The Economic Capital and Risk Adjustment Performance for VA with Guarantees with an example of GMAB

Transcription

1 h Economic Capia an Risk Ajsmn Pfomanc fo VA wih Gaans wih an xamp of GMAB Jn Zho Risk Anayics Pincipa, Qaniaiv Anaysis Gop, Amican Innaiona Gop, Inc. Song Won Pak 1 Managmn Associa, Qaniaiv Anaysis Gop, Amican Innaiona Gop, Inc. Songwon.pak@aig.com Absac Economic Capia EC an Risk Ajs Pfomanc Masmn RAPM a bcoming incasingy impoan ciia in poc vopmn an pfomanc vaaion fo insanc companis. h EC famwok can assis in sagic cision making an incas capia aocaion fficincy. Howv, h a vy fw pcis an ai finiions fo EC an RAPM in if insanc cas sis. his pap poposs a gna finiion fo EC ha can b appi o vaios yps of insanc pocs in a consisn mann. Aso, w wi isa how o caca RAPM by sing a fai va famwok. W appy h EC famwok sing a spcific xamp: a simp Vaiab Anniy wih Gaan Minimm Accon Bnfi GMAB. h pap aso isas how o ajs EC fo a ong im-hoizon an how hging affcs h conomic capia an ova pfomanc. W conc by showing how h EC-bas RAPM famwok fo h GMAB can b xn o vaa vaios yps of Vaiab Anniy VA gaans sch as a Gaan Minimm Incom Bnfi GMIB. W aso show how o incas h fficincy of capia sag. Ky wos: Vaiab Anniy, Gaan Minimm Accon Bnfi, Economic Capia, RAPM, Capia Managmn, Va-A-Risk 1 Goqiang Li of ERM QR povi caf viw an vaab inps. Mak Sny fom QR aso gav s sf commns.

2 1. Inocion Economic Capia EC an Risk Ajs Pfomanc Masmn RAPM a bcoming incasingy impoan ciia in poc vopmn an pfomanc vaaion fo insanc companis. Evn hogh h gaoy capia qimn povis sfficin pocion o poicyhos, i fais o cap a company's "" isk, which ss in infficin aocaion of capia. Compa wih gaoy capia qimns, h conomic capia can impov h fficincy of capia sag sinc: - EC caps h ivsificaion bnfis among bsinss. - EC aows h company o s fim-spcific aa an mo o vaa is own isk. - EC can fc a of h avaiab financia socs ha may no b cap n gaoy capia qimns, sch as hging insmn an h mak-o-mak fai va of iabiiis. - EC's isk hoizon cospons o h company's bging pocss which is say on annay. - EC povis a consisn famwok n which h pfomanc of iffn ins of bsinss can b vaa. In spi of h pomising bnfis of EC, h a vy fw ai cas sis of is s in if insanc. h main ppos of his pap is o povi a conc mhooogy fo conomic capia cacaion. W s an xamp of a VA wih a GMAB o isa h mhooogy. W aso fin a Risk-Ajs Pfomanc Masmn RAPM in his conx. inay, w iscss h ciica isss o b sov in o o appy h EC famwok in a pacica sing. 1

3 2. Economic Capia fo h Vaiab Anniy wih Gaans W s wo anom vaiabs o caca h conomic capia fo h VA wih gaans: 1 ai Va an 2 Loss fncion. Un h conomic famwok, boh asss an iabiiis sho b va n h mak-consisn picing mas. Dmining h mak va of mos asss is saighfowa, b h is no gna mhooogy fo cacaing h fai va of a iabiiy. W popos a mhooogy o caca h fai va fo conomic capia moing. An, hn, w wi show how o iv h oss isibion o caca EC. 2.1 Vaaion foma fo h VA wih GMAB h fai va is qivan o h mak-o-mak va a a cain im. o povi h xac foma fo i, w wi s h foowing assmpions fo h GMAB. Unying Sock oa Rn: S S = µ σ W wh µ : h xpc oa n σ : h voaiiy m fo h P-mas W : h sana Bownian Moion pocss. n Pocss q = 0 S0 S, wh fs a c popoionay fom h fn a h coninos a of q = m ε δ an m : h managmn an xpns f ε : h conomic cos fo h GMAB δ : h spa o h GMAB chag. h vas of m, ε, an δ a consans. 2

