Transcript Nonlinear Dynamics and Complex Systems

Stochastic Modeling:
A Research Experience for
Undergraduates
Rick Gorvett, FCAS, MAAA, ARM, FRM, Ph.D.
Director, Actuarial Science Program
State Farm Companies Foundation Scholar in Actuarial Science
University of Illinois at Urbana-Champaign
Actuarial Research Conference
RMU, PA
August 2007
Agenda
• Context and motivation
• U/G research opportunities at the University of
Illinois
• REU background
• Summer 2007 REU at Illinois
• Mentoring issues
• Conclusions: What I learned
• Discussion: Moving forward
Context and Motivation
• Yesterday’s “Applied Research” panel
discussion
– “Passion” for research
 Instill in act sci undergraduates
– Next problem is too interesting to spend time
writing up the last one
 U/G exposure to the research process 
relatively painless
Context and Motivation (cont.)
• If most act sci practitioners stop with a B.S. or
B.A. degree…
 U/G research is generally the only pre-career
opportunity for exposure to research for actuaries
• The actuarial profession will benefit from
pushing some level of research into the
classroom
University of Illinois
Undergraduate Research Opportunities
• In-class research
– E.g., “cocktail party papers”
• Actuarial call paper programs
• State Farm Research Center
• National Science Foundation (NSF) Research
Experiences for Undergraduates (REU)
NSF REU Program Description
• “The Research Experiences for Undergraduates (REU)
program supports active research participation by
undergraduate students… in ongoing research programs
or in research projects specifically designed for the REU
program… and to help ensure that they receive the best
education possible. ”
• “Research experience is one of the most effective
avenues for attracting talented undergraduates to, and
retaining them in careers in, science and engineering,
including careers in teaching and education research…
projects are also encouraged, when appropriate, to
involve students at earlier stages in their college
experience.”
UIUC REU Program
• “The University of Illinois at Urbana-Champaign
will host two 8 week summer programs, for 6
students each over the next 5 years.”
• Past topics:
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Evolutionary Game Theory
Geometric Group Theory
Number Theory
Harmonic Analysis
Visualization in Virtual Environments
UIUC REU Program (cont.)
• “Our students join the working environments of
faculty mentors, their postdocs, graduate students
and collaborators. Students work on individual or
small group projects in their mentor's current
research area.”
• Aspects of REU offerings
– Tutorials in relevant math areas, LaTeX, computer
packages.
– Topic development and literature searching
– Communication skills (writing, presenting)
Stochastic Modeling REU
• Stochastic Modeling in Actuarial Science and
Financial Mathematics
• Phase 1 – classroom instruction in:
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Probability and statistics
Markov chains and Poisson processes
Brownian motion
Simulation
Ruin theory
Interest rate modeling
Options and option pricing
Stochastic Modeling REU (cont.)
• Phase 2 – assignments for later discussion:
– The (Mis)behavior of Markets: A Fractal View of
Risk, Ruin, and Reward by Mandelbrot
– Various recent research papers
– TeX / LaTeX and Excel
– Data sources and modeling software
• Phase 3 – exposure to research:
– Development of a research topic
– Preparation of paper and presentation
Universities Represented by
the Selected Participants
• Baylor University
• Brown University
• Russell Sage College
• University of Illinois at Urbana-Champaign
• Washington University (St. Louis)
• Wheaton College (IL)
“A Model for Increasing Black Life
Expectancy by Reducing Infant
Mortality, Homicide, and HIV”
• A historical discrepancy of life expectancies between
white and black Americans
• Three top causes: infant mortality, homicide among
black males ages 14-24, and HIV deaths among blacks
males and females ages 45-64
• Model to examine potential impact of reductions in
these factors
• Model could serve as a basis for evaluation and costbenefit analysis of possible future public policy
initiatives
“A Markov Chain Approach to
Modeling Cancer Risk”
• Considers the conceptual and motivational
foundations for mathematical models of cancer
risk
• Develops and analyzes a Markov chain model
for the potential emergence of cancerous cells
• Model is based on the gene mutation theory of
cancer progression
• Reflects the likelihood over time that a cell will
mutate and become malignant
Nine-State
Model
Nine State Markov Chain
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“A Regime-Switching Model for
Foreign Exchange Rates”
• Develops a regime-switching model for foreign
exchange rates
• Model classifies FX rate movements according to low,
medium, and high volatility regimes
• Discrete Markov chain framework for modeling the
monthly likelihoods of regime movements
• Applied to 14 years of historical monthly U.S./U.K.
and U.S./Australia FX rates
• Simulations of one-step-ahead predictions compared
between regime-switching model and a simple AR(1)
model
“On the Statistical Properties of
Interest Rates and a Basic RegimeSwitching Model”
• Update of Becker (1991), “Statistical Basis of
the Lognormal Distribution as a Basis for
Interest Rate Changes,” Transactions of Society
of Actuaries 43
• Examines properties of the lognormal
assumption as applied to interest rate
movements
• Develops a regime-switching model of interest
rates
“Stochastic Modeling for Illinois
Prepaid Tuition Contracts”
• Economic Growth and Tax Relief Reconciliation Act
of 2001, Section 529 plans
• Examines the Illinois prepaid college tuition program,
CollegeIllinois!
• Several stochastic models are developed for
simulating, via Monte Carlo techniques, potential
future tuition inflation and trust fund investment
returns
• Evaluates the impact on trust fund solvency of these
stochastic financial variables
“A Fractal-Geometric Approach to
Risk Management and Mitigation”
• Examines the Multifractal Model of Asset
Returns (MMAR) framework
• Applies MMAR to financial time series,
particularly the S&P 500 index
• Considers philosophical issues associated with
MMAR
Mentoring Issues
• Mentor-suggested versus student-developed
research topics
• How hard to “push”
• High impact: discussing broad perspective and
context
• High impact: economic / financial details
(perhaps even more than mathematical /
statistical)
Conclusions: What I Learned
• Encourage “interdisciplinary” learning and
investigation
• Push students to identify topic early, and use interim
deadlines extensively
• Students are different from one another
– Researchers versus businesspeople
– Participant selection process
– Research topic / project
• Size and breadth of the REU topic are critical
• The best U/Gs are extremely capable
Note:
www.math.uiuc.edu/~gorvett/
Discussion
• How can we incorporate research into an
undergraduate actuarial science curriculum?
– Increase passion for research
– Promote a way of thinking
• What constitutes a reasonable research topic /
project for undergraduates?
• What is an appropriate level of mentoring?
• What efforts are currently underway?