Nested Stochastic Valuation of Large Variable Annuity Portfolios: Monte Carlo Simulation and Synthetic Datasets

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1 daa Aricle Nesed Socasic Valuaion of Large Variable Annuiy Porfolios: Mone Carlo Simulaion and Syneic Daases Guoun Gan * and Emiliano A. Valdez Deparmen of Maemaics, Universiy of Connecicu, 341 Mansfield Road, Sorrs, CT , USA; emiliano.valdez@uconn.edu * Correspondence: guoun.gan@uconn.edu; Tel.: Received: 11 July 218; Acceped: 3 Augus 218; Publised: 1 Sepember 218 Absrac: Dynamic edging as been adoped by many insurance companies o miigae e financial riss associaed wi variable annuiy guaranees. To simulae e performance of dynamic edging for variable annuiy producs, insurance companies rely on nesed socasic proecions, wic is igly compuaionally inensive and ofen proibiive for large variable annuiy porfolios. Meamodeling ecniques ave recenly been proposed o address e compuaional issues. However, i is difficul for researcers o obain real daases from insurance companies o es meamodeling ecniques and publis e resuls in academic ournals. In is paper, we creae syneic daases a can be used for e purpose of addressing e compuaional issues associaed wi e nesed socasic valuaion of large variable annuiy porfolios. Te runime used o creae ese syneic daases would be abou ree years if a single CPU were used. Tese daases are readily available o researcers and praciioners so a ey can focus on esing meamodeling ecniques. Keywords: Mone Carlo; regime-swicing mulivariae Blac Scoles; meamodeling; variable annuiy; porfolio valuaion 1. Inroducion A variable annuiy VA) is a popular life insurance produc creaed by insurance companies o address many people s concerns abou ouliving eir asses [1,2]. Under a VA policy, e policyolder agrees o mae a lump-sum or a series of purcase paymens o e insurer and in reurn e insurer agrees o mae benefi paymens o e policyolder, beginning eier immediaely or on a fuure dae. Policyolders coose o inves eir money in one or more invesmen funds provided by e insurance company. A main feaure of VAs is a ey come wi guaranees or riders, wic are designed o proec e policyolder s capial agains mare downurns. Tere are wo ypes of guaraneed benefis embedded in VA policies: dea benefis and living benefis. A guaraneed minimum dea benefi GMDB) guaranees a specified amoun o e beneficiary upon e dea of e policyolder regardless of e performance of e invesmen porfolio. Examples of living benefis include e guaraneed minimum widrawal benefi GMWB), e guaraneed minimum income benefi GMIB), e guaraneed minimum mauriy benefi GMMB), and e guaraneed minimum accumulaion benefi GMAB). A GMWB guaranees a e policyolder can ae sysemaic annual widrawals of a specified amoun from e policy over a period of ime, even oug e invesmen porfolio mig be depleed. A GMIB guaranees a e policyolder can conver e VA policy o an annuiy according o a specified rae. A GMMB guaranees a e policyolder can receive a specific amoun a e mauriy of e policy. A GMAB guaranees a e policyolder can renew e conrac during a specified window afer a specified waiing period. Daa 218, 3, 31; doi:1.339/daa3331

2 Daa 218, 3, 31 2 of 21 Due o ese aracive guaranees, many VA policies were sold in e pas wo decades. Figure 1 sows e annual VA sales in e Unied Saes during e period from 28 o 217. In e figure, we see a, excep for 217, e annual sales in all ese years were above $1 billion. Te guaranees embedded in VA policies are financial guaranees and canno be adequaely addressed by radiional acuarial meods [3]. To miigae e financial riss associaed wi e VA guaranees, many insurance companies wi a VA business ave adoped dynamic edging [4,5]. Sales in billions) $156 $128 $141 $158 $147 $145 $14 $133 $15 $ Figure 1. Variable annuiy sales in e Unied Saes from 28 o 217. Te numbers are obained from LIMRA Secure Reiremen Insiue. Year To simulae e performance of dynamic edging for VA producs, insurance companies rely on nesed socasic proecions [5]. Nesed socasic proecions are also referred o as socasic on socasic proecions. Figure 2 concepualizes e srucure of a ypical nesed socasic proecion, wic involves wo layers of socasic proecions. A eac node of an ouer socasic pa, a se of inner socasic pas is embedded. Usually e ouer socasic pas are real-world scenarios, wic reflec a realisic paern of underlying mare prices a are used o generae realisic disribuions of oucomes. In conras, e inner socasic pas are ris-neural scenarios, wic use unrealisic assumpions abou ris premiums for purposes of calculaing derivaive prices under e no-arbirage assumpion. Figure 2. Nesed socasic proecions.

