Internal model in life insurance : application of least squares monte carlo in risk assessment


 Erick Jordan
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1 Ieral model lfe surace : applcao of leas squares moe carlo rs assessme  Oberla euam Teugua (HSB)  Jae Re (Uversé yo, HSB)  rédérc Plache (Uversé yo, aboraore SA) 04. aboraore SA 50 Aveue Toy Garer yo cedex 07 hp://www.sfa.fr/la_recherche
2 ITERA ODE I IE ISURAE: APPIATIO O EAST SQUARES OTE ARO I RISK ASSESSET Oberla euam T. Verso.0 Jae Re rédérc Plache Uversé de yo  Uversé laude Berard yo ISA Acuaral School HSB Assuraces  Geeral Rs Assessme ad Asse ably Absrac I hs paper we show how prospecve modellg of a ecoomc balace shee usg he leas squares oe arlo (S) approach ca be mplemeed pracce. The frs am s o revew he covergece properes of he S esmaor he coex of lfe assurace. We pay parcular aeo o he praccales of mplemeg such a echque he real world. The paper also preses some examples of usg he valuao fuco calbraed hs way. KEYWORDS: Solvecy II, Ecoomc capal, Ecoomc balace shee, rs maageme, rs appee, lfe surace, sochasc models, leas squares oe arlo Résumé obje de ce arcle es de préseer la mse e œuvre opéraoelle de la modélsao prospecve d u bla écoomque e ulsa l approche de «eas Squares oe carlo» (S). e premer objecf de ce arcle es de rever sur les propréés de covergece de l esmaeur S das le coexe de l Assurace ve. e arcle sse sur les problémaques opéraoelles lées à la mse e œuvre d ue elle echque. Il présee auss des exemples d ulsao de la foco d évaluao as calbrée. OTSES: Solvablé II, apal écoomque, Bla écoomque, geso des Rsques, Rs Appee, Assurace Ve, odèles sochasques, eas Squares oe arlo. oac: oac: orrespodg auhor. oac: Isu de Scece acère e d Assuraces (ISA)  50 aveue Toy Garer yo edex 07 race. Geeral Rs Assessme ad Asse ably HSB Assuraces: 0 esplaade du Gééral de Gaulle mmeuble oeur Défese 9400 ourbevoe a Défese 4  race PUBI 
3 OTETS ITRODUTIO... 3 RO ESTED SEARIOS TO EAST SQUARES OTE ARO DESRIPTIO O THE S ETHOD EXAPES ISSUES WITH THE EAST SQUARES OTE ARO ETHOD OVERGEE O THE S HOIE O UBER O SIUATIOS AD UBER O REGRESSORS REGRESSIO BASIS UTIOS REGRESSIO ODE APPIATIO PRESETATIO REGRESSIO QUAITY AAYSIS O THE S UTIO OUSIO APPEDIES REEREES PUBI  APPIATIO O S I RISK ASSESSET Page
4 ITRODUTIO Oe of he ey advaces of Solvecy II over Solvecy I s ha surace compaes asses ad lables mus be valued a ecoomc or "far value" (see Solvecy II, arcle 75). ar value s he amou for whch a asse could be exchaged or a lably seled bewee owledgeable, wllg pares a arm's legh rasaco. The valuao prcples he surace coex are se ou by Wührch ad al. [008]. The Solvecy II sadards hus ae a ecoomc vew of he balace shee ad roduce a harmozed Europea vew of ecoomc capal ha represes he mmum capal requreme o gve a surace or resurace compay 99.5% cofdece of survvg a suao of ecoomc ru o a oeyear horzo. Ecoomc capal ca be esmaed usg eher a modular approach (sadard formula) or by (paral) eral modellg. The laer volves a fer aalyss of he compay s rss ad requres he dsrbuo of capal cosumpo o be defed over a oeyear horzo. Also, Solvecy II ecourages compaes o develop a more dealed approach o rs maageme. Arcle 45 of he Solvecy II Drecve ses he rules for hs eral rs maageme. The framewor for persoalsed rs maageme s he Ow Rs ad Solvecy Assessme (ORSA), whch s based o he defcao, defo ad moorg of he compay's ey rs dcaors. I addo, for facal reporg purposes, surace compaes value her busess usg he Prese Value of uure Profs (PVP). Ths s based o he porfolo of surace polces wre a he calculao dae, ag o accou all he coracual oblgaos ha flow from hem ad cludg he value of ay embedded opos. ally, surace compaes mus draw up a busess sraegy for a defed horzo. Ths sraegy mus projec over he forecas horzo, based o a realsc se of assumpos:  he ecoomc balace shee ad solvecy capal requreme,  IRS prof before ax ad IRS Balace