Actuarial Science and Financial Mathematics: Doing Integrals for Fun and Profit

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Actuarial Science andFinancial
MathematicsDoing Integrals for Fun and Profit

• Rick Gorvett, FCAS, MAAA, ARM, Ph.D.
• Presentation to Math 400 Class
• Department of Mathematics
• University of Illinois at Urbana-Champaign
• March 5, 2001

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(No Transcript)
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Presentation Agenda

• Actuaries -- who (or what) are they?
• Actuarial exams and our actuarial science courses
• Recent developments in
• Actuarial practice
• Academic research

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What is an Actuary?The Technical Definition

• Someone with an actuarial designation
• Property / Casualty
• FCAS Fellow of the Casualty Actuarial Society
• ACAS Associate of the Casualty Actuarial
  Society
• Life
• FSA Fellow of the Society of Actuaries
• ASA Associate of the Society of Actuaries
• Other
• EA Enrolled Actuary
• MAAA Member, American Academy of Actuaries

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What is an Actuary?Better Definitions

• One who analyzes the current financial
  implications of future contingent events
• - p.1, Foundations of Casualty Actuarial
  Science
• Actuaries put a price tag on future risks. They
  have been called financial architects and social
  mathematicians, because their unique combination
  of analytical and business skills is helping to
  solve a growing variety of financial and social
  problems.
• - p.1, Actuaries Make a Difference

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Membership Statistics (Nov., 2000)

• Casualty Actuarial Society
• Fellows 2,061
• Associates 1,377
• Total 3,438
• Society of Actuaries
• Fellows 8,990
• Associates 7,411
• Total 16,401

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Casualty Actuaries

• Insurance companies 2,096
• Consultants 668
• Organizations serving insurance 102
• Government 76
• Brokers and agents 84
• Academic 16
• Other 177
• Retired 219

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Basic Actuarial Exams

• Course 1 Mathematical foundations of actuarial
  science
• Calculus, probability, and risk
• Course 2 Economics, finance, and interest
  theory
• Course 3 Actuarial models
• Life contingencies, loss distributions,
  stochastic processes, risk theory, simulation
• Course 4 Actuarial modeling
• Econometrics, credibility theory, model
  estimation, survival analysis

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U of I Actuarial Science ProgramMath Courses
Beyond Calculus

• Exam
• Math 210 Interest theory 2
• Math 309 Actuarial statistics Various
• Math 361 Probability theory 1
• Math 369 Applied statistics 4
• Math 371 Actuarial theory I 3
• Math 372 Actuarial theory II 3
• Math 376 Risk theory 3
• Math 377 Survival analysis 4
• Math 378 Actuarial modeling 3 and 4

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U of I Actuarial Science ProgramOther Useful
Courses

• Math 270 Review for exams 1 and 2
• Math 351 Financial Mathematics
• Math 351 Actuarial Capstone course
• Fin 260 Principles of insurance
• Fin 321 Advanced corporate finance
• Fin 343 Financial risk management
• Econ 102 / 300 Microeconomics
• Econ 103 / 301 Macroeconomics

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CAS Exams -- Advanced Topics

• Insurance policies and coverages
• Ratemaking
• Loss reserving
• Actuarial standards
• Insurance accounting
• Reinsurance
• Insurance law and regulation
• Finance and solvency
• Investments and financial analysis

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The Actuarial Profession

• Types of actuaries
• Property/casualty
• Life
• Pension
• Primary functions involve the financial
  implications of contingent events
• Price insurance policies (ratemaking)
• Set reserves (liabilities) for the future costs
  of current obligations (loss reserving)
• Determine appropriate classification structures
  for insurance policyholders
• Asset-liability management
• Financial analyses

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Table of Contents From a Recent Actuarial Journal

• North American Actuarial Journal
• July 1998
• Economic Valuation Models for Insurers
• New Salary Functions for Pension Valuations
• Representative Interest Rate Scenarios
• On a Class of Renewal Risk Processes
• Utility Functions From Risk Theory to Finance
• Pricing Perpetual Options for Jump Processes
• A Logical, Simple Method for Solving the Problem
  of Properly Indexing Social Security Benefits

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Actuarial Science and Finance

• Coaching is not rocket science.
• - Theresa Grentz, University of Illinois
  Womens Basketball Coach
• Are actuarial science and finance rocket science?
• Certainly, lots of quantitative Ph.D.s are on
  Wall Street and doing actuarial- or
  finance-related work
• But.

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Actuarial Science and Finance (cont.)

• Actuarial science and finance are not rocket
  science -- theyre harder
• Rocket science
• Test a theory or design
• Learn and re-test until successful
• Actuarial science and finance
• Things continually change -- behaviors,
  attitudes,.
• Cant hold other variables constant
• Limited data with which to test theories

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Recent Developments inActuarial Practice

• Risk and return
• Pricing insurance policies to formally reflect
  risk
• Insurance securitization
• Transfer of insurance risks to the capital
  markets by transforming insurance cash flows into
  tradable financial securities
• Dynamic financial analysis
• Holistic approach to modeling the interaction
  between insurance and financial operations

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Dynamic Financial Analysis

• Dynamic
• Stochastic or variable
• Reflect uncertainty in future outcomes
• Financial
• Integration of insurance and financial operations
  and markets
• Analysis
• Examination of systems interrelationships

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DynaMo (at www.mhlconsult.com)
Outputs Simulation Results
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Key Variables

• Financial
• Short-Term Interest Rate
• Term Structure
• Default Premiums
• Equity Premium
• Inflation
• Mortgage Pre-Payment Patterns

