Acceleration Techniques for Life Cash Flow Projection Based on Many Interest Rates Scenarios Cash Flow Proxy Functions

Transcription

1 Acceleraion Techniques for Life Cash Flow Projecion Based on Many Ineres Raes Scenarios Cash Flow Proxy Funcions Auhor: Marin Janeček, Tools4F, s.r.o. and Economic Universiy in Prague, 207 Acknowledgmen: I wan o hank very much o my smar colleagues Lucie Fremundová and Dana Fidrmucová who kindly helped me wih he developmen of he approaches described here, especially wih heir implemenaion ino real calculaion ools accompanying by valuable remarks. Moivaion Majoriy of acuarial asks in life insurance is based on projecion of fuure cash flows from insurance conracs (exising conracs in-force as a he valuaion dae or/and fuure new conracs concluded afer he valuaion dae; valuaion dae is he dae when he projecion sars). Such asks are for example: Accouning purposes: o Liabiliy Adequacy Tes (LAT) i.e. esing of adequacy of he accouning value of life echnical provision, o IFRS7 where he echnical provision is based on he presen value of he fuure cash flows and also he expeced cash flows hemselves are crucial in he financial reporing. Solvency II: o valuaion of: bes esimae value of life insurance liabiliies (BEL), fair value of life insurance liabiliies, o sress esing of he BEL according o differen assumpions of moraliies, lapses, expenses, ineres raes, equiies, ec. all hese are used for deerminaion of SCR (Solvency Capial Requiremen), ORSA (Own Risk and Solvency Assessmen), inernal risk managemen and many oher applicaions. Oher value and risk managemen purposes: o Asses Liabiliy Managemen (ALM), o Embedded Value (EV) calculaion, o Deerminaion of he Value of New Business (VNB), o Profi Tesing of new producs, o Business Plans. Tradiionally, acuaries solve hese asks by running a projecion model ha calculaes a developmen of he cash flows (and oher variables) per each individual policy in scope o fuure periods based on defined inpus. Such inpus ypically are lis of conracs (exising and/or including fuure new business) and

2 assumpions of he fuure developmen (ypically probabiliies of deahs and lapse, uni coss, level of commissions, ineres raes, ec.). Le us call his deailed modelling approach as he "per-policy model for he purposes of his paper. Such calculaions (using he per-policy model) are quie exensive. Le us imagine, a mid-sized life insurance company: wih policies in he porfolio; for which we wan o projec nex 50 years (usually even more) on he monhly sep; herefore, 50 years represens 600 monhs; every monh, le us assume a calculaion of 500 variables (usually ). This already represens 50 x 0 9 ( x 600 x 500) calculaions of variables under one se of assumpions. We will call such one calculaion of he per-policy model (under one se of assumpions) as one scenario run or jus a scenario. Obviously, already his one scenario run akes some ime i srongly depends on powerfulness of he sofware and hardware used and opimizaion of he projecion model srucure. Therefore, usually only a limied number of scenarios can be processed in a reasonable ime by his per-policy model. However, acuaries are more and more asked (by managers, regulaors, audiors, ec.) o deliver resuls based on many more han unis of scenarios, ofen hundreds or even (many) housands. Typically, such asks are for example: valuaion of Time Value of Financial Opions and Guaranees (TVFOG) ofen included in life conracs, where resuls (usually he presen value of he fuure cash flows PVCF) of several hundreds of ineres raes scenarios are needed; dynamic and sochasic ALM, where iner-relaions beween asses and liabiliies are modelled on monhly (or annual) basis and specific invesmen sraegy is decided dynamically based on he asses cash flow, he cash flow from policies and he acual invesmen reurns. Due o his quie complex calculaion, usually even one dynamic scenario of he per-policy model described above is very ime-consuming. And again, differen sakeholders can ask for resuls under many scenarios e.g. for he purposes of TVFOG valuaion, opimizaion of he invesmen sraegy, empirical disribuion of he BEL and many ohers; inernal model (in a sense of Solvency II) where again housands of scenarios should be run. To obain resuls of hese asks, he runimes of he per-policy model migh be unaccepably long (several days, even weeks/monhs) even if he acuary employs professional sofware (as Prophe, Sophas, MoSes,...) and hardware (powerful compuers wih many cores ec.). This is why some alernaive soluion of how o obain he life cash flows projecion resuls (and oher resuls based on he cash flows projecion ypically presen value of he cash flows (PVCF)) based on 2

