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
Having successfully completed this module you will be able to:
- define a compound Poisson process and calculate various types of probabilities using simulation.
- 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
- to describe the effect on the probability of ruin, in both finite and infinite time, of changing parameter values by reasoning or simulation.
- analyse problems of pricing and reserving in relation to contracts involving several lives
- define the aggregate claim process and the cash-flow process for a risk.
- analyse problems of pricing and reserving in relation to multiple decrement tables
- describe and apply a basic chain ladder method for completing the delay triangle using development factors.
- 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.
- define and use standard actuarial functions involving several lives
- show how the basic chain ladder method can be adjusted to make explicit allowance for inflation.
- describe how a statistical model can be used to underpin a run off triangles approach.
- to value basic benefit guarantees using simulation techniques.
- 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.
- 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.
- 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.
- 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
|Total study time||150|
Resources & Reading list
Jordan CW (1975). Textbook on Life Contingencies. Society of Actuaries.
Benjamin B, Haycocks HW & Pollard JH (1980). The Analysis of Mortality and Other Actuarial Statistics. Butterworth-Heinemann,.
School of Actuaries and Institute and Faculty of Actuaries (2002). Formulae and Tables for Actuarial Examinations. School of Actuaries and Institute and Faculty of Actuaries.
Bowers NL et al (2007). Actuarial Mathematics. Society of Actuaries.
Neill A (1977). Life Contingencies. Heinemann.
Promislow SD (2005). Fundamentals of Actuarial Mathematics. John Wiley.
Dickson DCM et al (2009). Acutarial Mathematics for Life Contigent Risks. CUP.
Gerber HU (1997). Life Insurance Mathematics. Springer.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External