Replicating. embedded options

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1 CUING EDGE EMBEDDED OPIONS Replcng embedded opons Insurnce compnes nvesed hevly n sochsc models. he nex horzon s o embed hese models more deeply no sse lbly mngemen processes. he uhors beleve h he replcng porfolo s powerful ool o esblsh hese lnks BY JOACHIM OECHSLIN, OLIVIER AUBRY, MAHIAS AELLIG, ALEXANDER KÄPPELI, DANIEL BRÖNNIMANN, ARNAUD ANDONNE AND GUILLAUME VALOIS Inroducon he mngemen of opons nd gurnees embedded n lfe nsurnce polces s now brodly recognsed s cenrl o sse lbly mngemen of nsurnce compnes. he ndusry hs remendously mproved s cpbles n hs re over he ps couple of yers. he CRO Forum sudy on fnncl rsk mgon (CRO Forum (6)) shows h he ndusry s vew on how o mnge opons nd gurnees s convergng, nd sks regulors o provde ncenves for rsk mgng sreges nd remove legcy regulons hnderng menngful rsk mgon pproches. he emergence of Europen Embedded Vlues (EEV) hs furher brodened he wreness for opons nd gurnee mngemen n he nsurnce ndusry. However, here re sll vrey of dfferen pproches ppled under he erm EEV, whle cler rend owrds mrke-conssen vluon s vsble. Upon he relese of he EEV prncples n erly 4, he ndusry hs seen he fve erly mover dsclosures n 4, of whch none ws mrke conssen. In 6, ccordng o n Erns & Young survey (Erns & Young (6)), he ndusry hs lredy seen seven dsclosures bsed on mrke-conssen pproches. hs rcle s no nended o be n nroducon o mrke-conssen vluon echnques. here re vrous publcons h serve hs purpose; one of he mos recen nd helpful s Klberer s noe (Klberer (6)) n he Germn mgzne Der Akur, or EH lecure noes (Wührch & Furrer (6)) on he subjec. he m of hs rcle s o demonsre how mrke-conssen vluons cn be ulsed for more hn jus vluon, nmely for he mngemen of opons nd gurnees. Mrke-conssen vluon However, before gong no hs, we need o esblsh bsc ermnology o sr from. he her of mrke-conssen pproch o EEV s se of economc scenros, projeced over long perod of me (n lne wh he long-erm nure of lfe nsurnce busness), of relevn economc vrbles such s neres res, foregn exchnge res, cred spreds, equy nd rel ese ndces, nd ohers. hroughou mos of hs rcle, we ssume h he scenros used re rel world scenros h reflec bes-esme ssumpons of he fuure developmen of hese economc vrbles, complemened by se of deflors h llow for mrke-conssen vluon. We furhermore ssume h he number n of scenros s suffcenly lrge, nd h we do no need o worry bou convergence ssues. For ech of hose scenros, full prof nd loss semen nd blnce shee s deermned over he enre run-off of he porfolo. Mngemen needs o mke ssumpons on polcyholder nd mngemen behvor n cern suons. As resul of he projecon, se of csh flows cf, for ech me sep nd ech smulon s vlble, bsed on he rel world scenros consdered. ogeher wh he respecve deflors ϕ,, he mrke-conssen vlue of he lfe nsurnce porfolo s pproxmed hrough: V = cf cf n cf = ( ) n E ϕ E ϕ ϕ = = = =,, We cn decompose ech deflor no wo fcors, he scenro ndependen dscoun fcor (whch s ody s prce P of zero bond wh correspondng mury), nd ρ whch represens chnge of probbly mesure. In ech scenro we hve he relonshp ϕ, =P ρ,. echnclly, ρ represens he chnge (.e. Rdon-Nkodym dervve) from he rel world probbles o Februry 7 47

