A pricing model for the Guaranteed Lifelong Withdrawal Benefit Option

Transcription

1 A pricing model for he Guaraneed Lifelong Wihdrawal Benefi Opion Gabriella Piscopo Universià degli sudi di Napoli Federico II Diparimeno di Maemaica e Saisica

2 Index Main References Survey of he Variable Annuiy Producs and embedded opions Pricing model for GLWB opions Applicaion of he model o USA marke

3 Index Main References Survey of he Variable Annuiy Producs and embedded opions Pricing model for GLWB opions Applicaion of he model o USA marke

4 Main References Bauer D., Kling A., Russ J., A universal pricing framework for guaraneed minimum benefi in variable annuiies. ASIN Bullein 38 (2), Boyle P.P, Schwarz E., Equilibrium prices of guaranees under equiy-linked conracs. Journal of Risk and Insurance 44 (2), Clemens J., For a conservaive invesmen, variable annuiies are oo cosly. Wall Sree Journal C1, January 21 Dai M., Kwok Y.K., Zong J., Guaraneed minimum wihdrawal benefi in variable annuiies. Mahemaical Finance ( in press). Holz D., Kling A., Rub J., GMWB For Life. An Analysis of Lifelong Wihdrawal Guaranees. Working Paper ULM Universiy Milevsky M.A., Salisbury.S., Financial valuaion of guaraneed minimum wihdrawal benefis. Insurance: Mahemaics and Economics 38, 21-38

5 Index Main References Survey of he Variable Annuiy Producs and embedded opions Pricing model for GLWB opions Applicaion of he model o USA marke

6 Wha is a Variable Annuiy? As VAs are essenially a new produc class in he U.K., an indusry sandard definiion does no ye exis We shall define a VA as any uni-linked or manage fund vehicle wich offers opional guaranee benefis. ( Ledlie e al., 2008, Insiue of Acuaries, London) Uni Linked policy Embedded Opions Variable Annuiy

7 he embedded opions GMDB: he Guaraneed Minimum Deah Benefih opion guaranees a minimum reurn of he principal invesed upon he deah of he policyholder GMAB: he Guaraneed Minimum Accumulaion Benefih opion offers he policyholder a guaraneed minimum a mauriy if he is sill alive GMIB: he Guaraneed Minimum Income Benefih opion offers a minimum income sream from a specified fuure poin in ime. GMWB: he Guaraneed Minimum Wihdrawal Benefih opions gives he policyholder he possibiliy o wihdraw a pre-specified amoun annually unill he mauriy even if he fund value has fallen below his value.

8 Wha is a GLWB? he GLWB opion is a GMWB for Life opion: i offers a lifelong wihdrawal guaranee; herefore, here is no limi for he oal amoun ha is wihdrawn over he erm of he policy, because if he accoun value becomes zero while he insured is sill alive he can coninue o wihdraw he guaraneed amoun annually unil deah.

9 he aims of he work Develop a pricing model and define a fair price for he GLWB opion Verify if he curren GLWB price on he USA marke is fair or he marke seems o be underpriced or overpriced.

10 Index Main References Survey of he Variable Annuiy Producs and embedded opions Pricing model for GLWB opions Applicaion of he model o USA marke

11 he financial assumpion he single premium 0 paid by he policyholder is invesed in a fund. Following he sandard assumpion in he lieraure (Boyle and Schwarz 1997), we model he evoluion of he fund as: dw ( ) W d d W dz Where Z is a Brownian moion, he GLWB opion and approach where g is he guaraneed rae. is he insurance fee paid for is he wihdrawal a ime. In a saic G g0

12 he GLWB payoff he policyholder receives he amoun guaraneed unill he is sill alive; moreover, a he dae of deah he beneficiary will receive any remaining fund value. he discouned value a =0 of he GLWB V0 is he sum of he discouned values of living and deah benefis: V 0 LB0 DB0 LB 0 is he discouned value of a life annuiy. Since he random ime τ of deah and he fund value a his ime are independen DB 0 E E e r Max W ;0

13 he Deah Benefi (1) If we fix he dae, he deah benefi can be calculaed by Io s Lemma and rewrien as a payoff of a Quano Asian Pu (QAP) Opion: ;0 max d e G e DB Z Z ;0 1 max 0 ) 2 ( 0 ) 2 ( 2 2 d e G G e DB Z Z

14 he Deah Benefi (2) Using a sandard echnique in lieraure, he No-arbirage imezero value of he deah benefi a ime is: where he expecaion is under he risk neural measure Q. If we do no fix he dae of deah and consider boh he expecaions: where x is he policyholder s age and n is he final age Q Z r Z r Q QAP E DB d e G G e E DB ) ( ;0 1 max ) ( x n x x QAP E q p DB 0 0 0