3 Daa 218, 3, 31 3 of 21 Te compuaion of nesed socasic proecions for a large VA porfolio is igly compuaionally inensive and ofen proibiive because every policy in e porfolio needs o be proeced over many pas for a long ime orizon [6]. For example, if we use 1 real-world scenarios in e ouer layer and 1 ris-neural pas in e inner layer, and proec e cas flows a yearly seps for 3 years, en e oal number of proecions for eac policy is /2 = , wic is already a big number. For a porfolio of 1, conracs, e number of proecions would be Suppose a a single CPU can process 2, cas flow proecions in a second. Ten, i will ae is CPU , = 7.37 years o process all e cas flow proecions for e porfolio. Te amoun of ime sown in e above equaion is us e runime used o proec e cas flows once. To calculae e Grees, we need o proec e cas flows muliple imes a differen socs of e mare. Tis will increase e runime mulifold. Recenly, meamodeling approaces ave been proposed o address e compuaional issues associaed wi e valuaion of large VA porfolios see, for example, [7 16]). Te main idea of ese meamodeling approaces is o build a predicive model based on a se of represenaive VA policies and eir fair mare values or oer quaniies of ineres). Te predicive model is en used o esimae e fair mare values for all e policies in e porfolio. Tis can reduce e number of policies a are valued by Mone Carlo simulaion. Since predicive models are usually muc faser an Mone Carlo simulaion, e gain in valuaion runime is significan. However, i is difficul for academic researcers o obain real daases from insurance companies o assess e performance of meamodeling ecniques. In is paper, we creae syneic daases a can be used by researcers and praciioners o es meamodeling meods for e efficien valuaion of large VA porfolios under nesed socasic simulaion. In paricular, we implemen a nesed socasic valuaion engine a is used o calculae e Grees for VA policies along ouer layer pas. Te purpose of is wor is o relieve researcers from spending ime on creaing suc daases, wic can be exremely ime-consuming o creae. Tis paper differs from a previous paper [1] in a is paper focuses on developing compreensive syneic daases wile e previous paper focuses on meamodeling. Tis paper also differs from e paper [17] because is paper is abou creaing syneic daases under e socasic-on-socasic valuaion framewor wile e paper [17] is abou creaing syneic daases for valuaion only a ime zero. Te remaining par of is paper is srucured as follows. Secion 2 presens a nesed socasic simulaion engine for valuing e guaranees embedded in variable annuiies. In Secions 3 and 4, we presen syneic daases a can be used o es e performance of meamodeling ecniques. In Secion 5, we conclude e paper wi some remars. Te sofware a implemens e nesed Mone Carlo simulaion engine is described in Appendix A. 2. Nesed Socasic Valuaion In is secion, we describe e nesed socasic valuaion engine. In paricular, we inroduce e ris-neural scenario generaor, e real-world scenario generaor, and e cas flow proecions Ris-Neural Scenario Generaor Ris-neural scenarios are used in e inner loop o calculae e dollar Delas. We use a mulivariae Blac Scoles model inroduced by Carmona and Durrelman [18] o generae ris-neural scenarios. Tis model is also described in [17].

4 Daa 218, 3, 31 4 of 21 Le S 1), S 2),..., S ) be indices in e financial mare. Under e mulivariae Blac Scoles model, e ris-neural dynamics of e indices are given by [18]: or S ) d S ) S ) = r d + [ = exp r s d s 2 σ l d B l), S ) = 1, = 1, 2,..., 1) ) σl 2 + ] σ l B l), = 1, 2,...,, 2) were B 1), B 2),..., B ) are independen sandard Brownian moions, r is e sor rae of ineres, and e marix σ l ) is used o capure e correlaion among e indices. Le =, 1 =,..., m = m be ime seps wi equal space and suppose a e coninuous forward rae is consan wiin eac period. For = 1, 2,..., m, e accumulaion facor of e index for e period 1, ) can be calculaed as: A ) = S) S ) 1) = exp [ f 1 2 ) σl 2 + were f is e annualized coninuous forward rae for period 1, ) and ] l) σ l Z, 3) Z l) = Bl) Bl) 1). By e propery of Brownian moion, we now a Z l) 1, Zl) 2,..., Zl) m variables wi a sandard normal disribuion. Te coninuous reurn for e period 1, ) is calculaed as: Te marix ) = ln A ) = f 1 2 σl 2 + σ 11 σ 12 σ 1 σ 21 σ 22 σ σ = σ 1 σ 2 σ are independen random σ l Z l). 4) can be obained from e following Colesy decomposiion of e covariance marix Σ: σ σ = Σ = diagν)r diagν), 5) were ν = ν 1, ν 2,..., ν ) is a vecor of index volailiies, diagv) is a diagonal marix wi ν as diagonal elemens, and R is e correlaion marix. In marix form, Equaion 4) can be expressed as R 1) f R 2) 1 2 ν2 Z 1) 1 f =. 1 2 ν σ Z 2). 6). R ) f 1 2 ν2 Z )