3 many scenarios in reasonable ime wih accepable deviaion from he per-policy model resuls are o be invesigaed. Objecive Wihin his paper, we will focus on wo approaches how o deermine (esimae) he fuure life insurance cash flows based on many ineres raes scenarios in a reasonable runime. This is very imporan for many real life applicaions. In pracice, usually he firs ask where i is required is he valuaion of he TVFOG (required a leas by he Solvency II). We will call hese echniques as "Analyical" and "Inerpolaion" approximaions in his paper. Boh approaches have been developed wihin our eam in he Tools4F acuarial consuling company during he years 205 and 206. We have also esed hem on real life insurance porfolio wih very good (for he Inerpolaion approach even wih excellen) resuls. Noe: Noe ha in he real life pracice, we migh be ineresed in approximaion of several variables. Very ofen: a) average of PVCFs under many ineres raes scenarios o deermine BEL = E{PVCF}, b) PVCF under each ineres raes scenario individually (especially for he risk managemen purposes) o obain he full PVCF disribuion, c) cash flow a any ime in fuure under any ineres raes scenario his is wha we are focusing on here in his paper, d) oher variables projecion (premium, expenses, claims, echnical provisions, profis,...) under any ineres raes scenario. In his paper, we will focus on approximaion of he fuure cash flows ad c) as: if we are able o esimae he fuure cash flow a any ime under any ineres raes scenario, hen we are auomaically able o calculae he presen value hese cash flows (i.e. o obain PVCF (as ad b)) and naurally also o calculae he average of hese PVCFs o obain he esimaion of he BEL (as ad a)) and for ad d), he analogical approach as for he cash flows esimaion could be applied. Imporan noe! Noe ha he only ype of ineres raes ha affecs he fuure cash flows hemselves are he raes ha are he basics for he profi sharing decisions. The (cumulaive) profi sharing amoun is usually paid o he beneficiary a he policy erminaion (lapse, deah, mauriy, ec.). These raes follow he company profi sharing principles. Very ofen, and his is wha we expec here in his paper, hey depend on he marke ineres raes developmen. Typically, he profi sharing rae is based on he invesmen performance of he company asses porfolio (in accouning or marke value view) or migh be defined as he fuure risk free ineres rae (poenially plus some spread) for a given mauriy, ec. 3

4 There are oher ypes of ineres raes used in he life insurance calculaions. For example: for discouning of fuure cash flows (usually he risk free raes should be applied) or for projecion of fuure accouning profis invesmen reurns (following accouning principles) should be projeced. However, hese ineres raes do no affec he cash flow iself. Therefore, if no specified oherwise, he ineres raes in his paper from now on, will mean he ineres raes ha are credied o he policyholders funds as profi share plus a guaranee (if he guaranee is included in he policy definiion). Noaion We will use he following noaion in his paper. valuaion dae n represens he dae from which he projecion sars he ime in ime uni (in pracice usually monh) from he valuaion dae; =0 a he valuaion dae oal number of monhs projeced, =, 2,, n i (j) ineres raes scenario j, vecor of ineres raes i (j) = (i (j), i2 (j),, in (j) ) i (j) he ineres rae (of he scenario j a he ime ) ha represens he oal ineres rae ha is credied o he policyholder (profi share + poenial guaranee) I (j) (j) ) r I (j) = ( + ir+ r rεn and r < (j) (j) ( + i r+2) ( + i ) I (j) I (j) = I (j) 0. s CF m M CF oal number of ineres raes scenarios considered, j =, 2,, s cash flow of he policy m a he ime under ineres raes scenario j he policy number, m =, 2,, M oal number of policies wihin he porfolio cash flow of a porfolio of policies a he ime under ineres raes scenario j CF = CF 4

