# Statistics 470:   Actuarial Mathematics

Section 0101, MWF 2, Rm 0407                                Spring 2006

See below for:

Take-Home Exam Problems due 5/17/06
NOTE: do the most current versions of the Exam problems: a few of them
have been corrected to be clearer and easier than the ones handed out in class.

Instructor: Eric Slud, Statistics Program & Actuarial Advisor

Objective:  This course introduces several of the major mathematical ideas
involved in calculating life-insurance premiums, including: compound interest and
present valuation of future income streams; probability distributions and expected
values derived from life tables; the interpolation of probability distributions from values
estimated at one-year multiples; the `Law of Large Numbers' describing the regular
probabilistic behavior of large populations of independent individuals; and the detailed
calculation of expected present values arising in Insurance problems.

Prerequisite: Calculus through Math  240-241. Some Probability at the level of
Stat 400 would be helpful. Ideas from probability and statistics will be developed as
needed, through course notes and reference to the Stat 400 text, Introduction to Probability
and Statistics, 6th ed. (2004) by R. Devore.  However, this material may go a little quickly
if you have really never been exposed to it before.

Main Text:  Book notes (written by me) available here for download, one chapter at a time.
(It is currently in a single pdf-file. Individual chapters will be revised and placed in the
same directory, with revision dates, as the term goes on.
)

Recommended Texts:
(1)  Life Insurance Mathematics 3rd ed. (1997), by H. Gerber, with
Exercises by S.H. Cox, Springer-Verlag.
(2) Theory of Interest  and Life Contingencies With Pension Applications: A Problem
Solving Approach, 3rd ed. (1999) by Michael M. Parmenter, ACTEX Publications.

I will probably place these on reserve in EPSL.

Course format:  Graded homeworks (one every  1.5 weeks) , one in-class midterm just
before Spring Break in March, and a final exam which may take the form of a take-home final or
project. Homework counts 40%, and midterm and final/project each count 30%, toward the
course grade. Project and/or take-home topics will be distributed and discussed after the mid-term.

Homeworks:  will regularly be posted hereAny changes (such as changes in due-date,
corrections of misprints, etc.)  and hints will also be posted  to the same "HWx.txt" page
where the homework is assigned.

To see a sample of problems for the In-Class Test on Friday 3/31/06, click here.
This Sample was the Test I gave in Spring 2001. The Test Friday will not
necessarily follow exactly the same format or problem types, but the level
of difficulty will be about the same.

Project/Final:  in lieu of an in-class final exam, I have sometimes given
either a 10-12 problem take-home consisting of problems on the
course material from past actuarial exams, OR  of doing a   5-10 page
Project paper on some additional topic related to the course material.
Guidelines for Projects can be found here.

Take-Home Problems for Final would be handed out in class a little less than
2 weeks before the Final Exam date, and would be due at the scheduled time of the
Final Exam. I will make up a new set of problems to reflect the level of old and
recent "Course 150" Actuarial exams.
Click here for Take-Home. (But make sure to do the most current version.)

HANDOUTS & EXHIBITS

(1)  An R program  Balance.Discrete  to calculate accumulated balances from streams of
deposits (or withdrawals, treated as negative deposits) over a series of times, with possibly
time-varying but piecewise interest rates which can change just after each deposit.

(2)  A MATLAB program  RefExmp.m  to calculate quantities related to mortgage
refinancing. Once you understand what it calculates, you may (and probably should, unless
you prefer another programming platform like a spreadsheet) use it in Homework Set 3.

COURSE   OUTLINE

I. Overview of actuarial mathematical problems.
A. Theory of interest and actuarial notation.

II. Introduction to Life Tables & Mortality Measurements.
A. Probability densities, random variables, expectation, law of large numbers.
B. Relative frequencies and empirical death rates.  Connections with probabilities.
C. Survival curves. Force of mortality (hazard rates).
D. Theoretical survival models. Estimation from life-table data.
i. Stochastic simulation of insurance experience.
E. Actuarial approximations for survival probabilities.
F. Probability in demography: stationary populations and age-distributions.

III. Calculation of Insurance Premiums. Valuation of insurance contracts. Reserves.

### If you need help...

My office hours are Monday 11-12, Wednesday 3 and Friday 1-2.  I will often be available except on
Tuesdays, but please send an e-mail or arrange with me in class for an office appointment.

## Getting Started In R

For the systematic Introduction to R and R reference manual distributed with the R software,
and for software you can download free, visit the R website .

### Important Dates

• First Class: Wednesday, January 25
• Mid-Term Exam:   Friday, March 31, 2006.
• Final Exam:  Wednesday, 5/17/06:   Take-Home Exam Due, 3:30pm.