5 Daa 218, 3, 31 5 of 21 Algorim 1 sows e pseudo-code of e ris-neural scenario generaor. Once we ave index scenarios simulaed from Equaion 3), we can obain e invesmen fund scenarios by blending ese index scenarios as follows: F ) i = were g is e number of invesmen funds and w l A l) i, = 1, 2,..., g, w 11 w 12 w 1 w 21 w 22 w W = w g1 w g2 w g is e fund mapping a maps e indices o e g invesmen funds. Algorim 1: Pseudo-code of e Ris-neural Scenario Generaor. Inpu: Forward raes f, volailiies ν, correlaion marix R, seed s, n, m, Oupu: Scenario marices A 1),..., A ) 1 Se e seed of e random number generaor o be s; 2 Calculae covariance marix Σ; 3 Calculae e Colesy decomposiion σ; 4 for i = 1 o n do 5 for = 1 o m do 6 Generae a vecor of normal random number z 1, z 2,..., z ); 7 Ge a vecor of index reurns R 1, R 2,..., R ) by Equaion 6); 8 Le A ) i 9 end 1 end expr ), were A ) i is e i, )- enry of A ) ; 11 Save e scenario marices ino files Real-World Scenario Generaor Real-world scenarios are used in e ouer loop o simulae e movemens of e mare. Ris-neural scenarios are prospecive and parameers of e ris-neural scenario generaor are calibraed o mare daa. Real-world scenarios are rerospecive and e parameers of a real-world scenario generaor are calibraed o isorical daa. In pracice, e regime-swicing model [19] is ypically used o generae real-world scenarios. Here, we inroduce a mulivariae wo-regime regime-swicing model for generaing correlaed real-world scenarios for muliple indices. Wiin a regime and a ime period, e evoluion of e indices follows e mulivariae log-normal model. Le ρ denoe e regime a ime and M denoe e ransiion marix, i.e., ) p M = 1,1 p 1,2, p 2,1 p 2,2 were p s,r = Pρ +1 = r ρ = s),, s, r = 1, 2. Le π = π 1, π 2 ) be e uncondiional probabiliy disribuion of e regime-swicing process. Ten, we ave πm = π,

6 Daa 218, 3, 31 6 of 21 wic gives [19]: π 1 = p 2,1 p 1,2 + p 2,1, π 2 = p 1,2 p 1,2 + p 2,1. 7) Le S 1), S 2),..., S ) be indices in e financial mare. Under e mulivariae wo-regime regime-swicing log-normal model, e ris-world dynamics of e indices are given by: S ) [ = exp µ ρ ) ρ + ] σ ρ ) l B l), S ) = 1, = 1, 2,...,, 8) were B 1), B 2),..., B ) are independen sandard Brownian moions, µ ρ ) is e geomeric mean of e index in e regime ρ, e marix σ ρ ) l ) is used o capure e correlaion among e indices in e regime ρ, and ρ {1, 2} is e regime number. Le =, 1 =,..., m = m be ime seps, were is e ime sep. Ten, for = 1, 2,..., m, e accumulaion facor of e index for e period 1, ) in e regime ρ can be calculaed as A ) = ρ S ) ρ S ) 1) ρ = exp [ µ ρ ) + σ ρ ) l ] l) Z, 9) were Z l) = Bl) Bl) 1). In marix form, e reurns can be expressed as R 1) R 2). R ) ρ ln A 1) ln A 2) =. ln A ) ρ = µ ρ ) 1 µ ρ ) 2. µ ρ ) + σ ρ ) Z 1) Z 2). Z ), 1) were e marix σ ρ ) can be obained from e following Colesy decomposiion of e covariance marix Σ ρ ) : σ ρ ) σ ρ ) = Σ ρ ) = diagν ρ ) )R ρ ) diagν ρ ) ), 11) were ν ρ ) is a vecor of index volailiies for regime ρ, diagv ρ ) ) is a diagonal marix wi ν ρ ) as diagonal elemens, and R ρ ) is e correlaion marix for regime ρ. Le ρ be e iniial regime. Ten, for = 1, 2,..., m, e regime for period can be deermined by generaing a uniform random number u as follows: ρ = 2, if ρ 1 = 1 and u p 1,2 ; 1, if ρ 1 = 1 and u > p 1,2 ; 1, if ρ 1 = 2 and u p 2,1 ; 2, if ρ 1 = 2 and u > p 2,1. Te coninuous reurn of e index for e period 1, ) in e regime ρ is calculaed = ln A ) ρ = µ ρ ) ρ + 12) σ ρ ) l) l Z. 13)

