A Set of new Stochastic Trend Models

Transcription

1 A Set of new Stochastc Trend Models Johannes Schupp Longevty 13, Tape, 21 th -22 th September

2 Introducton Uncertanty about the evoluton of mortalty Measure longevty rsk n penson or annuty portfolos wth stochastc mortalty models Parametrc mortalty models: Lee-Carter model, Carns-Blake-Dowd model, APC model, etc. Reduce the nformaton about exposures and deaths to a few parameters: CBD: Two tme dependent parameter processes (Carns et al. (2006)): log q x,t 1 q x,t = κ t 1 + κ t 2 x x Parameter processes calbrated for Englsh and Welsh males older than 65 years L κ t 1, κ t 2 max wth the assumpton of D x,t ~Po(E x,t m x,t ) or D x,t ~Bn(E x,t, q x,t ) 2 September 2017 A Set of new Stochastc Trend Models

3 Introducton Popular choce: a (multvarate) random walk wth drft (RWD) for stochastc forecasts Backtestng n 1963 based on a 10-year calbraton: Future observatons far outsde the 99% quantle Hstorc trend changed once n a whle Only a pecewse lnear trend Random changes n the trends slope Random fluctuaton around the prevalng trend In prncple, our approach can be appled to any parametrc mortalty model Extrapolatng only the most recent trend, systematcally underestmates future uncertanty, see e.g. Sweetng (2011), L et al. (2011), Börger et al. (2014) 3 September 2017 A Set of new Stochastc Trend Models

4 Agenda Introducton Specfcaton of a Stochastc model Trend component Drft component Parameter estmaton Three alternatve approaches Open ssues 4 September 2017 A Set of new Stochastc Trend Models

5 Stochastc Trend model Contnuous pecewse lnear trend, wth random changes n the slope and random fluctuaton around the prevalng trend Model the trend process wth random nose κ t = κ t + ε t ; ε t ~f Extrapolate the most recent actual mortalty trend κ t = κ t 1 + d t In every year, there s a possble change n the mortalty trend wth probablty p In the case of a trend change λ t = M t S t Wth absolute magntude of the trend change M t ~h Sgn of the trend change S t bernoull dstrbuted wth values -1, 1 each wth probablty 1 2 d t = d t 1 + λ t, where λ t = 0 wth probablty 1 p M t S t wth probablty p In prncple, also other dstrbutons are possble (Pareto, Normal, t-dstrbuton, ) We propose to use: f = N 0, σ 2 ε,t, h = LN(μ, σ 2 ) ~g Parameters to be estmated for projectons startng n t=0 (typcally latest 2 observaton, case: f = N 0, σ ε,t, h = LN(μ, σ 2 )): p, σ 2 ε,t, μ, σ 2, d 0, κ 0 5 September 2017 A Set of new Stochastc Trend Models

6 Parameter estmaton Alternatve I Calbraton based on hstorc trends Use hstorc trends/drfts to estmate parameters (see e.g. Hunt and Blake (2014), Sweetng (2011), Börger and Schupp (2015)). Choose optmal hstorc trends/drfts based on some optmzng crteron (OLS, Lkelhood, ). Advantage: Intutve hstorc curves Börger and Schupp (2015): For k 0,, m fnd trend process d 0, κ 0, λ N+2,, λ 0 where exactly k of λ N+2,, λ 0 are unequal to zero (trend curve wth k trend changes). Update σ ε 2 teratvely. Choose optmal trend process wth AIC/BIC/MBIC. Example: Random Walk wth changng drft (n the sprt of Hunt and Blake (2014)) Possble Problems: hstorc observatons are unlkely to be generated wth the drft change densty ( nconsstent predcton possble), only few observatons. Outlers can have a huge nfluence 6 September 2017 A Set of new Stochastc Trend Models

7 Parameter estmaton Alternatve II Calbraton based on hstorc trends wth a combned lkelhood Include the dstrbuton of the trend changes used for smulatons n the optmzaton crteron Calbrate optmal hstorc trends based on f N κ N,,0 σ 2 ε, d 0, κ 0, λ N+2,, λ 0 g λ N+2,, λ 0 μ, σ 2, p For k 1,, m fnd trend process d 0, κ 0, λ N+2,, λ 0 that maxmzes f N κ N,,0 σ 2 ε, d 0, κ 0, λ N+2,, λ 0 g λ N+2,, λ 0 μ, σ 2, p, where exactly k of λ N+2,, λ 0 are unequal to zero (trend curve wth k trend changes). Update σ 2 ε, p, μ, σ 2 teratvely. Based on optmal goodness of ft (f N κ N,,0 σ 2 ε, d 0, κ 0, λ N+2,, λ 0 ) choose optmal hstorc trend Advantages: Consstency between hstorc trends and stochastc smulaton, avod rather subjectve selecton wth nformaton crtera The parameters requred for stochastc forecasts are part of the calbraton: p, σ ε 2, μ, σ 2, d 0, κ 0 7 September 2017 A Set of new Stochastc Trend Models