5 f F F value of he fund of policy m, a he ime under a given scenario j, if no decremens (deahs, lapses, ec.) are assumed. value of he fund of policy m, a he ime under a given scenario j, aking ino accoun he decremens. value of he fund of he whole porfolio of policies, a he ime under ineres raes scenario j, aking ino accoun he decremens. F = F l (m) l pf number of policies in force a he ime, for a policy m number of policies in force a he ime, for he whole porfolio; l pf = l (m) There also migh be some addiional noaions defined if necessary for he specific purpose in he following ex. Analyical approach Usually, o obain CF, he cash flow for he whole porfolio, a per-policy model calculaes cash flows CF per each individual policy m and hen sums i up (according o each projecion monh ). I is, CF = CF () Idea and Objecive The Analyical approach aims o find he analyical formula of CF where: he only variable inpus are he ineres raes; More specifically, for a deerminaion of he CF a he ime, he ineres raes (j) (j) (j) (i, i2,, i ) i.e. unil he ime are he inpus. They all are known a he ime. oher pars of he formula are coefficiens ha are no dependen on ineres raes. I is, o find he formula for CF in he following form: CF = CFfix pf + [coef (q) fn (q) (i (j), i2 (j),, i (j) )] q (2) 5

6 where CFfix pf = (m) CFfix and CFfix (m) represens he value of he par of he cash flow a he ime for he policy m, ha does no depend on he ineres raes assumpion; CFfix (m) εr; coef (q) is a coefficien (coef (q) εr) relevan o he ime deermined for he whole porfolio. Noe ha boh, CFfix (m) as well as coef (q) can be deermined from resuls of one run of he per-policy model (no maer wha ineres raes scenario is assumed). fn (q) is a funcion of ineres raes a he ime ; same funcion for every policy a he ime and q idenifies he pair of relaed coef (q) and fn (q), q =, 2,. If we are able o find such an analyical formula for CF, hen we can calculae i ou of he per-policy model (for any ime and scenario j). This may cause he calculaion of he cash flows projecion faser han when he per-policy model is used. Example: To be more illusraive, le us look closer a a specific example. Le us consider: universal life ype of policy where in case of deah he insurance company pays o fixed sum assured (SA) plus o he value of he fund a he ime of deah including he cumulaive profi share (f ). Le us invesigae he formula for he cash flow of he whole policy porfolio a ime = 2 (i.e. second projecion monh) CF 2. A he ime = 2, we know he ineres raes of he specific scenario for he firs wo projeced monhs i is (i (j), i2 (j) ). Remember ha boh raes are he ineres raes ha are credied o he policyholder (profi share + guaranee (if exiss)) as menioned above (Imporan noe on he page 3). I is, CF 2 = [l2 (m) P2 (m) l2 (m) E2 (m) d2 (m) (SA (m) + f 2 ) w2 (m) f2 ma 2 (m) f2 ] (3) Where (on op of he noaion defined above in he par Noaion): 6

7 d 2 (m)... number of policies considered o die wihin he second monh, of he policy m; w 2 (m)... number of policies considered o lapse wihin he second monh, of he policy m; ma 2 (m)... number of policies ha are a mauriy in he second monh, of he policy m; P 2 (m)... premium of policy relaed o he second monh, of he policy m; E 2 (m)... expenses of policy relaed o he second monh, of he policy m; SA (m)... sum assured relaed o he policy m. Le us now coninue adjusing he formula (3) in order o separae ineres rae sensiive and non-sensiive pars: (m) (m) (m) (m) (m) CF 2 = [l2 P2 l2 E2 d2 SA (m) ] + [(d 2 (m) + w2 (m) + ma2 (m) ) f2 ] (4) The firs sum of he formula (4) is usually no dependen on he specific ineres raes scenario le us denoe i CFfix 2. I is, CFfix 2 pf = [l 2 (m) P2 (m) l2 (m) E2 (m) d2 (m) SA (m) ]. (5) Le us now look o he second sum of he formula (4) and remember ha he fund value a specific ime (here = 2) could be deermined as: where f 2 = (f0 + pr0 saving,(m) ) ( + i (j) ) ( + i2 (j) ) + pr saving,(m) ( + i2 (j) ) (6) pr 0 saving,(m) is he saving par of he premium ha is paid by policy m a he ime = 0 and analogically pr saving,m relaes o he ime =. f 0 is he saring fund value (i.e. he fund value a he valuaion dae) of policy m. This naurally does no depend on he ineres raes developmen in fuure and hus f 0 = f0 (m). Then, we can wrie he formula (4) in he following form: CF 2 = CFfix2 pf + [(d 2 (m) + w2 (m) + ma2 (m) ) (f0 (m) + pr0 saving,(m) ) ( + i (j) ) ( + i2 (j) )] + [(d2 (m) + w2 (m) + ma2 (m) ) pr saving,(m) ) ( + i2 (j) )] Le us now denoe: coef 2 () = [(d 2 (m) + w 2 (m) + ma 2 (m) ) (f 0 (m) + pr 0 saving,(m) )] (7) 7