2 CUING EDGE EMBEDDED OPIONS he rsk-neurl -forwrd mesure or equvlenly, he probbly densy of he rel world scenros n he rsk neurl -forwrd mesure. As consequence, he mrke-conssen vlue of he lfe nsurnce porfolo cn lso be expressed s he sum of he presen vlues of he expeced vlues of ll csh flows under he respecve rsk-neurl forwrd mesure : V = P E ( cf ) E wh ( cf ) n cf, = n = ρ, I s convenen o ssume h he scenro s dsnc scenro, nmely he cerny equvlen scenro. In he cerny equvlen scenro, ll sses re ssumed o ern he correspondng rsk free re of reurn ny pon n me, nd hence ll economc prmeers (such s neres res) ssume her forwrd vlue. he deflor n he cerny equvlen scenro ssumes he vlue of he zero-bond wh he correspondng mury. he cerny equvlen vlue of he lfe nsurnce porfolo s:, = V C = cf P nd denoes he vlue of he lfe nsurnce porfolo n cpl mrke wh no volly. he me vlue of he porfolo s defned s he dfference beween he cerny equvlen vlue nd he mrke-conssen vlue. Some people prefer o work wh rsk-neurl scenros, rher hn wh rel world scenros. In hs cse, reference sse, clled numerre N, s chosen, nd ll oher sses re clbred round he numerre. All consderons n hs rcle cn lso be ppled n such se-up s suffcen o consder he nverse of he numerre s he deflor (ϕ =N - ) nd swch o he ppropre rsk-neurl mesure for generng he scenros nd he csh-flows. Replcng lfe nsurnce porfolo wh fnncl nsrumens Whle he mrke-conssen vlue nd he me vlue of opons nd gurnees per se s relevn nformon, he sochsc smulons even conn subsnlly more nformon. However, hs nformon s somewh hdden n he sheer moun of csh flow d. he objecve s o lern more bou he srucure of hese csh flows. How cn we beer undersnd he srucure of hese csh flows, f no by mens of csh flows of smple, well-undersood fnncl producs wh whch we re fmlr? Le us ssume we hve chosen se of K bsc fnncl nsrumens, whch we undersnd well nd whch cn be vlued esly nd wh whch we beleve we cn, hrough lner combnon, fnd resonble pproxmon of he orgnl csh flows. We cn esly deermne he csh flow CF k for ech of hose bsc fnncl, nsrumens k, me sep, nd scenro. Gong forwrd, we vew he csh flows of he orgnl porfolo s vecor cf nd he csh flows of he bsc fnncl nsrumens s vecors CF k n he vecor spce of ll csh flows wh me seps nd n scenros. hs vecor spce s of he fne dmenson n. Our objecve s o fnd weghs w k for ech of hose bsc fnncl nsrumens such s o cheve good mch. Le us cll hs lner combnon: K R = w k CF k= of he bsc fnncl nsrumens, he replcng porfolo. In order o defne wh good mch mens, we deermne sclr produc, nd he correspondng merc =, on he vecor spce of ll csh flows. Fnlly, k we deermne he weghs such h: R cf = mn Noe h he replcng porfolo depends on he bsc fnncl nsrumens nd he mesure chosen, nd hence s no unquely deermned. I lso depends on he se of scenros chosen; however, hs dependency loses s mpornce f suffcenly hgh number of scenros re used. Fndng n ppropre merc here s no sngle soluon o hs merc. Le us herefore sr o consder wh s probbly he mos srghforwrd sclr produc, bsed on he sum of he expeced vlues of he produc of he wo csh flows (regrded s rndom vrbles) under he rsk-neurl forwrd mesures: =,, whle, b = E ( b ) n, b, ρ, = n = = hs produc fulflls ll crer of sclr produc, specfclly symmery, lnery, nd posve defneness. Furhermore, he sclr produc s ndependen from he rel world probbles of he scenros, snce he rskneurl forwrd mesures re unquely deermned n complee mrkes (cenrl heorem of sse prcng). 