## 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 Advanced discrete distributions **

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

18.4 The credibility premium

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.