Monitoring actuarial assumptions in life insurance

Transcription

1 Monitoring actuarial assumptions in life insurance Stéphane Loisel ISFA, Univ. Lyon 1 Joint work with N. El Karoui & Y. Salhi IAALS Colloquium, Barcelona, 17 LoLitA

2 Typical paths with change of regime at date 3 t t Vt N t V t Nt Yt N t Y t 15 5 Nt t t (a) Processes N and V t (b) Processes N and Y t Figure: Sample paths, for ρ = 1.5, of the cusum processes N, V ρ (left) and N, Y ρ t for ρ =.5 (right ). 16/36 NEK, IWSM Columbia University New York, June 15

3 Quick Outline Why is quick detection important in insurance? Quick version of quickest detection Monitoring populations in practice Monitoring other insurance portfolios (quickly if time permits)

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6 institut de science financiere et d assurances laboratoire saf Longevity risk components 2/4 The trend The mortality improvement is not a diversifiable risk : it a ects the whole portfolio and can thus not be managed using the law of large numbers π Actual mortality estimation Estimation error for an estimate of mortality based on actual experience :the error is larger for small populations (or for poorly represented age groups) Doubled improvements pension value interest rate females Mortality level at 8% of the expected pension value interest rate Doubled improvements pension value interest rate males Mortality level at 8% of the expected pension value interest rate % +32bp +3.1% +19bp +6.7% +42bp +3.7% +24bp % +43bp +4.7% +36bp +7% +57bp +5.7% +48bp % +55bp +7.6% +8bp +6.3% +74bp +9.1% +7bp % +6bp +13.2% +7bp +4.3% +84bp +15.4% +281bp Table: TGH5/TGF5 with flat interest rate of 3% Slide 13/117 Stéphane LOISEL et Yahia SALHI Biometric and Surrender Risks oct. 15

7 Bayesian setup for random change-point Brownian framework with abrupt change in the drift Based on the conditional distribution of the time of change, Formulated as an optimal stopping problem Page(1954), Shiryaev(1963), Roberts(1966), Beibel(1988), Moustakides (4), and many others... Poisson framework with abrupt change in intensity Based on the conditional distribution of the time of change, with exponential or geometric prior distribution More recent studies : Gal (1971), Gapeev (5), Bayraktar (5, 6), Dayanik (6) for compound Poisson, Peskir, Shyriaev(9) and others 9/38 Nicole El Karoui Tunis CREMMA

8 Mathematical settings We consider a portfolio of insured population: Let N = (N t ) t be a counting process indicating the deaths of policyholders and λ = (λ t ) t its intensity. The counting process N t, is available sequentially through the filtration F t = σ{n s, < s t}. We suppose that the insurance company relies on a Cox-like model to project her own experienced mortality: λ t = ρλ t, λ t is a reference intensity and ρ is a positive parameter. λ is considered deterministic and may refer whether to a projection of national population/best estimate... Model risk/parameter uncertainty: Change-point λ t = 1 {t<θ} ρλ t + 1 {t θ} ρλ t. Without loss of generality we can assume that ρ = 1 and let ρ = ρ > 1. 3 / 18

9 Probabilistic formulation Let P θ (resp. E θ [ ]) be the probability measure (resp. expectation) induced when the change takes place at time θ Example For θ =, the process is out-of-control For θ =, the process is in-control Detect the change-point θ as quick as possible while avoiding false alarms Optimality Criteria, Lorden (1971)-like [ ] The detection delay E θ (N τ N θ ) + Fθ The frequency of false alarm E [N τ ] 4 / 18

10 Optimization Problem Optimization Problem Find τ such that C(τ ) = inf τ sup θ [, ] ess supe θ [ (N τ N θ ) + Fθ ] subject to E [N τ ] = ω. Assumption t 1 λ sds <, P, P -a.s. 2 N = P, P -a.s. 5 / 18

11 Optimality of the Cusum procedure (1/7) Let the Radon-Nikodym density of P with respect to P be defined as dp dp Ft = exp U t, where U t = log(ρ)n t + (1 ρ) t λ s ds is the log-likelihood ratio. Let V (x) be the cusum process; with head-start x < m; defined as V t (x) = U t ( x) U t (1) where U t is the running infimum of U, i.e. U t = inf s t U s. The process V (x) measures the size of the drawup, comparing the present value of the process U to its historical infimum U. Let τ m (x) be the fist hitting time of V (x) of the barrier m, i.e. Theorem τ m (x) = inf{t, V t (x) m}. If E [N τm()] = ω then τ m () is optimal, i.e. inf τ C(τ) = C(τ m ()) 6 / 18

12 Typical paths with change of regime at date 3 t t Vt N t V t Nt Yt N t Y t 15 5 Nt t t (a) Processes N and V t (b) Processes N and Y t Figure: Sample paths, for ρ = 1.5, of the cusum processes N, V ρ (left) and N, Y ρ t for ρ =.5 (right ). 16/36 NEK, IWSM Columbia University New York, June 15