15 he pricing formula he zero-value of he GLWB opion if he policyholder assumes a saic sraegy is: V n x r 0 pxge pxqxe0( QAP ) 0 Our main conribuion lies in bifurcaing he GLWB opion ino a life annuiy plus a porfolio of QAP opions wih decreasing srikes and increasing mauriies, where he weighs of composiion are he deferred probabiliies of deah

16 Index Main References Survey of he Variable Annuiy Producs and embedded opions Pricing model for GLWB opions Applicaion of he model o USA marke

17 Applicaion: he USA marke he guaraneed raes offered for GLWB opions are: Guaraneed Rae Policyholder s age able 1: he guaraneed rae in he Usa Marke he average of he sub-accoun volailiy for he universe of VA producs is 0.18 (Morningsar Principia Pro) We use he laes USA moraliy able (Human Moraliy Daabase); we consider x=60 and n=110 We se and r

18 he ruin probabiliy Le P(ξ ) he probabiliy ha W his zero a some poin < ω-x. We compue P(ξ ) wih Mone Carlo simulaions if g is 5% and he insurance fee is 60 b.p., which are hypohesis consisen wih he curren marke. P(ξ) μ = 4% μ = 6% μ = 8% μ = 10% μ = 12% σ = 10% 52.1% 45.1% 39.2% 30.9% 24,6% σ = 15% 58.5% 52.9% 48% 43.7% 39,5% σ = 18% 61.9% 57% 52.1% 48.1% 44,5% σ = 20% 64.3% 60% 54.9% 51% 48% able 1: he probabiliy ha he insurer has o pay he guaraneed amoun

19 Mone Carlo simulaion We have carried ou many Mone Carlo Simulaion under differen scenarios generaing for each of hem 1000 pahs of evoluion of he fund We have calculaed he fair insurance fee: once r, g and σ have been fixed we have searched he fair value of he fee making equal he iniial premium o he zero value of he fuure cash flows V 0.

20 Numerical Resuls Guaraneed Rae σ = 0.18 σ = 0.20 σ = b.p b.p. 83 b.p b.p b.p. 138 b.p b.p. 167 b.p. 226 b.p b.p. 308 b.p. 389 b.p. able 3: he fair fees for a policyholder aged 60

21 Remarks We pay aenion o he fair insurance fee under he hypohesis g=5% and σ=18%, which are consisen wih he marke. In his case he fair insurance is equal o 79 b.p., whereas he curren marke fee ranges beween 60 b.p. and 70 b.p. Alhough here is a common belief ha he guaranees embedded in variable annuiy policies are overpriced (see Clemens (2004)), our analysis shows ha he USA marke of GLWB is underpriced, in line wih he resuls obained by Milevsky and Salisbury (2005) for he GMWB marke.

22 Furher Seps Dynamic Sraegy he policyholder can wihdraw more or less han G. We analyze wo cases: 1) o widraw less han G or an amoun equal o G 2) o wihdraw more han G

23 Furher Seps 1) In conras o a GMWB, for a GLWB wihdrawing nohing or less han G can never be opimal. In fac, for a GMWB his sraegy exends he life of guaranee; insead, in a GLWB here is a lifelong guaranee and no adjusmens are made for fuure guaraneed wihdrawals. Hence, when he policyholder wihdraws less han G, he fuure guaranees are he same, bu heir values F s are lower because W is greaer. In addiion, we have o consider ha wihdraw less han G involves a smaller LB and a greaer DB. However, due o he maringale propery of he fund process and he fee deduced from he accoun value, he expeced value of he addiional deah benefi is never greaer han he wihdrawal amoun. So, he raional policyholder wihdraws a leas G.

24 Furher Seps 2) In his case, we have o balance wo effec: on one hand, wihdraw more han G involves a greaer LB and a smaller D. However, due o he maringale propery of he fund process and he fee deduced from he accoun value, he expeced value of he decrease of he deah benefi is never greaer han he increase of he wihdrawal amoun. So, if i is opimal for he policyholder o wihdraw more han G, han i has o be opimal o wihdraw he mos i is possible, i.e. compleely surrender he conrac. hus, we have o define a decision rule in order o esablish he surrender ime: for each possible scenario he raional policyholder would wihdraw exacly he annual guaraneed amoun unil he value of he fund less he penaly exceeds he value of fuure benefis; hen, he would surrender he conrac.