio o fiace, we may refer Bicheler e al. (987), Fourie e al. (2, 999), a Nualar (995). I is well ow ha, usig Malliavi calculus, he iffereial of V() ca be eresse as V [π ψ (X( ),, X( m )) X() ], (5.4) where π is a raom variable o be eermie. There are may beefis of he above formula. We oly ee oe ime Moe Carlo simulaio o calculae he sesiiviy of V(). So we ca save simulaio a comuig ime. A we o o ee he arallel shif of, + ε, wih arbirary small value of ε. We ca oice ha he weigh π oes o ee o he ay off fucio ψ, which is also a imora avaage of he above formula. Le us cosier a robabiliy sace (Ω, F, P), a a se C of raom variables o he Wieer-sace Ω of he form F F(ω ) f h (( ) W ( ),..., h (( ) W (, (5.5) where ω is a ah i he Wieer-sace Ω, f S( ), S( ) is he se of ifiiely iffereiable fucios o, a h,, h L 2 ( Ω + ). For F C, we efie he Malliavi erivaive DF of F by ) 7
f D F h ( ) W ( ),..., h ( ) W ( ) hi ( ) i, i a.s. (5.6) We efie he orm of F by / 2 F,2 ([F 2 ]) /2 2 + ( ) D F. (5.7) We eoe he Baach sace which is he comleio of C by D,2, wih he orm,2. Le U be a sochasic rocess, U() U(ω,) L 2 ( Ω + ). For ay φ D,2 a fie ω, boh U a Dφ are i L 2 ( + ) H. We wrie U, Dφ U ( ) Dφ, (5.8) for he saar ier rouc i L 2 ( + ). Noe ha his eressio is sochasic, U, Dφ U, Dφ (ω ). (5.9) We calculae [ U, Dφ ] by iegraig over all ω Ω. We efie δ (U), so calle he Soroho iegral, as ajoi o D by [ U, Dφ ] [δ (U) φ]. (5.) Oe of he ieresig facs of Malliavi Calculus is ha he ivergece oeraor δ coicies wih he saar Iô iegral. Le U be a aae sochasic rocess i L 2 ( Ω + ). The we have δ (U) U ( ) W ( ). (5.) For a aae raom variable F D,2, usig he chai rule for Malliavi erivaives Dψ (F) ψ (F) DF. (5.2) By iegraio by ars, we have [ψ (F)] Dψ ( F), DF DF, DF ψ ( F) δ DF DF, DF [ ψ (F)π ], (5.3) where π δ DF DF, DF is a raom variable o be eermie a ψ is a ay off fucio. Noe ha (5.3) is a way of he erivaio of (5.4). For T >, we assume ha F X(T) is a soluio o he sochasic iffereial equaio, X() β (X()) + σ (X()) W(). X ( ) We efie he age rocess {Y() ; } of X(), as he associae firs X () variaio rocess efie by he sochasic iffereial equaio, 8
Y() β (X())Y() + σ (X()) Y() W(), Y(), (5.4) where rimes eoe erivaives. The he Malliavi erivaive of F X(T) is give by D s X(T) σ (X(s)) Y(s) - Y(T), (5.5) for s T a zero oherwise. For X() a F X(T), we wa o calculae he rice sesiiviy wih resec o, [ ψ ( X ( T ))] [ ψ ( X ( T )) Y ( T )]. (5.6) Usig Malliavi calculus, we wa o fi he weigh π δ ( u), for some aae rocess u, such ha [ ψ ( X ( T ))] [ ψ ( X ( T )) π ] [ ψ ( X ( T )) δ ( u) ] [ Dψ ( X ( T )), u ], by (5.) [ ψ ( X ( T )) DX ( T ), u ], by (5.2) ψ ( X ( T )) T ( D X ( T )) u( ), by (5.8) T ψ ( X ( T )) Y ( T ) Y ( ) σ ( X ( )) u( ), by (5.5). (5.7) Comarig (5.6) a (5.7), we ee T Y ( ) σ ( X ( )) u( ), (5.8) a we have a soluio Y ( ) u(). (5.9) Tσ ( X ( )) So we ca fi a weigh π δ ( u) whe F X(T), π δ ( u) T Y ( ) W ( ), by (5.). (5.2) T σ ( X ( )) By he similar way, we ca fi a weigh π δ ( u) whe F is he mea value of he rocess {X(); T}, F T X ( ). (5.2) A weigh π δ ( u) is give by π ( u) 2 δ 2Y ( ) T δ Y ( s) s. (5.22) σ ( X ( )) 6 A Hegig Sraegy usig BG Sochasic Moraliy Moel a Malliavi Calculus Now we begi wih a sale of erm isurace. Uer a sochasic force of moraliy moel, he e sigle remium of -year erm life isurace wih face amou ayable a he e of he year whe () ies is 9
