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Math 478 / 568: Actuarial Modeling

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Math 478 / 568:Actuarial Modeling

Professor Rick Gorvett

Spring 2014

Office Hours: 3-4 pm Tuesdays, 3-4 pm Wednesdays, or by appointment

Textbook: Klugman, Panjer, and Willmot, 4thedition

Exam dates: 3 exams, per syllabus

Grades: Exams, homeworks, project

- 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

- 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

- How do we represent the potential for events?
- Frequency × severity = aggregate loss
- Statistical distributions
- Frequency – e.g.,
- Poisson
- Negative binomial

- Severity – e.g.,
- Lognormal
- Exponential
- Gamma
- Pareto

- Frequency – e.g.,

How do we select amongst alternative models?

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
- How do we “blend”:
- Old and new data?
- Group versus individual data?
- E.g., { Z•New + (1-Z) •Old }

- How do we “blend”:
- Simulation
- How do we use models to estimate the impact of potential future scenarios?

- “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 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

Two real-world examples

http://www.youtube.com/watch?v=AfnvFnzs91s

Space Shuttle Challenger Explosion

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 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.”

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.”

How predictable was it?

Data:

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

Or:

Taco Bell

and

The Mir Space Station

- 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

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?

Aggregate loss = frequency times severity

What is the probability of Mir hitting the target?

What will it cost Taco Bell if it does?

- Potential cost:
- Population of United States?
- Cost of a Crunchy Beef Taco?
- Potential cost?

- Probability of a hit?
- Indicated premium?

- Issues:
- Uniform distribution across debris field?
- How many will cash in?
- What about expenses / fixed costs?

- Random variables
- Conditional probabilities and Bayes Theorem.
- Survival functions.
- Hazard rates.