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School of Mathematics

# MATH69511 - 2012/2013

General Information
• Title: Actuarial Models
• Unit code: MATH69511
• Credits: 15
• Prerequisites:
• Co-requisite units: None
• School responsible: Mathematics
• Member of staff responsible: Dr. R Loeffen
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## Specification

### Aims

The unit aims to provide an understanding of the use of mathematical models of mortality, illness and other life history events in the study of processes of actuarial interest and to develop the skills of estimating the parameters in these models including the skills of how to apply methods of smoothing observed rates of mortality and how to test for goodness-of-fit of the models.

### Brief Description of the unit

This course unit is primarily concerned with models for life contingent and life related contingent risks and the statistical estimation of transition rates in such models. The course starts with an introduction of Markovian models for transition between multiple states (e.g. alive, ill, dead ) and their properties. It then proceeds to develop and use such models in the context of life related insurance and other actuarial statistics, looking at the principles of valuation of benefits, policies and reserves and the calculation of premiums. The estimation of and statistical tests of mortality rates within such models are then looked at.

### Intended Learning Outcomes

Upon successful completion, the students are expected to be able to

• demonstrate a good understanding of
1. the basic properties of and results in Markov Chains,
2. the probabilistic and statistical methodology of multiple state models for life and other contingencies,
3. the principles of valuation of benefits and reserves and calculation of premiums in multi state models,
4. the principles behind the graduation of transition intensities;
• formulate, solve and interpret an appropriate multi-state Markov model in an actuarial context and show judgement in the selection and application of the appropriate tools. Construct the equation of values of benefits and premiums for a life contingent insurance policy within a multi-state Markov model;
• fit data to an actuarial models, make statistical inferences and interpret the results;
• apply graduation methods on estimated mortality rates;
• calculate pure premiums and valuation of life contingent insurance policies;
• Demonstrate a capacity for analytical thinking and ability to use stochastic modelling concepts in real life problems.

### Syllabus

1. Discrete time Markov Chains (MC): Transition probabilities, Chapman-Kolmogorov equations, time homogeneous MC. Classification of states of a MC. Limiting behaviour and stationary distribution of MC. [4]
2. Continuous time MC: Transition rates and the generator matrix. Chapman-Kolmogorov equations. The forward and backward differential equations. Occupancy times and their distribution. Time homogeneous MC. Time inhomogeneous MC, age related transition rates, residual occupancy times and integrated form of backward equations. Applications to multi-state actuarial models. [9]
3. Valuation, under fixed interest rates, of
1. state dependent and transition dependent benefits,
in a multi-state Markov model. The equivalence principle and the equation of value of an insurance policy. Examples from life insurance, disability insurance and pensions. Policy values and reserves. Recursive calculation of reserves. Thiele's differential equation. [8]
4. Estimation within Markov Models: The two state model of mortality, the general state model of mortality; the Binomial and Poisson mortality models; Estimation of mortality rates by age. [7]
5. Methods of Graduation of crude estimates and comparative properties. Statistical tests assessing success of graduation. [5]

### Teaching and learning methods

1. Lectures and Feedback Tutorials: There are three lectures and one tutorial class each week.
2. Private Study: In addition, students should expect to spend at least four hours each week on private study for this course unit.

### Assessment

Coursework 20%
Examination at the end of the semester, two and a half hours duration, 80%.