8 coef 2 (2) = [(d 2 (m) + w 2 (m) + ma 2 (m) ) pr saving,(m) ] fn 2 (),(j) = ( + i (j) ) ( + i2 (j) ) fn 2 (2),(j) = ( + i 2 (j) ) and remember ha: fn 2 (),(j) and fn2 (2),(j) do no depend on he specific policy, i.e. are he same for all he policies and hus can be exraced from he sum and coef 2 () and coef2 (2) do no depend on ineres raes and can be obained from one run of he per-policy model (no maer of wha ineres raes scenario is assumed). We herefore finally obain he formula (4) in he form: CF 2 = CFfix2 pf + coef 2 () fn2 (),(j) + coef2 (2) fn2 (2),(j) = ha already is he desired form as in he formula (2). Pracical process: In pracice, he calculaion process is as follows: = CFfix pf (q) (q),(j) q= coef 2 fn2 (8) ) Derive he formulas for he funcions fn (q) and CFfix pf and coef (q). This is wha is necessary o do jus once for each of he insurance produc unil he per-policy model of he produc is changed. 2) Every valuaion dae (usually every monh) hen: a. Run he per-policy model once (wih whaever ineres raes scenario) and derive he values of he coefficiens CFfix pf and coef (q) and save hem. b. Now, follow his cyclical calculaion: FOR j = o s FOR = o n (j) (j) (j) i.take he ineres raes (i, i2,, i from he given ineres raes scenario j (hese are he inpus) ii. Calculae CF by he formula (8). I is, CF = CFfix pf + [coef (q) fn (q) (i (j), i2 (j),, i (j) )] Resul of his cycle is he projecion of CF from = o he final fuure ( = n) for one specific ineres raes scenario j. Resuls of his cycle is he CF projecion for s ineres raes scenarios. q 8

9 Pracical issues: Alhough his approach could be seen as quie simple and clear wih no deviaions beween he CF from he per-policy model and from he approximaion, quie ofen i is no easy o deermine he analyical formula CF exacly. Typical issues for example are: producs where he deah benefi is defined as a maximum of he specific sum assured (SA) and he curren (a he ime ) value of he clien s fund ( f ). I is, he deah benefi equals o max (SA, f ). The complicaion here is ha o esimae fuure deah benefi par of he cash flow, we need o sum his max values per each individual policy. Bu for some policies f < SA, for some he opposie holds f SA. A he same ime, naurally, f is very much dependen on ineres raes; funcion fn (q),(j) migh be quie complex if (wha is usually he case in pracice) he profi share value is added o he fund value once per 2 monhs (usually a he end of he year); o derive he values of he parameers CFfix pf and coef (q) he per-policy model ofen has o be exended heavily and also, number of parameers CFfix pf and coef (q) migh be huge (also ens of housands). This is why he firs run of he per-policy model from which all he parameers are derived migh be quie ime-consuming; he firs ime preparaion is usually exremely demanding. To succeed, i is required o: o undersand all he deails of he modeled life producs; o undersand he per-policy model formulas in deails; o be advanced in mahemaics o be able o derive he correc formula in he desired form; o be exremely paien he formulas usually include many indices, sub-pars, members, ec. All (00%) of hem mus be correc, even one misake in one index makes all he resuls incorrec; o have experience in life modelling and be smar o solve he pracical issues; o be advanced and very careful in programming he approximaion formulas again, no misake is allowed. High-level resuls Naurally, he resuls of his Analyical approach very much depends on specific company siuaion and canno be generalized correcly. Here, we commen briefly jus basic resuls from our case sudy ha was a mid-sized life insurance company, using Prophe ool, having large variey of life produc ypes (wih many insured persons, riders, funds, ec.). 9