4 Acaia Assmpion G : h gaan va : h maiy fo h GMAB : h coninos foc of aps : h coninos foc of moaiy No ha h is a spa δ o h GMAB chag. If h GMAB is pic wihin a no-abiag famwok, δ sho b zo; howv, in pacic, GMAB wis say commi a spa ov h mak pic of h GMAB as a cshion fo mak ncainy. Un h assmpion scib abov, h fai va V fo h VA wih gaans a im is caca as Q Q f G E PV E V = wh Expc va n h mak consisn picing mas o Q-mas E Q : : Risk a f : GMAB Maiy : Psn Va of h incom vaa a im. PV o any yp of gaan, h anayica foma fo h xpc fs is: PV E Q τ τ δ ε x x s s s Q p s p E = s s s Q s E δ ε = = s - Q E 0 δ ε = Q E 0 δ ε = q 0 δ ε = 3

5 q 0 δ ε = q 0 δ ε = q q 1 δ ε = wh h pobabiiy ha a poicyho ag x wi say in-foc a im x. : τ P x o h bnfi pa, a VA wih GMAB can b pic wih h Back-Shos-Mon opion picing foma bcas is payoff xcing h cmns is qivan o a p opion. Wih h cmn assmpion, h GMAB can b pic wih h foowing foma: Q f G E = q G P,,, wh is h Back-Shos-Mon pic foma givn a sock pic S, h xcis pic X, h im o maiy an ivin yi.,,, X S P I is impoan o sss ha h V of a GMAB ypicay cass wih im. As im-o-maiy cass, cash infows GMAB fs an ofows boh cas, b h cash ofows cass a a sow a. Sinc h GMAB has no inim cash ofows bfo maiy, h xpc cash ofow fom h GMAB may xc h xpc cash infows. As an xm xamp, h V immiay bfo maiy sho b ss han o qa o zo bcas h a no ong fs o chags b h company may n o pay h GMAB bnfis. h foowing ab shows his naa cas in h V of a VA wih GMAB. Each va is h xpc va a ach im bas on 5,000 simaions. Ya V PVGMAB PVGMAB Ya Ya Ya Ya Ya Ya ab Expc ai Va a vaios im hoizons$, hosans 4

6 2.2 Loss isibion an Economic Capia h gna finiion of EC is a capia amon o cov f ncainy ov a isk hoizon wih a cain confinc v. o caca EC, w hav o fin a anom vaiab ha can cap h f s ncainy o gna h isibion of isks. Givn h isibion fo h anom vaiab, w can min h conomic capia bas on a van isk mas sch as Va-a-Risk VaR o Coniiona ai Expcaion CE. o xamp, h VaR can b fin as: ~ Loss R < 1α pob 1 wh Loss: a oss anom vaiab a im vaa n a aisic mas P-mas R ~ : a ag isk oanc o Va-a-Risk α : a confinc v Loss isibion can b iv fom h n isibion sinc h oss is qivan o h ngaiv n. h n, by finiion, is h sm of h incom gain an h capia gain. o h GMAB, incom gain is fin as: Incom Gain = Incom Expns Incom = GMAB Chag Inim Cash Expns = GMAB Commission & Expnss Rn = Incom Gain V V = Chang in ai Va Hnc, oss can b caca as Loss = - Incom Gain V = - Incom Gain V V0 = V0 Incom Gain V Aso, in his qaion, Emb Va EV a im can b fin as EV = Incom Gain V 5

7 Wih h finiion of EV, h qaion 1 bcoms pob ~ Loss R < 1α = pob V 0 Incom Gain V R < 1α = pob V 0 EV R < 1α ~ ~ his foma is xacy sam as ha fo VaR cacaion. Hnc, h conomic capia can b caca bas on VaR as: pob EV 0 EV EC, α < 1α wh EV0 = V0 Simiay, h conomic capia bas on h CE can b caca as EC, α VaR = f 1α wh VaR: Va-a-Risk a a confinc v α : h oss anom vaiab a im f: h isibion fncion fo h oss anom vaiab No ha sinc incom gain fom h GMAB is pah-pnn vaiab, h is no anayica foma o caca h conomic capia fo any yp of gaan, incing GMAB. o his ason, a aisic simaion P-mas is qi o caca h conomic capia. Givn h conomic va isibion, w can sima h VaR fo h VA wih GMAB a ach isk hoizon. h isk hoizon wi b min pning on a company's cision making pocss. Insa of iscssing which isk hoizon is van, w inoc an iss a o isk hoizons. Risk, by na, os incas as h isk hoizon gs ong. o xamp, h ncainy ov n yas is cay bigg han ha ov on ay o on ya. o his ason, h 6

