The University of Southampton
Courses

# MATH3066 Actuarial Mathematics II

## Module Overview

Synopsis: The module extends the mathematical framework developed in MATH3063 in order to enable modelling of long term financial transactions where the various cash flows are contingent on the death or survival of several lives, or where there are several competing sources of decrement present. Having extended this framework, we can address pricing and reserving issues for contracts on a pair of lives, such as a husband and wife. The module begins by extending the notion of a life table to several lives, as a precursor to examining assurances on a pair of lives where the benefit is paid on the first (joint life assurance) or last (last survivor assurance) of the pair to die. Assurances payable only on a specified ordering of the deaths (contingent assurances) are also covered, as are correspondingly ordered annuities (reversionary annuities). Attention then turns to a single life subject to several competing sources of decrement who may leave the population of active members by age retirement, by ill-health retirement, by death, or by leaving employment covered by the scheme. Both multiple-state and multiple-decrement models are employed in such contexts, and both are examined. The concepts of aggregate claim and cash-flow process are explained. Then Poisson process are used to model the number of claims, and the distribution of inter-arrival claims are discussed. The concept of ruin probability is covered where compound poison processes and simulation techniques are covered to calculate various type of finite and infinite horizon ruin probabilities. Next, the method of chain ladder and their application in delay trainable are studied where inflation is allowed and statistical models are applied. Finally, Bornhuetter-Ferguson method for estimating outstanding claim amounts are investigated.

### Aims and Objectives

#### Module Aims

To provide students with an advanced understanding of actuarial mathematics and experience of its application to life insurance and claim analysis.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• define and use standard actuarial functions involving several lives
• to define a development factor and show how a set of assumed development factors can be used to project the future development of a delay triangle.
• describe and apply a basic chain ladder method for completing the delay triangle using development factors.
• show how the basic chain ladder method can be adjusted to make explicit allowance for inflation.
• describe and apply the average cost per claim method for estimating outstanding claim amounts.
• describe and apply the Bornhuetter-Ferguson method for estimating outstanding claim amounts.
• describe how a statistical model can be used to underpin a run off triangles approach.
• to value basic benefit guarantees using simulation techniques.
• analyse problems of pricing and reserving in relation to contracts involving several lives
• show how a multiple state model or multiple decrement model may be used to describe the evolution of a population subject to several sources of decrement
• analyse problems of pricing and reserving in relation to multiple decrement tables
• define the aggregate claim process and the cash-flow process for a risk.
• use the Poisson process and the distribution of inter-event times to estimate the number of events in a given time interval and waiting times.
• define a compound Poisson process and calculate various types of probabilities using simulation.
• define the probability of ruin in infinite/finite and continuous/discrete time and state and explain relationships between the different probabilities of ruin, and calculate them by simulation.
• to describe the effect on the probability of ruin, in both finite and infinite time, of changing parameter values by reasoning or simulation.

### Syllabus

- Joint life functions. Probabilities of death or survival of either or both of two lives. Joint life and last survivor assurance and annuity functions, corresponding present values, means, and variances. Extension to consideration of continuous and mthly frequencies, and to functions dependent on term as well as age. Application to pricing and reserving problems. - Multiple state models. Probabilities of transfer and forces of transition between states. Kolmogorov equations. Application to death/sickness model. - Multiple decrement models and multiple decrement tables. Independent and dependent rates of decrement, corresponding single decrement tables, and relationships. Evolution of a population subject to several sources of decrement. Application of multiple decrement models to pricing and reserving problems. Poisson process, Compound Poisson process, Simulation, Ruin probability, Delay triangle, Chain ladder Method, Development factor, Bornhuetter-Ferguson method.

### Learning and Teaching

#### Teaching and learning methods

Lectures, tutorials, office hours, assigned problems, private study

TypeHours
Teaching60
Independent Study90
Total study time150

Neill A (1977). Life Contingencies.

Bowers NL et al (2007). Actuarial Mathematics.

Benjamin B, Haycocks HW & Pollard JH (1980). The Analysis of Mortality and Other Actuarial Statistics.

Gerber HU (1997). Life Insurance Mathematics.

School of Actuaries and Institute and Faculty of Actuaries (2002). Formulae and Tables for Actuarial Examinations.

Promislow SD (2005). Fundamentals of Actuarial Mathematics.

Jordan CW (1975). Textbook on Life Contingencies.

Dickson DCM et al (2009). Acutarial Mathematics for Life Contigent Risks.

### Assessment

#### Summative

MethodPercentage contribution
Assignment 20%
Class Test 10%
Exam  (2 hours) 70%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

Pre-requisite: MATH3063

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.