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.