UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Math 478 / 568
Actuarial Modeling
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Rick
Gorvett |
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Spring, 2012 |
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374
Altgeld Hall |
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Phone: 244-1739 |
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213 Gregory Hall 9:30-10:50 am TuTh |
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Office
Hours: 3-4 pm Tues,
3-4 pm Wed, or by
appointment |
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Class Website: |
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E-mail:
gorvett@illinois.edu |
Required Texts
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Loss Models: From Data to Decisions, by Klugman, Panjer, and Willmot, 3rd
edition |
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Chapter
8, Credibility, Foundations of Casualty
Actuarial Science, Fourth Edition, 2001, Casualty Actuarial Society
(FCAS) http://www.soa.org/files/pdf/C-21-01.pdf |
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Topics in Credibility Theory http://www.soa.org/files/pdf/c-24-05.pdf |
Course Overview
In the course of her/his work, an
actuary frequently must develop mathematical models to describe the insurance
loss process. Such modeling is performed under a variety of conditions: with
large or small amounts of data, for different types of insurance coverages, etc. This course is intended to introduce the
student to the intuitive underpinnings, the mathematical foundations, and the
real-world applications of actuarial modeling theory.
This course is essentially divided into
thirds:
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Topic |
Anticipated Timing |
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Loss
modeling |
First third of course |
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Model
selection and parameter estimation |
Second third of course |
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Credibility
theory and simulation |
Final third of course |
In the Loss modeling section, we
will study how to mathematically model the severity (size), the frequency
(number), and the aggregate value (total dollar amount) of claims sustained by
the policyholders of an insurance company. Such mathematical models are
critical for developing appropriate prices for insurance policies, and for
determining the level of loss reserve liabilities to be entered on the
financial statements of an insurer. This section will also address risk
measures. The Estimation section will involve the determination and estimation
of actuarial models and their parameters based on data. The Credibility
section will examine how to determine the relative weights to be given to
various sources of data in calculating premium rates and loss reserve levels.
All of these three general topics are essential to the effective performance of
the actuarial function, and are found on professional actuarial Exam 4/C.
Accessibility
Statement
To
ensure that disability-related concerns are properly addressed throughout the semester,
students with disabilities who require reasonable accommodations to participate
in this class are asked to contact me within the first two weeks of class.
Honors Projects
For those involved in a Campus or
College honors program, such as James Scholar, you are welcomed to perform an
honors project in association with this class. Such a project would involve an
opportunity for undergraduate research relevant to actuarial science. Please
let Prof. Gorvett know of your interest in the first few weeks of the class.
Some key dates
for the class are noted below:
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January 17
(Tuesday) |
First day of
class |
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February 16
(Thursday) |
Exam # 1 |
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March 15
(Thursday) |
Exam # 2 |
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May 1
(Tuesday) |
Last day of
class |
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May 9
(Wednesday, 1:30 pm) |
Exam # 3
(Final Exam) |
Practicing
actuaries may be invited to give guest presentations to the class on several
occasions during the semester.
Grading
Course grades will
be determined based on the following weights:
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Homework and
other assignments |
25 % |
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Project |
10 % |
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Exams 1 and 2 |
40 % (20%
each) |
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Final Exam |
25 % |
Exams may not be
made up or taken at different times, except in extremis. In general, the
grade weight for a missed exam (provided there is a valid excuse) will accrue
equally to any remaining exams. Homework assignments will be approximately
weekly, and are due at the beginning of the class on the due
date. Late homework assignments will not be accepted. A
missed homework assignment may not be made up. However, in calculating
your course score and grade, your lowest homework assignment score will be
dropped. Class attendance is expected; serious attendance problems may result in a
grade lower than that indicated by the weighting system above.
Please note that
the final exam schedule is prescribed by the university; in general,
instructors are not permitted to change the final exam timing of their courses.
Graduate
students taking this class (Math 568) for four hours of credit will be required
to complete significant additional work. Performance on this additional work
will impact the overall grade of a graduate student.
This class
employs a zero-tolerance approach to cheating. In the event that a
student cheats (cheating includes, but is not limited to, giving or receiving
aid on an exam, or participating in the submission of an in-class assignment by
one student on behalf of another student) that student will be penalized in
accordance with the 2011-12 UIUC Student Code (particularly, Article I, Section
403, Penalties for Infractions of
Academic Integrity).
