Surveillance of BiometricsAssumptions

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1 Surveillance of BiometricsAssumptions in Insured Populations Journée des Chaires, ILB 2017 N. El Karoui, S. Loisel, Y. Sahli UPMC-Paris 6/LPMA/ISFA-Lyon 1 with the financial support of ANR LoLitA, and Chair Risques financiers ILB, 20 October /27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

2 Figure: Longevity Risk Thank you, Alexandre Boumezoued

3 Longevity purpose Mortality evolution Mainly concerned with mortality evolution which may change over a projection period Mortality profile/level/trend : Heterogeneity and other factors Interest in experienced mortality : Deaths can be observed sequentially Model risk, parameter uncertainty: Long-term projection Risk Management of Mortality evolution Sequential information on death occurrences Monitoring and surveillance of mortality dynamics Updating mortality assumptions 3/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

4 What is the Disorder Problem? The Poisson disorder problem is less formally stated as follows Observe a trajectory of the Poisson process (N t ) whose intensity changes from λ to ρλ at some (unknown) time θ. The problem is to find a rule to detect θ as quickly as possible with a limited number of false alarms 0 θ Disorder time 4/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

5 Where do the disorder problems arise Insurance compagnies: Recalculate the premiums for the future sales of insurance policies when the risk structure changes (ρ > 1) Pension funds: Large exposure to change in mortality risk : ρ < 1 Quality control and maintenance: Inspect, recalibrate, or repair as soon a manufacturing process goes out of control Fraud and computer intrusion detection: Alert the inspectors for an immediate investigation as soon as abnormal activity in credit card, cell phone calls, or computer network traffic are detected. 5/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

6 How to choose the parameter ρ for pensions 6/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

7 Overview of change detection The observation process Let N = (N t ) t 0 be a counting process (of claims, or deaths) with stochastic intensity λ = (λ t ) t 0, Cox process A change in the intensity occurs at an unobservable date θ, from λ t to ρλ t, ρ > 0. Proportional hazard models Change of Point θ=model-risk Random with known prior (Bayesian) Deterministic but unknown (Non-Bayesian but Robust) Change of Probability measures: Known statistics P and P Under P, no change, (λ t ) holds (θ = ) Under P, immediate change, (ρλ t ) holds (θ = 0) Under P θ, λ θ t = 1 t<θ λ t + 1 t θ ρλ t 7/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

8 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 for partially observable process Page(1954), Shiryaev(1963), Roberts(1966), Beibel(1988), Moustakides (2004), and Dayanik (2006),... 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 (2005), Bayraktar (2005, 2006), Dayanik (2006) for compound Poisson, Peskir, Shyriaev(2009) and others New methods using particle filters Andrieu, Legland, Zhang ( 2005) 8/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

9 Robust Detection Problem, Page (1954) Robust Detection Non-Bayesian framework, mainly motivated by the lack of any priors on the statistical behavior of change-point. Concerned with general counting process, in particular inhomogeneous Poisson process The problem Detect the change-point θ as quick as possible τ While avoiding false alarms Optimality criteria, Lorden (1971)-Like Detection delay estimate: E θ [ (Nτ N θ ) + F θ ] Frequence of false alarm E [ Nτ ] 9/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

10 The robust problem Robust criterium C(τ) = A min-max problem under constraint [ sup esssup ω E θ (Nτ N θ ) + ] Fθ θ [0, ] Find τ such that C = C(τ ) = min C(τ), under control of the error of the second type, ( sound the alarm when there is not change (θ = ) ) is small or equivalently if E [N τ ] π. The rule is stable by time rescaling and so the same for all intensity structure such that 0 λ s ds = and N =, a.s.. 10/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

11 Likelihood formulation Conditional probability ratio between P =no change, P= immediate change. The conditional probability ratio process of P w.r to P is d P/dP = E t = exp ( log(ρ)n t (ρ 1)Λ t ) Put U ρ t = N t β(ρ)λ t, so ρ Uρ t is a P-martingale. Then the CPR of P w.r to P is : ρ Uρ t = (1/ρ) Uρ t The β coefficient where β(ρ) = ρ 1 log(ρ), with β(ρ) = 1 0 ρu du.= Laplace transform of U[0, 1]. Put β(ρ) = β(1/ρ) = β(ρ)/ρ, and ρ = 1/ρ. 11/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

