ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Duration Models


LOPEZ Olivier

Département : Finance


In this course, we introduce the main statistical tools used to model and infer duration phenomenons commonly encountered in actuarial science. Elaborating lifetables, designed to reflect the mortality of an insured population, is the main example that we consider, although additional examples coming from both life and non-life insurance are also proposed. In the last part of this course, we consider prospective models used to analyze and forecast the evolution of mortality through time, which is a key issue in evaluating the longevity risk.

Experience obtained: at the end of this course, the students will know how to

- construct models adapted to describing duration phenomenon

- perform statistical estimation in these models, taking their specificities into account

- design lifetables reflecting the mortality of an insured portfolio

- use prospective models to explain and forecast the evolution of mortality


  1. General concepts in survival analysis. Characterizations of the distribution of lifetimes. Incomplete observations: censoring and truncation.
  2. Non parametric estimation. Kaplan-Meier estimator of the survival function. Nelson-Aalen estimator of the cumulative hazard rate. Estimation of death rates.
  3. Smoothing techniques. Moving averages, splines. Whittaker-Henderson smoothing. Bayesian smoothing.
  4. Parametric modelling. Maximum likelihood estimation for censored and truncated lifetimes. Estimation from death rates. Classical distributions used to model lifetimes. Goodness-of-fit procedures.
  5. Prospective models. Lee-Carter model. Cairns-Blake-Dowd model. Extensions.


Delwarde, A., Denuit, M. (2006) Construction de tables de mortalité périodiques et prospectives, Paris, Ed. Economica.

Kalbfleisch, J.D., Prentice, R.L. (2002) The statistical analysis of failure time data, Second Ed. New-York, Wiley.

Planchet, F., Thérond, P. (2011) Modélisation statistique des phénomènes de durée, applications actuarielles, Paris,Ed. Economica.