3 he wy we hve been seng up hs vecor spce nd s sclr produc nd merc, csh flow hs mesure zero f nd only f he csh flow s zero for ll me seps nd ll scenros. In oher words, wo csh flows re consdered dencl f nd only f hey mch n ech ndvdul me sep nd scenro. hs s rher rgd defnon of mesure. In prcce, hs works fne for lfe nsurnce porfolos whose csh flow n ny me sep cn be resonbly pproxmed hrough smple fnncl nsrumens pyng csh flow n precsely hs me sep. hs s no he cse for mny lfe nsurnce porfolos. For exmple, sngle good equy yer does no men h he exr performnce s mmedely shred wh polcyholders. In mos cses, only fer seres of good equy yers does he nsurnce compny shre sgnfcn pr of he exr performnce wh he polcyholder. he sme holds n he oppose cse. No only he nsurnce compny, bu lso he polcyholder s smoohng. Afer sngle yer of hgh neres res, no every polcyholder wll mke use of he surrender opon n order o nves s money n lernve sse clsses or n new lfe nsurnce polces wh hgher gurneed reurns. As consequence of hs smoohng, he csh flow of lfe nsurnce porfolo me s ofen drven by cpl mrke prmeers before me. Le us herefore hnk bou loosenng he rgdy of he bove mesure lle, n order o beer llow for such smoohng processes. Idenfyng clsses of equvlen csh flows he ccrul mehod: he mng of csh flow clerly mers. A csh flow of $ s worh more ody hn n yers. However, nsed of cshng n he dollr ody, you my choose o connuously renves for one yer on rollng bss over yers. Wh you ge n yers hs go he sme presen vlue for you ody, nmely one dollr. In lne wh hs, le us ssume we connuously renves every csh flow Noe h ρ s dfferen for ech. See Wuhrch nd Fuller (6), secon 4.3., for more deled explnons E denoes he expeced vlue under rsk-neurl probbles 3 Provded he Economc Scenro Generor s bsed on complee model 48 Lfe & Pensons

3 we receve for one yer on rollng bss unl he ermnl yer, nd only hen csh he money n. he ermnl csh flows cn be deermned hrough ccrung he neres on csh flow up o he ermnl yer for ech scenro. For csh flow, hs mens: N erm, =,, wh N, = exp rs, N ( ) s= =, N, denoes he end vlue, or prce, of he ccrued un csh ccoun me n scenro, nd r, denoes he one yer re me n scenro. Le us now consder wo csh flows nd b s equvlen f hey resul n he sme ermnl csh flow: b erm =b erm for ll scenros. he vecor spce of he resulng csh flow equvlence clsses hs now dmenson n, rher hn n. Clerly, wo equvlen csh flows nd b lwys hve he sme presen vlue boh n erms of mrke-conssen nd cerny-equvlen vlue snce he ccrul does no chnge he vlue of csh flow. Bu le us prove hs semen more formlly, snce he formly wll help us n oher respecs ler on. he prce process ϕ N of he ccrued csh ccoun, jus s ny sse prce process, s mrngle under he rel-world mesure, h s, fulflls he condon: ϕ N = E ( ϕ N I ) for ll, nd ll s > s s In oher words, he condonl expeced vlue of he deflor-weghed sse prce me s, gven he nformon I known me, equls he deflorweghed sse prce me. For sngle csh flow occurrng me, nd he correspondng ermnl csh flow erm, equon () nd ssumng s= mples: N = ϕ E( ϕ I ) = ϕ E( erm ϕ I ) N From hs, we cn esly derve he presen vlues: 3 erm V = E( ϕ ) = E( ϕ ϕ E( ϕ I )) = E( erm ϕ ) = V whch mens h he orgnl csh flow occurrng me nd he ermnl csh flow erm occurrng me hve he sme presen vlue. kng he sum over vrous me seps, shows h he presen vlues of he orgnl csh flow nd he correspondng ermnl csh flow re dencl. Gven h nd b re equvlen, he presen vlues of he correspondng ermnl csh flows re dencl, nd fnlly, he presen vlues of nd b re dencl. he fc h wo equvlen csh flows nd b hve he sme presen vlue s cernly helpful, bu do hey lso hve he sme sensvy owrds chnges n cpl mrke prmeers s me goes by? From he consderons ledng o equon (3), we cn esly confrm h hs s ndeed he cse. Le us look he vlue of n orgnl csh flow occurrng me, nd he correspondng ermnl csh flow erm occurrng me fer one yer from now: V I = ϕ E( ϕ I ) = ϕ E( ϕ ϕ E( e