The University of Southampton
Courses

# MATH3063 Actuarial Mathematics I

## Module Overview

This subject arises through a fusion of compound interest theory with probability theory, and provides the mathematical framework necessary for analysing such contracts, which are essentially long term financial transactions in which the various cash flows at different times are contingent on the death (life assurance) or survival (life annuities) of one or more specified human lives. Having developed this framework, we can address issues such as how to determine the premium that should be charged for a certain life assurance contract, including allowance for expenses and/or profit, and how to determine the value that should be represented in the balance sheet of a life assurance company in respect of the policies that it has sold. These examples reflect the two main traditional areas of actuarial activity within a life assurance company: pricing and reserving. The module begins with an examination of the various factors that affect mortality, and of how risk classification may be used to address the heterogeneity within a given population. Next, probabilities of survival and death are introduced, and it is shown how these may be represented within and extracted from life tables. Compound interest theory is then combined with such probabilities to analyse and evaluate both life assurance benefits and life annuity benefits. With the relevant theory fully developed, the module then becomes somewhat more applied. Premium calculation is explored in detail first, followed by the determination and application of reserves, and, in both areas, the theory is applied to quite realistic and complex problems. Finally, the alternative perspective of cash-flow analysis, or profit-testing, is introduced and applied to assess the emergence of profit from, and overall profitability of, a life contract. Pre-requisite for MATH3066

### Aims and Objectives

#### Module Aims

To provide students with a solid grounding in the subject of life contingencies for a single life, and experience of its application to the analysis of life assurance and life annuity (including pension) contracts.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Describe the various factors that affect mortality, and how risk classification may be used to address the heterogeneity within a given population
• conduct a cash-flow analysis and profit test for a unit-linked contract or complex conventional contract.
• analyse the liabilities of simple assurance and annuity contracts, both in terms of emerging costs and present values, in relation to product pricing, reserving, and their applications
• show how the relationship between the pricing basis and reserving basis affects the emergence of profit for simple assurance and annuity contracts
• Show how a single decrement model may be used to describe the evolution of a population subject to a single source of decrement
• Define terms associated with single decrement models, such as radix, decrement rate (failure rate, mortality rate), probability of survival, expectation of life, hazard rate (force of mortality
• Analyse simple problems of emerging costs using simple single decrement models
• calculate present values and accumulated values of cash flows at a specified rate of interest, using a single decrement model to take account of the probability of the payments being made;
• define and use standard annuity and assurance functions for a single decrement model, including both select and ultimate functions, and demonstrate the relationships between corresponding annuity and assurance functions
• analyse straightforward problems involving an equation of value, using a single decrement model to take account of the probability of the payments being made;

### Syllabus

Factors affecting human mortality. Selection. Risk classification. Survival models, survival functions, survival tables. Temporary initial selection in mortality tables. Select, ultimate, and aggregate mortality rates. Single decrement functions and inter-relationships. Radix, decrement rate (failure rate, mortality rate), probability of survival, expectation of life, hazard rate (force of mortality). Human mortality functions. Probability (density) functions, mean, and variance for complete future lifetime and curtate future lifetime. Hazard rate functions due to Gompertz and Makeham. Laws of mortality. Cash flows under financial contracts dependent on death or survival. Concept and calculation of present value and accumulated value of a contingent stream of payments. Mean and variance of such a present value. Equations of value for given contingent payment streams and methods of solution for unknown payment amounts, times of payment, and yields. Cash flows for pure endowments, level and varying immediate annuities, level and varying whole life, endowment, and term assurances, including critical illness insurance, for a single life, and corresponding standard annuity and assurance functions. Present values and accumulated values for such contracts. Mean and variance of such present values. Extension to annuities payable more frequently than once per year and to assurances payable immediately on death. Definition and use of corresponding commutation functions, and their relationships to annuity and assurance functions. Equations of value for net and office premiums for assurance and annuity contracts, and their solution. Reserves and calculation of policy values by both retrospective and prospective methods. Expected and actual death strain, mortality profit. Paid-up policy values, surrender values, alterations. Emergence of profit from a contract, for a given reserving basis and discount rate. Profit testing, profit vector, profit signature, and measures of profitability. Choice of pricing and reserving bases, and why they may differ. Effect of choice of basis on emergence of profit. Cash-flow modelling and profit testing for unit-linked contracts and for complex conventional contracts.

### Learning and Teaching

#### Teaching and learning methods

Lectures, assigned problems, private study

TypeHours
Teaching42
Independent Study108
Total study time150

Gerber HU (1997). Life Insurance Mathematics.

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

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

Jordan CW (1975). Textbook on Life Contingencies.

Bowers NL et al (1997). Actuarial Mathematics.

Promislow SD (2005). Fundamentals of Actuarial Mathematics.

D.C.M. Dickson, M.R. Hardy, and H.R. Waters (2013). Actuarial Mathematics for Life Contingent Risks.

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

Neill A (1977). Life Contingencies.

### Assessment

#### Summative

MethodPercentage contribution
Assignment 20%
Class Test 10%
Exam 70%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

Prerequisites: (MATH2011 AND MATH2040) OR (MATH2040 AND ECON1011)

### 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

An essential purchase is the set of tables produced by the profession, which will be used extensively and it will be necessary for students to bring their own copy to both Class Tests and to the Final Examination. Faculty of Actuaries and Institute and Faculty of Actuaries - Formulae and Tables for Actuarial Examinations (Faculty of Actuaries and Institute and Faculty of Actuaries, 2002

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.