actuarial science and financial mathematics: doing integrals for fun and profit
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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? - PowerPoint PPT PresentationTRANSCRIPT
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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
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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 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
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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
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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 JournalJuly 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
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….
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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
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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 system’s interrelationships
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U/WInputs
Investment& Economic
Inputs
U/W GeneratorPayment Patterns
U/W Cycle
CatastropheGenerator
InvestmentGenerator
U/WCashflows
InvestmentCashflows
TaxOutputs
& SimulationResults
DynaMo (at www.mhlconsult.com)
Interest RateGenerator
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Key VariablesFinancial
• 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
Distribution for SURPLUS /Ending/I115
PR
OB
AB
ILIT
Y
Values in Hundreds
0.00
0.03
0.06
0.10
0.13
0.16
6.8 13.9 21.1 28.2 35.4 42.5 49.7
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Year 2004 Surplus DistributionOriginal Assumptions
0
0.05
0.1
0.15
0.2
0.25-3
2.9
1.3
35.5
69.7
103.
913
8.2
172.
420
6.6
240.
827
5.0
309.
2
Millions
Pro
babi
lity
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Year 2004 Surplus Distribution Constrained Growth Assumptions
0
0.05
0.1
0.15
0.2
0.2567
.7
94.4
121.
114
7.8
174.
620
1.3
228.
025
4.7
281.
430
8.1
334.
8
Millions
Pro
babi
lity
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Model Uses
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
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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
ActuarialScience
Finance
Stochastic Calculus /Ito’s Lemma
Financial Mathematics
PortfolioTheory
ContingentClaimsAnalysis
Fuzzy SetTheory
Markov ChainMonte Carlo
Chaos Theory /Fractals
Theoryof Risk
DynamicFinancialAnalysis
InterestRateModeling
InterestTheory
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Financial Mathematics
Interest Rate Generator
Cox-Ingersoll-Ross One-Factor Model
dr = (-r) dt + r0.5 dZ
r = short-term interest rate = speed of reversion of process to long-run mean = long-run mean interest rate = volatility of processZ = standard Wiener process
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Financial Mathematics (cont.)
Asset-Liability Management
Duration
D = -(P / r) / P
Convexity
C = P / r2
r
P
Price-YieldCurve
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Stochastic Calculus
Brownian motion (Wiener process)
z = (t)0.5
z(t) - z(s) ~ N(0, t-s)
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Stochastic Calculus (cont.)
Ito’s Lemma
Let dx = a(x,t) + b(x,t)dz
Then, F(x,t) follows the process
dF = [a(F/x) + (F/t) + 0.5b2(2F/x2)]dt + b(F/x)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)+(r+0.52)t] /t0.5
d2 = d1 - t0.5
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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 = Adt + AdzA
dL/L = Ldt + LdzL
Correlation = AL
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Stochastic Calculus (cont.)
Mathematical DFA Model (cont.)
• In a risk-neutral valuation framework, the interest rate cancels, and x=A/L follows:
dx/x = xdt + xdzx
where
x = L2 - AL AL
x2 = A
2 + L2 - 2AL AL
dzx = (AdzA - LdzL ) / x
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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))+x-(x2 /2), x
2 )
or
ln(x(t)) ~ N(ln(x(t-1))+(L2 /2)-(A
2 /2), A2+L
2-2AL
AL )
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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 ==> [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/