8 confinc v fo ach iffn isk hoizon sho b ajs accoing o ag aings an hisoica fa pobabiiis. h foowing ab shows h hisoica fa pobabiiis by ach aing gop. Risk Hoizon 1-Ya 3-Ya 5-Ya AAA 0.00% 0.04% 0.12% AA 0.01% 0.08% 0.26% ab Avag Dfa Ras 1981 o 2004% 2 h company may o may no icy s hisoica fa pobabiiis as h confinc inva. Howv, i is ca ha confinc inva sho b ajs fo ach isk hoizon; ohwis isk may b ih ovsima o nsima compa o h "" isk. Anoh iss on h confinc inva is how w can sima an xm f ai va. As sn in h ab, h hisoica confinc inva fo AA gop ov 1-ya isk hoizon is 99.99%. If w wan o s his confinc inva an sima h f-ai va bas on h simaions, a as 100,000 simaions a qi o sima h f-ai va, qiing hg amon of comping socs an im. Bcas of h socs qi, w n o vop an anaiv mhooogy o c h nmb of simaions. Exm Va hoy o Vaianc Rcion chniqs can b s fo his ppos. 2.3 Economic Capia wih Hging Pofoio A hging pofoio os no c qi gaoy capia; howv a hging pofoio can sbsaniay c Va a Risk, so i can aso c conomic capia qimns. Wih a hging pofoio, h oss is fin as: Loss = - Incom Gain V MV of Hging = V0 Incom Gain V MV of Hging 2 Soc: Sana & Poo s Goba ix Incom Rsach 7

9 = V0 EV MV of Hging wh MV of Hging = Chang in Mak Va of Hging Pofoio Wih h oss isibion, EC wih a hging pofoio can b sima bas on a van mas as xpain in h pvios scion. 2.4 Risk-Ajs Pfomanc Masmn As xpain in Scion 2.2, EV is h va ca fo h bsinss a a cain im. Hnc, vas ca fom h poc can b fin as Incom Gain = Incom Expns EV = Incom Gain V Economic Va AEVA = EV Cos of Capia c wh Cos of Capia = EC 1 c: cos of capia% h mb va is somims ca Avaiab Capia o h amon of avaiab financia socs. I is mainy compos of wo pas 1 aiz incom gain ov h pas an 2 naiz picing magin which wi b aiz ov h f. In h conomic viwpoin, h wo vas a aiiv an a gain o shahos. I is impoan o nsan how EV affcs pfomanc masmns. W wi monsa his wih a simp xamp. Ya V V Incom Gain SmIncom Gain Rn=IG V EV=V SmIG ab Bsinss Uni 1, which has iniia picing magin 8

10 Ya V V Incom Gain SmIncom Gain Rn=IG V EV=V SmIG ab Bsinss Uni 2, which has no iniia picing magin wo abs abov show ha h fai va cass wih h passag of im, casing capia oss bwn ach im. D o h capia oss, oa n fo ach im pio bcoms zo; fo xamp, oa n bwn im 0 an im 1 is compos of incom gain1 Capia Loss-1. If w vaa h pfomanc js bas on incom gain, boh bsinsss ca xacy h sam va; howv, in h conomic sns, h wo Bsinss Unis hav iffn EV a ach im 10 vs. 0, vn hogh hy hav h sam n an incom gain. Acay, h fis bsinss has h posiiv financia socs a ach im. his is bcas Bsinss Uni 1 so h sam poc a h high pic han Bsinss Uni 2 i. h EV concp givs appopia ci o Bsinss Uni 1 fo is spio pfomanc. Hnc, if w vaa h pfomanc bas on h EV concp, w can g consisn pfomanc masmn sinc w can vaa how mch va is a by a paica poc gass of im hoizons. h concp of EV can b shown as: V Incom Gain Accma Incom Gain Discon Va EV = Incom Gain V Accoingy, h concp of ai Va is ccia o cacaing h isk-ajs n. V xiss o h xcss of fs ov xpc bnfis an is gaay as as incom ov h maining if of h poc. An iss of pfomanc masmn is how o assss V. Un h fai va famwok, his ss in an iniia naiz capia gain an pocs high 9