7 Daa 218, 3, 31 7 of 21 Algorim 2 sows e pseudo-code of e wo-regime regime-swicing real-world scenario generaor. Algorim 2: Pseudo-code of e Two-regime Regime-swicing Real-world Scenario Generaor. Inpu: p 1,2, p 2,1, n, m,, seed s, regime-1 volailiies v 1), regime-2 volailiies v 2), regime-1 correlaion R 1), regime-2 correlaion R 2), regime-1 geomeric means µ 1), regime-2 geomerix means µ 2) Oupu: Scenario marices A 1),..., A ) 1 Se e seed of e random number generaor o be s; 2 Calculae π 1 and π 2 by Equaion 7); 3 Calculae covariance marices Σ 1) and Σ 2) ; 4 Calculae e Colesy decomposiions σ 1) and σ 2) ; 5 for i = 1 o n do 6 Generae a uniform random number u; 7 Le ρ = 1 if u π 1, oerwise ρ = 2; 8 for = 1 o m do 9 Generae a uniform random number u; 1 Deermine e regime ρ according o Equaion 12); 11 Generae a vecor of normal random number z 1, z 2,..., z ); 12 Ge a vecor of index reurns R 1, R 2,..., R ) by Equaion 1); 13 Le A ) i 14 end 15 end expr ), were A ) i is e i, )- enry of A ) ; 16 Save e scenario marices ino files. From e above equaion, we can derive e expecaions and covariances of e condiioned reurns as follows: [ ] E = µ ρ ) ρ and Cov ), R m) = ρ ρ σ ρ ) l σ ρ ) ml. Te reurn of e index for e period 1, ) can be expressed as = I {ρ ρ =1 =1} + I {ρ ρ =2 =2}, 14) were I is an indicaor funcion. Te expeced reurn for e period 1, ) can be calculaed as p177 in [2]: [ E ] [ = E From Equaion 14), we ave R m) ρ =1 ] [ Pρ = 1) + E ρ =2 ] Pρ = 2) = µ 1) π 1 + µ 2) π 2. 15) = ρ =1 Rm) I {ρ ρ =1 =1} + ρ =2 Rm) I {ρ ρ =2 =2},

8 Daa 218, 3, 31 8 of 21 wic gives [ E R m) ] = π 1 π 2 µ 1) µ1) µ 2) µ2) m 2 + m 2 + ) σ 1) l σ 1) ml + σ 2) l σ 2) ml ). Terefore, we ave Cov, R m) ) [ = E R m) ] [ E = π 1 π 2 2 µ 1) µ 2) π 1 σ 1) l σ 1) ml + π 2 ] [ E R m) ] ) µ 1) m µ 2) m σ 2) l σ 2) ml ) +. 16) Leing m = in e above equaion, we ge e variance of e reurn as follows: were ν ρ ) = E Var [ ) ) ] 2 [ E ]) 2 = π 1 π 2 2 µ 1) µ 2) ) 2 + π1 = π 1 π 2 2 µ 1) µ 2) ) 2 + π1 σ 1) l ν 1) is e volailiy of e index in e regime ρ, i.e., ν ρ ) = ) 2 σ ρ ) l. ) 2 + π2 ) 2 + π2 ν 2) σ 2) l ) 2 ) 2, 17) Equaions 15) and 17) can be used o validae e real-world scenarios. Tese equaions can also be used o specify parameers for e wo-regime regime-swicing model if we wan o conrol e overall mean reurns and e overall volailiies of e indices Nesed Socasic Valuaion To describe ow nesed socasic proecions are done, we le N 1 be e number of ouer loop pas and le T 1 be e number of ime nodes in e ouer loop. Le X = {x 1, x 2,..., x n } be a porfolio of n VA policies. Algorim 3 sows a ig-level sec of e nesed socasic valuaion engine. A eac node along eac ouer loop pa, we calculae e fair mare values of eac policy using e ris-neural scenarios. For deails abou ow policies are aged along a real-world pa and ow e cas flows are proeced along a ris-neural pa, readers are referred o [17].

9 Daa 218, 3, 31 9 of 21 Algorim 3: A Hig-level Sec of e Nesed Socasic Valuaion Engine. Inpu: A porfolio X of VA policies, ris-neural scenarios, real-world scenarios, fund mapping, moraliy ables Oupu: Marices of parial dollar delas 1 for p = 1 o N 1 do 2 for = 1 o T 1 do 3 for i o n do 4 Age x i o ime along e p real-world pa; 5 Calculae e fair mare value of e aged policy wi all indices soced up 1%; 6 Calculae e fair mare value of e aged policy wi all indices soced down 1%; 7 Calculae e oal dollar dela of e aged policy; 8 Le M p,) i, be e parial dollar dela of x i on e index a ime along e p real-world pa; 9 end 1 end 11 end 12 Reurn e marices of parial dollar delas M p,). To assess e performance of dynamic edging, parial dollar delas are required as edging is done by individual radable indices. Te parial dollar dela on e index is normally calculaed as follows: FMV..., AV 1, 1.1AV, AV +1,...) FMV..., AV 1,.99AV, AV +1,...),.2 were AV denoes e accoun value invesed in e index. However, calculaing parial dollar delas using e above equaion requires proecing cas flows a many index socs. Tis is proibiive under e nesed socasic valuaion framewor. To reduce e runime, we only calculae oal dollar dela a eac node along an ouer loop pa as follows: = FMV1.1AV 1,..., 1.1AV ) FMV.99AV 1,...,.99AV ),.2 were is e number of indices. Ten, we approximae e parial dollar delas as follows: ) = AV AV AV. 18) Te relaion given in Equaion 18) can be derived as follows. Suppose a e fair mare value of e guaranees embedded in a VA policy is a funcion of e oal accoun value, i.e., FMV = f AV 1 + AV AV ). Ten, e parial dollar dela on e index is calculaed as ) = f AV AV = f TA TA AV AV = f TA TA AV TA = AV TA, were TA = AV 1 + AV is e oal accoun value. 3. Syneic Porfolio and Payoffs In is secion, we describe e syneic porfolio and e payoffs of e guaranees embedded in e VA policies.