E( ) erm erm = ϕ ϕ I = I V rm erm ϕ I ) I ) = hs mens h he vlues me, gven he nformon I vlble me, of he orgnl csh flow nd he correspondng ermnl csh flow re dencl. hs mens h hey hve he sme sensves o cpl mrke developmens over one yer. hs lso holds for ny me horzon exceedng one yer. hs s n mporn chrcersc h we expec n equvlence relonshp o fulfll n order o be useful for he purposes of replcon. If hs condon s no fulflled, he resulng replcng porfolo my be useless, snce one nsrumen my be replced by n nsrumen wh compleely dfferen sensves. Noe h he oppose s no necessrly rue, for nsnce, wo fnncl nsrumens wh he sme presen vlue nd he sme sensves re no necessrly equvlen under hs relonshp. 4 hs ccrul equvlence relonshp llows us o defne sclr produc on he vecor spce of csh flow equvlence clsses, whch s only bsed on he ermnl csh flows: 4 erm erm =,, whle, = E ( ) er b b n m erm b ρ, he bove consderons on presen vlues nd sensves hve been derved from he mrngle propery of he ccrued csh ccoun. However, ny sse prce process fulflls he mrngle propery, nd hence ny sse cn be used o defne n equvlence relonshp. Deermnng he weghs Le us wre he erm n equon () we re supposed o mnmse s funcon q of he weghs. Due o he lnery of he sclr produc, q s qudrc form n he weghs: K n = k l k q( w) = w w CF, CF w CF, cf + cf, cf k l k, l= k= Alernvely, we cn wre q(w)=w Δ w-w μ+δ, where Δ s symmercl nd posve-defne mrx mde up of he pr-wse sclr producs of he bsc fnncl nsrumens, μ s vecor mde up of he sclr produc beween ll bsc fnncl nsrumens nd he orgnl csh flow, nd δ, sclr, s he sclr produc of he orgnl csh flow wh self. he vecor w s mde up of he ndvdul weghs. Due o he fc h q s posve-defne qudrc form, he mnmum s unquely deermned hrough he weghs for whch he dervve of q wh respec o he weghs s zero: K q = mn wq = w µ = or equvlenly, w=δ - μ. hs s he soluon o sysem of lner equons n dmenson K. he complexy of he clculon s hence solely drven by he number of bsc fnncl nsrumens chosen, no by he number of smulons or he number of me seps. Assessng he quly of replcon A frs mpresson of he quly of replcon comes from comprson of he vlue of he replcng porfolo wh he vlue of he orgnl porfolo. Noe h boh he mrke-conssen nd he cerny-equvlen vlue should be compred. he closer hese vlues re, he beer. Provded hese vlues mch resonbly, here s noher good ndcor of he quly of he replcon, he R, or he squre of he correlon coeffcen beween he replcng porfolo nd he orgnl csh flow n he rsk-neurl -forwrd mesure: 4 Consder he followng wo nsrumens: (A) Insrumen h pys n equy ndex fer one yer; (B) Insrumen h pys n equy ndex fer yers k Februry 7 49

4 Replcng porfolo CUING EDGE EMBEDDED OPIONS Fgure. ermnl csh flows -, R cov ( R, cf ) = correl ( R, cf ) = vr ( R) vr ( cf ) An R of mens perfec replcon, close o mens good replcon, nd mens no replcon ll. Smple lgebr llows us o express R by mens of sclr produc nd expeced vlues of csh flows: R = ermnl csh flows, 5,, 5, -5, 5,, 5, -5, -, Orgnl porfolo Fgure. Zero coupon bonds (underlyng n M) Fgure 3. Recever swpons (underlyng n M), - -, -, -3, -4, R, cf E ( R) E ( cf ) ( ) E ( ) R E R cf cf In cses where he number K of bsc fnncl nsrumens consdered s comng close or s exceedng he number of smulons, R mgh no be good ndcor of he quly of replcon. he reson for hs s h for suffcenly lrge number K of bsc fnncl nsrumens s lwys possble o fnd perfec replcon. In some cses, he quly of replcon s good n ll respecs, bu he replcng porfolo conns lrge long-shor posons. hs mgh occur f some bsc fnncl nsrumens