11 voaiiy in h isk-ajs pfomanc masmn. o sov his iss, w n o vis a smoohing mhooogy o fc h conomic sns of n. Rca ha h EV is compos of 1 aiz incom gain an 2 naiz picing magin. I is ca ha aiz incom gain sho b a o h n fom h bsinss; h naiz picing magin wi b aiz ov h maining ifim. hfo, w can appy h amoizaion concp o fc h maining picing magin as h n fo ach im hoizon by: V on Risk-Ajs CapiaVORAC 1 τ = 1 V/EC 1 Wh τ: im-o-maiy = his is an anna a of n w can xpc fo h maining if of im of h poc. Wih h concp of VORAC, vaios Risk-Ajs Pfomanc Masmns a fin as: Rn on Risk-Ajs CapiaRORAC = Incom Gain / EC,α Ajs Rn on Risk-Ajs CapiaAj. RORAC = RORAC VORAC Risk-Ajs Rn on Risk-Ajs CapiaRARORAC = Aj. RORAC Cos of Capia % 10

12 3. Cas Sy: VA wih GMAB In his scion, w wi povi h isaiv xamp fo h GMAB. o povi a conc xamp abo EC an RAPM, w compa wo bsinsss: 1 on wih iniia spa 1% o h GMAB f an 2 h oh wih no spa. o simaion, w assm ha h is no gain o osss fom managmn an xpns fs, so ha w a vaaing isks ony fom h GMAB bsinss. In aiion, incom gain fo ach pio is accma a h isk-f a. Unying Ass µ: 10%, σ: 15%, ivin: 1.5%, isk f a: 5%, Impi Vo.: 20% Pmim: $1,000,000 GMAB Gaan: 100%, Maiy: 10-ya, M&E : 1.5% Bsinss 1 Bsinss 2 Picing Assmpion GMAB : 1.31%, Spa o GMAB Chag: 1% GMAB Spa= 2.31% oa : 3.81% GMAB : 0.98%, Spa o GMAB Chag: 0% GMAB Spa= 0.98% oa : 2.48% Dcmns Moaiy: 1%, Laps: 2% Expns Iniia Commission: 5%, M&E : 1.5% Nmb of simaions 5,000 aisic scnaios fo h nying sock inx ab3.1. Vaaion assmpions fo ach bsinss ni. Iniia Picing Magin Bas on h assmpion abov, ach bsinss has an iniia picing magin as foows: PV Incom PVGMAB Picing Magin o V0 Bsinss Bsinss ab 3.2. Iniia picing magin fo ach bsinss$, hosans. 11

13 Economic Capia an RAPM As xpain abov, h conomic capia can b s bas on h oss n isibion. h foowing cha shows an xamp fo h BU1 ov a 1-ya isk hoizon. Disibion fo Rn a 1-Ya fo h VA wih GMAB % 25% 38% 50% 75% 90% 99% -100 Economic Capia -150 Cha 3.1. Rn isibion a 1-ya hoizon fom h BU1. h foowing ab smmaizs EC an RAPM fo ach bsinss ni. o simpiciy, w s confinc invas as foows insa of sing an xmy high pcni va. Aso, h s is bas on 5,000 simaions an EC is min by VaR mas. Incom gain an ai Va a h xpc vas acoss a of h 5,000 simaions. Risk Hoizon 1-Ya 3-Ya 5-Ya C.L. % 99.00% 98.00% 97.00% Bsinss Uni BU1 BU2 BU1 BU2 BU1 BU2 EC Incom Gain RORAC 17.60% 5.11% 12.14% 3.80% 11.10% 3.78% V

