# Syllabus Math 568

## Math 568. Actuarial Loss Models Instructor Syllabus

This course focuses on applications of statistical techniques to actuarial models. Most examples are tailor-made for applications to property/casualty insurance or life insurance products.

Textbook: S.A. Klugman, H. H. Panjer, G.E. Willmot. Loss Models: From Data to Decisions, 4th Edition

PART I INTRODUCTION (5 hours)
1 Modeling
1.1 The model-based approach
1.2 Organization of this book
2 Random variables
2.1 Introduction
2.2 Key functions and four models
3 Basic distributional quantities
3.1 Moments
3.2 Percentiles
3.3 Generating functions and sums of random variables
3.4 Tails of distributions
3.5 Measures of Risk

PART II ACTUARIAL MODELS (12 hours)
4 Characteristics of Actuarial Models
4.1 Introduction
4.2 The role of parameters
5 Continuous models
5.1 Introduction
5.2 Creating new distributions
5.3 Selected distributions and their relationships
5.4 The linear exponential family
6 Discrete distributions
6.1 Introduction
6.2 The Poisson distribution
6.3 The negative binomial distribution
6.4 The binomial distribution
6.5 The (a, b, 0) class
6.6 Truncation and modification at zero
7.1 Compound frequency distributions
7.2 Further properties of the compound Poisson class
7.3 Mixed frequency distributions
8 Frequency and severity with coverage modifications
8.1 Introduction
8.2 Deductibles
8.3 The loss elimination ratio and the effect of inflation for ordinary deductibles
8.4 Policy limits
8.5 Coinsurance, deductibles, and limits
9 Aggregate loss models
9.1 Introduction
9.2 Model choices
9.3 The compound model for aggregate claims
9.4 Analytic results
9.5 Computing the aggregate claims distribution
9.6 The recursive method

PART III CONSTRUCTION OF EMPIRICAL MODELS (6 hours)
10 Review of mathematical statistics
10.1 Introduction
10.2 Point estimation
10.3 Interval estimation
11 Estimation for complete data
11.1 Introduction
11.2 The empirical distribution for complete, individual data
11.3 Empirical distributions for grouped data
12 Estimation for modified data
12.1 Point estimation
12.2 Means, variances, and interval estimation
12.3 Kernel density models

PART IV PARAMETRIC STATISTICAL METHODS (9 hours)
13 Frequentist estimation
13.1 Method of moments and percentile matching
13.2 Maximum likelihood estimation
13.3 Variance and interval estimation
13.5 Maximum likelihood estimation of decrement probabilities
14 Frequentist Estimation for discrete distributions
14.1 Poisson
14.2 Negative binomial
14.3 Binomial
14.4 The (a, b, 1) class
14.5 Compound models
14.6 Effect of exposure on maximum likelihood estimation
15 Bayesian estimation
15.1 Definitions and Bayes’ theorem
15.2 Inference and prediction
15.3 Conjugate prior distributions and the linear exponential family
15.4 Computational issues
16 Model selection
16.1 Introduction
16.2 Representations of the data and model
16.3 Graphical comparison of the density and distribution functions
16.4 Hypothesis tests
16.5 Selecting a model

PART V CREDIBILITY (9 hours)
17 Introduction and Limited Fluctuation Credibility
17.1 Introduction
17.2 Limited fluctuation credibility theory
17.3 Full credibility
17.4 Partial credibility
17.5 Problems with the approach

18 Greatest accuracy credibility
18.1 Introduction
18.2 Conditional distributions and expectation
18.3 The Bayesian methodology
18.5 The Buhlmann model
18.6 The Buhlmann-Straub model
18.7 Exact credibility

If time permits, the following topics can be covered.

19 Empirical Bayes parameter estimation
19.1 Introduction
19.2 Nonparametric estimation
19.3 Semiparametric estimation

PART VI SIMULATION
20 Simulation
20.1 Basics of simulation
20.2 Simulation for specific distributions
20.3 Determining the sample size
20.4 Examples of simulation in actuarial modeling

Midterm Exams (2 hours)

Total: 43 hours

Additional work required beyond the work required of undergraduates in the paired course:

Graduate students are expected to complete an extra research-oriented project or case study. A typical project involves reading and reporting on an academic paper or doing independent research on a relevant subject. A case study may require applications of theory and methodologies taught in class with substantial work on programming in software platforms such as R, Matlab or Python, etc.