Key elements Construction process Application and results LTCI: a multi-state semi-Markov model to describe the dependency process for elderly people Guillaume Biessy Friday, April 4th 2014 Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 1 / 8
Key elements Construction process Application and results Same old people, brand new model A simple 3 state model: The new model: s : age. Properties of the new model x : time spent in the current state. 4 states of dependency. GIR 4 to 1: levels of dependency continuous time scale. used for the french public aid. semi-Markov model. GIR 1 is the most severe state. Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 2 / 8
Key elements Construction process Application and results A few definitions Definition (Markov process) The future of the process only depends on its past through the current state . Definition (semi-Markov process) The future of the process depends on its past through both the current state and the time spent in the current state . Definition (semi-Markov kernel) A semi-Markov process is entirely determined by its semi-Markov kernel Q i , j ( s , x ) with: i : departure state. s : age at entry in state i . j : arrival state. x : duration. Fundamental relation Q i , j ( s , x ) = p i , j ( s ) × F i , j ( s , x ) . � �� � � �� � probability duration law Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 3 / 8
Key elements Construction process Application and results Jump probabilities and duration laws Duration laws (cdf, index not displayed) Jump probabilities F ( s , x ) = ( 1 − λ ) W 1 ( s , x ) + λ W 2 ( s , x ) p i , j ( s ) = a i , j × s + b i , j . W ( s , x ) = 1 − e − σ x ν e β s . Jump probabilities from state GIR 4 Duration law for the transition between GIR 4 and GIR 0 100 0.3 75 jump probabilities next state GIR 3 0.2 GIR 2 50 density GIR 1 death 25 0.1 0 0.0 60 70 80 90 100 age in years 0 1 2 3 4 5 time in years Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 4 / 8
Key elements Construction process Application and results Estimation of parameters The likelihood function has the following expression: � n p − 1 � � N � 2 ) δ p 2 ) δ p k + 1 ( t p k , t p × c 1 ( t p n p , T p 1 × c 2 ( t p n p , T p 1 , T p L = c X p k + 1 ) k , X p 2 X p X p np np � �� � p = 1 k = 1 � �� � specific censoring terms observed transitions k + 1 ( t p k , t p k + 1 ( t p k + 1 ( t p k , t p k + 1 − t p c X p k + 1 ) = p X p k ) × f X p k ) . k , X p k , X p k , X p � �� � � �� � jump probability density of duration law ( t p n p , T p np ( t p n p , T p 2 − t p c 1 2 ) = S X p n p ) . X p np � �� � marginal survival function c 2 ( t p n p , T p 1 , T p np , 0 ( t p np , 0 ( t p n p , T p 1 − t p np ( t p n p , T p 2 − t p 2 ) = p X p n p ) × F X p n p ) + S X p n p ) . X p np � �� � � �� � � �� � jump probability cds of duration law marginal survival function Optimization performed using the Nelder-Mead algorithm. Selection of the best sub-model according to the BIC. Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 5 / 8
Key elements Construction process Application and results Statistics on simulated trajectories Number of dependent people in the population, by age Distribution of survival time in dependency 60000 0.2 number of individuals state 40000 GIR 4 density GIR 3 GIR 2 GIR 1 0.1 20000 0 0.0 50 55 60 65 70 75 80 85 90 95 100 105 0 1 2 3 4 5 6 7 8 9 10 11 12 age in years time survived in dependency Graphs generated using 1 million trajectories with initial age of 50. Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 6 / 8
Key elements Construction process Application and results Pricing methodology for pure level premium The required pure level premium is the value p ∗ such that: E [ NPV ( p ∗ × I P )] = E [ NPV ( B )] . NPV: Net Present Value. B : benefit cash flows. I P : premium unit cash flows. p n = � µ B ( n ) n → + ∞ p ∗ a. s. by the law of large numbers. Estimator of the premium: � → � µ P ( n ) µ B ( n ) (resp. � � µ P ( n ) ): estimator of empirical mean of NPV ( B ) (resp. NPV ( I P ) ). σ B ( n ) (resp. � � σ P ( n ) ): estimator of empirical mean of NPV ( B ) (resp. NPV ( I P ) ). We show that with n large enough: p n | ≤ � � σ P ( n ) � Φ − 1 ( 1 − α 2 ) | p ∗ − � σ B ( n ) + � p n � µ P ( n ) √ n � with a level of confidence of 1 − α Φ : cumulative distribution function of the standard normal law. Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 7 / 8
Key elements Construction process Application and results Conclusion Summary: A multi-state continuous time model based on semi-Markov process. We define it through jump probabilities and duration laws. Calibration uses the Maximum Likelihood method. Pricing relies on Monte Carlo simulations. Long-term objectives: Assess the sampling error (Bootstrap methods). Study causes of dependency. Take into account other covariates (using e.g., Cox model). Find trends for model parameters. Thank you for your attention ! Friday, April 4th 2014 Guillaume Biessy LTCI: dependency as a 4-state semi-Markov process 8 / 8
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