Shee. There are, he, may ad varous ssues relag o a beer udersadg of rs. Valuaos a =0 ca be based o he oe arlo approach ad geerally pose o major echcal problem. However, he forwardloog projeco s much rcer ad rases real challeges (see Deveau ad osel [009]). There s o closed formula for valug opos surace lables ad he ecoomc value of he balace shee depeds o he formao avalable a he me of valuao: s herefore radom. The purpose of hs paper s o show how prospecve modellg of a ecoomc balace shee usg he leas squares oe arlo (S) approach s mplemeed pracce, mag possble o esmae he prospecve value of s compoes. The S echque s already used he facal world o value exoc opos (see ogsaff ad Schwarz [00]). The frs am s o aalyse he covergece properes of he S esmaor he coex of surace as dscussed by Bauer ad al. [00]. We pay parcular aeo o he EIOPA al Repor o Publc osulao o. 3/009 o he Proposal for Gudeles o orward oog Assessme of Ow Rss (ref.: EIOPA/3/44) PUBI  APPIATIO O S I RISK ASSESSET Page 3
5 praccales of mplemeg such a echque he real world. The paper also preses examples of he use of he evaluao fuco calbraed hs way. Seco revews he dffcules of mplemeg esed scearos ad summarses possble soluos, cludg S echques. Seco 3 descrbes he S mehod, dscusses he covergece properes of he approach ad emphasses he ssues wh praccal mplemeao. Seco 4 preses a applcao of S o he mos commo savgs corac sold race: he euro fud. PUBI  APPIATIO O S I RISK ASSESSET Page 4
6 Techcal provsos Basc Ow uds RO ESTED SEARIOS TO EAST SQUARES OTE ARO A beer udersadg of porfolo rs meas beg able o acpae how he ecoomc balace shee wll reac o cera defed rs facors. or sace, for ay compoe (e asse value, bes esmae, ec.) of he ecoomc balace shee oppose, value a me s wre: where: T E f D, u Q u u represes he vecor of rs facors (yeld curve, equy dex, lapses rae, ec.) a me, Q s a rseural measure a me. f u s he cash flow assocaed wh compoe a he me u u, D, s he dscou facor for cashflows over he perod u. Asse Asse Realsc balace shee ables Shareholders' fuds Prese value of fuure profs Rs arg Opos ad Guaraees Bes Esmae (ceray) Deferred ax e Asse value Oher ables I pracce: here s o aalyc formula for hs codoal expecao as he erms of surace coracs coa a umber of embedded opos (rae guaraee, prof sharg, surreder opo, ec.): The flows f u are pahdepede, There are eracos bewee lables ad asses. ca be esmaed usg a oe arlo approach by: K f D, u K K T u u Iferga [03] shows ha K s a covergece esmaor for calculao of he bes esmae. =0 =. =T real world Scéaro () Prcg Scearos Usually, hs mehod s used o value he balace shee a =0: The ecoomc value of he asses s observed o he mare, PUBI  APPIATIO O S I RISK ASSESSET Page 5
7 The lables are valued by oe arlo smulao usg he valuao prcples defed by EIOPA, IRS or eral gudace. Whe, ad become radom varable. To deerme he dsrbuo of, we ca use he esed Scearos approach: Whe usg oe arlo mehod o esmae a large, umber of,... we al abou esed Scearos. real world scearos (ouer scearos) K prcg scearos (er scearos) =0 = =T Advaages: Gves a al esmae of he emprcal dsrbuo of he ecoomc balace shee, Gves precse resuls for he realworld suaos aalysed. Dsadvaages: Heavy demads o processg resources ( x K smulaos):. calculag me: moh for,000 x,000 a 3 secods per scearo,. sorage space: 50 GB for,000 x,000, Robusess of al dsrbuo: jus 5 scearos deerme VAR 99.5% for,000 x,000 scearos o formao o pos bewee he ouer scearo. Several soluos have bee developed o ge roud hese dffcules:  Revele [0] ses ou he replcao approaches. These echques sruggle wh he complexy of lfe surace coracs (log durao, redempo opos, profs sharg, ec.),  Deveau ad osel [009] descrbe a accelerao algorhm ha ca be appled whe usg esed scearos o calculae Solvecy II ecoomc capal. The algorhm wors by reducg he umber of ouer scearos. They are parcularly eresed al dsrbuo. The focus of hs mehod s o esmag ecoomc capal ad s hard o apply o he mechacs of rs maageme ad porfolo valuao. PUBI  APPIATIO O S I RISK ASSESSET Page 6
