## Actuarial Models 1

 Unit code: MATH69511 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

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#### 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.

#### Overview

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.

#### Learning outcomes

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

• Given a description in words of a particular application where things change randomly over time, be able to construct a Markov chain that serves as a model for this application.
• Be able to derive and/or compute probabilities, expectations and distributions associated with a Markov chain given a description of these quantities in words.
• Given some data of a time homogeneous Markov chain be able to estimate its transition intensities and probabilities via maximum likelihood.
• Be able to use the census approximation in order to estimate mortality rates given census data.
• Be able to carry out certain tests commonly used in practice in order to verify whether a given graduation of mortality rates is successful.
• Be able to compute expected present values of cash flows associated with a Markov chain or a life insurance policy given a description of such cash flows in words.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework 20%
• Examination at the end of the semester, three hours duration, 80%.

#### Syllabus

• 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]
• 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]
• 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]
• 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]
• Methods of Graduation of crude estimates and comparative properties. Statistical tests assessing success of graduation. [5]

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Ronnie Loeffen - Unit coordinator