8 Parameter estmaton Alternatve III Calbraton based on MLE Stochastc forecasts requre: μ, σ 2, p, σ ε 2, κ 0, d 0. Not necessarly a hstorc trend requred. The focus here wll be solely on forecasts! Idea: Classc MLE: L μ, σ 2, p, σ ε 2, κ 0, d 0 κ max = 1,2 Example: Consder last three years and one ndex: κ 2 κ 1 κ 0 d 1 d 0 d 1 Known trend n 0, unknown trend n -1 (possble trend change λ 0 ) L μ, σ 2, p, σ ε 2, κ 0, d 0 κ 2, κ 1, κ 0 = f N κ 0 κ 0 σ ε 2, κ 0 f N κ 1 (κ 0 d 0 ) σ ε 2, κ 0, d 0 f N g κ 2 μ, σ 2, p, σ ε 2, κ 0, d 0 = f N ε 0 σ ε 2, κ 0 f N ε 1 σ ε 2, κ 0, d 0 g λ 0 μ, σ 2, p f N κ 2 κ 0 d 0 d 1 σ ε 2, κ 0, d 0 dλ 0 R = f N ε 0 σ ε 2, κ 0 f N ε 1 σ ε 2, κ 0, d 0 g λ 0 μ, σ 2, p f N κ 2 κ 0 d 0 (d 0 λ 0 ) σ ε 2, κ 0, d 0 dλ 0 R Knowng μ, σ 2, p, σ ε 2, κ 0, d 0, we can gve a lkelhood functon for the hstorc data max max 8 September 2017 A Set of new Stochastc Trend Models

9 Parameter estmaton Alternatves III Consder the complete hstory: L θ κ N,,0 max wth θ μ, σ 2, p, σ 2 ε, κ 0, d 0 We can calculate the trend process recursvely κ s = κ 0 sd 0 + s 1 l=1 l λ (s 1 l), 0 s L μ, σ 2, p, σ 2 ε, κ 0, d 0 κ N,,0 = f N ε 0 σ 2 ε, κ 0 f N ε 1 σ 2 ε, κ 0, d 0 N s 1 R N 1 s=2 g λ (s 2) θ f N κ s (κ 0 sd 0 + l λ s 1 l l=1 ) θ dλ N+2,,0 max Challenge: In parameter calbraton, we need to solve and optmze ths N-1 dmensonal ntegral f N s 1 κ s (κ 0 sd 0 + l λ s 1 l l=1 ) θ f N κ s s 1 (κ s+1 d 0 λ s 1 l l=1 ) θ 9 September 2017 A Set of new Stochastc Trend Models

10 Parameter estmaton Alternatves III Lkelhood of the trend model L μ, σ 2, p, σ 2 ε, κ 0, d 0 κ N,,0 = f N ε 0 σ 2 ε, κ 0 f N ε 1 σ 2 ε, κ 0, d 0 R N 1 N s=2 g λ (s 2) θ f N s 1 κ s (κ 0 sd 0 + l λ (s 1 l) l=1 θ dλ N+2,,0 max Use Monte-Carlo ntegraton to calculate and optmze the N-1 dmensonal ntegral. Basc dea: I = f x g x dx Smulate x 1,, x m wth x ~g I = 1 m m =1 f(x ) Here: Smulate x 1,, x m trends accordng to x l = λ N+2,, λ 0 l wth λ j ~g Calculate I = 1 m m 0 j= N l=1 f N (ε j θ, x l ) for = 1,2 Startng n t = 0 we smulate hstorc trends. The estmated parameters can be used for projectons drectly. 10 September 2017 A Set of new Stochastc Trend Models

11 Parameter estmaton Alternatves III A frst example: NLD-males (constant volatlty) wth 1.4 Mo trals μ = 5, σ 2 = 0.7, p = , σ ε 2 = , κ 0 = 2.266, d 0 = Startng n 2012 we smulate hstorc paths Advantages: Maxmum of consstency n forecasts, flexblty on dstrbutonal assumptons Dsadvantages and open ssues: No hstorc trends Trends x l wth a hgh lkelhood ( 0 j= N f N ε j θ, x l ) are extremely rare Huge number of smulatons necessary Domnated by very few smulatons 11 September 2017 A Set of new Stochastc Trend Models

12 Lterature Börger, M., Flescher, D., Kuksn, N., Modelng Mortalty Trend under Modern Solvency Regmes. ASTIN Bulletn, 44: Börger, M., Schupp, J., Modelng Trend Processes n Parametrc Mortalty. Workng Paper, Ulm Unversty. Carns, A., Blake, D., Dowd, K., A Two-Factor Model for Stochastc Mortalty wth Parameter Uncertanty: Theory and Calbraton. Journal of Rsk and Insurance, 73: Hunt, A. and Blake, D. (2014). Consstent mortalty projectons allowng for trend changes and cohort effects. Workng Paper, Cass Busness School L, J. S.-H., Chan, W.-S., and Cheung, S.-H. (2011). Structural changes n the Lee-Carter ndexes: detecton and mplcatons. North Amercan Actuaral Journal, 15(1): Sweetng, P., A Trend-Change Extenson of the Carns-Blake-Dowd Model. Annals of Actuaral Scence, 5: September 2017 A Set of new Stochastc Trend Models

13 Contact Johannes Schupp(M.Sc.) +49 (731) September 2017 A Set of new Stochastc Trend Models