Transcript Casualty Actuarial Society Seminar on Dynamic Financial

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
• Society of Actuaries:
– Fellows:
– Associates:
– Total:
8,990
7,411
16,401
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
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•
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•
•
•
•
•
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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:
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
CAS Exams -- Advanced Topics
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•
•
•
•
•
•
•
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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
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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 finance-related
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
Tax
Interest Rate
Generator
Investment
& Economic
Inputs
Investment
Generator
Investment
Cashflows
Outputs
& Simulation
Results
Key Variables
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•
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Financial
Short-Term Interest Rate
Term Structure
Default Premiums
Equity Premium
Inflation
Mortgage Pre-Payment
Patterns
Underwriting
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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
P R O B A B IL IT Y
Distribution for SURPLUS /
Ending/I115
0.16
0.13
0.10
0.06
0.03
0.00
6.8
13.9
21.1
28.2
Values in Hundreds
35.4
42.5
49.7
Year 2004 Surplus Distribution
Original Assumptions
0.25
0.15
0.1
0.05
Millions
.2
30
9
.0
27
5
.8
24
0
.6
20
6
.4
17
2
.2
13
8
.9
10
3
.7
69
.5
35
1.3
0
-32
.9
Probability
0.2
Year 2004 Surplus Distribution
Constrained Growth Assumptions
0.25
0.15
0.1
0.05
Millions
.8
33
4
.1
30
8
.4
28
1
.7
25
4
.0
22
8
.3
20
1
.6
17
4
.8
14
7
.1
12
1
.4
94
.7
0
67
Probability
0.2
Model Uses
Internal
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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
Markov Chain
Monte Carlo
Stochastic Calculus /
Ito’s Lemma
Financial Mathematics
Theory
of Risk
Interest
Theory
Chaos Theory /
Fractals
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:
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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/