8  euam ad Plache [0] are eresed cug he umber of er scearos ad show ha valuao error ca be cu o less ha 5% whe secodary (or er ) scearos are replaced by a few wellchose compose scearos.  Bo ad al. [04] ry o approxmae he mare value of lables usg aalycal formulas,  Bauer ad al. [00] are he frs o se ou a dealed ad documeed applcao of S lfe surace. However, he auhors ecouer problems showg he covergece of he S esmaor due o he chage probables a =. The auhors also measure he mpacs o a fcoal porfolo ad fal o address he problems applyg hese echques o a realworld porfolo (complexy of lables ad asses, processg me, sorage space, calculao ools, ec.). ally, he mechasm for selecg he regresso base s o explaed deal.  Aoher approach ha could overcome some of he problems wh esed sochascs s o erpolae he resuls of he esed scearos: socalled curve fg. Of all hese echques, he S approach s emergg as he sadard for eral modellg he lfe surace dusry, for several reasos:  ca esmae he value of ecoomc capal,  ca be used o maage rs (ORSA): rs hedgg, calculao of rs appee dcaors,  s used A sudes: o deerme opmal allocao, projec porfolos, ec. Below, we prese he applcao of S for a porfolo of lfe surace coracs. PUBI  APPIATIO O S I RISK ASSESSET Page 7
9 3 DESRIPTIO O THE S ETHOD The core dea of he leas squares oe arlo (S) mehod s o mmc he behavour of he lables usg a fuco ha cludes all argeed rs facors as pus (ecoomc ad/or oecoomc varables). The precse behavour (or prcg) fuco of he lables s uow. I s approxmaed by a approach based o he Taylor seres mehod. Ths echque approxmaes he behavour fuco by a lear combao of he bass fucos appled o he argeed rs facors: * s he umber of regressors,,,, represes he vecor for he rs facors a me, *, oal umber of argeed rs facors,, bass ),, represes a seres of fucos (he regresso represes he mpac of he erm o he quay. The heorecal jusfcao for hs approxmao derves from he properes of codoal T expecaos Hlber space. EQ f u D, u s a eleme a Hlber u space ca be expressed as a lear combao of a couable se of orhogoal, ad fucos measurable where: :,,, are real umbers,,, a orhogoal bass. * If we choose a srcly posve aural eger, we ca wre: or,..., s defed by : PUBI  APPIATIO O S I RISK ASSESSET Page 8
10 PUBI  APPIATIO O S I RISK ASSESSET Page 9 Wh: , , ,,,,  R,,, We have 0 whe, we arrve a he approxmao. The coeffces are he esmaed wo sages:. Usg he oe arlo mehod we geerae realsaos of he radom varables 3...,,,. we calculae by eas squares regresso of arg : The resul of hs opmsao programme s he OS esmaor:,,, Wh: ,,...,, ,...,, The S fuco ca be wre S, 3 The are realsaos of he radom varable T u u u D f z, such ha Q z E
11 3. EXAPES Example : S prcg a Europea opo: Here, we cosder a ahemoey Europea call opo wh maury T=. We assume he uderlyg s a geomercal Browa moo, he rsfree rae s cosa a 3.5% ad he uderlyg s mpled volaly s 30%. The prce of hs Europea opo s derved by he BlacScholes formula: S d K exp rt d d avec d d T S l r K T T The able shows wo examples of S fucos for he opo prce a =: (x) S S (=500,=3) (=000,=) x^ x^ x^ The chars below compare he S resuls o he BlacScholes calculao. 50% g : Prce equy opo (=500,=3) 50% g : Prce Equy Opo (=000,=) 00% 00% g Daa S (=3) g Daa Real Prce S (=) 50% 50% Real Prce Opo Prce 00% Opo Prce 00% 50% 50% 0% Uderlyg Asse 0% Uderlyg Asse oe he greaer precso of he S esmaor whe he umber of fg daa s large. Example : ogsaff ad Schwarz [00] use S o value Amerca opos. They show he covergece of he S approach valug complex opos. PUBI  APPIATIO O S I RISK ASSESSET Page 0
