More Detailed STAT 470 syllabus

This is a more detailed version of the STAT 470 syllabus based on a previous offering.

OBJECTIVE OF THE COURSE: 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. 

Course 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 from scratch, as needed, through course notes and reference to the Stat 400 text (recommended for this course as well), Introduction to Probability and Statistics by R. Devore. (If you do not have any background in probability and statistics, there are a number of basic books which contain good basic discussions of random variables and probability at the level of the second Actuarial Exam. A few standard ones are: Ross, S., Intro.to Probability Theory (used for Stat 410); Hoel, Port, and Stone, Intro. to Probability Theory; Larson, R., Intro. to Probability Theory and Statistical Inference; Larsen and Marx (currently used for Stat 400); Hogg, R. and Craig, A., Intro. to Mathematical Stat.; and many others. 

TEXT:  Life Insurance Mathematics (1990), by H. Gerber, Springer-Verlag;  plus Notes may be handed out in class. 

Recommended: (1) Actuarial Mathematics (1986), N. Bowers, H. Gerber,  J. Hickman, D. Jones, & C. Nesbitt, Society of Actuaries.  (2) Life Contingencies (1975), C.W. Jordan, Society of Actuaries; reprinted by Mad River Books/ACTEX. 

Course format: Graded homeworks (one every week or two, counting up to one-half of the course grade), an in-class midterm, and a  take-home final or project.   Project and/or take-home topics will be distributed and discussed after the mid-term. 

COURSE OUTLINE 

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

I. Introduction to Life Tables.  Mortality Measurements.  Relative frequencies and empirical death rates.  Connections with  probabilities. 
      A. Basics of probability densities, random variables, expectation: the law of large numbers. 
      B. Survival curves. Force of mortality (hazard rates). 
      C. Theoretical survival models. Estimation from life-table data. 
      D. Actuarial approximations for survival probabilities. 
      E. Probability calculations related to demography: stationary populations and age-distributions. 

II. Calculation of Insurance Premiums. Loading, valuation of insurance contracts. Reserves. 

III. Life tables and survival probabilities for multiple lives or multiple causes of death. 
      A. Multiple Decrement ("competing risk") models. 
 

Final-Project Topics 
      Each student will have the choice of doing an extended problem-set (such as a selection of problems of the sort that appeared on old Actuarial (SOA) Exams) OR doing a 5-8 page paper on a mathematical actuarial topic.  The following is a list of suggested topics for such a final paper or project. Some of the topics will be absorbed into the course, on a rotating basis, covered near the end of the semester. 

      The paper is to be based on at least one source such as a journal article or textbook, and is to represent some additional reading beyond the assigned reading in the course. 

      The list below is not exhaustive: a student can work out his/her choice of topic with the instructor. 

 (A) Statistical goodness of fit of a hypothesized "theoretical" lifetime distribution (with or without fitted parameters).  This topic would involve reading about chi-squared goodness-of-fit tests and implementing one on a survival-time data set. 

 (B) Estimating a survival function with the "Kaplan-Meier Estimator": this is a standard biostatistics topic, extending life-table ideas to the case where some individuals are lost to observation or die from an uninteresting (e.g., accidental) cause. 

 (C) "Graduation", or numerical interpolation and smoothing of life-table age-specific death-rates. This topic is related to smoothing splines, and would involve implementation using illustrative life-table data. 

 (D) "Leslie matrices": biological/demographic models of population size, from which one can see whether a small population is likely to die out. (This topic is related to the probability topic of (Multitype) Branching Processes.) 

 (E) Risk-and-ruin theory, or Reinsurance. This topic involves simulation or mathematical calculations of the probability over some longer time-horizon of disastrous adverse fluctuations in survival which would cause an insurer to go bankrupt. 

      Of these topics, (A) & (B) are primarily statistical; (C) involves some numerical analysis; (D) involves linear algebra and a small amount of Markov chain theory; and (E) involves either a little more Markov chain and stochastic-process theory or a willingness to learn a small amount about simulating random insurance portfolios on the computer. 

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