13 V t U t

14 Detection Procedure Algorithm Step 1: Fix the input parameters: The post-change intensity through the specification of ρ and the false alarm constraint ω. Step 2: Determine the threshold m as the solution of the equation E [N τm ] = ω. Step 3: For each new observation at time t compute the value of the CUSUM process V given by the iterative relation V t+1 = (V t 1 + U t ) +. Step 4: Compare the current value of V to the threshold m and stop the procedure once V t m and sound an alarm. Hence τ m () = t. 13 / 18

15 Detection Procedure Real World (1/4) We consider the Continuous Mortality Investigation assured lives dataset and England & Wales national population. We split data into two periods: We consider the period as a training period. The Cox model is estimated over this period using the MLE. Hence we monitor the sequentially the dataset over the period and look for changes on the mortality of assured lives. 14 / 18

16 Detection Procedure Real World (2/4) Cusum process Cusum process Time in year Time in year Figure: Detection scheme for age groups 5 59 (right) and 8 89 (left). The post-change is set to ρ = 15% and the false alarm constraint to ω = λ. 15 / 18

17 Detection Procedure Real World (4/4) τ m Age ρ = 1.5 ρ = 1.15 Observed Table: Detection of mortality change with a post-change ratio of ρ = 1.15 and an average run length (false alarm) constraint of. The right column reports the detected change-point using an off-line procedure. 17 / 18

18 i n s t i t u t d e s c i e n c e f i n a n c i e r e e t d a s s u r a n c e s l a b o r a t o i r e s a f Monitoring Mortality Sounding an alarm for the change ρ Hyp ρ Targer We simulate deaths on the portfolio with different levels ρ Targer = 95%, 9% and 85% s.t. D(x, t) Pois(ρ Targer L(x, t) µ ERM (x, t)) We suppose that the actuary made an assumption of ρ Hyp = % We set-up the monitoring/surveillance on the observed deaths and try to detect a change from ρ Hyp = % to ρ Targer = 95%, 8% and 85% respectively. We test different sizes of the portfolio small sized, 5 and a (relatively) large and compare the results Slide 64/69 Stéphane LOISEL et Yahia SALHI Solutions de Transfert du Risque de Longévité pour les Rentes Genevoises oct. 15

19 i n s t i t u t d e s c i e n c e f i n a n c i e r e e t d a s s u r a n c e s l a b o r a t o i r e s a f Monitoring Mortality Sounding an alarm for the change ρ Hyp = % ρ Targer = 95% Threshold Cusum Process, V(t) Dates, t Slide 66/69 Stéphane LOISEL et Yahia SALHI Solutions de Transfert du Risque de Longévité pour les Rentes Genevoises oct. 15

20 i n s t i t u t d e s c i e n c e f i n a n c i e r e e t d a s s u r a n c e s l a b o r a t o i r e s a f Detection Delay Impact of Portfolio Size and Age Tranches Size 5 Ages Hyp deaths time % 95% % 9% % 85% % 95% % 9% % 85% Slide 67/69 Stéphane LOISEL et Yahia SALHI Solutions de Transfert du Risque de Longévité pour les Rentes Genevoises oct. 15

21 Detection Delay 15 5 ρ =.75 ρ =.85 ρ =.8 ρ =.7 False Alarm Detection Delay 15 5 ρ =1.15 ρ =1.2 ρ =1.25 ρ = False Alarm

22 4 35 ρ =.99 3 ρ =.95 Detection Delay False Alarm

23 P1 P2 P3 P4 P P6 P7 P8 P9 P 9 Integer Age P11 P12 P13 P Gender Female Male Exposures

24 P1 P2 P3 P4 P P6 P7 P8 P9 P Value P11 P12 P13 P14.8 All Surv Train Gender Female Male.... All Surv Train All Surv Train All Surv Train All Surv Train Type

25 P1 P2 P3 P4 P5 Target = 1.8 Real = 1.29 Target = 1.8 Real = 1.21 Target =.92 Real = Target =.92 Real =.56 Target = 1.8 Real = P6 P7 P8 P9 P Target =.92 Real = Target = 1.8 Real = 1.3 Target =.92 Real =.28 Target =.92 Real =.46 Target = 1.8 Real = 1.8 Cusum Process P11 P12 P13 P14 Jan 11Apr 11Jul 11Oct 11Jan 1 3 Target =.92 Real =.42 6 Target =.92 Real =.66 3 Target =.92 Real =.55 4 Target =.92 Real = Jan 11Apr 11Jul 11Oct 11Jan 12Jan 11Apr 11Jul 11Oct 11Jan 12Jan 11Apr 11Jul 11Oct 11Jan 12Jan 11Apr 11Jul 11Oct 11Jan 12 Time

26 12 8 Claims Date

27 Detection Time 26 Dec 3 Cusum Process, V Time, t

28 Claims Date

29 Cusum Process, V Time, t

30 Perspectives LoLitA closing international conference, Paris, Jan , 18 LoLitA Lecture Notes Own Longevity of LoLitA: after 18