12 3. ISSUES WITH THE EAST SQUARES OTE ARO ETHOD I pracce, applyg S o lfe surace rases a umber of ssues: Idefyg characersc rs facors : ay rs facors may affec a surace frm s ecoomc balace shee: o Ecoomc rss: rae rs, equy rs, propery rs, cred rs, volaly, ec. o oecoomc rss: apse rs, moraly rs, expese rs, ec. However, s eough o arge a lmed umber of rs facors. The S fuco derved hs coex gves a esmae of lables whe oly he argeed rs facors are radom: s herefore ofe called a paral eral model. Wha regresso bass o use? The choce of regresso bass fuco ca have oeglgble effecs o al dsrbuo. I a oedmesoal evrome, here are may fucos ha have he propery of orhogoaly: hebyshev, Herme, aguerre, egedre, ec. (see Abramowz ad Segu [964]). I geeral, we aalyse he mpac of > rs facors. To correcly measure he eraco of rs facors o he balace shee s mpora o specfy he form of he orhogoal fucos defed muldmesoal space. How o choose he opmal dmeso o gve he bes approxmao of? I pracce, may prove mpossble o calbrae he S fuco whe he umber of regressors s very hgh. Wha mehods ca be used o deerme opmal coeffces? Ths queso ouches o wo ssues: o Defo of fg daa or calbrao: how may calbrao scearos does ae o ge a bes approxmao of? geerao of he fg daa: oe arlo or quasoe arlo o accelerae covergece of he S smulao, o Opmsao of he regresso fuco: bacward, forward, sepwse, ec. PUBI  APPIATIO O S I RISK ASSESSET Page
13 3.3 OVERGEE O THE S 3.3. OVERGEE O THE S UDER THE RISKEUTRA EASURE Q S We show ha, coverges The covergece s demosraed wo sages: S, owards uder he rseural measure The frs sage s based o he covergece of he oe arlo esmaor. I ca be smply demosraed usg he law of large umbers ad properes of orhogoal fucos (see appedx). Q. The secod sage s obvous. See also Bauer e al [00] seco OVERGEE O THE S UDER THE HISTORIA EASURE P I surace, he dsrbuo of rs facors ha he dsrbuo of s obaed uder s uder he hsorcal measure P. P, mplyg T E f D, u Q u u f f P Q+ hage of measure Q Q+, f f PUBI  APPIATIO O S I RISK ASSESSET Page
14 We herefore eed o esablsh he covergece properes uder he hsorcal measure. Bauer ad al. [00] specfy ha because of he chage measure a me, he covergece of S, oward uder he measure P cao be guaraeed. S I hs seco we wll show ha, coverges probably oward uder he measure P. True, covergece probably s weaer ha covergece, bu s sll sroger ha covergece dsrbuo ad eough o demosrae he relevace of he S approach o valug ecoomc capal. S Propery:, coverges probably oward uder he hsorcal measure P Proof: S,, I seco 3.3., we showed ha S mples ha, coverges probably oward S, uder he measure Q. Ths Q. uder he measure, S, Q lm Q A 0 wh A w; w w ad 0., The probably measures Q ad P are equvale f, f 0, dp f dq a radom varable of expecao ( f dq ): we have, A f A dq P.,, f f, ebesgue s domaed covergece heorem mples ha: A, lm P A, lm 0 f lm f 0 dq S, coverges probably oward A, A, dq dq uder he measure P, ad f 3.4 HOIE O UBER O SIUATIOS AD UBER O REGRESSORS *.e. e : he S fuco ha approxmaes he value of compoe of he ecoomc balace shee a me s: S, S We showed he prevous seco ha, coverges oward ow, we are eresed fxed. * S e, wh a error bewee., ad PUBI  APPIATIO O S I RISK ASSESSET Page 3
15 S,,. I absolue erms, hs fuco does o have a opmum. Also, excep parcular cases (facal asses) we do o ow, so, o quafy 4 S. We ca esmae by measurg he devao bewee s hard, ad he resuls of esed scearos. Bu hs approach suffers from he major dsadvaages of he esed scearos approach (see seco ). S I pracce, we measure he devao bewee he, fuco ad a seres of values for : we call hese valdao scearos. The valdao scearos are chose from he dsrbuo se of. I geeral, some wey pos are eough o measure he qualy of he S fuco. The char below shows he esmao error (sum of square of devaos) for he prce of a Europea opo from usg he S mehod: SSE o valdao scearos bps = =8 =3 =0.5 = =8 =5 =30 =8 oe ha he covergece s faser whe s very large:  Whe =0,000, he sum of he square of prcg errors falls o 0.04% for polyomal degree =3.  Whe =,000, he sum of he square of errors s always more ha 0.3% rrespecve of he degree of he polyomal. 4 The value T s o a esmaor of, as of he radom varable z f D, u ad o values E z u u Q are realsaos PUBI  APPIATIO O S I RISK ASSESSET Page 4