10 Daa 218, 3, 31 1 of Syneic Porfolio We adoped a subse of e syneic VA porfolio creaed in [17]. Ta syneic porfolio conains 19 ypes of producs, eac of wic as 1, policies. We seleced 2 policies from eac produc ype. Te subse conains 38, policies. Readers are referred o [17] for a descripion of e feaures or variables of e VA policies. Table 1 sows e number of policies in eac produc ype by gender. Abou 4% of e policies in eac produc are female. Table 2 sows e summary saisics of some numerical fields. In e able, we see a all funds ave many zeros. Tis is because many policies generally do no inves in all e funds. Te age is e number of years beween e bir dae and e curren dae. Te ime o mauriy is calculaed from e curren dae and e mauriy dae. Table 1. Disribuion of gender by produc ype. Gender ABRP ABRU ABSU DBAB DBIB DBMB DBRP F M Gender DBRU DBSU DBWB IBRP IBRU IBSU MBRP F M Gender MBRU MBSU WBRP WBRU WBSU F M Table 2. Summary saisics of some fields. Noe a age and m are calculaed from e bir dae, valuaion dae, and mauriy dae. Min 1s Q Mean 3rd Q Max gbam. 187, , , ,6, gmwbbalance.. 35, ,78.73 widrawal.. 22, , FundValue1.. 33, , , FundValue2.. 38, , ,26, FundValue3.. 26, , , FundValue4.. 26, , , FundValue5.. 22, , , FundValue6.. 35, , ,42, FundValue7.. 29, , ,54.12 FundValue8.. 3, , ,72.85 FundValue9.. 29, , ,37.63 FundValue1.. 3, , ,822.7 age m Guaranee Payoffs We calculaed e payoffs of e guaranees for e porfolio along eac of e 1 real-world pa. Te payoff is calculaed as e sum of e dea benefi and e living benefi. Figure 3 sows e guaranee payoffs of e porfolio a eac mon along e 1 real-world pa. In e figure, we see a ere are some relaively large guaranee payoffs afer e 3 mon a some real-world pas. Te large payoffs are caused e GMAB producs, wic allow policyolders o renew. A isogram of e presen values of e guaranee payoffs along e real-world pas is sown in Figure 4. From e isogram, we see a e disribuion of ese presen values is posiively sewed.

11 Daa 218, 3, Guaranee payoff in billions) of Mon Figure 3. Guaranee payoffs along e 1 real-world pas PV of guaranee payoffs in billions) Figure 4. Presen values of e guaranee payoffs. We also calculaed e presen values of e guaranee payoffs along eac of e 1 real-world pas. Figure 5 sows a isogram of ese presen values. Te isogram sows a e disribuion of e presen values is posiively sewed. A some bad real-world pas, e guaranee payoffs are muc larger an ose a oer real-world pas. Table 3a sows some summary saisics of ese

12 Daa 218, 3, of 21 presen values. A e bes real-world pa, e guaranee payoff of is porfolio is 11,421 millions. If e wors real-world pa occurs, e guaranee payoff of is porfolio is 168,574 millions. Guaranee payoff in millions) Mon Figure 5. Guaranee payoffs along e wors e dar line) and e bes e gray line) real-world pas. Table 3. Summary saisics and condiional ail expecaions of e presen values of e guaranee payoffs. Te numbers are in millions. a) Min. 1s Qu. Median Mean 3rd Qu. Max. 11,421 3,393 4,556 45,535 54, ,574 b) CTE5 CTE75 CTE95 61, , , Table 3b sows e condiional ail expecaions CTEs) of e presen values of e guaranee payoffs a ree differen levels. Te CTE75 is calculaed as e mean of presen values from e wors.25 1 = 25 real-world pas. In e able, we see a e CTE75 is around 75,595 millions. Figure 6 sows e 1 real-world pas of e five indices, wic are e large cap equiy, e small cap equiy, e inernaional equiy, e fixed income, and e money mare. Te dar ic line in eac subfigure corresponds o e wors real-world pa, wic produces e larges presen value of e guaranee payoffs. Te gray ic line in eac subfigure corresponds o e bes real-world pa, wic produces e lowes presen value of e guaranee payoffs. In e figure, we see a e bes pa is above e wors pa.