consdered re very smlr n cern mesure chosen. Such suon cn be mged hrough eher reducon of bsc fnncl nsrumens consdered, or hrough nroducon of cern ddonl consrns n he opmson. Reflecng consrns A replcon s lwys n pproxmon, nd he resulng replcng porfolo wll never excly mch he orgnl csh flow. However, n some nsnces, s convenen o force some chrcerscs of he replcng porfolo o excly mch hose of he orgnl csh flow. For exmple, s cernly desrble h he mrke-conssen nd he cerny-equvlen vlues of he replcng porfolo mch he respecve vlues of he orgnl csh flow (whch s generlly no excly he cse). hs prculr consrn of mchng prces s cully esy o cheve snce s lner consrn. We express hs consrn by mens of he smple formul: w p = P whle he vecor p s mde of he prces of ech ndvdul bsc fnncl nsrumen, nd P s he vlue of he orgnl csh flow, whch s o be mched. Noe h hs works he sme wy for he mrke-conssen nd for he cerny-equvlen vlue. In order o mnmse q under he bove consrn, we cn drw on Lgrnge mulplers. he correspondng Lgrnge funcon s: Q = q + λ ( w p P) Accordng o Lgrnge s mulpler mehod, he mnmum of q under he bove consrn s unquely deermned hrough he weghs for whch he dervve of Q wh respec o he weghs nd λ s zero, h s: = p w µ = w λq p λ P, or ˆ wˆ ˆ µ = ledng o he weghs ŵ=δˆ μˆ. hs s gn se of lner equons, whose dmenson s now K+, rher hn K whou he consrn. Any number of lner consrns cn be dded n he sme wy. Cse exmple Replcng rel lfe nsurnce porfolo he purpose of hs secon s o pply he bove replcon echnques o rel lfe nsurnce porfolo. We hve chosen Europen ndvdul lfe porfolo, for whch ll sses nd lbles re denomned n EUR. he porfolo mosly conns regulr premum endowmens nd deferred nnues, bu lso sngle premum endowmens nd erm nsurnce. Mos of he busness s subjec o prof shrng wh polcyholders. he porfolo nlysed n hs cse exmple s llusrve n he sense h he ndvdul produc lnes hve been weghed rbrrly such s o cheve mx h s dfferen from he mx of ny exsng porfolo. he porfolo s modeled over 4 yers usng sndrd curl projecon sofwre. 5, smulons hve been used. he bsc fnncl nsrumens used for he replcon re zero bonds nd recever swpons wh vrous mures, equy posons wh vrous holdng perods, nd equy pu opons wh vrous mures. A ol of 54 5 Lfe & Pensons

5 ble : Bsc fnncl nsrumens Insrumen Mury/holdng Srkes perod EUR zero bonds,3,5,7,,5,,3,5,7,3,35,4 EUR recever swpons on yer swp,5,7,,,5,7,,, 5,7 Equy poson,3,5,7,,5,,3,5,7,3,35,4 Equy pu opons,3,5,7,,5,,3,5,7,3,35,4 Beween 3.% (one yer mury) nd.5% (7 yer mury).9% (one yer mury) nd.7% (4 yer mury) Fgure 4. Sensves of replcng porfolo Vluon Bse cse Mrkeconssen Ineres res +bp.58 Source of volles Impled vols - pcp 3.3 Cernyequvlen 39 Ineres res.4 me vlue -7 Eques.9 ble : Replcng porfolo Replcon E mllon Underlyng Mrkeconssen vlue EUR zero bonds EUR recever swpons Equy poson Equy pu opons Cernyequvlen vlue me vlue , ol 39-7 ndvdul nsrumens hve been chosen. ble gves n overvew on he bsc fnncl nsrumens: he srkes of he recever swpons nd he equy pu opons hve been chosen ou-of-he-money s per he blnce shee de, reflecng he esmed verge gurneed re of he underlyng lfe nsurnce porfolo. he vrous holdng perods used for he equy poson hve no economc menng (n equy poson s n equy poson ndependen from he holdng perod). he need o consder vrous csh flow represenons for eques rses from he fc h he ccrul wh he one yer re resuls n dfferen ermnl csh flows for he sme equy ndex. hs hs been dscussed n he consderons ledng o formul (4). he replcon hs been crred ou under he consrn h boh he mrke-conssen vlue nd