16 Sum of errors So, we fx a maxmum accepable error S, Emax Emax E max E, ad ry o deerme max such ha E, max Emax. To acheve hs error we mus fx as hgh as possble:  Depedg o calculao ad sorage capacy: geeral 00,000 scearos provde good covergece of he S resul,  We combe hs wh varace reduco echques o acheve a beer covergece propery. Havg fxed he umber of smulaos, we choose he opmal polyomal degree by measurg he sum of he squares of errors observed he valdao scearos: "Opmal" polyomal degree Polyomal degree 3.5 REGRESSIO BASIS UTIOS 3.5. OEDIESIOA ORTHOGOA BASIS UTIOS ogsaff ad Schwarz [00] propose usg orhogoal polyomals such as aguerre, egedre or hebyshev polyomals, weghed wh a fallg expoeal erm (o preve he polyomals explodg o fy). A sysem of x fucos s orhogoal over he erval b w x f verfes he followg propery: b a w x x x m 0, s m dx, s m a, wh weghg fuco PUBI  APPIATIO O S I RISK ASSESSET Page 5
17 Orhogoal bass Herme hebyshev, egedre, Table : Examples of orhogoal fucos a, b w x x, exp x / x / x / x He x T e d dx e cos cos d P x x aguerre 0, x : m 0 : m 0!! dx x e d x exp x e x dx Abramowz ad Segu [964] prese oher examples of orhogoal fucos. The bass fucos used our sudy are weghed by her weghg fuco o preve hem explodg o fy UTIDIESIOA ORTHOGOA BASIS UTIOS I geeral, we aalyse he mpac of > rs facors. I hs coex, s mpora o specfy he form of he orhogoal fucos defed. I a Browa evrome where he rs facors beg aalysed are depede, a smple approach s o geeralse he orhogoal fucos defed. Gve a sysem of orhogoal fucos x x, x over b *,,..., we have: I dmeso, s clear ha: b a P P, m ' ', m P x wx wx,,... dx dx : a, b P,,... 0, f, f o ' ', m, m a,, for I he Browa uverse he weghed Herme fuco whe x He x verfes exp x /. he orhogoal propery by vrue of he form of he weghg fuco PUBI  APPIATIO O S I RISK ASSESSET Page 6
18 I remas o be show wheher hs resul ca be geeralsed, as he rss aalysed are o always muually depede ad he evrome s o ecessarly Browa. I our sudy, we are however gog o use he se of fucos P,...,, j for j * ad a fxed oedmesoal bass fuco x x,, x. The able below shows a example applcao dmeso 3 for he frs 3 erms he uvarae fuco. Table 3: orhoormal bass fuco of dmeso 3 3 Rs facors ew orhoormal bass fuco, up o x x x3 3 degrees 0 0 P_0_0_(x,x,x3)=(x3) 0 0 P_0 0(x,x,x3)=(x) 0 0 P 0_0(x,x,x3)=(x) 0 P_0 (x,x,x3)=(x)(x3) 0 P 0_(x,x,x3)=(x)(x3) 0 P 0(x,x,x3)=(x)(x) 0 0 P_0_0_(x,x,x3)=(x3) 0 0 P_0 0(x,x,x3)=(x) 0 0 P 0_0(x,x,x3)=(x) 0 P_0 (x,x,x3)=(x)(x3) 0 P_0 (x,x,x3)=(x)(x3) 0 P 0_(x,x,x3)=(x)(x3) 0 P 0_(x,x,x3)=(x)(x3) 0 P 0(x,x,x3)=(x)(x) 0 P 0(x,x,x3)=(x)(x) P_0_0_3(x,x,x3)=3(x3) P_0_3_0(x,x,x3)=3(x) P_3_0_0(x,x,x3)=3(x) P (x,x,x3)=(x)(x)(x3) We fd ha here are 9 erms hs muldmesoal fuco. The able below shows he umber of erms he muldmesoal fuco as a fuco of he umber of rs facors ad degree of he oedmesoal fuco: PUBI  APPIATIO O S I RISK ASSESSET Page 7
19 Table 4: maxmum umber of regressors axmum umber of regressors Rs acor Polyomal degree REGRESSIO ODE I he prevous seco, we saw ha he umber of erms of he S fucos could be very hgh he surace coex. There are varous ecoomerc echques for selecg he bes model from a se of possble caddaes. or sace (see Hocg [976]):  Seleco (forward): Sar wh a model coag oly he cosa, he add oe varable a each sage: o a each sage, selec he mos sgfca varable, o repea ul all he mos sgfca varables have bee seleced.  Elmao (bacward): Sar wh a model coag all regressors ad elmae oe a each sage. o a each sage, elmae he leas sgfca varable, o repea ul all he leas sgfca varables have bee elmaed.  bdrecoal: a combao of forward/bacward approaches (sepwse). Sar wh a model coag oly he cosa. o arry ou a forward seleco, leavg ope he possbly of droppg ay of he varables ha becomes sgfca a each sage. o Repea ul all he varables seleced are sgfca ad all he elmaed varables are sgfca. There are may crera for sgfcace ( R, AI, BI, p, ec.). Bauer ad al. [00] show ha allow s crero wors well a S coex as gves he bes resuls he p eve of heerosedascy of resduals. The chars below show he umber of regressors obaed usg he dffere cofguraos aalysed: PUBI  APPIATIO O S I RISK ASSESSET Page 8