13 Daa 218, 3, of 21 Figure 6. Real-world pas of e indices. Te dar ic line is e wors real-world pa. Te gray ic line is e bes real-world pa. Noe a e bes real-world pa is no e one a e very op and e wors real-world pa is no e one a e very boom. Tis is because e payoffs of GMAB producs in bull mare are large. In oer words, if e real-world pa a e very op occurs, e GMAB producs will incur large payoffs because e policyolders can renew by reseing e benefi o e iger of e accoun value and e exising benefi base. Figure 5 sows e guaranee payoffs a monly seps along e bes and e wors real-world pas. In e figure, we see a in general e payoffs along e bes real-world pa are iger an ose along e wors real-world pa. A a few mons near e end of e proecion orizon, e payoffs a e bes pa are iger an ose a e wors pa. Tis is caused by e GMAB producs, wic ave iger payoffs a beer mares due o e renew feaure. 4. Parial Dollar Delas In is secion, we presen e parial dollar delas calculaed by e nesed Mone Carlo simulaion meod described in Secion 2. As discussed in Secion 2, e nesed socasic valuaion program produces many marices of e parial dollar delas. In fac, e program produces N 1 H marices of parial dollar delas, were N 1 is e number of real-world pas and H is e number of indices. For p = 1, 2,..., N 1 and = 1, 2,..., H, le M p,) be e marix of e parial dollar delas on e index:

14 Daa 218, 3, of 21 M p,) = M p,) 1,1 M p,) M p,) 2,1 M p,). 1,2 M p,) 1,T 2,2 M p,) 2,T., 19).... M p,) n,1 M p,) n,2 M p,) n,t were n is e number of policies in e porfolio, T is e number of ime poins were parial dollar delas are calculaed, and M p,) i, denoes e parial dollar dela of e i policy on e index a evaluaion ime poin along e p real-world pa. Since we used N 1 = 1 real-world pas and T = 31 evaluaion ime poins and e number of indices is H = 5, e number of marices we produced is 5. Eac marix as a size of 38, 31. We saved all e marices o CSV files wi only six decimal places. If zip all e CSV files, e size of e zip file is around 2 GB Aggregae Resuls Te aggregae parial dollar delas along a real-world pa are calculaed as follows: M p,) n = M p,) i,. 2) i=1 In oer words, e aggregae parial dollar delas are e parial dollar delas of e wole porfolio. Te aggregae oal dollar delas are calculaed as M p) = H =1 n i=1 M p,) i,. 21) Figure 7 sows e aggregae parial dollar delas and aggregae oal dollar delas along e 1 real-world pas. In e figure, we ave e following observaions: Te aggregae parial dollar delas do no approac zero a e end of e proecion orizon. Tis is caused by e GMAB producs, wic beave similar o call opions. Te guaranees are more sensiive o indices wi iger volailiies. For example, e magniudes of e aggregae parial dollar delas on e small cap equiy are larger an ose on oer indices. Te aggregae oal dollar delas along e bes and e wors real-world pas ave similar magniudes. Tis is because e dollar delas of e GMAB produc offse ose of oer producs. For equiy indices, wic ave ig volailiies, e aggregae parial dollar delas ave similar magniudes along e bes and e wors real-world pas. For non-equiy indices, wic ave low volailiies, e aggregae parial dollar delas along e wors real-world pa ave iger magniudes an ose along e bes real-world pa. Figures 8 and 9 sow e isograms of e aggregae parial dollar delas a e Year 1 and e Year 3, respecively. In e figures, we see a e disribuions of e aggregae parial dollar delas a e Year 3 is more sewed a ose a e Year 1.

15 Daa 218, 3, of 21 Figure 7. Aggregae parial dollar delas and aggregae oal dollar dela along e 1 real-world pas. Te dar ic and e gray ic lines correspond o e wors and e bes real-world pas, respecively Dollar Dela Dollar Dela Dollar Dela Dollar Dela Dollar Dela 5 Figure 8. A isogram of e aggregae parial dollar delas along e 1 real-world pas a e end of Year 1. Te numbers are in millions.