10 Cash flow fi The qualiy of he cash flow fi, i.e. he difference beween he CF from he approximaion and he per-policy model very much depends on he produc complexiy. There are producs (like in he example on he page 6 above) where he cash flow could be calculaed exacly wih no difference. However, for some producs simplificaions and parial alernaive soluions mus be developed. In our case sudy, we managed o obain he difference in vas mas majoriy of ineres raes scenarios far below % every year, even for complex producs and even if he esed ineres raes scenarios were exreme (from -40% o +60%). Presen value of he cash flows fi PVCF differences (per-policy model vs. approximaion) in each of he individual scenarios were up o 0.5% (±0.5%). Average of he PVCFs (BEL esimaion) from he approximaion were very much close o he value from he per-policy model (usually, he differences were below 0.%). Acceleraion resuls In our case, he calculaion runime decreased from 5 o 40 imes (for 000 o scenarios). More for higher number of scenarios as he mos ime consuming operaion is o calculae he parameers of he Analyical approach ha are based on one run of he per-policy model, bu he model ha is heavily exended from he sandard one o obain all he deailed resuls necessary for he parameers deerminaion. We also implemened he analyical formula calculaions in MS Excel. The speed would be definiely be much higher if more proper ool would be used (e.g. R, Pyhon, MaLab). Conclusion o he Analyical approach We see he Analyical approach as quie powerful wih he following feaures: replicaes very well he per-policy model cash flows and heir presen values; runime migh be significanly shorer if proper calculaion ool (R, Pyhon, Malab, ) is used; however, he pracical difficuly is ha he firs ime preparaion (deerminaion of he formulas for all he parameers and exending he exising per-policy model) is usually exremely demanding. 0

11 Inerpolaion approach This approach again (same as for he Analyical approach) aims on finding an esimaion of cash flow a specific ime under given ineres raes scenario j for given porfolio of policies (e.g. one produc) CF. Inerpolaion approach ideas: The main ideas of his approach are he following: ) Imagine now ha we already have a se of s ineres raes scenarios i (), i (2),, i (s) for which we inend o calculae/esimae CF. (i (j) = (i (j), i2 (j),, in (j) ) has he same inerpreaion as defined in he par Noaion) 2) Le us build a new (ficive) scenario of ineres raes i max = (i max, i 2 max,, i n max ) ha is deermined for every ime as a maximum of i (j) from all he scenarios j. I is, for every, i max = max i (j) j. Under his ficive i max scenario, he projeced fuure profi share credied o cliens a each and every ime (based on i max ), will obviously be higher or equal o i j under any scenario j (wihin our se of scenarios). Therefore: => he fuure benefis paymens (ha include he cumulaive profi share) will be he highes of all from he given se of scenarios and => he relevan cash flow, le us denoe i CF pf,i max (if defined as incomes minus oucomes as usual), creaes he lower boundary of all cash flows form all he scenarios wihin our se. I is, CF pf,i max CF pf,i (j) for any and any j. 3) And vice versa. If we build a ficive scenario of ineres raes i min = (i min, i 2 min,, i n min ) ha is deermined for he ime as a minimum of i (j) from all he scenarios j; i is, for every, i min = min i (j) j, hen => he fuure benefi paymens (ha include he cumulaive profi share) will be he lowes of all from he given se of scenarios and => he relevan cash flow, le us denoe i CF pf,i min, creaes he upper boundary of all cash flows form all he scenarios wihin our se. I is, CF pf,i min CF pf,i (j) for any and any j.

