LTCI: a multi-state semi-Markov model to describe the dependency - - PowerPoint PPT Presentation

SLIDE 1

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

SLIDE 2

Key elements Construction process Application and results

Same old people, brand new model

A simple 3 state model: The new model: s: age. x: time spent in the current state. GIR 4 to 1: levels of dependency used for the french public aid. GIR 1 is the most severe state. Properties of the new model 4 states of dependency. continuous time scale. semi-Markov model.

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SLIDE 3

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 Qi,j(s, x) with: i: departure state. j: arrival state. s: age at entry in state i. x: duration. Fundamental relation Qi,j(s, x) = pi,j(s)

probability

× Fi,j(s, x)

duration law

.

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SLIDE 4

Key elements Construction process Application and results

Jump probabilities and duration laws

Jump probabilities pi,j(s) = ai,j × s + bi,j.

25 50 75 100 60 70 80 90 100 age in years jump probabilities next state GIR 3 GIR 2 GIR 1 death

Jump probabilities from state GIR 4

Duration laws (cdf, index not displayed) F(s, x) = (1 − λ)W1(s, x) + λW2(s, x) W(s, x) = 1 − e−σxνeβs.

0.0 0.1 0.2 0.3 1 2 3 4 5

time in years density

Duration law for the transition between GIR 4 and GIR 0

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SLIDE 5

Key elements Construction process Application and results

Estimation of parameters

The likelihood function has the following expression: L =

N

• p=1

     np−1

• k=1

cXp

k ,Xp k+1(tp

k , tp k+1)

• bserved transitions

× c1

Xp

np

(tp

np, T p 2 )δp

1 × c2

Xp

np

(tp

np, T p 1 , T p 2 )δp

2

• specific censoring terms

    

cXp

k ,Xp k+1(tp

k , tp k+1) = pXp

k ,Xp k+1(tp

k )

• jump probability

× fXp

k ,Xp k+1(tp

k , tp k+1 − tp k )

• density of duration law

. c1

Xp

np

(tp

np, T p 2 ) = SXp

np (tp

np, T p 2 − tp np)

• marginal survival function

. c2

Xp

np

(tp

np, T p 1 , T p 2 ) = pXp

np ,0(tp

np)

• jump probability

× FXp

np ,0(tp

np, T p 1 − tp np)

• cds of duration law

+ SXp

np (tp

np, T p 2 − tp np)

• marginal survival function

. Optimization performed using the Nelder-Mead algorithm. Selection of the best sub-model according to the BIC.

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Key elements Construction process Application and results

Statistics on simulated trajectories

20000 40000 60000 50 55 60 65 70 75 80 85 90 95 100 105

age in years number of individuals

state GIR 4 GIR 3 GIR 2 GIR 1

Number of dependent people in the population, by age

0.0 0.1 0.2 1 2 3 4 5 6 7 8 9 10 11 12

time survived in dependency density

Distribution of survival time in dependency

Graphs generated using 1 million trajectories with initial age of 50.

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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∗ × IP)] = E [NPV(B)] . NPV: Net Present Value. B: benefit cash flows. IP: premium unit cash flows. Estimator of the premium: pn = µB(n)

• µP(n)

→

n→+∞ p∗ a. s. by the law of large numbers.

• µB(n) (resp.

µP(n)): estimator of empirical mean of NPV(B) (resp. NPV(IP)).

• σB(n) (resp.

σP(n)): estimator of empirical mean of NPV(B) (resp. NPV(IP)). We show that with n large enough: |p∗ − pn| ≤ σB(n) + pn σP(n) Φ−1(1 − α

2 )

• µP(n)√n

with a level of confidence of 1 − α Φ: cumulative distribution function of the standard normal law.

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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 !

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