20 umber of emrs umber of erms umber of erms umber of erms S  ull odel S  Bacward Polyomal degree Polyomal degree ordare legedre wag herm chebyshev ordare legedre wag herm chebyshev S  orward S  Sepwse Polyomal degree Polyomal degree ordare legedre wag herm chebyshev ordare legedre wag herm chebyshev Table 5: umber of regressors of he fg fuco based o he opos aalysed or hese four model seleco mehods, he umber of regressors creases wh he maxmum degree of he oedmesoal bass fuco. The sepwse mehod resuls he fewes regressors. The bacward ad full model regresso models explode f he polyomal degree ad umber of rs facors are hgher ha 7. PUBI  APPIATIO O S I RISK ASSESSET Page 9
21 4 APPIATIO I hs seco we loo a he praccal mplemeao of S mehod. 4. PRESETATIO albrao of he S fuco s doe by deermg he coeffces: regresso bass,,, fxed such ha for a. I pracce, hs s a mulsage process: Sage : smulae a umber of ouer scearos: fg scearos, Sage : smulae oe er scearo for each ouer scearo ( pracce, for faser covergece of Ss, we smulae ahecal er scearos). real world scearos (ouer scearos) (or ) prcg scearos (er scearos) =0 = =T Sage 3: usg he A model, value each balace shee em by D for each er scearo, Sage 4: choose a regresso bass ad carry ou a lear regresso ha gves leas squares bewee he A resuls ad he seres of rs facors seleced, Sage 5: es he fuco's valdy agas he valdao scearos. Schemacally, he process has he followg archecure; Regresso aalyss g scearo A resuls : g scearo ESG A odel Regresso model S fuco Valdao scearo A resuls : Valdao scearo g qualy: Valdao scearo PUBI  APPIATIO O S I RISK ASSESSET Page 0
22 4.. DESRIPTIO O THE PORTOIO: HOIE O RISK ATORS TO AAYSE I race he lfe surace mare geeraed reveue of 08.8 bllo 0 mag he fourhlarges he world ad he secodlarges Europe 5. The oal value of lfe coracs ousadg race was,458.3 bllo a 3//0. ore ha 85% of hs s made up of euro fuds. Euro coracs are savgs coracs ha coracually guaraee he capal vesed. The sums pad cao fall value ad are creased each year by a reur, he mmum guaraeed rae (Rae Guaraee) plus a prof sharg bous (based o he echcal ad facal reurs o he asses represeg regulaed commmes). The surer mus pay ou a leas 85% of facal gas ad 90% of echcal profs o polcyholders (Profsharg opo). I addo, come eared each year s defvely accrued. The surer effecvely guaraees he accrued value of capal a all mes (Surreder opo). To cover hese regulaed commmes, rech lfe surers are vesed he followg asse classes (Source: SA 6 ):  OED sovereg deb: 3%,  orporae bods: 37%,  Eques, propery, vesme fuds ad oher asses: 5%,  oey mares: 6%. Euro coracs are affeced by mare ad echcal rss. I hs paper, we aalyse he mpac of he followg mare rss:  Rae rs, rse or fall,  Rs of a fall eques mares,  Rs of a fall he propery mare. oe ha s smple o exed he echque preseed here o oecoomc rss operaoally. However, he covergece properes have ye o be show, oably because of he chage probably a me ad he defo of he rseural dsrbuo of oecoomc facors. 4.. A ODEIG Resuls are based o he A model used by HSB Assuraces Ve. Ths sofware mees he surer s am of havg a powerful sochasc modellg ool wh easly audable resuls. Ths model s he referece ool used all A wor whch allows us o value he ecoomc balace shee ad s varous sesves. Besdes sochasc smulaos, provdes he followg fucoales: 5 see. 6 hps://www.ffsa.fr/ses/jcms/p_9377/fr/lassuracefracasee03?cc=f_7345 PUBI  APPIATIO O S I RISK ASSESSET Page
