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
•
•
•
•
•
•
•
•
•
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
•
•
•
•
•
•
•
•
•
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 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
•
•
•
•
•
•
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
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
35.4
Values in Hundreds
42.5
49.7
Year 2004 Surplus Distribution
Original Assumptions
0.25
0.15
0.1
0.05
Millions
30
9.2
27
5.0
24
0.8
20
6.6
17
2.4
13
8.2
10
3.9
69
.7
35
.5
1.3
0
-32
.9
Probability
0.2
Year 2004 Surplus Distribution
Constrained Growth Assumptions
0.25
0.15
0.1
0.05
Millions
33
4.8
30
8.1
28
1.4
25
4.7
22
8.0
20
1.3
17
4.6
14
7.8
12
1.1
94
.4
0
67
.7
Probability
0.2
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
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:
–
–
–
–
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/