he cernyequvlen vlue 5 mch he correspondng vlues of he underlyng porfolo. he resul from he replcon s summrzed n ble. he replcon quly s 93.4% mesured n erms of R, ndcng good resul. Fgure shows he ermnl csh flows of he orgnl porfolo gns he ermnl csh flows of he replcng porfolo for ll 5, scenros. he chr grphclly llusres he hgh quly of he replcon cheved n hs cse exmple. Fgures nd 3 llusre he mury profle of he zero bonds nd he underlyng of recever swpons n he resulng replcng porfolo. In erms of zero bonds, he replcng porfolo conns boh long nd shor posons. In erms of recever swpons, here s cler shor poson over he enre run-off. he re n fgure 3 represens he unhedged renvesmen volume nheren n hs porfolo. Mny lfe nsurnce porfolos conn embedded opons h re ph dependen, nd so does he porfolo consdered n hs cse exmple. In order o keep hs exmple smple, we hve only consdered few opons n he bsc Eques +% Fgure 5. Sensvy of replcng porfolo o prllel neres re shfs % - -.%.%.%.% fnncl nsrumens, ll of whch re Europen ype (h s, no ph dependen). hs mens h he replcng porfolo derved here s jus he bes represenon of he underlyng porfolo hrough Europen ype opons nsrumens. However, nohng prevens exendng he se of bsc fnncl nsrumens wh ph dependen opons, such s Amercn nd Asn ype opons. Clerly, he selecon of he ppropre se of bsc fnncl nsrumens s where he scenfc specs end nd he judgemen nd experence of he rsk mnger becomes relevn. Applcons of he replcng porfolo Sensvy nlyss: he sensvy of he mrke-conssen vlue wh respec o cpl mrke prmeers such s neres res, foregn exchnge res, equy ndces, nd volles s key elemen of ny dsclosure. he EEV prncples explcly requre he dsclosure of such sensves. Sensves ofen lso form he bss for ALM cves, s shown n he followng secon. he mos srghforwrd pproch o sensvy clculon s he brue force pproch,.e. usng mulple ses of slghly djused scenros. Whle hs pproch s concepully smple, ofen nvolves merl compuon power, lmng he number of sensves deermned o only few, whle 5 And hus lso he me vlue Februry 7 5

6 CUING EDGE EMBEDDED OPIONS mny more would be of neres from mngemen perspecve. Once replcng porfolo hs been deermned, he sensves come lmos for free. he replcng porfolo consss of lmed number of smple fnncl nsrumens, for whch he sensves ofen cn be deermned explcly. Fgure 4 shows he sensves of he mrke-conssen vlue for he cse exmple nroduced n he prevous secon, deermned on he bss of he replcng porfolo. he sensves of he replcng porfolo hve lso been ssessed for lrger vrons of neres res, volles, nd equy ndex levels, nd hve been compred wh he resuls of brue force revluon. he comprson showed encourgng resuls. he expeced quly of he replcon depends on he purpose of he replcon. he uhors recommend h such comprsons re lwys performed o vlde he quly of he replcng porfolo. Specfclly n lfe nsurnce, porfolos wh merl me vlue of opons nd gurnees, s crucl o undersnd he convexy h my resul hereof. In he prevous cse exmple, we hve seen merl shor poson n recever swpons. hese posons ndeed resul n srong neres re convexy. hs resul hs lso been derved, pplyng dfferen echnques (Oechsln, e l (4)). Asse-lbly mngemen (ALM): ALM s n cvy h kes plce n he nerfce beween cures/rsk mngers, he fnnce funcon, nd he nvesmen mngemen funcon. I hence seems nevble h some sor of common lnguge exss n whch he nvolved pres exchnge qunve nformon h llows proper mch or conscous nd delbere msmch beween sses nd lbles. Before mrke-conssen vluon echnques hve been esblshed, such common lnguge ws lrgely mssng. he nsghs