12 4) Therefore, for every ineres rae scenario j, i (j) = (i (j), i2 (j),, in (j) ) from he given se of scenarios and every ime, he relevan CF will be posiioned somewhere beween CF pf,i max and CF pf,i min. I is, CF pf,i max CF CF pf,i min for every. 5) However, he range beween CF pf,i max and CF pf,i min could be quie wide. Therefore, beer specificaion of where o find CF wihin his range is necessary. 6) Le us build a grid of z (also ficive) ineres raes scenarios ha is defined in he following way. For every ime spli he inerval beween i min and i max ino z- equidisan inervals. Le us idenify he specific grid scenario g k where k =, 2,, z. Then for every ime he difference i g k i g k+ is consan for all k. (Noe ha i g = i min and i g z = i max.) Noe: There migh be oher spli of he inerval beween i min and i max, e.g. seing hem according o quanile values (e.g. 0 h, 20 h,, 90 h ) for every ime. However, based on he experimenal resuls, boh opions give similar resuls and hus he spli approach is no crucial. 7) Then calculae he cash flows projecion (cash flow for every ime ) wihin he per-policy model for each of he grid scenarios of ineres raes - CF pf,g k. I means, he per-policy model will run z scenarios. 8) For every ineres raes scenario j (j =, 2,, s) and every ime we would like o find he CF beween he values of he cash flow a he ime of he closes grid scenarios g k and g k+. However, he following quesions remain: a) How o find, for each ime individually, he closes g k and g k+? b) If we find hem, how o inerpolae beween heir cash flows o esimae properly he CF? I is, wha proporion of CF pf,g k and CF pf,g k+ o ake? 9) To solve hese quesions, le us find an alernaive variable ha saisfies boh of he following wo condiions: a) I is highly correlaed (under ineres raes scenarios) o CF (condiion ) b) I is possible o be calculaed easily and quickly ou of he per-policy model. (condiion 2) 0) If we find such a variable le us denoe i V (j) for now hen: a) calculae he V g k for every grid scenarios (k =, 2,, z) and every ime (ou of he per-policy model); b) for every scenario j and for every ime : i. calculae V (j) and find he grid scenarios gk and g k+ where V g k V (j) V g k+ ; 2

13 ii. see where he V (j) ranks beween he V g k and V g k+. iii. I is, deermine he raio p = V g k+ (j) V g V k+ g V k ; deermine he esimaion of he CF as being beween he grid values CF pf,g k and CF pf,g k+ using he same inerpolaion as for he value of V (j) ranking beween V g k and V g k+. I is, CF = p CF pf,g k + ( p) CF pf,g k+. ) Wha could be he variable V (j)? Le us consider he value of he cliens fund o be paid a he ime for he given scenario j for he considered porfolio of policies le us denoe i F paid,(j). For his variable, we expec high correlaion wih he CF as he fund value paid is usually a significan par of he cash flow ha is paid a he ime and is naurally dependen on he ineres raes. This saisfies he condiion as defined above on he page 2. 2) F paid,(j) could be deermined by he following calculaion: Le us define and remind (some of he definiions have already been se above in he par Noaion): f = F l (m) gr... guaraneed ineres rae (heoreically could depend on ime, bu usually i does no). I equals o 0 for he producs where minimum ineres rae is no guaraneed. f (m),gr = F (m),gr (m)... fund value of policy m a he ime wih no decremens, for a scenario l where all he fuure ineres raes i (for every ) equals o he rae gr. (Noe ha f 0 (m) = f0 (m),gr as his is he saring value of he fund a he =0 (he valuaion dae)) pf,gr gr f F = (m),gr pf... is he average value of f l (j) F f = pf... is he average value of f l F = F pf,gr = F (m),gr F f (m),gr = f (m),gr + f (m),gr +gr (Noe ha he f (m),gr represens he saving par of he premium.) 3