• Underwriting
• Loss Freq. / Sev.
• Rates and Exposures
• Expenses
• Underwriting Cycle
• Loss Reserve Dev.
• Jurisdictional Risk
• Aging Phenomenon
• Payment Patterns
• Catastrophes
• Reinsurance
• Taxes

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Sample DFA Model Output
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Year 2004 Surplus DistributionOriginal
Assumptions
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Year 2004 Surplus Distribution Constrained
Growth Assumptions
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Model Uses

• Internal
• Strategic Planning
• Ratemaking
• Reinsurance
• Valuation / MA
• Market Simulation and Competitive Analysis
• Asset / Liability Management

• External
• External Ratings
• Communication with Financial Markets
• Regulatory / Risk-Based Capital
• Capital Planning / Securitization

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Recent Areas of Actuarial Research

• Financial mathematics
• Stochastic calculus
• Fuzzy set theory
• Markov chain Monte Carlo
• Neural networks
• Chaos theory / fractals

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The Actuarial ScienceResearch Triangle
Mathematics
Stochastic Calculus / Itos Lemma
Fuzzy Set Theory
Markov Chain Monte Carlo
Financial Mathematics
Theory of Risk
Interest Theory
Chaos Theory / Fractals
Dynamic Financial Analysis
Interest Rate Modeling
Actuarial Science
Finance
Portfolio Theory
Contingent Claims Analysis
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Financial Mathematics

• Interest Rate Generator
• Cox-Ingersoll-Ross One-Factor Model
• dr a (b-r) dt s r0.5 dZ
• r short-term interest rate
• a speed of reversion of process to long-run
  mean
• b long-run mean interest rate
• s volatility of process
• Z standard Wiener process

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Financial Mathematics (cont.)

• Asset-Liability Management
• Duration
• D -(dP / dr) / P
• Convexity
• C d2P / dr2

Price-Yield Curve
P
r
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Stochastic Calculus

• Brownian motion (Wiener process)
• Dz e (Dt)0.5
• z(t) - z(s) N(0, t-s)

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Stochastic Calculus (cont.)

• Itos Lemma
• Let dx a(x,t) b(x,t)dz
• Then, F(x,t) follows the process
• dF a(dF/dx) (dF/dt) 0.5b2(d2F/dx2)dt
  b(dF/dx)dz

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Stochastic Calculus (cont.)

• Black-Scholes(-Merton) Formula
• VC S N(d1) - X e-rt N(d2)
• d1 ln(S/X)(r0.5s2)t / st0.5
• d2 d1 - st0.5

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Stochastic Calculus (cont.)

• Mathematical DFA Model
• Single state variable A / L ratio
• Assume that both assets and liabilities follow
  geometric Brownian motion processes
• dA/A mAdt sAdzA
• dL/L mLdt sLdzL
• Correlation rAL

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Stochastic Calculus (cont.)

• Mathematical DFA Model (cont.)
• In a risk-neutral valuation framework, the
  interest rate cancels, and xA/L follows
• dx/x mxdt sxdzx
• where
• mx sL2 - sAsL rAL
• sx2 sA2 sL2 - 2sAsL rAL
• dzx (sAdzA - sLdzL ) / sx

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Stochastic Calculus (cont.)

• Mathematical DFA Model (cont.)
• Can now determine the distribution of the state
  variable x at the end of the continuous-time
  segment
• ln(x(t)) N(ln(x(t-1))mx-(sx2 /2), sx2 )
• or
• ln(x(t)) N(ln(x(t-1))(sL2 /2)-(sA2 /2),
  sA2sL2-2sAsL rAL )

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Fuzzy Set Theory

• Insurance Problems
• Risk classification
• Acceptance decision, pricing decision
• Few versus many class dimensions
• Many factors are clear and crisp
• Pricing
• Class-dependent
• Incorporating company philosophy / subjective
  information

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Fuzzy Set Theory (cont.)

• A Possible Solution
• Provide a systematic, mathematical framework to
  reflect vague, linguistic criteria
• Instead of a Boolean-type bifurcation, assigns a
  membership function
• For fuzzy set A, mA(x) X gt 0,1
• Young (1996, 1997) pricing (WC, health)
• Cummins Derrig (1997) pricing
• Horgby (1998) risk classification (life)

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Markov Chain Monte Carlo

• Computer-based simulation technique
• Generates dependent sample paths from a
  distribution
• Transition matrix probabilities of moving from
  one state to another
• Actuarial uses
• Aggregate claims distribution
• Stochastic claims reserving
• Shifting risk parameters over time

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Neural Networks

• Artificial intelligence model
• Characteristics
• Pattern recognition / reconstruction ability
• Ability to learn
• Adapts to changing environment
• Resistance to input noise
• Brockett, et al (1994)
• Feed forward / back propagation
• Predictability of insurer insolvencies

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Chaos Theory / Fractals

• Non-linear dynamic systems
• Many economic and financial processes exhibit
  irregularities
• Volatility in markets
• Appears as jumps / outliers
• Or, market accelerates / decelerates
• Fractals and chaos theory may help us get a
  better handle on risk

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Conclusion

• A new actuarial science paradigm is evolving
• Advanced mathematics
• Financial sophistication
• There are significant opportunities for important
  research in these areas of convergence between
  actuarial science and mathematics

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Some Useful Web Pages

• Mine
• http//www.math.uiuc.edu/gorvett/
• Casualty Actuarial Society
• http//www.casact.org/
• Society of Actuaries
• http//www.soa.org/
• Be An Actuary
• http//www.beanactuary.org/