from he mrke-conssen vluons for he frs me provded powerful common lnguge nmely he sensves of he mrke-conssen vlue o vrous economc prmeers. However, he sensvy nlyss hs ofen no nswered ll quesons relevn n he conex of ALM, for nsnce exposures o yeld curve roon, o neres re convexy, nd mpled volles. Here s where he replcng porfolo cn be helpful. Imgne wo rsk mngers lkng o he nvesmen mngers: he frs sys, We hve now nlysed he sensves of our mrke-conssen vlue under vrous cpl mrke scenros. We found we hve go problem n cse of prllel yeld curve declne or n cse of generl ncrese n mpled volles. Here s he ls of he sensves he second sys, We hve now nlysed our ne sse-lbly poson on he bss of he models for mrke-conssen vluons. We hve pproxmed he ne poson wh se of rher smple fnncl nsrumens. We found h we re shor merl poson n neres re opons, whch we expressed hrough srp of recever swpons wh hese mures, underlyngs, nd srkes. I s hs second sor of communcon we beleve rsk mngers, fnnce people nd sse mngers should hve. he replcng porfolo s hence mos nsrumenl for he purposes of ALM n he lfe nsurnce ndusry. A crcl lmon for he use of replcng porfolos n ALM s he fc h he lbly vlue nd s dynmcs my, o cern degree, depend on he llocon of sses coverng he lbly. 6 In such cses, he replcng porfolo only represens he underlyng lfe nsurnce porfolo for he specfc sse llocon consdered n he replcon. Any chnge n sse llocon requres he compuon of new replcng porfolo. Rsk Mngemen: Vlue--rsk mesures or reled mesures such s expeced shorfll re wdely ppled for rsk mngemen purposes n he nsurnce ndusry. Solvency regmes re ncresngly bsed on such pproches, he Swss Solvency es (Swss Federl Offce of Prve Insurnce (3)) beng only one exmple. he objecve s o deermne he wors developmen of he mrke-conssen vlue of n nsurnce porfolo o cern confdence level (ofen 99% or more) over cern perod n me (ofen one yer), bsed on rel world probbles. Crryng ou hese clculons srcly ccordng o he defnon ofen requres mmense compuon power. I leds o nesed smulons. Here s where he replcng porfolo cn help. For ech of he rel-world scenros, one mgh pproxme he mrke-conssen vlue of sses nd lbles on he bss of he replcng porfolo, whch s generlly much cheper n erms of compuon power. Clerly, he quly of he replcng porfolo n exreme cpl mrke envronmens mus be esed by mens of comprson wh he resuls of brue force re-vluon (see secon on sensves). he uhors could lso mgne h mercs dfferen from he one proposed n (4) my led o beer replcon n exreme scenros. hs subjec shll be ddressed hrough furher reserch n hs re. L&P Jochm Oechsln s depuy chef rsk offcer nd Olver Aubry s hed of rsk modellng AXA Group. Mhs Aellg s former chef cury lfe of Wnerhur Group. Alexnder Käppel nd Dnel Brönnmnn re senor rsk modellers AXA Group. Arnud ndonne nd Gullume Vlos re senor ALM mngers AXA Group. We would lke o hnk grn Klberer for hs vluble npu o hs rcle nd for beng n excellen sprrng prner n numerous dscussons. References Fnncl Rsk Mgon n Insurnce me for Chnge, CRO Forum, 6 ( Europen Embedded Vlue Resuls 5 Yer End, Erns&Young, 6. Klberer Mrke conssen vluon of nsurnce lbles Bsc pproches nd oolbox, Der Akur (6) Hef M. Wührch nd HJ Furrer Lecure Noes on Mrke-Conssen Acurl Vluon, EH Zürch, SS6, Verson.3, 6 J. Oechsln e l: Incorporng polcyholder expecons no Asse Lbly Mngemen, Rsk Mgzne, November 4 Swss Solvency es Whe Pper, Swss Federl Offce of Prve Insurnce, 3 6 hs s specfclly he cse when prof shrng depends on he performnce of he specfc sses coverng he lbles 5 Lfe & Pensons