16 Daa 218, 3, of Dollar Dela Dollar Dela Dollar Dela Dollar Dela Dollar Dela 5 Figure 9. A isogram of e aggregae parial dollar delas along e 1 real-world pas a e end of Year 3. Te numbers are in millions Seriaim Resuls Tere are many seriaim resuls, maing i difficul o sow all e resuls in deail. In is secion, we only sow e seriaim resuls from e bes and e wors real-world pas idenified before. Figure 1 sows a isogram of e seriaim parial dollar delas a e end of Year 1 if e bes real-world pa occurs. Figure 11 sows a similar isogram if e wors real-world pa occurs. Bo figures sow a e disribuions of e seriaim parial dollar delas are igly sewed. In addiion, some policies ave posiive dollar delas if e bes real-world pa occurs. Tis is caused by e GMAB producs as a bull mare can rigger e renew opion embedded in suc producs. Figures 12 and 13 sow e box plos of seriaim parial dollar delas by produc ype a e end of Year 1 along e bes and e wors real-world pas, respecively. In ese figures, we see a e GMAB, GMIB, and GMMB producs are more sensiive an e GMDB and GMWB producs in erms of e magniudes of e delas. In addiion, more policies ave posiive delas wen e bes real-world pa occurs an e case wen e wors real-world pa occurs.

17 Daa 218, 3, of Dela 1 in ousands) Dela 2 in ousands) Dela 3 in ousands) Dela 4 in ousands) Dela 5 in ousands) Figure 1. Hisograms of e seriaim parial dollar delas a e end of Year 1 of e bes real-world pa Dela 1 in ousands) Dela 2 in ousands) Dela 3 in ousands) Dela 4 in ousands) Dela 5 in ousands) Figure 11. Hisograms of e seriaim parial dollar delas a e end of Year 1 of e wors real-world pa.

18 Daa 218, 3, of 21 ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 1 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 2 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 3 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 4 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 5 in ousands) Produc ype Figure 12. Box plos of e parial dollar delas by produc ype a e end of Year 1 of e bes real-world pa.

19 Daa 218, 3, of 21 ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 1 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 2 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 3 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 4 in ousands) Produc ype ABRP ABRU ABSU DBAB DBIB DBMB DBRP DBRU DBSU DBWB IBRP IBRU IBSU MBRP MBRU MBSU WBRP WBRU WBSU Dela 5 in ousands) Produc ype Figure 13. Box plos of e parial dollar delas by produc ype a e end of Year 1 of e wors real-world pa Runime We implemened e nesed socasic valuaion engine as a disribued muli-reading program in Java. We used e HPC Hig Performance Compuing) cluser ps://pc.uconn.edu/) a e Universiy of Connecicu o run e program. In paricular, we used eig insances of e program wi 2 cores for eac insance o calculae e parial dollar delas for e porfolio. Eac insance of e program andles one ouer loop pa a a ime. Te coordinaion beween differen insances is done via e mecanism of file locing. Even wi 16 cores, i oo abou wo wees o ge all e calculaions done.

20 Daa 218, 3, 31 2 of 21 For e convenience of comparison, we accumulae e runime used by all reads o ge e runime a would be used by a single core. Figure 14 sows a isogram of e runime used o calculae e parial dollar delas for an ouer loop pa. In e figure, we see a, if a single core is used, i would ae e core abou 2 32 o finis e calculaion for a single ouer loop pa. If we aggregae e runime used o process all 1 ouer loop pas, e runime is 93,722,2.966 s or 2.97 years. In oer words, if we used a single CPU o calculae e parial dollar delas for e porfolio of 38, VA policies wi 1 real-world pa and 1 ris-neural pas, i would ae is CPU abou 2.97 years o finis e calculaion. Noe a we only calculaed e parial dollar delas a 3 ime poins along e ouer loop pas. If we wan o calculae e delas a 36 ime poins along e ouer loop pas, i would ae a single core abou 36 years Hours Figure 14. Disribuion of e runime for e 1 real-world pas. 5. Concluding Remars Meamodeling ecniques ave been proposed o address e compuaional issues associaed wi e nesed socasic valuaion of large VA porfolios. However, i is difficul for researcers o obain real daases from insurance companies o es e meamodeling ecniques and publis e resuls in academic ournals. I is e primary purpose of is paper o creae syneic daases o address compuaional issues. Tese syneic daases can be used by researcers and praciioners o es ecniques, especially meamodeling ecniques, o speed up e nesed socasic valuaion of large VA porfolios. Tese syneic daases ave some limiaions. Firs, e syneic VA policies are simpler an VA policies sold in e real-world. Second, e Mone Carlo simulaion is also simpler an e one used in pracice. For example, we did no consider e policyolder beavior in e cas flow proecions. Aloug e syneic daases ave limiaions, we can sill use em o es meamodeling ecniques. If a meamodeling ecnique does no wor for e syneic daases, en i is unliely o wor for real daases. Auor Conribuions: Bo G.G. and E.A.V. conceived e primary idea of e meodology used. G.G. wroe e Java code o implemen e meodology and creae e syneic daa. In consulaion wi E.A.V., G.G. analyzed e daa and produced e oupu. Bo auors ad periodic meeings o discuss e resuls and e wriing of e manuscrip. Funding: Tis wor is suppored by a CAE Ceners of Acuarial Excellence) gran p://acscidm.ma.uconn. edu) from e Sociey of Acuaries. Conflics of Ineres: Te auors declare no conflic of ineres.

