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Math 478 / 568:Actuarial Modeling
Professor Rick GorvettSpring 2015
Syllabus
• Office Hours: 3-4 pm Tuesdays, 3-4 pm Wednesdays, or by appointment
• Textbook: Klugman, Panjer, and Willmot, 4th edition
• Exam dates: 3 exams, per syllabus
• Grades: Exams, homeworks, project
Syllabus (cont.)
• Graduate Students: Do Math 478, plus an extra project– Project is potentially semester-long
• U/G Honors Students: Project alternatives will be handed out ~ half-way through the semester
Class Objectives
• Understand the mathematical foundations of actuarial modeling– Loss modeling
–Model selection and parameter estimation
– Credibility theory and simulation
• Appreciate this material in a multi-disciplinary context
• Learn Exam 4/C material
Loss Modeling• How do we represent the potential for
financial consequences of events?• Frequency × severity = aggregate loss• Statistical distributions– Frequency – e.g.,
• Poisson• Negative binomial
– Severity – e.g., • Lognormal• Exponential• Gamma• Pareto
Model Selection and Parameter Estimation
• How do we select amongst alternative models and parameters?
• How do we use empirical data to determine the characteristics of distributions?
• In what sense are some models and parameters “better” or “optimal” in a given situation?
Credibility Theory and Simulation
• Credibility theory– How do we “blend”:• Old and new data?• Group versus individual data?• E.g., { Z•New + (1-Z) •Old }
• Simulation– How do we use models to estimate the
impact of potential future scenarios?
Actuarial Science and Finance
• “Coaching is not rocket science.” - Theresa Grentz, former University of
Illinois Women’s Basketball Coach
• Are actuarial science and financial mathematics “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
Motivation
Two real-world examples
Example # 1
Space Shuttle Challenger Explosion
http://www.youtube.com/watch?v=AfnvFnzs91s
Facts Leading Up to Launch…
• 23 successful launches prior to January 28, 1986
• Previous launches at temperatures from 53°F to 81°F
• Challenger launch on morning of 1/28/86 was at 31°F – far below previous launches
Launch / O-Ring Information
• Launch vehicle configuration:– Orbiter– External fuel tank– Two solid rocket boosters, manufactured
by Morton Thiokol (MT)• Sections sealed with O-rings, whose
performance is sensitive to temperature
• But: MT’s recommendation stated that “Temperature data (are) not conclusive on predicting primary O-ring blowby.”
The Result• Vehicle exploded 73 seconds after launch• Cause (per Rogers Commission): gas leak
in SRB, caused by failure or degredation of O-ring, led to weakening or penetration of external fuel tank
• Rogers Commission conclusion: “A careful analysis of the flight history of O-ring performance would have revealed the correlation of O-ring damage in low temperature.”
Statistical Analysis
• How predictable was it?• Data:
Statistical Analysis (cont.)
• Or:
Charts from “Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure,” by Dalal, et al, Journal of the American Statistical Association, December 1989
Example # 2
Taco Belland
The Mir Space Station
Taco Bell and Mir
• Space Station Mir– In orbit for 15 years– Expected to crash back to earth on March 24,
2001, in the Pacific Ocean– Size of projected debris field: 200 km × 6,000
km
• Taco Bell– Offered a free Crunchy Beef Taco to every U.S.
resident if the core of Mir hit a 144 square-meter target 15 km off Australian coast
Taco Bell and Mir (cont.)
• Suppose you are an actuary, working for an insurance firm
• Your firm has been approached by Taco Bell to insure against the potential financial loss associated with their possible Mir-related payout
• What’s a reasonable price for such coverage?
Taco Bell and Mir (cont.)
• Aggregate loss = frequency times severity
• What is the probability of Mir hitting the target?
• What will it cost Taco Bell if it does?
Taco Bell and Mir (cont.)
• Potential cost:– Population of United States?
– Cost of a Crunchy Beef Taco?
– Potential cost?
• Probability of a hit?
• Indicated premium?
Taco Bell and Mir (cont.)
• Issues:– Uniform distribution across debris field?
– How many will cash in?
–What about expenses / fixed costs?
Next Time
• Random variables– Conditional probabilities and Bayes
Theorem.– Survival functions.– Hazard rates.
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