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Casualty Actuarial Society Seminar on Dynamic Financial Analysis

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Casualty Actuarial Society Seminar on Dynamic Financial Analysis Powered By Docstoc
					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? • Actuarial exams and our actuarial science courses

• Recent developments in
– Actuarial practice – Academic research

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

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

Membership Statistics (Nov., 2000)
• Casualty Actuarial Society:
– Fellows: – Associates: – Total: 2,061 1,377 3,438 8,990 7,411 16,401

• Society of Actuaries:
– Fellows: – Associates: – Total:

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

“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

U of I Actuarial Science Program: Math Courses Beyond Calculus
• • • • • • • • • Math 210: Math 309: Math 361: Math 369: Math 371: Math 372: Math 376: Math 377: Math 378: Interest theory Actuarial statistics Probability theory Applied statistics Actuarial theory I Actuarial theory II Risk theory Survival analysis Actuarial modeling
Exam # 2 Various 1 4 3 3 3 4 3 and 4

U of I Actuarial Science Program: Other Useful Courses
• Math 270: Review for exams # 1 and 2

• Math 351:
• Math 351: • Fin 260: • Fin 321: • Fin 343:

Financial Mathematics
Actuarial Capstone course Principles of insurance Advanced corporate finance Financial risk management

• Econ 102 / 300: Microeconomics • Econ 103 / 301: Macroeconomics

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

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

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

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 financerelated work • But….

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

Recent Developments in Actuarial 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

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

DynaMo (at www.mhlconsult.com)
Catastrophe Generator

U/W Inputs

U/W Generator Payment Patterns U/W Cycle

U/W Cashflows Outputs & Simulation Results

Tax

Interest Rate Generator Investment & Economic Inputs Investment Generator

Investment Cashflows

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

Sample DFA Model Output
Distribution for SURPLUS / Ending/I115
P R O B A B IL IT Y
0.16 0.13 0.10 0.06 0.03 0.00 6.8 13.9 21.1 28.2 35.4 42.5 49.7

Values in Hundreds

Year 2004 Surplus Distribution
Original Assumptions
0.25 0.2

Probability

0.15 0.1 0.05 0

3.9

8.2

2.4

6.6

0.8

5.0 27

-32 .9

1.3

35

69

10

13

17

20

24

Millions

30

9.2

.5

.7

Year 2004 Surplus Distribution
Constrained Growth Assumptions
0.25 0.2

Probability

0.15 0.1 0.05 0

1.1

7.8

4.6

1.3

8.0

4.7

1.4

8.1 30

67

94

12

14

17

20

22

25

28

Millions

33

4.8

.7

.4

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 / RiskBased Capital • Capital Planning / Securitization

Recent Areas of Actuarial Research
• Financial mathematics • Stochastic calculus • Fuzzy set theory

• Markov chain Monte Carlo
• Neural networks

• Chaos theory / fractals

The Actuarial Science Research Triangle
Mathematics
Fuzzy Set Theory Stochastic Calculus / Ito’s Lemma

Markov Chain Monte Carlo
Chaos Theory / Fractals

Financial Mathematics
Theory of Risk Interest Theory

Actuarial Science

Dynamic Financial Analysis Portfolio Theory

Interest Rate Modeling Contingent Claims Analysis

Finance

Financial Mathematics
Interest Rate Generator

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

Financial Mathematics (cont.)
Asset-Liability Management
P

Price-Yield Curve

Duration D = -(dP / dr) / P Convexity
r

C = d2P / dr2

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

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

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

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

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

Stochastic Calculus (cont.)
Mathematical DFA Model (cont.) Can now determine the distribution of the state variable x at the end of the continuoustime 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 )

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

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)

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

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

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”

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

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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