A : Ω + v ( ω ) q+ ( ω), (6.) where v /( + i), i aual effecive ieres rae, Ω is he se of scearios, a ω is a sceario, ω Ω. For each >,, a ω Ω, he survival robabiliy (ω ) is (ω ) e µ ( s, ω) s, (6.2) where µ (s,ω ) is he force of moraliy a age +s o he sceario ω, a he eah robabiliy q + (ω ) is q + (ω) + (ω ). (6.3) We rewrie (6.) as Ω + + A : v ( ω ) v + ( ω). (6.4) As a hegig sraegy, we cosier he sales of -year ure eowme o offse ay losses from -year erm isurace. The e sigle remium of he -year ure eowme issue o () is A : Ω [ v ( ω)]. (6.5) We wa o offse he losses from -year erm isurace sales wih he sales of - year ure eowme. Le us efie he hege raio,, o be he umber of -year ure eowme o be sol o offse he losses from he sales of -year erm isurace olicies. Le us eoe he liabiliy o be L, L A + :. (6.6) We wa o fi he hege raio such ha he sesiiviy of he liabiliy wih resec o he moraliy rae chages equals, L, (6.7) µ where µ µ (,ω ). The liabiliy L is L A + : Ω + + v ( ω ) v + ( ω) + Ω [ v ( ω)]. (6.8) The sesiiviy of he liabiliy wih resec o he moraliy rae chages is L Ω + + µ µ v ( ω ) v + ( ω) + Ω [ v ( ω)]. (6.9) µ 2
For a fie age, we assume ha µ (,ω ) µ (+,ω ) a he yamics of he force of moraliy are give by5 µ () β ( µ ()) + σ ( µ ()) W() { σ 2 + b l µ () b l µ ()} µ () + σ µ () W(). (6.) 2 µ () The age rocess {Y() ; } of µ () is he associae firs variaio µ rocess efie by he sochasic iffereial equaio, Y() β ( µ ())Y() + σ ( µ ()) Y() W(), Y(), (6.) where rimes eoe erivaives. For a give >, he eece value of he survival robabiliy is Ω Ω [ (ω)] e µ ( s, ω) s Ω [ ψ ( F(, ω)) ], (6.2) where F (, ω) µ (, s ω) s, (6.3) a ψ ( F (, ω)) e( F (, ω) ). (6.4) Usig he resul of Malliavi Calculus, we have Ω [ ψ ( F(, ω)) ] Ω [ ψ ( F(, ω)) π (, ω) ], (6.5) µ where he weigh π δ ( u) is give by 2 π (, ω) δ ( u) 2Y ( s, ω) δ Y ( l, ω) l σ ( µ (, )) s ω 2 2Y ( s, ω) Y ( l, ω) l W ( s). (6.6) σ ( µ ( s, ω)) Now he hege raio such ha he sesiiviy of he liabiliy wih resec o he L moraliy rae chages equals,, is eresse as follows, µ Ω v + { ( ω) π (, ω) ( ω) π ( +, ω) } Ω [ v + ( ω) π (, ω)]. (6.7) 7 Numerical amles 5 Here we cosier he yamics of he force of moraliy uer he Browia Gomerz (BG) moel. The yamics may chage accorig o he choice of he moraliy rae moels. 2
We have couce a simulaio o calculae he hege raio uer he BG sochasic moraliy moel. We use he arameers b.5, i 5% for he aual Ω effecive ieres rae, a σ.2 a.23. We use µ () [ µ (, ω)] as he iiial values. For he year erm isurace, we show he hege raios, (6.7), of he ure eowme i he Table 6. Table 6. Hege aios σ \Age 35 4 45 5 55 6 σ.2.8378.9778.595.3239.76686 2.72489 σ.23.8965.9727.8998.299.65499 2.42923 Noe ha he hege raios i Table 5 a Table 6 are iffere. We o o ee o use ay aroimaios so here are o resiual riss whe we use a sochasic moraliy moel a Malliavi calculus. This is a imroveme of he hegig sraegies. From Table 6 we ca oice ha he hege raios grow raily as age icreases, so we ee more ure eowmes o hege he moraliy rae riss i he moifie eowmes for ol ages. 8 Coclusio We have mae a observaio ha moraliy imroveme has bee eeriece esecially for he mile-age males for he las ecaes. If here eis moraliy rae shocs he isurace comay may face losses from life isurace sales. As a hegig sraegy, he isurace comay may evelo moifie eowmes. We rese he hege raios of ure eowmes o offse he losses from erm life isurace i eveloig moifie eowme olicies. We also show he hegig sraegy uer he Browia Gomerz sochasic force of moraliy moel usig he resuls from Malliavi Calculus. efereces Beema, J.A., a Fuellig, C.P. (99). Ieres a Moraliy aomess i Some Auiies, Isurace: Mahemaics a coomics, Vol. : 275-287. Bejami, B., a Pollar, J.H. (98). The Aalysis of Moraliy a Oher Acuarial Saisics, Heiema, Loo. Bicheler, K., Gravereau, J., a Jaco, J. (987). Malliavi Calculus for Processes wih Jums, Goro a Breach Sciece Publishers. Biffis,. (25). Affie Processes for Dyamic Moraliy a Acuarial Valuaios, Isurace: Mahemaics a coomics, Vol. 37: 443-468. 22
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