23 complace wh surace rules by carryg ou accoug closes, reproducg he surer s arges (cludg paymes o polcyholders), he opo of geerag sresses ha mpede he surers arges, he opo of usg sochasc scearos o value opos embedded he coracs ESG: ITTIG AD VAIDATIO SEARIOS We deermed 00,000 fg scearos. To accelerae covergece of S fucos we sued Sobol s quasoe arlo (Q) echque o smulae 50,000 ouer scearos coupled wh ahecal varables for he er scearos. Sascs for he fg daa are summarsed below: Sascs daa Yeld urve shoc (bps) apal Idex shoc (%) fg Shor rae og rae Equy Idex Propery Idex I % 00% AX % 50% EA 3 4 0% 30% STD % 70% The valdao scearos were chose o measures he dvdual effecs ad eracos of he arge varables o he ecoomc balace shee. They are show below: Valdao Yeld urve shoc (bps) apal Idex shoc (%) scearo Shor rae og rae Equy Idex Propery Idex % 0% % 50% % 80% % 80% % 50% % 0% % 0% % 0% % 50% % 50% % 70% % 80% % 80% % 0% 4..4 REGRESSIO TOOS The S fuco was calbraed usg 00,000 resuls ae from he A model. The regresso ool used s he reg procedure he SAS sofware pacage. PUBI  APPIATIO O S I RISK ASSESSET Page
24 We aalysed 0 S approaches, composed of he followg combaos of regresso bass fucos ad seleco mehods: Regresso bass/seleco mehod Ordary aguerre egedre Herme hebyshev ull odel x x x x x orward x x x x x Bacward x x x x x Sepwse x x x x x 4. REGRESSIO QUAITY 4.. DETAIED EXAPES aguerre fuco/full model The able below shows a example of he S fuco obaed. The oedmesoal bass for he regresso s he aguerre fuco. The maxmum degree s ad he balace shee em beg modelled s e asse value (AV). Degree of aguerre fuco Term Shor rae og rae Equy Idex Propery Idex oeff. Iercep P P P P P P P P P P P P P P o seleco mehod was used o he regressors ad he S fuco herefore has 4 erms. oe, however, ha he coeffces of erms P5 ad P are zero. The char shows he qualy of he AV fg o he valdao scearos. PUBI  APPIATIO O S I RISK ASSESSET Page 3
25 Valdao Scearos g uco The average valuao error o AV for hs S fuco s 0.7%. The maxmum error s 6%. Ordary polyomal/sepwse The char below shows he qualy of he regresso for aoher S fuco used o smulae AV. Here, we used a sepwse seleco mehod ad he oedmesoal regresso bass fuco was a ordary polyomal wh a maxmum degree of 5: Valdao Scearos g uco We fd hs gves a beer qualy of regresso. The average relave error s 0.0 %. The maxmum relave error s.8% of he value of he valdao scearo. PUBI  APPIATIO O S I RISK ASSESSET Page 4
26 The char below shows how hs S fuco wors for ecoomc capal: 0% g desy of dela AV 5% Valdao scearos 0% 5% Ecoomc apal 0% oe ha he valdao scearos are well spread ou alog he AV dsrbuo. Ecoomc capal exposed o rae, equy ad propery rs s esmaed by hs S fuco. PUBI  APPIATIO O S I RISK ASSESSET Page 5
27 Axmum error axmum error axmum error Axmum error Average error Average error Average error Average error 4.. STATISTIS OR THE TEST ASES I hs seco, we prese he sascs for he resuls of all he cases we esed. We esed 0 cofguraos of possble regresso models (see seco 3.6) wh oedmesoal fucos of degrees bewee ad 6. The chars below show average ad maxmum fg errors for AV for each opo aalysed: S  ull odel.50%.00% 0.50% 0.00% 0.50% % .50% .00% .50% Polyomal degree ordare legedre wag herm chebyshev S  orward.50%.00% 0.50% 0.00% 0.50% % .50% .00% .50% Polyomal degree ordare legedre wag herm chebyshev S  Bacward.50%.00% 0.50% 0.00% 0.50% % .50% .00% .50% Polyomal degree ordare legedre wag herm chebyshev S  Sepwse.50%.00% 0.50% 0.00% 0.50% % .50% .00% .50% Polyomal degree ordare legedre wag herm chebyshev Table 5: Average error of he fg fuco from he opos aalysed.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% S  ull odel Polyomal degree S  Bacward 4.00%.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% Polyomal degree ordare legedre wag herm chebyshev ordare legedre wag herm chebyshev.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% S  orward Polyomal degree.00% 0.00% 8.00% 6.00% 4.00%.00% 0.00% S  Sepwse Polyomal degree ordare legedre wag herm chebyshev ordare legedre wag herm chebyshev Table 6: axmum error of he fg fuco from he opos aalysed PUBI  APPIATIO O S I RISK ASSESSET Page 6