12 Likelihood formulation and Cusum process Sequential conditional probability ratio: The log ρ ratio of density processes on F t of P θ w.r to P is log ρ (dp θ /dp) = U ρ t θ Uρ θ where U ρ t = N t β(ρ)λ t The Cumulative Sum rule (cusum)= Log Max Likelihood in time Based on max s t (ρ Ut Us ), ( sign of ln(ρ)) Depend on the sign of ρ 1 Cusum stopping time, τ m = inf{t ; cusum t m} Main question: The optimality of τ m, if the false alarm constraint is achieved, E(N τ π m ) = π. 12/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

13 Reflected Counting process with drift Running supremum Running supremum Z t = sup s t Z s Put X t = U t = N t + βλ t. So, X t is continuous. Ū t is not continuous and increases only at jumps of N, such that Ūt = U t Reflected processes and Cusum processes Cusum rule based on max s t ρ Ut Us ρ > 1: Reflected process X at its maximum, V t = sup θ t (U t U θ ) = X t X t ρ < 1 Reflected process U process at its maximum: Y t = sup θ t (X t X θ ) = Ū t U t, 3/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

14 Typical paths for ρ > 1 Figure: Sample paths of the processes U(x), V (x) and L(x) when λ is time-homogeneous. 14/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

15 Typical paths with change of regime at date 3 t t Vt N t Nt Vt N t Y t 10 5 Nt 1 V t t (a) Processes N and V ρ t t (b) Processes N and Y ρ t Figure: Sample paths, for ρ > 1, of the cusum processes N, V ρ (left) and N, Y 1/ρ t (right ). 15/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

16 Detection Procedure Algorithm Step 1: Fix the input parameters: The post-change intensity through the specification of ρ and the false alarm constraint π. Step 2: Determine the threshold m as the solution of the equation E [N τm ] = π. 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 (0) = t. 16/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

17 Optimality results Optimality in decrease in intensity Let T be a stopping times with finite cusum bound, such that E(N T ) = E(N τ Y m ) = g m (0)., then E ( T 0 ρ1 Y s dn s ) h m (0) E ( ρ Y T ), Optimality in increase in intensity Let T be a stopping times with finite cusum bound,such that E(N T ) = E(N τ V m ) = h m (0)., then E ( T 0 ρvs dn s ) h m (0) E ( ρ V T ), 7/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

18 MonteCarlo simulation Evolution of the global population λ t = a ( 1 + exp( (t b)/c) ) 1, t 0, where a, b and c are some constant parameters given in a b c σ simulations for ρ = 1.1, 1.5, 2 ρ m = 5 h 0 (0) () h (0) () () m = 10 h 0 (0) () () h (0) () () /27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

19 Detection Procedure Real World 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. It is not easy to estimate the post change coefficient ρ Hence we monitor sequentially the dataset over the period and look for changes on the mortality of assured lives. 19/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

20 Detection Procedure Real World Cusum process Cusum process Time in year Time in year Figure: Detection scheme for age groups (right) and (left). The post-change is set to ρ = 1.15% and the false alarm constraint to π = 100 λ. 20/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

21 Detection Procedure Real World τ m Age ρ = 1.50 ρ = Table: Detection of mortality change with a post-change ratio of ρ = 1.15 and an average run length (false alarm) constraint of 100. The right column reports the detected change-point using an off-line procedure. When ρ is larger the detection delay is longer Difficile to disctinct effective change or false alarm Other results mention that a breakpoint on the data on the /27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

22 Monitoring Mortality 22/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

23 Monitoring MortalityII 23/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

24 Monitoring Mortality III 24/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

25 Monitoring Mortality 25/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

26 Age pyramids multi-portolios 26/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

27 Cusum multi-portolios 27/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

28 Conclusion The cusum detection rule is optimal in the case of non-homogeneous Poisson process with a modified Lorden criterion Very easy to implement Non asymptotic criterium Based on fine properties of scale functions, easy to extend to Levy processes The proof provide lower bound for some conditional ratio (Basseville) Applications in non-life insurance Further research on possible extensions 28/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017

29 With the indulgence of Sau, Gold of health and longevity Thank you 29/27 N.El Karoui-S.Loisel-Y.Salhi ILB, 20 Oct 2017