# Math 178B: General Course Outline

## Course Description

178B. Foundations of Actuarial Mathematics: Further Topics in Long Term Actuarial Mathematics. (4). Lecture, three hours; discussion, one hour. Requisite: 170S (or 170B or Statistics 100B), 178A. The second of the three quarter sequence 178ABC. Multiple state models, pensions, health insurances, profit testing. Topics in statistics used in actuarial work: methods of estimation and probability distributions. Letter grading.

## Course Information:

Mathematics 178A and the first half of Mathematics 178B will almost completely cover the syllabus of the Long-Term Actuarial Mathematics exam by the Society of Actuaries. At the end of Mathematics 178A, students learned to value and set premiums for different types of insurances using traditional actuarial models. They were also exposed to typical models and calculations used in life contingencies. Mathematics 178B first extends this work to multistate models and then covers pensions, health insurances, and profit-testing. The last three weeks of the course will cover the probability distributions employed in most common actuarial theory and begins the study of the Short Term Actuarial Mathematics syllabus by the Society of Actuaries.

## Textbook(s)

(DHW)
Dickson, David C.M., Hardy, Mary R. and Waters, Howard R., Actuarial Mathematics for Life Contingent Risks. 2nd ed., Cambridge University Press, 2013.

(Hardy)
Hardy, Mary R., Long-Term Actuarial Mathematics Study Note. Society of Actuaries, 2017. Education and Examination Committee of the Society of Actuaries - Long Term Actuarial Mathematics Supplementary Note.
https://www.soa.org/Files/Edu/2018/2018-ltam-supplementary-note.pdf

(KPW)
Klugman, Stuart A., Panjer, Harry H. and Willmot, Gordon E., Loss Models: From Data to Decisions. 3rd Edition, Wiley, 2012.

## Schedule of Lectures

Lecture Section Topics

1

DHW 8.1 - 8.3

Examples, assumptions and notations of multiple state models.

2

DHW 8.4 - 8.6

Probability formulae and computations, Kolmogorov equations, premiums.

3

DHW 8.7

Policy values, Thiele?s Differential Equation

4

DHW 8.8 - 8.9

Multiple decrement models

5

DHW 8.10 - 8.12

Multiple decrement tables

6

DHW 8.13

Discrete time models

7

Hardy 2

Disability income, long term care, critical illness insurance, continuing care communities

8

Hardy 3

Policy value recursions

9

Hardy 4.1 - 4.3

Mortality improvement modelling

10

DHW 9.1 - 9.4

Joint life and last survivors benefits, independent future lifetimes

11

DHW 9.4 - 9.5

12

DHW 9.6 - 9.8

Model with dependent future lifetimes, common shock model

13

DHW 10.1 - 10.4

Salary scale function, DC contribution

14

DHW 10.5

Service table

15

DHW 10.6

Benefit valuation

16

DHW 10.7

Funding benefits

17

Review/Leeway

18

Midterm

19

DHW 12.1 - 12.4

Introduction to profit testing and principles

20

DHW 12.5 - 12.8

Profit measures. Using profit test to calculate premium and reserves, case of multiple state models

21

KPW 3.1 - 3 3

Introduction to tails

22

KPW 3.1 - 3.3

Basic distributions (moments, percentiles, generating functions and sums of random variables)

23

KPW 3.4

Tails and their classifications

24

KPW 3.5

Measures of risk (value at risk, tail value at risk)

25

KPW 4.1 - 4.2

Continuous actuarial models, background probability

26

KPW 5.1 - 5.2

Examples of continuous models, creating new distributions

27

KPW 5.3 - 5.4

Relations between distributions. Linear exponential family

28

KPW 6.1 - 6.3

Poisson and negative binomial distributions

29

KPW 6.4 - 6.5

Binomial distributions and (a, b,0) class

30

KPW 6.6

Truncation and modification at 0