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

Actuarial Science and Financial Mathematics: Doing 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. Presentation Agenda. Actuaries -- who (or what) are they?

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

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  1. Actuarial Science andFinancial Mathematics:Doing 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

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

  3. 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

  4. 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

  5. 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

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

  7. “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

  8. U of I Actuarial Science Program:Math 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

  9. U of I Actuarial Science Program:Other 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

  10. 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

  11. 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

  12. 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

  13. Actuarial Science and Finance • “Coaching is not rocket science.” - Theresa Grentz, University of Illinois Women’s 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….

  14. Actuarial Science and Finance (cont.) • Actuarial science and finance are not rocket science -- they’re harder • Rocket science: • Test a theory or design • Learn and re-test until successful • Actuarial science and finance • Things continually change -- behaviors, attitudes,…. • Can’t hold other variables constant • Limited data with which to test theories

  15. 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

  16. Dynamic Financial Analysis • Dynamic • Stochastic or variable • Reflect uncertainty in future outcomes • Financial • Integration of insurance and financial operations and markets • Analysis • Examination of system’s interrelationships

  17. DynaMo (at www.mhlconsult.com) Catastrophe Generator U/W Inputs U/W Cashflows U/W Generator Payment Patterns U/W Cycle Tax Investment Cashflows Interest Rate Generator Investment & Economic Inputs Investment Generator Outputs & Simulation Results

  18. 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 Key Variables

  19. Sample DFA Model Output

  20. Year 2004 Surplus DistributionOriginal Assumptions

  21. Year 2004 Surplus Distribution Constrained Growth Assumptions

  22. Internal Strategic Planning Ratemaking Reinsurance Valuation / M&A Market Simulation and Competitive Analysis Asset / Liability Management External External Ratings Communication with Financial Markets Regulatory / Risk-Based Capital Capital Planning / Securitization Model Uses

  23. Recent Areas of Actuarial Research • Financial mathematics • Stochastic calculus • Fuzzy set theory • Markov chain Monte Carlo • Neural networks • Chaos theory / fractals

  24. The Actuarial ScienceResearch Triangle Mathematics Fuzzy Set Theory Stochastic Calculus / Ito’s Lemma 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

  25. 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

  26. Financial Mathematics (cont.) Asset-Liability Management Duration D = -(dP / dr) / P Convexity C = d2P / dr2 Price-Yield Curve P r

  27. Stochastic Calculus Brownian motion (Wiener process) Dz = e (Dt)0.5 z(t) - z(s) ~ N(0, t-s)

  28. Stochastic Calculus (cont.) Ito’s 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

  29. Stochastic Calculus (cont.) Black-Scholes(-Merton) Formula VC = S N(d1) - X e-rt N(d2) d1 = [ln(S/X)+(r+0.5s2)t] / st0.5 d2 = d1 - st0.5

  30. 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

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

  32. 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), sA2+sL2-2sAsL rAL )

  33. 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

  34. 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 ==> [0,1] • Young (1996, 1997): pricing (WC, health) • Cummins & Derrig (1997): pricing • Horgby (1998): risk classification (life)

  35. 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

  36. 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

  37. 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”

  38. 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

  39. 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/

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