Stochastic Modeling: A Research Experience for Undergraduates

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Stochastic ModelingA 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

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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

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Context and Motivation

• Yesterdays 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

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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

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University of IllinoisUndergraduate 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)

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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.

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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
• Evolutionary Game Theory
• Geometric Group Theory
• Number Theory
• Harmonic Analysis
• Visualization in Virtual Environments

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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)

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Stochastic Modeling REU

• Stochastic Modeling in Actuarial Science and
  Financial Mathematics
• Phase 1 classroom instruction in
• Probability and statistics
• Markov chains and Poisson processes
• Brownian motion
• Simulation
• Ruin theory
• Interest rate modeling
• Options and option pricing

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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

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Universities Represented bythe Selected
Participants

• Baylor University
• Brown University
• Russell Sage College
• University of Illinois at Urbana-Champaign
• Washington University (St. Louis)
• Wheaton College (IL)

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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
  cost-benefit analysis of possible future public
  policy initiatives

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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

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Nine-State Model
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Nine State Markov Chain
x0
x1A
x1B
x2
y0
y1A
y1B
y2
w
x0
x1A
x1B
x2
y0
y1A
y1B
y2
w
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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

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On the Statistical Properties of Interest Rates
and a Basic Regime-Switching 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

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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

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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 SP 500 index
• Considers philosophical issues associated with
  MMAR

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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)

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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/

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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?