The
topic list for CAS/SOA Exam 4/C can be found at
http://www.beanactuary.org/exams/preliminary/exams/syllabi/2012-05-exam-c.pdf
Class Summaries
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Date |
General Topic |
Specific Class Topics |
Assignments |
Textbook Reading |
Other Items / Links |
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Jan. 17 |
Introduction
and motivation. |
Review of
syllabus. Motivation via
modeling examples |
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Ch. 1 |
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Jan. 19 |
Random
variables |
Conditional
probabilities and Bayes Theorem. Survival
functions. Hazard rates. |
Ch. 2 |
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Jan. 24 |
Statistical
quantities |
Moments. Truncation and
censoring. Deductibles
and limits. Per loss and
per payment. |
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Ch. 3.1 to 3.2 |
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Jan. 26 |
Losses and
distributions |
Limited
expected values. Mean excess
loss. Per loss and per
payment random variables. |
Ch. 8.1 to 8.3 |
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Jan. 31 |
Losses
and distributions (cont.) |
Allocating
losses to parties. Using limited
expected values. Loss
elimination ratio. |
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Ch.
8.3 to 8.5 |
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Feb. 2 |
Losses
and distributions (cont.) |
Parametric and
scale distributions. New
distributions from old. Mixing
distributions. Frequency
distributions. |
Ch.
4 Ch.
5.1 to 5.2 Ch.
6.1 to 6.5 |
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Feb. 7 |
Aggregate loss
models |
Combining
frequency and severity. Convolutions. Moments of
aggregate distributions. |
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Ch. 9.1 to 9.5 |
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Feb. 9 |
Aggregate loss
models (cont.) |
Stop-loss
insurance. (a,b,0) and (a,b,1) distributions. |
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Feb. 14 |
Review for
Exam # 1 |
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Feb. 16 |
Exam # 1 |
114
David Kinley Hall. Regular
class time. 3x5-inch
notecard allowed. |
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Feb. 21 |
Empirical
models |
Handed back
and discussed exam. Severity
inflation and leveraged effect. Types of risks
in modeling. |
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Feb. 23 |
Empirical models (cont.) |
Quality of
estimators. Point and
interval estimation. Hypothesis
testing. |
Ch. 12 |
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Feb. 28 |
Empirical models (cont.) |
Complete vs incomplete data. Individual vs group data. Complete data
analytics. |
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Ch. 13 |
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Mar. 1 |
Empirical
models (cont.) |
Modified data
analytics. Kaplan Meier. Nelson Aalen. Kernel models. |
Ch. 14 |
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Mar. 6 |
Parameter
estimation |
Method of
moments. Percentile
matching. Maximum
likelihood. |
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Ch. 15 |
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Mar. 8 |
Parameter
estimation (cont.) |
MM and PM
problems. MLE with
modified data. |
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Ch. 15 |
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Mar. 13 |
Parameter
estimation (cont.) |
Asymptotic
variance. Information. |
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(Answers to be
provided in class on March 13.) |
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Mar. 15 |
Exam # 2 |
141
Wohlers Hall. Regular
class time. 3x5-inch
notecard allowed. |
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Mar. 27 |
Model
Selection |
Handed back
and discussed Exam # 2. Graphical
model evaluation techniques. |
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Ch. 16 |
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Mar. 29 |
Model
Selection (cont.) |
Hypothesis
tests: KS, AD, Chi square, likelihood ratio. |
Ch. 16 |
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Apr. 3 |
Model
Selection (cont.) |
Finish up hyp tests. Ranking
models. VaR and TVaR. |
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Ch. 16, Ch. 3 |
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Apr. 5 |
Credibility |
Motivation. Limited
fluctuation or classical credibility. |
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Ch. 20 |
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Apr. 10 |
Credibility
(cont.) |
Classical
(cont.). |
Class project
description and data have been distributed to teams. |
Ch. 20 |
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Apr. 12 |
Credibility
(cont.) |
Buhlmann credibility |
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Ch. 20 |
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Apr. 17 |
Credibility
(cont.) |
Buhlmann credibility
(cont.) |
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Ch. 20 |
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Apr. 19 |
Credibility
(cont.) |
Bayesian
credibility |
Ch. 20 |
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Apr. 24 |
Credibility (cont.) |
Bayesian
credibility (cont.) |
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Ch. 20 |
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Apr. 26 |
Credibility (cont.) |
Buhlmann vs Bayesian credibility. Buhlmann-Straub
credibility. |
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Ch. 20 |
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May 1 |
Credibility (cont.) |
Empirical
Bayes estimation. |
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Ch.
20 |
(Answers to be
provided in class on May 1.) |
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May 9 |
Final Exam |
213
Gregory Hall. 1:30
pm start time. 3x5-inch
notecard allowed. |
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