14 Risk Hoizon 1-Ya 3-Ya 5-Ya C.L. % 99.00% 98.00% 97.00% Bsinss Uni BU1 BU2 BU1 BU2 BU1 BU2 VORAC 6.09% 0.39% 5.42% 0.84% 5.85% 1.23% EV Aj. RORAC 23.69% 5.50% 17.57% 4.63% 16.95% 5.01% Cos a 10% EVA RARORAC 13.69% -4.50% 7.57% -5.37% 6.95% -4.99% ab 3.3. EC an RAPM fo ach bsinss a sva isk hoizons $, hosans. Economic Capia wih Hging h foowing ab smmaizs hging anaysis fo h bsinss ni 1 a 1-ya isk hoizon. h hging amon is fin as h pcnag of h iniia GMAB pic. In his xamp, h iniia mak va of h GMAB is $95.28 hosans. Aso, fo h hging insmns, w s h A-h-Mony opion wih a consan impi voaiiy of 20%. h capia gain fom h hging insmns is a o h incom gain. Hging % of GMAB 0% 10% 20% 30% 40% EC Incom Gain Cap Gain fom Hg RORAC 17.60% 17.45% 17.27% 17.06% 16.86% V VORAC 6.09% 6.43% 6.81% 7.24% 7.74% EV Aj. RORAC 23.69% 23.88% 24.08% 24.29% 24.60% Cos a 10% EVA RARORAC 13.69% 13.88% 14.08% 14.29% 14.60% ab 3.4. Hging Anaysis fo BU1 wih iffn hging bgs $, hosans. 13

15 h hging anaysis shows: - Hging is ffciv n h conomic capia famwok sinc hging pofoio incass capia fficincy; fo xamp, ov h 1-ya isk hoizon, h RARORAC wiho hging is 13.69% whi ha wih 40bps hging is 14.60%. - Sinc h hging sagy qis a sma amon of conomic capia, h company can iiz his xcss capia o gna mo gains. - If w s RORAC as h pfomanc masmn, hn h no-hging sagy ominas oh sagis; fo xamp, RORAC wih no-hging is 17.60% whi ha wih 40-hging is ony 16.86%. his xamp monsas h impoanc o a VORAC o h pfomanc masmn. Ajs RORAC o RARORAC can fcs h company s conomic va ca fom h bsinss. 4. Economic Capia fo oh yps of Gaans hs fa, w hav vaa conomic capia an hav psn isk-ajs pfomanc masmns fo h VA wih GMAB. h conomic capia mo fo h GMAB can b xpn o mo compica yps of gaans sch as sch as Gaan Minimm Wihawa Bnfis GMWB o Gaan Minimm Incom Bnfis GMIB. Rca ha h conomic capia fo h VA wih gaans is fin as pob V 0 Incom Gain V EC, α < 1 α wh Incom Gain is vaa n a aisic masp-mas an V is h fai va a im vaa n a isk-na masq-mas. o h GMAB, w i a simaion sing h P-mas an caca V bas on an anayica foma. h basic ia fo oh yps of gaans is ssniay sam; howv, h changing iss fo oh yps of gaans is ha h is no anayica soion fo h fai va cacaion bcas of h pah-pnn popy of bnfi o-fows GMDB o GMWB 14

16 an h ncainy fom h f ins as cv GMIB. h fai va can ony b caca wih a isk-na simaion Q-mas. As a s, a wo-i simaion is qi. h fis i ss aisic scnaios ov h isk hoizon an h scon i ss isk na scnaios fom h isk hoizon p o h poc maiy o h vaaion hoizon. h cha bow shows his concp gaphicay. If w wan o o isk anaysis, sch as a pofi s bas on h conomic capia, w n on mo o aisic scnaio fo simaions. Raisic Simaion Risk Na Simaion EEV0 EC = VaR EEV,α -50 Risk Hoizon Vaaion Hoizon EEV Disibion fo VA wih GMAB oay Cha 4.1. wo-i simaion famwok On ifficy wih h wo-i simaion famwok is ha picing a a cain isk hoizon qis a s of mak consisn isk-na scnaios a h isk hoizon. h pobm is ha w may no hav mak insmns a h isk hoizon, so i s iffic o caiba an gna isk na scnaios; hnc, w hav o vis a mho o caiba h isk-na scnaio a ach isk hoizon. 15

17 5. Concsion his pap inocs a mhooogy o vaa EC fo VA wih gaans. Loss an fai va anom vaiabs a ccia o h cacaion of conomic capia. Using hs wo ky anom vaiabs, w inoc how o vaa EC fo VA wih GMAB an w fin vaios RAPM. Aso, w iscss som ciica isss ha ms b sov in o o appy h EC famwok o a bsinss. h mo inoc in his pap can b xn o oh if insanc pocs, sch as vaiab anniis wih oh yps of gaans. 16