**Course co-ordinator(s):** Prof Angus Macdonald (Edinburgh), Dr Alastair Wallis (Malaysia).

**Aims:**

- To understand the use of mathematical models of mortality, illness and other life history events in the study of processes of actuarial interest.
- To be able to estimate the parameters in these models, mainly by maximum likelihood.
- To apply methods of smoothing observed rates of mortality and to test the goodness-of-fit of the models.

**Summary:**

- Estimation for lifetime distributions: Kaplan-Meier estimate of the survival function, estimation for the Cox model for proportional hazards.
- Statistical models for transfers between multiple states (e.g., alive, ill, dead), the multi-state Markov model, relationship between probabilities of transfer and transition intensities, estimation for the parameters in these models. The binomial and Poisson models of mortality.
- Methods of projecting future mortality rates to allow for improving longevity.
- Methods of graduation: parametric and standard table. Tests of consistency of crude estimates of rates of mortality and their graduated values.
- Computing facilities, especially R, will be used extensively and this work will be assessed by practical assignments.

## Detailed Information

**Pre-requisite course(s):** F78AB Actuarial and Financial Mathematics B & F78PB Probability and Statistics B .

**Location: **Edinburgh, Malaysia.

**Semester: **2.

**Syllabus:**

- Introduction, Notation and Revision:
- life time distributions, survival functions, rates and forces of mortality

- Estimating the Lifetime Distribution:
- cohort studies
- censoring
- Kaplan-Meier estimate of the survivor function
- Cox regression model, partial likelihood, estimation

- Markov Models: Theory:
- computation of
_{t}p_{x} - multi-state Markov models
- Kolmogorov forward equations

- computation of
- Markov models: Data and Estimation:
- 2-state model
- maximum likelihood estimate (MLE) of the force of mortality
- score function and the maximum likelihood theorem
- properties of the MLE of the force of mortality
- likelihood and estimation in the multi-state model

- Binomial and Poisson Models of Mortality
- binomial model
- two assumptions: uniform distribution of deaths, constant force of mortality
- likelihood and estimation for the binomial model
- actuarial estimate of
*q*_{x} - Poisson model

- Graduation and Statistical tests:
- graduation process
- testing adherence to data
*χ*test, standardised deviations test, sign test, change of sign test, grouping of signs test, serial correlation test^{2}

- Exposed to Risk
- Calculation of exact exposed to risk.
- Calculation of approximate exposed to risk using census data.

- Mortality Projection
- Approaches to projecting mortality
- The Lee-Carter model
- The Cairns-Blake-Dowd model
- The P-spline model
- Age-period-cohort models
- Sources of forecast error

- The course F79SU is synoptic with F79SP Stochastic Processes.

**Learning Outcomes: Subject Mastery**

At the end of studying this course, students should be able to:

- Estimate a survival function using the Kaplan-Meier method.
- Find the partial likelihood function in the Cox model.
- Use the partial likelihood to estimate the parameters (with standard errors) in the Cox model.
- Write down an appropriate Markov multi-state model for a system with multiple transfers.
- Obtain the Kolmogorov forward equations in a Markov multi-state model.
- Derive the likelihood function in a Markov multi-state model with data.
- Use the likelihood function to estimate the parameters (with standard errors) in a Markov multi-state model with data.
- Obtain the likelihood function in the 2-state model with states
*Alive*and*Dead*under the binomial or Poisson models. - Use any of two assumptions (uniform distribution of deaths, constant force of mortality) to reduce the binomial likelihood to a function of a single parameter, and estimate the parameter.
- Obtain graduations of mortality data using generalized linear models in R, and interpret the results.
- To apply the
*χ*test, the standardised deviations test, the sign test, the change of sign test, the grouping of signs test, the serial correlation test to testing the adherence of a graduation to data.^{2} - Calculate exposed to risk using both exact and census methods.
- Fit stochastic models (Lee-Carter, Cairns-Blake-Dowd, P-spline) to mortality data

and make projections of future mortality.

**Learning Outcomes: Personal Abilities**

At the end of the course, students should be able to:

- Demonstrate the ability to learn independently
- Manage time, work to deadlines and prioritise workloads
- Present results in a way which demonstrates that they have understood the technical and broader issues of modelling mortality and morbidity data
- Communicate findings effectively in the actuarial and financial services industry

**Reading list:**

The course book is: I D Currie, *Survival Models*. The book is essential reading and is available from the department. It contains outline copies of the lecture material and all tutorial material. Copies of past examination papers and illustrative R code will be available through the course web site.

Supplementary reading is provided in Macdonald, A.S.,

Richards, S.J & Currie, I.D. (2018). *Modelling Mortality with Actuarial *

*Applications*. Cambridge University Press.

**Assessment Methods:**

2 hour exam (80%-90%), project work (10%-20%).

**SCQF Level: **9.

**Credits:** 15.

## Other Information

**Help:** If you have any problems or questions
regarding the course, you are encouraged to contact the lecturer

**VISION:** further information and course materials
are available on VISION