Module Directory

Calculations in clinical trials, pensions and life and health insurance require reliable estimates of transition intensities/survival rates. This module covers the estimation of these intensities.

The module also covers estimation procedures for lifetime distributions including censoring, Kaplan-Meier estimate, Nelson-Aalen estimate and Cox model; statistical models of transfers between multiple states; maximum likelihood estimators for the transition intensities; Binomial and Poisson models of mortality; estimation of age-dependent transition intensities; the graduation process; testing of graduations and measuring the exposed-to-risk.

The aims of this module are:

• to critically outline the distinctive characteristics of non-parametric estimation procedures for the lifetime distribution;


• to critically analyse and derive maximum likelihood estimators for the transition intensities;


• to examine in detail the Binomial and Poisson models of mortality and compare with the Markov models;


• to analytically describe the estimation procedure for transition intensities depending on age;


• to critically analyse and carry out tests for the consistency of crude estimates with a standard table or a set of graduated estimates;


• to describe in detail the process of graduation and the advantages and disadvantages of the various methods.


• to understand copula in bivariate survival models.


• to implement survival analysis models (e.g. Cox regression models and parametric regression models) using R.

By the end of the module, students will be expected to be able to:

1. Describe the principles of actuarial modelling


2. Describe non-parametric estimation procedures for the lifetime distribution, including censoring, the Kaplan-Meier estimate, Nelson-Aalen estimate and Cox regression model (proportional hazards model);


3. Derive maximum likelihood estimators (and hence estimates) for the transition intensities in models of transfers between states with piecewise constant transition intensities;


4. Describe the Binomial and Poisson models of mortality, deriving maximum likelihood estimators for the probability/force of mortality and compare with the Markov models;


5. Describe how to estimate transition intensities depending on age, exactly or using the census approximation, including calculation of exposed to risk and specification of census formulae based on various age definitions;


6. Describe and carry out tests for the consistency of crude estimates with a standard table or a set of graduated estimates;


7. Describe the process of graduation and the advantages and disadvantages of the various methods.

This module covers the related units of CS2 (Risk Modelling and Survival Analysis, Core Principles), Institute and Faculty of Actuaries CS2 syllabus.

The syllabus includes the following: Concepts underlying actuarial modelling; distribution and density functions of the random future lifetime, the survival function and the force of hazard.

Indicative syllabus:

1. Concepts of actuarial modelling: Describe why and how models are used, their benefits and limitations. Explain the concept of survival models, describe the model of lifetime or failure time from age x as a random variable, state the consistency condition between the random variable representing lifetimes from different ages, define the distribution and density functions of the random future lifetime, the survival function, the force of mortality or hazard rate, and derive relationships between them. State the Gompertz and Makeham laws of mortality. Compute life tables and define the expected value and variance of the complete and curtate future lifetime and derive expressions for them. Define the curtate future lifetime from age x and state its probability function.

2. Binomial and Poisson models of mortality Describe an observational plan in respect of a finite number of individuals observed during a finite period of time, and define the resulting statistics, including the waiting times, derive the likelihood function for constant transition intensities in a Markov model of transfers between states described. Derive maximum likelihood estimators for the transition intensities and state their asymptotic joint distribution. Describe the Poisson approximation to the estimator. Describe the Binomial model of the mortality of a group of identical individuals subject to no other decrements between two given ages. Derive the maximum likelihood estimator for the rate of mortality in the Binomial model and its mean and variance and describe the advantages and disadvantages of the multiple state model and the Binomial model, including consistency, efficiency, simplicity of the estimators and their distributions, application to practical observational plans and generality.

3. Exact or approximate estimations of transition intensities. Describe the principle of correspondence and explain its fundamental importance in the estimation procedure. Specify the data needed for the exact calculation of a central exposed to risk (waiting time) depending on age and sex and calculate a central exposed to risk. Explain how to obtain estimates of transition probabilities, including in the single decrement model the actuarial estimate based on the simple adjustment to the central exposed to risk. Explain the assumptions underlying the census approximation of waiting times. Explain the concept of rate interval. Describe how to test crude estimates for consistency with a standard table or a set of graduated estimates, and describe the process of graduation by parametric formula, standard table or graphical and state the advantages and disadvantages of each.

4. Explain the concept of survival models. Recognize the characteristics of survival data, e.g. censoring and truncation. Describe the various ways in which lifetime data might be censored. Describe the Kaplan-Meier (or product limit) estimate of the survival function and the Nelson-Aalen estimate of the cumulative hazard rate compute it from typical data and estimate its variance. Determine the proper method to be used in analyzing time-to-event data (e.g., parametric, semi-parametric or non-parametric method). Describe the Cox proportional hazard model, derive the partial likelihood estimate, and state its asymptotic distribution. Perform survival analysis using a computer statistical software package. Interpret computer outputs.

5 . Explain the basics of bivariate survival models (copulas) Explain that bivariate survival data are usually modelled via copulas. Describe how a copula can be characterised as a multivariate distribution function which is a function of the marginal distribution functions of its variates and explain how this allows the marginal distributions to be investigated separately form the dependency between them. Explain the meaning of the terms: dependence or concordance, upper and lower tail dependence; and state in general terms how tail dependence can be used to help select a copula suitable for modelling particular types of risk. Describe the form and characteristics of different types of copulas.

6. Introduction to extreme value theory:

Recognise extreme value distributions, suitable for modelling the distribution of severity of loss and their relationships.

Calculate various measures of tail weight and interpret the results to compare the tail weights.

7. Method of graduation and statistical tests:

Describe and apply statistical tests of the comparison of crude estimates with a standard mortality table testing for:
a. the overall fit
b. the presence of consistent bias
c. the presence of individual age where the fit is poor
d. the consistency of the 'shape' of the crude estimates and the standard table

For each test describe:
a. the formulation of the hypothesis
b. the test statistic
c. the distribution of the test statistic using approximations where appropriate
d. the application of the test statistic
e. how tests should be amended to compare crude and graduated sets of estimates
f. how tests should be amended to allow for the presence of duplicate policies

Describe the reasons for graduating crude estimates of transition intensities or probabilities and state the desirable properties of a set of graduated estimates.

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.