14 paid (m)... is he oal number of policies ha are expeced o obain benefi paymens a ime relaed o he policy m; usually i is a sum of number of policies expeced o die, lapse or maure a ime. Then he policyholder fund o be paid a he ime for he whole policy porfolio is: F paid,(j) = (),gr () = paid (),gr [(f0 + f (),gr 0 ) I + f (),gr I + f (),gr 2 2 I + + f I ] + +paid (2) [(f0 (2),gr + f 0 (2),gr ) I + f (2),gr +paid (M) [(f0 (M),gr + f 0 (M),gr ) I + f (M),gr () = paid (),gr [f0 I + ( f (),gr + gr f (),gr 0 ) I + ( f (),gr 2 + gr f (),gr ) I + + ( f (),gr + gr f (),gr 2 ) I ] + (2) +paid (2),gr [f0 I + ( f (2),gr + gr f (2),gr 0 ) I + ( f (2),gr 2 + gr f (2),gr ) I + + ( f (2),gr + gr f (2),gr 2 ) I ] + + (2),gr I + f (2),gr 2 2 I + + f I ] + + (M),gr I + f (M),gr 2 2 I + + f I ] = (M) +paid (M),gr [f0 I + ( f (M),gr + gr f (M),gr 0 ) I + ( f (M),gr 2 + gr f (M),gr ) + ( f (M),gr 3 + gr f (M),gr 2 ) I + ( f (),gr 3 + gr f 2 I + ( f (2),gr 3 I I + + ( f (M),gr + gr f (M),gr 2 ) I ] = + gr f 2 () = paid [I f (),gr ( + gr ) + I + i f (),gr 2 ( + gr ) + i 2 + I f (),gr 3 ( + gr 2 ) + + I + i f (),gr ] gr (2) +paid [I f (2),gr ( + gr ) + I + i f (2),gr 2 ( + gr ) + i 2 + I f (2),gr 3 ( + gr 2 ) + + I + i f (2),gr ] gr (M) +paid [I f (M),gr ( + gr (M) ) + I + i f 2, gr ( + gr ) + i 2 + I f (M),gr 3 ( + gr 2 ) + + I + i f (M),gr ] 3 + gr Noe ha: (),gr ) (2),gr ) for producs wih he ineres rae guaranee, i gr for any and herefore ( ) 0; +gr +i for uni-linked producs wih no ineres rae guaranee his boundaries do no exis. 4

15 (m),gr gr 3) Le us now replace f wih he average value f. paid,(j) I is, we consider ha all he policies are he average ones. Rewrie he formula for F under paid,(j) his (a his momen quie srange) assumpion and denoe i F. I is: = F paid,(j) () = paid [I f gr ( + gr ) + I + i gr f 2 ( + gr ) + i 2 + I gr f 3 ( + gr 2 ) + + I + i gr f ] gr (2) +paid [I f gr ( + gr ) + I + i gr f 2 ( + gr ) + i 2 + I gr f 3 ( + gr 2 ) + + I + i gr f ] gr (M) +paid [I f gr ( + gr ) + I + i gr f 2 ( + gr ) + i 2 + I gr f 3 ( + gr 2 ) + + I + i gr f ] 3 + gr Therefore, we obain he final form: paid,(j) F = paid I F pf,gr pf ( ) + I l +gr +i F pf,gr 2 pf ( ) + I l +gr +i F pf,gr 3 pf ( 2 ) l +gr +i 3 3 [ where paid = (m) paid. I F pf,gr l +gr ] (9) Noe ha all he variables in he formula are known a he ime. paid,(j) 4) Noe ha (and we are aware of ha) he F given by he formula (9) does no express he fund value o be paid a he ime for he whole porfolio considered and he scenario j exacly. I is even very likely ha i does no fi o he real fund value o be paid a he ime a all. However, i does no maer much. Remember wha were our condiions ha should be fulfilled for he variable V (j) he Condiion and Condiion2 on he page 2. paid,(j) I seems (based on real case sudies) ha F esimaion mees boh condiions very well. 5

16 paid,(j) 5) Alernaively, and more pracically wih he same resul, we can derive he F his process: paid,(j) F (j) = paid f paid,(j) (j) F F 0 = 0, f0 = 0 pf l 0 (j) (j) f gr gr f = (f + + gr f gr ) ( (j) F + i ) = (f + l pf ( + gr) F gr pf ) ( + i ) l formula following In summary, paid,(j) (j) F F 0 = 0, f0 = 0 pf l 0 paid,(j) F (j) (j) = paid f = paid (f + gr F F gr l (+gr) Where again, all he variables in he formula (0) are known a he ime. l ) ( + i ) for > 0 (0) Pracical process: The real process in pracice is he following: ) Deermine he ficive grid scenarios i g k, k =, 2,, z following he approach as described in he poin 6) on he page 2. Typically, 5 z 20, usually z = 0 showed o us o be he mos effecive for mos of he producs and sill wih a very good fi. 2) Calculae CF pf,g k for every grid scenario (k =,2,,, z) wihin he per-policy model. paid,g 3) Calculae F k (0) ou of he per-policy model. for every grid scenarios (k =,2,,, z) and every ime using he formula (9) or 4) Then we have prepared everyhing for he fas esimaion of CF for any ineres raes scenario j following his procedure: FOR j = o s FOR = o n paid,(j) ) Calculae F following he formula (9) or (0) paid,g 2) Find he closes values of F k from he grid scenarios; paid,g I is, find k where F k paid,(j) paid,g F k+ F. 3) Calculae raio p = F paid,g k+ paid,(j) F paid,g F k+ paid,g F k () 6

