University of Texas, Austin Course Syllabus
Department of Mathematics
M 339J (56140): Probability Models with Actuarial
Applications - Spring 2011
I. Course Title: M 339J Probability Models with Actuarial Applications (Unique 56140)
II. Location, Days and Time: Ernest Cockrell Jr. Hall room 1.204 (ECJ 1.204)
Whiteboard, document camera, seven rows of stadium seating, 75 seats.
Tuesdays and Thursdays 8:00 am – 9:15 am
III. Faculty: Mark M. Maxwell, PhD, ASA
Clinical Professor of Mathematics
Paul V. Montgomery Fellow of Mathematics
Program Director of Actuarial Studies
Editor of the E&R section of the SOA newsletter Expanding Horizons
Office: RLM 11.168
Office Hours: Tuesday and Thursday 9:30 am – 11:00 am
Friday 9:00 am – 11:00 am
Additional hours available by appointment or by chance
E-mail: [email protected]
Telephone: (512) 471-7169 – Work
(412) 716-5528 – Cellular
IV. Grader or Teaching Assistant: None
V. Prerequisites: M 362K (Probability) with a grade of ‘C-’ or better and at least one of
M 358K (Applied Statistics) or M 378K (Mathematical Statistics) with a grade of ‘C-’ or
better.
VI. Description of the Course: Introductory actuarial models for life insurance, property
insurance, and annuities. With M 349P (Actuarial Statistical Estimates), this course
covers the syllabus for the professional actuarial exam on model construction - SOA
Exam C / CAS Exam 4. This class may be counted toward the quantitative reasoning flag
requirement. Offered every spring semester only. This is a 3-credit course.
VII. Course Objectives: An introduction to modeling and important actuarial methods
useful in modeling. A thorough knowledge of calculus, probability, and mathematical
statistics is assumed. Students will be introduced to useful frequency and severity models
beyond those covered in SOA Exam MLC. The student will be required to understand
the steps involved in the modeling process and how to carry out these steps in solving
business problems. Students should be able to: 1) analyze data from an application in a
business context; 2) determine a suitable model including parameter values; and 3)
provide measures of confidence for decisions based upon the model. Students will be
introduced to a variety of tools for the calibration and evaluation of the models.
Page -1-
VIII. Learning Outcomes: Students will become familiar with survival, severity,
frequency and aggregate models, and use statistical methods to estimate parameters of
such models given sample data. Students will be able to identify steps in the modeling
process, understand the underlying assumptions implicit in each family of models,
recognize which assumptions are applicable in a given business application, and
appropriately adjust the models for impact of insurance coverage modifications.
Specifically, after taking both M 339J AND M 349P, students will be expected to
perform the tasks listed below:
A. Severity Models
1. Calculate the basic distributional quantities:
a) Moments
b) Percentiles
c) Generating functions
2. Describe how changes in parameters affect the distribution.
3. Recognize classes of distributions and their relationships.
4. Apply the following techniques for creating new families of distributions:
a) Multiplication by a constant
b) Raising to a power
c) Exponentiation
d) Mixing
5. Identify the applications in which each distribution is used and reasons
why.
6. Apply the distribution to an application, given the parameters.
7. Calculate various measures of tail weight and interpret the results to
compare the tail weights.
B. Frequency Models: For the Poisson, Mixed Poisson, Binomial, Negative
Binomial, Geometric distribution and mixtures thereof:
1. Describe how changes in parameters affect the distribution.
2. Calculate moments.
3. Identify the applications for which each distribution is used and reasons
why.
4. Apply the distribution to an application given the parameters.
5. Apply the zero-truncated or zero-modified distribution to an application
given the parameters.
C. Aggregate Models
1. Compute relevant parameters and statistics for collective risk models.
2. Evaluate compound models for aggregate claims.
3. Compute aggregate claims distributions.
D. For severity, frequency and aggregate models
1. Evaluate the impacts of coverage modifications:
a) Deductibles
b) Limits
c) Coinsurance
2. Calculate Loss Elimination Ratios.
3. Evaluate effects of inflation on losses.
E. Risk Measures
1. Calculate VaR, and TVaR and explain their use and limitations.
Page -2-
F. Construction of Empirical Models
1. Estimate failure time and loss distributions using:
a) Kaplan-Meier estimator, including approximations for large data sets
b) Nelson-Åalen estimator
c) Kernel density estimators
2. Estimate the variance of estimators and confidence intervals for failure time
and loss distributions.