28 oe ha:  The shape of he curves s comparable whaever regresso echque s used (alhough he bacward, forward ad sepwse mehods resul fewer regressors ha he full model, see seco 3.6): valuao error falls wh he degree of he polyomal. However, mplemeg S echque pracce becomes mpossble wh a polyomal degree of more ha.  oe a subsaal error marg (maxmum error of over % whe he bass polyomal fuco has degree less ha 3, all cases). Valuao error sars o sablse a 4. The choce of opmum degree mus herefore be eher 4 or 5,  The ordary ad egedre polyomal boh gve very smlar resuls because of he cosa weghg fuco (see appedx). The aguerre polyomal sablses more qucly. The Herme ad hebyshev polyomals are he leas sable. The chars below show he value of ecoomc capal (RB) esmaed usg he dffere opos aalysed: S  ull odel S  Bacward RB (Sadard ormula) 30 5 RB (Sadard ormula) Polyomal degree Polyomal degree Ordare egedre aguerre Herme hebyshev Ordare egedre aguerre Herme hebyshev S  orward S  Sepwse RB (Sadard ormula) 30 5 RB (Sadard ormula) Polyomal degree Polyomal degree Ordare egedre aguerre Herme hebyshev Ordare egedre aguerre Herme hebyshev Table 7: value of rs based capal usg he opos aalysed g qualy s comparable whaever regresso mehod s used. Ths s because he use of opmsao echques for he S fucos does o reduce he precso of he resuls. The ma dfferece s he regresso bases, wh greaer covergece from he aguerre PUBI  APPIATIO O S I RISK ASSESSET Page 7
29 polyomal whch sablses more qucly. oe ha he S smulao gves a lower value of ecoomc capal ha he sadard formula for all bass fucos of degree 4 or more. 4.3 AAYSIS O THE S UTIO I hs seco, we exame he characerscs of he S fuco. The fuco aalysed here s obaed from he degree 4 aguerre polyomal bass fuco usg a sepwse seleco mehod IPAT O RISK ATORS The purpose of hs seco s o chec ha he behavour of he S fuco s cosse wh wha we ow abou he porfolo. The char below plos he value of he S fuco as a fuco of logerm raes (all else beg equal). AV og rae shoc Table 8: AV as a fuco of logerm yelds The log durao of her lables maes lfe surace coracs hghly sesve o movemes logerm eres raes. The curve s a bell curve reflecg he opposg effecs of he rae guaraee ad surreder opo o AV. A srogly egave chage log raes meas ha he rse he guaraee rae ouweghs he fall value of he redempo opo. Vceversa, a drasc rse he log rae would drve sharply up he value of he surreder opo ouweghg he mpac of he fall he rae guaraee. PUBI  APPIATIO O S I RISK ASSESSET Page 8
30 AV AV AV The curve below plos he value of he S fuco as a fuco of shorerm eres raes (all else beg equal) shor rae shoc Table 9: AV as a fuco of shorerm raes The curve s fallg. Ths reflecs he mpac of a progressve verso of he rae curve. The followg chars show he value of he S fuco as a fuco of equy ad propery dces (oly a equy rs shoc s modelled): Equy Idex 0 70 Propery Idex 0 Table 0: AV as a fuco of equy rs Table : AV as a fuco of propery rs The curves are rsg, reflecg he beefcal mpac o AV of rsg equy mares ad propery prces. All he dvdual effecs aalysed above are cosse wh wha we ow abou porfolo rss. The char below shows he S fuco as a fuco of log raes ad he equy dex. PUBI  APPIATIO O S I RISK ASSESSET Page 9
31 Table : AV as a fuco of log rae rs ad equy rs The char shows he combed mpac of log raes ad eques o AV AUATIO O EOOI APITA: ODUAR APPROAH VS S ODEIG The sadard formula for valug ecoomc capal s o apply shocs o he balace shee a =0. The value of ecoomc capal resuls from he combao of dvdual shocs ad a marx aggregag dvdual ems cosumpo of capal. The char below compares he values of ecoomc capal derved usg he sadard formula ad he S fuco: PUBI  APPIATIO O S I RISK ASSESSET Page 30
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