Analyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective

Transcription

1 Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive AFIR Colloquium Madrid, Spain June 22, 2 Alexander Bohner and Nadine Gazer Universiy of Erlangen-Nürnberg

2 AFIR 2 6/22/2 2 Inroducion Moivaion Paricipaing life insurance conracs: Imporan produc design in German life insurance marke Include ineres rae guaranees and bonus mechanisms hrough which profis are disribued and appropriaed o he policyholders Focus on: Analysis of surplus appropriaion schemes, i.e. differen ways of how a given amoun of surplus, deermined by a reserve based disribuion sysem, can be credied o he policyholders conracs

3 AFIR 2 6/22/2 3 Inroducion Aim of paper Examine surplus appropriaion schemes ofen inheren in paricipaing life insurance conracs: Bonus sysem: surplus increases deah and survival benefi Ineres-bearing accumulaion: accumulaes surplus on a separae accoun, deah benefi is kep consan Shorening he conrac erm: deah and survival benefi is kep consan, survival benefi is paid earlier Wih respec o heir impac on: Insurer s shorfall risk Ne presen value from a policyholder s viewpoin Conduc his analysis by considering moraliy risk as well as marke risk

4 AFIR 2 6/22/2 4 Model framework Insurance conrac and modeling moraliy probabiliies Pool of radiional paricipaing life insurance producs: Acuarially priced based on a moraliy able (DAV 28 T) Consan annual premium is given by equivalence principle: n n k n k : wih and x n x: n x: n k x x k n x x: n k x k k P ä S A A v p q v p ä v p Acual moraliy raes for risk measuremen derived by Lee- Carer (992) model: x x x, ln a b k e x x x x, x a b k Modificaion by Brouhns, Denui, and Vermun (22): ax x D ~ Poisson E wih e x, x, x x b k

5 AFIR 2 6/22/2 5 Model framework Policy reserves Acuarial reserve for individual conrac is given by Vx S A P ä x : n x : n Toal porfolio policy reserve is deermined by PR N d V i i x Developmen of paymens over ime n- + - n + ime x x+ x+ x+n- x+n P =P S P =P D Dec. 3 s where N = iniial number of conracs sold, S i P =P D d i Jan. s = number of deahs unil year S n- age sum insured premium dividend P n- =P D n- S n D n

6 AFIR 2 6/22/2 6 Model framework Developmen of he asse base Asse porfolio follows a geomeric Brownian moion P da( ) A( ) d A( ) dw ( ) Porfolio is composed of bonds and socks, wih a coninuous one-period reurn of he porfolio, given by r a rs a rb, wih E( r ) m.5 2 Asses a he end of year, afer accouning for decremens in he porfolio of policyholders due o deah, resuls o A A exp r S d, wih A, A E P N paymen of deah benefis, S = sum insured, depends on surplus scheme, d = number of deahs in year

7 AFIR 2 6/22/2 7 Model framework Surplus appropriaion schemes Acual policy ineres rae credied o he policyholders for period - unil, based on a smoohing scheme by Grosen and Jørgensen (2), is given by r P G max r, B PR IA RD where = surplus disribuion raio = arge buffer raio r G = guaraneed ineres rae Surplus for he -h year resuls o PR r r amoun is used differenly wihin each of he 3 companies depending on he concree appropriaion scheme P G

8 AFIR 2 6/22/2 8 Model framework Appropriaion scheme: bonus sysem. Bonus sysem: Surplus is used o increase he iniially guaraneed sum insured S (deah and survival benefi) Done by using he surplus as a single premium for an addiional conrac of he same ype wih same mauriy: Δ-values original conrac n ime Surplus per insured resuls in an addiional sum insured of P G n x : n i i S A PR r r N d increased sum insured is given by S S S

9 AFIR 2 6/22/2 9 Model framework Appropriaion scheme: ineres-bearing accumulaion 2. Ineres-bearing accumulaion: Sum insured is kep consan, i.e. S,,, S T Surplus is accumulaed on a separae accoun, IA Forward projecion of he ineres-bearing accumulaion accoun is given by IA P G i i IA IA r d N d PR r r Adjusmen for deah: funds ha belonged o policyholders ha died wihin he -h year, are passed o he colleciviy of policyholders

10 AFIR 2 6/22/2 Model framework Appropriaion scheme: shorening he conrac erm 3. Shorening he conrac erm: Surplus is used o decremen he remaining years o mauriy (conrac erm n() is a funcion of ime ) Reduce he conrac erm for full years only RD P G i i RD RD r d N d PR r r, RD Policy reserve incl. surplus for an individual insured V surplus n x V n x RD N d i i Deermine he years o reduce he conrac erm surplus kmax max k : V x n V x n k k K wih K,, n new policy period is given by n n k max

11 AFIR 2 6/22/2 Model framework Evaluaing he surplus appropriaion schemes Shorfall probabiliy (asses no sufficien o cover liabiliies): SP P Ts T, wih Ts inf : A PR IA RD,,..., T Ne presen value from a policyholder s viewpoin = expeced value of insurance benefis - premiums T Q rf rf NPV E px qx S e px P e Ts T Q Trf E T px ST IA RD TBT e T T T T s T N d i i T r r x x N d i i Q r f f E p A e c e p P e T s

12 AFIR 2 6/22/2 2 Numerical resuls Inpu parameers Asses socks bonds Expeced one-period reurns 8.% 6.2% Volailiy 2.95% 3.3% Correlaion beween socks and bonds Sock porion % Risk-free rae 3% Liabiliies r G r IA r RD Rae of ineres 2.25% % % Number of conracs sold, Sum insured in = Level premium for T = Conrac erm 3 Age of he policyholders in = 35

13 AFIR 2 6/22/2 3 Numerical resuls Shorfall risk SP as a funcion of sock porion and shock o moraliy SP =.5 for: Bonus sysem a = 9.8%.5 Ineres-bearing accum. a = 2.%. Shorening conrac erm a = 2.54%.5 shorfall probabiliy shorfall probabiliy + 5%. % 5% % 5% 2% 25% sock porion a shock o moraliy bonus sysem ineres-bearing accumulaion shorening he conrac erm

14 AFIR 2 6/22/2 4 Numerical resuls Shorfall risk SP as a funcion of conrac erm T shorfall probabiliy for a = % shorfall probabiliy for a = 25% conrac erm T conrac erm T bonus sysem ineres-bearing accumulaion shorening he conrac erm

15 AFIR 2 6/22/2 5 Numerical resuls Ne presen value NPV as a funcion of sock porion and shock o moraliy ne presen value ne presen value % 5% % 5% 2% 25% sock porion a shock o moraliy bonus sysem ineres-bearing accumulaion shorening he conrac erm

16 AFIR 2 6/22/2 6 Summary Resuls show: Even if he smoohing surplus disribuion scheme is he same, he impac of he concree surplus appropriaion (wih respec o guaraneed deah/survival benefis) differs subsanially: Insurer s risk siuaion, from highes o lowes: ) bonus sysem 2) shorening conrac erm 3) ineres-bearing accumulaion Ne presen value from policyholder s viewpoin: ) shorening conrac erm 2) ineres-bearing accumulaion 3) bonus sysem Increasing gap in shorfall risk beween 3 schemes for higher sock porions and higher disribued surplus In conras: shock o moraliy implies similar increase in risk Risk reducion for longer conrac periods no as effecive in case of he (mos common) bonus sysem, especially for high sock porion

17 Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Thank you very much for your aenion! AFIR Colloquium Madrid, Spain June 22, 2 Alexander Bohner and Nadine Gazer Universiy of Erlangen-Nürnberg