**Course Code: MATH 2115**

**Course Title: Life Contingencies I**

**Course Type: Core **

**Level: 2**

**Semester: 1 **

**No. of credits: **3

**Prerequisite(s): MATH 2274, MATH 2211**

**Course Rationale**

** **To introduce students to fundamental mathematical concepts in Actuarial Science.

This course provides the basic tools that are used in the analysis and valuation of instruments, such as insurance policies and annuities which are influenced by the (uncertain) length of the policyholder’s life. These are, of course, major areas of actuarial practice. The course also provides excellent examples of mathematical modelling of real situations and thus may serve as part of the education of both actuarial science students and mathematics majors in problem solving and modern applied mathematics.

This is the first part of a two semester sequence in Life Contingencies that prepares students for the appropriate actuarial examination.** **

** **

**Course Description**

This course is an introduction to life contingencies as applied in actuarial practice. Topics include present value random variables for contingent annuities and insurance, their distributions and actuarial present values, equivalence principle, and other principles for determining premiums and reserves.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through marked assignments supplemented by problem papers, allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the marked assignments and in-course tests followed by a final examination based on the whole course. ** **

** **

**Course Content**

- Role of an actuary

Survival distributions and life tables This course provides the basic tools that are used in the analysis and valuation of instruments, such as insurance policies and annuities which are influemced by the (uncertain) length of the policyholder’s life. These are, of course, major areas of actuarial practice. The course also provides excellent examples of mathematical modelling of real situations and thus may contribute to the education of mathematics majors.

This is the first part of a two semester sequence in Life Contingencies that prepares students for the appropriate actuaria examination.

- Utility theory
- Life insurance
- Life annuities
- Benefit premiums and reserves
- Introduction to multiple life functions

** ****Learning Outcomes**

On completion of these modules the student should be able to:

__Role of an Actuary__

- Explain the role of an actuary in insurance, pension, financial and non-traditional fields.
- Identify the tasks required by actuaries in applying the concepts of life contingencies theory in actuarial practice's.
__Survival Distributions and Life Tables__- Define survival-time random variables:

a) For one life, both in the single- and multiple-decrement models.

b) For two lives, where the lives are independent or dependent (including the common stock model).

- Calculate the expected values, variances, probabilities, and percentiles for survival-time random variables.
- Define the continuous survival-time random variable that arises from the discrete survival-time random variable using one of the following:

a) A uniform distribution.

b) A constant force of mortality.

c) A hyperbolic assumption.

__Utility Theory__

- Define:
- Expected value principle.
- Actuarial or fair value.
- Utility functions.
- Jensen’s inequalities.
- Stop-loss or excess loss insurance.
- Calculate for defined utility functions:
- Expected loss.
- Insurance premiums.

__Life Insurance__

- Define present-value-of-benefit random variables defined on survival-time random variables:
- Define the expected values, variances and probabilities for:

a) Present-value-of-benefit random variables.

b) Present-value-of-loss-at-issue random variables, as a function of the considerations (premiums).

c) Present-value-of-loss random variables, as a function of the considerations (premiums).

- Calculate the expected values, variances and probabilities for:

a) Present-value-of-benefit random variables.

b) Present-value-of-loss-at-issue random variables, as a function of the considerations (premiums).

c ) Present-value-of-loss random variables, as a function of the considerations (premiums).

__Life Annuities__

- Calculate considerations (premiums) for life annuities using:

a) The Equivalence Principle.

b) Percentiles.

- Calculate liabilities, analyzing the present-value-of-future-loss random variables using:

a) The prospective method.

b) The retrospective method.

c) Special formulas.

__Benefit Premiums and Reserves__

- Calculate:

a) Gross considerations (expense-loaded premiums).

b) Expense-loaded liabilities (reserves).

c) Asset shares.

__Introduction to Multiple Life Functions__

- Define:
- Joint distributions of future lifetimes.
- Joint life status.
- Last survivor status.
- Dependent life models (including common shock).

**Cognitive skills, Core skills and Professional Awareness**

- Awareness of the principal statistical distributions and models used in calculating expected actuarial values of a defined random variable.
- Possession of the knowledge required to work in the area of risk management in the actuarial context.
- Applicaton of the appropriate and rigorous use of mathematical modeling to formulate workable solutions to important financial problems.

**Teaching Methodology**

__Lectures__**:** Two lectures per week (50 minutes each)

Lab: One weekly two-hour lab session based on material covered during lectures and involving the use of Excel, Matlab, Maple and Axis to make Actuarial calculations.

.

**Assessment Criteria**

Mathematics of Finance is assessed by combination of coursework (50%) and a single 2-hour written exam at the end of the semester (50%).

**Assessment:** In-course Tests 40%

Assignments 10%

Final Exam 50%

**In-course Tests:** Two 50-minute written papers (20% each) consisting of compulsory questions of varying length

**Assignments:** Two papers to be submitted: one paper on the first part of the course and the other on the second part of the course. Each assignment is worth 5%. Tutorial practice papers will be given every week to be submitted the following week. Tutorial papers are not graded as part of the course work.

**Quizzes**: Six quizzes each worth 2%.

**Exam Format:** One two-hour written paper with compulsory questions.

**Course Calendar**

** **

Week |
Topic to be taught |
Assessment |
---|---|---|

1 |
Introduction/Course Overview
Role of an actuary |
Tutorial #1 is given |

2 |
Role of an actuary. Survival distributions and life tables |
Tutorial #2 is given and Tutorial #1 is corrected |

3 |
Survival distributions and life tables. |
Tutorial #3 is given and Tutorial #2 is corrected |

4 |
Utility theory |
Tutorial #4 is given and Tutorial #3 is corrected. |

5 |
Utility theory. Life insurance |
First coursework test is given |

6 |
Life insurance |
Tutorial #5 is given and Tutorial #4 is corrected |

7 |
Life annuities |
Tutorial #6 is given and Tutorial #5 is corrected |

8 |
Life annuities. |
Tutorial #7 is given and Tutorial #6 is corrected |

9 |
Loss-at-issue random variable |
Tutorial #8 is given and Tutorial #7 is corrected. |

10 |
Benefit premiums |
Second coursework test is given |

11 |
Benefit Reserves |
Tutorial #9 is given and Tutorial #8 is corrected |

12 |
Introduction to multiple life functions |
Tutorial #10 is given and Tutorial #9 is corrected |

13 |
Revision |
Revision |

**Required Reading **

The prescribed text is “*The Actuarial Mathematics -* Newton Bowers, James Hickman, Cecil Nesbitt, Donald Jones, Hans Gerber 2^{nd} Edition 1997

**Additional Readings**

Other recommended texts that students may find useful are:

*Actuarial Mathematics for Life Contingency Risks*– Mary R. Hardy, David C.M. Dickson, Howard R. Waters, Cambridge University Press 2009*Fundamentals of Actuarial Mathematics –*S. David Promislow 2006*Life Contingencies:**A Logical Approach to Actuarial Mathematics*- Robert W. Batten 2005 Edition