# MA545 Short-Term 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 short-term insurance as well.

Prerequisites: MA540, MA541
Cross-listing: 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 short-term insurances
• Premium and reserve for short-term insurance coverages

Instructors

Teaching Associate Professor Email Office
Xiaohu Li
xli82@stevens.edu Kidde 113A

Campus Fall Spring Summer
On Campus X

# 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 zero-truncated and zero-modified 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 Value-at-Risk and Tail-Value-at-Risk, 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 Buhlmann-Straub 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 run-off 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: 978-1625424747.
• Syllabus and study materials, SOA
Exam STAM: Short-Term Actuarial Mathematics

 Weights 1 Homework 50% 2 Mid-term 20% 3 Final 30% Maximum Possible 100%

Lecture Outline

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 3-9
Week 8 Risk measures: Coherent risk measures, Value-at-Risk, Tail-Value-at-Risk 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, goodness-of-fit test and likelihood ratio test, score-based 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 11-16
Week 14 Final exam Chapters 5-16