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MATH6130 Actuarial Mathematics II

Module Overview

Synopsis: The module extends the mathematical framework developed in MATH6129 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, and contribution rate and liability valuation issues for pension schemes. 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, as occurs, for example, in the case of an active member of a pension scheme, 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. In particular, the application of multiple-decrement table methods to the analysis of pension scheme funding is explored in some depth. Finally, the techniques of cash flow analysis and profit testing introduced in MATH6129 are extended to permit application to unit-linked contracts and to more complex conventional contracts.

Aims and Objectives

Module Aims

To provide students with an advanced understanding of actuarial mathematics and experience of its application to life insurance and pensions contracts

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • Define and use standard actuarial functions involving several lives
  • 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 and use commutation functions appropriate for the valuation of pension fund benefits and contributions
  • Conduct a cash-flow analysis and profit test for a unit-linked contract or complex conventional contract.


• 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 and to pension 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. • Pension scheme benefits. Main assumptions used in valuation: valuation interest rate, salary scale, in-service mortality table, post-retirement mortality table for age and ill-health retirement, rates of withdrawal, age retirement, ill-health retirement, etc. Derivation of commutation functions for valuation of contributions and benefits, and applications. • Cash-flow modelling and profit testing for unit-linked contracts and for complex conventional contracts.

Learning and Teaching

Teaching and learning methods

Lectures, tutorials, assigned problems, private study

Independent Study108
Total study time150

Resources & Reading list

Neill A (1977). Life Contingencies. 

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

Promislow SD (2005). Fundamentals of Actuarial Mathematics. 

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

Dickson, D.C.M, Hardy, M.R., Waters, H.R. (2009). Actuarial Mathematics for Life Contingent Risks. 

Bowers NL et al (1997). Actuarial Mathematics. 

Gerber HU (1997). Life Insurance Mathematics. 

Jordan CW (1975). Textbook on Life Contingencies. 



MethodPercentage contribution
Class Test 20%
Exam 80%


MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisites: MATH6129 AND MATH6131


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

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