MA545 ShortTerm Actuarial Mathematics
Course Catalog Description
Introduction
This course is tailored for the validation of Short Term Actuarial Mathematics (STAM) exam by Society of Actuaries (SOA). The main materials include (i) probability models commonly used for severity, frequency, risk aggregations, coverage modifications, and (ii) risk indications commonly used in credibility theory and reinsurance. The aim is to build up probability and statistical foundation for graduate students with concentration in actuarial science, and the focus is put on calculations concerned with coverage modifications and risk measures, constructing and estimation of parametric models. Also, the attention is paid to the basics of insurance and reinsurance, the calculation of premiums and reserve for the shortterm insurance as well.
Prerequisites: MA540, MA541
Crosslisting: None
Course Objective
The students will learn and understand the following actuarial models and the concerned associated probability and statistical methods:
• Commonly used severity and frequency models
• Modification of insurance coverage
• Most popular risk measures
• Parametric statistical models
• Credibility models and criteria
• Insurance and reinsurance coverages for shortterm insurances
• Premium and reserve for shortterm insurance coverages
Instructors
Teaching Associate Professor  Office  

Xiaohu Li

xli82@stevens.edu  Kidde 113A 
Campus  Fall  Spring  Summer 

On Campus  X 
More Information
Course Outcomes

• Compare two distributions based on tail characteristics, including moments, ratios of moments, limiting tail behavior, hazard rate function, and mean excess function.
• Derive and perform calculations with the zerotruncated and zeromodified versions of frequency distributions.
• Calculate aggregate loss distribution in case of various severity distributions.
• Evaluate the coverage modifications, including deductibles, coverage limits, and coinsurance.
• Calculate ValueatRisk and TailValueatRisk, and explain the desirable properties of a risk measure and determine whether a risk measure has these properties.
• Use the delta method to estimate the variance of the maximum likelihood estimator of a function of the parameter(s), and utilize Bayesian method to estimate the parameters for severity, frequency, and aggregate distributions.
• Utilize Buhlmann and BuhlmannStraub models and understand their relationship to Bayesian models, and apply empirical Bayesian method in nonparametric and semiparametric cases.
• Describe the types of policy limits and coverage modifications, the operation of basic forms of proportional and excess of loss reinsurance, and derive the distribution of claim amounts paid by the insurer and reinsurer under various forms of reinsurance.
• Describe the underlying statistical models and apply techniques for estimating unpaid losses from a runoff triangle, using chain ladder, average cost per claim and Bornhuetter Ferguson.
Course Resources
Textbook

Required textbook Loss Models: From Data to Decisions， Fourth Edition, 2012

Recommended texts
Introduction to Ratemaking and Loss Reserving for Property and Casualty Insurance Fourth Edition, 2015, Brown and Lennox, ACTEX, ISBN: 9781625424747.

Syllabus and study materials, SOA
Exam STAM: ShortTerm Actuarial Mathematics
Grading
Grading Policies
Weights  
1  Homework  50% 
2  Midterm  20% 
3  Final  30% 
Maximum Possible  100% 
Lecture Outline
Topic  Reading(s)  Homework  

Week 1  Severity models: basic probability distribution quantities  Chapters 3, 4  Exercises: 3.7, 3.8, 3.14, 3.17, 3.19, 3.22, 3.23, 3.29 
Week 2  Severity models: basic characteristics for models and variants of basic distributions  Chapters 4, 5  Exercises: 4.3, 4.7, 4.8, 5.6, 5.8, 5.9, 5.11, 5.16, 5.19, 5.21, 5.23, 5.25 
Week 3  Frequency models: Poisson, negative Binomial, (a,b,0) class and truncation  Chapter 6, 6.5, 6.6  Exercises: 6.1, 6.2, 6.4, 6.5, 6.6 
Week 4  Frequency models: compound, mixture and effect of exposure on frequency  Chapter 7  Exercises: 7.3, 7.4, 7.5, 7.8, 7.10 
Week 5  Coverage modification: deductible, coverage limit, coinsurance and their impact  Chapter 8  Exercises: 8.4, 8.5, 8.8, 8.9, 8.10, 8.12, 8.13, 8.15, 8.18, 8.20, 8.22, 8. 24, 8.26, 8.28, 8.30, 8.33 
Week 6  Aggregation loss models: compound models for aggregation claims, recursive method, and individual risk models  Chapter 9  Exercise: 9.9, 9.12, 9.15, 9.18, 9.21, 9.24, 9.27, 9. 29, 9. 31, 9.33 
Week 7  Midterm exam  Chapters 39  
Week 8  Risk measures: Coherent risk measures, ValueatRisk, TailValueatRisk  Chapter 3  Exercises: 3.32, 3.33, 3.34, 3.35, 3.36, 3.37 
Week 9  Construction and selection of parametric models: empirical distribution, point estimation, kernel density estimation, approximation  Chapters 11, 12  Exercises: 11.3, 11.5, 11.7, 11. 8, 11.9, 11.10, 11.11, 12.5, 12.6, 12.8, 12.10, 12.12, 12. 20, 12. 22, 12.24, 12.26, 12.35 
Week 10  Construction and selection of parametric models: frequentist estimation, moment and percentile matching, MLE for complete and incomplete data  Chapters 13, 14  Exercises: 13.7, 13.10, 13.13, 13.15, 13.21, 13.29, 13.32, 13.35, 13.38, 13.41, 13.44, 13.47, 13.50, 13.53, 13.56, 13.65, 13.66, 13.70, 13.71, 14.7, 14.8 
Week 11  Bayesian methods: inference, prediction and conjugate distribution  Chapter 15  Exercises: 15.11, 15.13, 15.15, 15.17, 15.21, 15.23, 15.25, 15.27 
Week 12  Model selection: tests for normality, goodnessoffit test and likelihood ratio test, scorebased approaches  Chapter 16  Exercises: 16.5, 16.6, 16.9, 16.10, 16.11, 16.12, 16.13, 16.24, 16.25, 16.27 
Week 13  Review  Chapters 1116  
Week 14  Final exam  Chapters 516 