3. Apply the following concepts in estimating failure time and loss
distribution:
a) Unbiasedness
b) Consistency
c) Mean squared error
G. Construction and Selection of Parametric Models
1. Estimate the parameters of failure time and loss distributions using:
a) Maximum likelihood
b) Method of moments
c) Percentile matching
d) Bayesian procedures
2. Estimate the parameters of failure time and loss distributions with censored
and/or truncated data using maximum likelihood.
3. Estimate the variance of estimators and the confidence intervals for the
parameters and functions of parameters of failure time and loss distributions.
4. Apply the following concepts in estimating failure time and loss
distributions:
a) Unbiasedness
b) Asymptotic unbiasedness
c) Consistency
d) Mean squared error
e) Uniform minimum variance estimator
5. Determine the acceptability of a fitted model and/or compare models using:
a) Graphical procedures
b) Kolmogorov-Smirnov test
c) Anderson-Darling test
d) Chi-square goodness-of-fit test
e) Likelihood ratio test
f) Schwarz Bayesian Criterion
H. Credibility
1. Apply limited fluctuation (classical) credibility including criteria for both
full and partial credibility.
2. Perform Bayesian analysis using both discrete and continuous models.
3. Apply Bühlmann and Bühlmann-Straub models and understand the
relationship of these to the Bayesian model.
4. Apply conjugate priors in Bayesian analysis and in particular the Poissongamma model.
5. Apply empirical Bayesian methods in the nonparametric and semiparametric cases.
I. Simulation
Page -3-
1. Simulate both discrete and continuous random variables using the inversion
method.
2. Estimate the number of simulations needed to obtain an estimate with a
given error and a given degree of confidence.
3. Use simulation to determine the p-value for a hypothesis test.
4. Use the bootstrap method to estimate the mean squared error of an
estimator.
5. Apply simulation methods within the context of actuarial models.
IX. Instructional Materials:
A. Textbook: Loss Models: From Data to Decisions, (Third Edition), 2008, by
Klugman, S.A., Panjer, H.H. and Willmot, G.E., ISBN 978-0-470-18781-4.
B. Calculator: Currently the Society of Actuaries (SOA) approves the following
calculators: Texas Instruments BA-35, BA II plus, BA II plus Professional, 30X,
and/or 30Xa. It is my strongest recommendation that you donate your graphing
utility to charity and rely on the TI BA II plus professional calculator as your
only calculator.
C. Other Study Materials: The Actex Study Manual or the CSM Study Manual
are available at www.actexmadriver.com. These are optional. Get with some
peers and obtain as many practice problems as you are able.
D. Study Notes Available from the Society of Actuaries: See www.soa.org.
http://soa.org/files/pdf/edu-2010-spring-exam-c.pdf
http://www.soa.org/education/exam-req/syllabus-study-materials/edu-multiplechoice-exam.aspx
E. Other Resources: Tables for Exam C/Exam 4
http://www.soa.org/files/pdf/edu-2009-fall-exam-c-table.pdf
X. Delivery System: This is your class. The responsibility of learning the course
objectives (section VI.) and attaining your learning outcomes is entirely your
responsibility. I imagine the first 15 – 30 minutes of each class being devoted to
reviewing assigned homework and 45 minutes of presentation on new content.
Classes typically begin by answering homework questions posed by the students.
Maxwell Presentations: My plan is to provide a fairly traditional lectureoriented class and presenting course material at least 75% of the time. I will
provide opportunities for students to take more ownership of being exposed to
actuarial model content.
B. Student Presentations: Students (individually or in a group) wishing to
present material to the class may be allowed up to 25% of class lecture time.
Such individuals will be required to meet with me 2+ days prior to the class
presentation. Presenting, or not, will have no direct impact on your course
grade. Presenters will have the opportunity to practice public-speaking
(employers value this), to have additional access to me (for whatever that is
worth), to have more investment in course content, and have the ability to
demonstrate personal responsibility and initiative.
A.
Page -4-
XI. Instructor Specific Course Policies:
A.
Make-up work: Make-up work is a rare event. If you must miss a
scheduled exam, you must make alternative accommodations with me (typically
taking the exam before it is scheduled). You need to expect at most one
opportunity to complete missed work, ever.