17 4) Calculae he CF esimaion as: CF = p CF pf,g k + ( p) CF pf,g k+. Resul of his cycle is he projecion of CF from = o he final fuure (=n) for one specific ineres raes scenario j. Resuls of his cycle is he CF projecion for s scenarios. Pracical noe - exrapolaion Noe ha up o now we have assumed ha he se of ineres raes scenarios is known from he beginning. Based on hem, we creae he grid scenarios (including he minimum (firs) and maximum (las) scenarios). In pracice, his is usually no he case. However, a possible soluion is: a) The range of he possible fuure ineres raes could usually be (a leas roughly) esimaed. For example based on he las periods raes or based on some heoreical ineres raes scenario generaor calibraed o he curren marke siuaion, ec. paid,(j) paid,g b) Even, if for some ime and some scenario j he F is ou of inerval paid,g (F, F z ) from he grid scenarios, he proporion p could be calculaed by he same (or similar formula) as (). In his case, i represens no he inerpolaion bu exrapolaion and abs(p) >. Resuls: From pracical poin of view, we recommend no o use jus he las wo values of F paid,g k from he grid scenarios bu more (usually hree is fine) and use a regression for he exrapolaion. The proxy resuls hen fi wih he per-policy model much beer. Cash flow fi The qualiy of he cash flow fi, i.e. he difference beween he CF from he approximaion and he per-policy model showed similar or even beer resuls comparing o he Analyical approach. I is, ha in our case sudy, he differences on annual cash flows were in vas majoriy imes and scenarios far below % each year, even if he esed complex producs on exreme ineres raes scenarios (from -40% o + 60%). Presen value of he cash flows fi PVCF differences in each of he individual scenarios were mosly up o 0.2% (±0.2%) and he average of he PVCFs (BEL) from he approximaion were very much close o he value from he per-policy model (usually <±0.05%). These resuls we consider as excellen. 7

18 Correlaions paid,(j) As menioned above, we expeced ha he F will be highly correlaed wih he CF. The experimenal resuls confirmed ha in vas majoriy cases he correlaion was higher ha 95% (very ofen higher han 99%). Acceleraion resuls The calculaion runime (in our case sudy) decreased considerably: for 00 scenarios he Inerpolaion echnique acceleraed he runime 0 imes; for 000 scenarios, i was 60 imes; for scenarios, 30 imes. Conclusion o he Inerpolaion approach We see he Inerpolaion approach as very powerful and effecive. I has he following feaures: replicaes he per-policy model cash flows and heir presen values excepionally well; runimes are significanly shorer; he principles of he Inerpolaion approach are clear and simple, herefore, is implemenaion is fas; This is a significan benefi comparing o Analyical approach. he approach is universal in a sense ha is does no depend much on he specific produc (and is complexiy). Therefore, he analys does no need o know all he produc deails (wha on he opposie is vial when implemening he Analyical approach). Conclusion We inroduced wo echniques ha migh be used for esimaing he cash flows projecion for a given ineres raes scenario wih a very good fi o he per-policy model resuls and wih significanly shorer runimes Analyical and Inerpolaion approach. We believe ha boh echniques are powerful and effecive. Especially he Inerpolaion approach shows excellen resuls in all he crieria we focused on: excellen fi; much shorer runime; he approach is very easy and herefore can be undersood and implemened wihin he real pracice in shor period of ime. 8