21 Daa 218, 3, of 21 Appendix A. Sofware and Daases We implemened e nesed socasic valuaion engine as a disribued muli-reading program in Java. Te daases and e sofware code can be downloaded from p:// ~gan/sofware.ml. References 1. Ledlie, M.C.; Corry, D.P.; Finelsein, G.S.; Ricie, A.J.; Su, K.; Wilson, D.C.E. Variable Annuiies. Br. Acuarial J. 28, 14, [CrossRef] 2. Te Geneva Associaion Repor. Variable annuiies An analysis of financial sabiliy. Available online: ps:// ga213-variable_annuiies_.pdf accessed on 3 Augus 218). 3. Boyle, P.; Hardy, M. Reserving for mauriy guaranees: Two approaces. Insurance Ma. Econ. 1997, 21, [CrossRef] 4. Copra, D.; Erzan, O.; de Ganes, G.; Grepin, L.; Slawner, C. Responding o e Variable Annuiy Crisis. McKinsey Woring Papers on Ris. 29. Available online: ps:// ris/our-insigs/responding-o-e-variable-annuiy-crisis accessed on 3 Augus 218). 5. Inernaional Acuarial Associaion. Socasic Modeling: Teory and Realiy from an Acuarial Perspecive; Inernaional Acuarial Associaion: Onario, Canada, Dardis, T. Model Efficiency in e U.S. Life Insurance Indusry. Model. Plaform 216, 3, Gan, G. Applicaion of daa clusering and macine learning in variable annuiy valuaion. Insurance Ma. Econ. 213, 53, [CrossRef] 8. Gan, G.; Lin, X.S. Valuaion of large variable annuiy porfolios under nesed simulaion: A funcional daa approac. Insurance Ma. Econ. 215, 62, [CrossRef] 9. Gan, G. Applicaion of Meamodeling o e Valuaion of Large Variable Annuiy Porfolios. In Proceedings of e Winer Simulaion Conference, Huningon Beac, CA, USA, 6 9 December 215; pp Gan, G.; Lin, X.S. Efficien Gree Calculaion of Variable Annuiy Porfolios for Dynamic Hedging: A Two-Level Meamodeling Approac. N. Am. Acuarial J. 217, 21, [CrossRef] 11. Gan, G.; Valdez, E.A. An Empirical Comparison of Some Experimenal Designs for e Valuaion of Large Variable Annuiy Porfolios. Depend. Model. 216, 4, Heazi, S.A.; Jacson, K.R. A neural newor approac o efficien valuaion of large porfolios of variable annuiies. Insurance Ma. Econ. 216, 7, [CrossRef] 13. Gan, G.; Huang, J. A Daa Mining Framewor for Valuing Large Porfolios of Variable Annuiies. In Proceedings of e 23rd ACM SIGKDD Inernaional Conference on Knowledge Discovery and Daa Mining, Halifax, NS, Canada, Augus 217; pp Heazi, S.A.; Jacson, K.R.; Gan, G. A Spaial Inerpolaion Framewor for Efficien Valuaion of Large Porfolios of Variable Annuiies. Quan. Financ. Econ. 217, 1, Gan, G.; Valdez, E.A. Regression Modeling for e Valuaion of Large Variable Annuiy Porfolios. N. Am. Acuarial J. 218, 22, [CrossRef] 16. Xu, W.; Cen, Y.; Coleman, C.; Coleman, T.F. Momen macing macine learning meods for ris managemen of large variable annuiy porfolios. J. Econ. Dyn. Conrol 218, 87, 1 2. [CrossRef] 17. Gan, G.; Valdez, E.A. Valuaion of Large Variable Annuiy Porfolios: Mone Carlo Simulaion and Syneic Daases. Depend. Model. 217, 5, [CrossRef] 18. Carmona, R.; Durrelman, V. Generalizing e Blac-Scoles Formula o Mulivariae Coningen Claims. J. Compu. Financ. 26, 9, [CrossRef] 19. Hardy, M. A Regime-Swicing Model of Long-Term Soc Reurns. N. Am. Acuarial J. 21, 5, [CrossRef] 2. Gan, G.; Ma, C.; Xie, H. Measure, Probabiliy, and Maemaical Finance: A Problem-Oriened Approac; Jon Wiley & Sons, Inc.: Hoboen, NJ, USA, 214. c 218 by e auors. Licensee MDPI, Basel, Swizerland. Tis aricle is an open access aricle disribued under e erms and condiions of e Creaive Commons Aribuion CC BY) license p://creaivecommons.org/licenses/by/4./).