B.
Cheating: It is bad, do not do it. Cheating during the final examination
will result in a course grade of ‘F’ and being placed on double-secret probation in
perpetuity.
C. Class Distractions: You will make the necessary arrangements so that cell
phones, pagers, watch alarms, mechanical erasers and the like do not disturb class.
D. Learning Situations Outside of Class: Following presentations in class is a
good start to understanding, being able to complete problems on your own shows
a higher level of awareness, and being able to explain solutions to others
demonstrates exceptional insight. Therefore, you are encouraged to form study
groups. I am available during class, during scheduled office hours, and by
appointment. I hope that you feel comfortable receiving help from me. I look
forward to helping those motivated students who have attempted their homework.
It is ineffective to learn a large amount of mathematics in a short period of time.
If you are having difficulty, see me immediately. Note: I will not re-present class
content that you miss. You are entirely responsible for your actions.
E. Extra Credit: None. Extra work is not a substitute to learning the material in
a timely fashion. It is inappropriate for you to request extra credit work.
F. Professionalism: Students are expected to maintain appropriate behavior in the
classroom and other activities that reflect the actuarial program and university.
G. Course Philosophy: Expectations, execution, no excuses, no exceptions. –
Tony Dungy.
XII: University Policies and Services
A. Students with Disabilities: The University of Texas at Austin provides upon
request appropriate academic accommodations for qualified students with
disabilities. For more information, contact the Office of the Dean of Students at
(512) 471-6259, 471-4641 TTY.
B. Policy on Academic Dishonesty: Students who violate university rules on
scholastic dishonesty are subject to disciplinary penalties, including the
possibility of failing in the course and/or dismissal from the University. For
Page -5-
further information, visit the Student Judicial Services web site at
www.utexas.edu/depts/dos/sjs/.
C. The UT Learning Center: Jester Center A332, (512) 471-3614.
D. Counseling and Mental Health Center
E. Computer Labs: RLM 8.118 and RLM 7.122.
XIII: Grading Information
A. Definition of Letter Grades:
A
B
C
D
F
Achievement of distinction with an unusual degree of intellectual
initiative. I would expect ‘A’ students to pass Exam C/4.
Superior work. Students earning a ‘B’ could pass C/4, but
I would think that they would have to prepare quite a bit more.
Average knowledge attainment. The Bob Beaves’ 2 things.
Unsatisfactory, but passing
Failing
B. Assessment During the Term: From the teacher - students will receive
feedback on their projects, while working in groups, during question and answer
periods, during office hours, and during competency examinations. From other
students - during study sessions and projects. From oneself – while working on
homework problems, in-class examinations, while discussing these concepts with
others, while presenting material to students, and on the comprehensive multiplechoice final examination.
C. Grade Factors: Your grade will be entirely determined by your scores
earned on homework quizzes, pop quizzes, in-class examinations, and any other
graded work. If you miss graded work, then you are responsible for the effect on
your grade. No other factors enter into determining/assigning your grade. Note that
students may be adversely affected by 25-point syllabus understanding penalties.
See section XIII. H.
D. Homework Notebook: As mentioned previously, my goal is expose topics of
life contingencies to University of Texas, Austin students. I trust it is our goal to
demonstrate content proficiency by obtaining a passing score on SOA Exam C /
CAS Exam 4. We consider the prompt and accurate completion of homework to be
the single most important factor in student learning. It is my expectation that
students study for this class (and the professional examination) as a model for future
study. All students are to keep (and bring to class) a homework notebook of all
assigned problems. You may choose to keep some notes, other exercises, sample
examinations, projects, etcetera with the study aid.
Assigned Problems: One of your goals should be to attempt and solve all
appropriate homework problems (from this text and elsewhere). If specific
exercises will be collected, they will be noted in class.
Page -6-
Scoring Rubric: Your homework notebook may be collected and graded at
random times throughout the term.
E. Final Examination: The comprehensive final examination will be designed in
consultation with the actuarial faculty and knowledgeable others. Your examination
will be scored and your grade assigned based upon the following rubric:
Assigned
Grade
93-100
90-92
87-89
83-86
80-82
77-79
70-76
60-69
0
Final Exam
Score
Faculty Prediction
90% confident that student
will pass SOA/CAS exam
now
50% chance to pass
SOA/CAS now, can
eventually pass
10% chance now, 75%
eventual
50% chance of eventually
passing
25% chance of eventually
passing
10% chance of eventually
passing
No chance, some
understanding
Minimal understanding
No understanding
Cheating on the final
Uses: Data will be kept, tracked, and compared to actual professional
examination results. These results will be used to modify/improve the course,
will be components in annual reports about the program, and will be included
in a faculty member’s promotion dossier.
