Stochastic Modeling:A Research Experience for

Undergraduates

Rick Gorvett, FCAS, MAAA, ARM, FRM, Ph.D.

Director, Actuarial Science ProgramState Farm Companies Foundation Scholar in Actuarial Science

University of Illinois at Urbana-Champaign

Actuarial Research ConferenceRMU, 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 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)

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:– 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:

– 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 bythe 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 cost-benefit 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 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

“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?