F. Typical Point Scale and Examination Dates:
Examination 1 (February 10th)
100 points
th
Examination 2 (March 10 )
100 points
Examination 3 (April 21st)
100 points
Comprehensive Final Examination (May 17th)
200 points
Graded Homework (random)
25 points each
Homework Notebook
Up to 100 points
Projects (Random)
approximately 10 points each
Pop Quizzes
approximately 20 points each, up to 100 points total
Penalties:
Syllabus Understanding
-25 points for failure to understand this
contract
Late work (if allowed)
25% if complete within one day
50% complete within a week, but after a day
100% if complete after one week
Page -7-
G. Letter Grade Ranges: The following scale will be used to assign grades at the
end of the term. Be careful using this scale on any individually scored work. Some
examinations are easier (most students score substantially higher) than other
examinations. It is your job to maximize your total points.
[90%-100%]
[80%-90%)
[70%-80%)
[60%-70%)
[0%-60%)
A/A- range
B+/B/B- range
C+/C/C- range
D+/D/D- range
Failing
H. Syllabus Understanding Penalty: Students WILL be assessed a 25-point
syllabus understanding penalty for failure to understand this syllabus contract.
Some common examples are listed below in HOPE that you WILL NOT repeat.
1. Immaturity (e.g., acting like you are 5 years old). Examples include
pouting, crying, whining, feeling sorry for oneself, saying “It is not fair
that …” or “But it’s not my fault that …”
2. Not taking responsibility for your own actions:
a) If you miss a class, do NOT ask me for to provide material that
you missed including: homework assigned, representing material
to you, if there will be an unannounced pop quiz, etcetera.
b) Excuses. Common former excuses include: (1) the student is a
graduating senior, (2) the student is not a good test taker, (3) the
student has a plane ticket departing prior to a scheduled exam, (4)
the student will lose their scholarship, (5) the student has a job
lined-up, (6) the student missed class in order to attend a job
interview, 7) - ﾠ ), and etcetera ad infinitum.
c) Other: Your parent contacts me. Almost anything a student does
AFTER the final examination has been given. Student asks me to
believe something that I know to be false.
ﾠ
XIV. Homework: The following is a partial list exercises should be understood. Yes,
all of them, AND others
1/18
1/20
1/25
1/27
2/1
2/3
2/8
Read chapter 1, Chapter 2 exercises: 2.1, 2.3, 2.4, 2.5
Exercises 3.1, 3.3, 3.4, 3.6, 3.7, 3.11, 3.13, 3.14, 3.16, 3.17, 3.19, 3.20
Exercises 3.21, 3.22, 3.23 3.27, 3.29
Exercises 3.34, 3.35, 3.36, Sample Exam #87, #89
Exercises 4.3, 4.4, 4.7, 4.9
Exercises 5.1, 5.2, 5.3, 5.6, 5.8, 5.10, 5.19
Exercise 5.21 5.22
XV. Changes: This syllabus is subject to modification. Any changes will be announced
in class.
©-2011 M. M. Maxwell. This syllabus is for the use of spring 2011 University of Texas, Austin students enrolled in M 339J.
Page -8-
M 339J(56140) Spring 2011 Course Calendar
January 17
Rev. Martin
Luther King Jr.
holiday
18
Spring 2011 Classes Begin
Syllabus and 1st Day handout
Chapter 1 – Read on Own
2.1: Introduction to Random Variables
2.2: Key functions, 4 models
OH 9:30A-11:00A
19
25
26
3.3: Moment Generating
Functions
3.4: Tails of Distributions
(sections 3.4.1-3.4.6)
OH 9:30A-11:00A
21
OH 9:00A-11:00A
OH 9:30A-11:00A
24
31
20
3.1: Moments
3.2: Quantiles
Last day to add/drop
27
3.5: Measures of Risk
(sections 3.5.1-3.5.5)
28
OH 9:00A-11:00A
OH 9:30A-11:00A
February 1
2
12th day of class –
OH 9:30A-11:00A
7
8
3
5.1: Intro to Continuous Models
5.2: Creating New Distributions
(sections 5.2.1-5.2.7)
Not as important
Ungraded homework #1 due
OH 9:30A-11:00A
4.1: Intro to Actuarial Models
4.2: The Role of Parameters
(sections 4.2.1-4.2.5)
9
5.3: Selected Distributions and Their
Relationships
5.4: Linear Exponential Family
Review for Examination 1
OH 9:30A-11:00A
4
OH 9:00A-11:00A
10
Examination 1 – Chapters 1-5
11
OH 9:00A-11:00A
OH 9:30A-11:00A
14
15
16
6.1: Intro to Discrete Distributions
and Processes
6.2: The Poisson Distribution
OH 9:30A-11:00A
17
6.3: The Negative Binomial Distribution
6.4: The Binomial Distribution
18
OH 9:00A-11:00A
OH 9:30A-11:00A
21
22
23
6.5: The (a,b,0) class
24
25
6.7: Truncation and Modification at Zero
OH 9:00A-11:00A
OH 9:30A-11:00A
28
OH 9:30A-11:00A
March 1
2
4
OH 9:00A-11:00A
Ungraded homework #2 due
OH 9:30A-11:00A
7
3
8.4: Policy Limits
8.5: Coinsurance, Deductibles, and Limits
8.6: Impact of Deductibles on Claim
Frequency
8.1: Intro to Frequency and Severity
With Coverage Modifications
8.2: Deductibles
8.3: Loss Elimination Ratio
OH 9:30A-11:00A
8
9
9.1: Introduction to Aggregate Loss
9.2: Models Choices
9.3: Compound Model for Aggregate Claims
10
Examination 2 – Chapters 6, 8, and 9.1-3
11
No office hours
OH 9:30A-11:00A
OH 9:30A-11:00A
14
15
16
17
18
21
22
23
24
25
9.4: Analytic Results
9.5: Computing the Aggregate Claims
Distribution
OH 9:30A-11:00A
OH 9:30A-11:00A
28
29
30
9.6: The Recursive Method
(sections 9.6.2-9.6.6)
9.7: Impact of Individual Policy
Modifications on Aggregate Payments
Last day to withdraw
OH 9:00A-11:00A
31
9.11: Individual Risk Model
(sections 9.11.1-9.11.4)
9.12: TVaR for Aggregate Losses
(sections 9.12.1-9.12.5)
April 1
OH 9:00A-11:00A
Good Friday
OH 9:30A-11:00A
4
OH 9:30A-11:00A
5
6
Chapter 12: Review of Mathematical
Statistics (sections 12.1-12.4)
Chapter 13: Estimation for Complete Data
(sections 13.1-13.3)
OH 9:30A-11:00A
11
18
13
19
20
21
OH 9:30A-11:00A
26
27
22
OH 9:00A-11:00A
28
15.2: Maximum Likelihood Estimation
(sections 15.2.4-15.2.5)
OH 9:30A-11:00A
3
4
15.5: Bayesian Estimation
(sections 15.5.1-15.5.3)
OH 9:30A-11:00A
9
15
OH 9:00A-11:00A
Examination 3 – Chapters 9, 12, 13, and 14
15.2: Maximum Likelihood Estimation
(sections 15.2.1-15.2.3)
OH 9:30A-11:00A
May 2
14
14.3: Kernal Density Models
14.4: Approximations for Large Data Sets
(sections 14.4.1-14.4.3)
Ungraded homework #3 due
OH 9:30A-11:00A
Academic advising
for summer/fall
April 13-15,18-22
15.1: Method of Moments and Percentile Match
Review for Examination 3
OH 9:30A-11:00A
25
8
OH 9:00A-11:00A
OH 9:30A-11:00A
12
14.1; Point Estimation
14.2: Means, Variances, and Interval Estimation
OH 9:30A-11:00A
Registration for summer
and fall semesters 4/18-4/29
7
5
15.5: Bayesian Estimation
(sections 15.5.4-15.5.5)
Review for Final
OH 9:30A-11:00A
10
29
OH 9:00A-11:00A
11
6
OH 9:00A-11:00A
Last Day of Class
12
13
19
20
M339V=M389V 2:00P-5:00P
16
17
M339J Multiple Choice Final
9:00A-noon
18