An Introductory Guide in the Construction of Actuarial ...
An Introductory Guide in the Construction of Actuarial Models:
A Preparation for the Actuarial Exam C/4
Marcel B. Finan Arkansas Tech University
c All Rights Reserved Preliminary Draft Last Updated
November 4, 2017
To my son Amin
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Preface
This is the fifth of a series of lecture notes intended to help individuals to pass actuarial exams. The topics in this manuscript parallel the topics tested on Exam C of the Society of Actuaries exam sequence. As with the previous manuscripts, the main objective of the present manuscript is to increase users' understanding of the topics covered on the exam. The flow of topics follows very closely that of Klugman et al. Loss Models: From Data to Decisions. The lectures cover designated sections from this book as suggested by the 2012 SOA Syllabus. The recommended approach for using this manuscript is to read each section, work on the embedded examples, and then try ALL the problems given in the text. An answer key is provided by request. Email:mfinan@atu.edu. Problems taken from previous SOA/CAS exams will be indicated by the symbol . This manuscript can be used for personal use or class use, but not for commercial purposes. If you find any errors, I would appreciate hearing from you: mfinan@atu.edu
Marcel B. Finan Russellville, Arkansas February 15, 2013.
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Contents
Preface
iii
Actuarial Modeling
1
1 Understanding Actuarial Models . . . . . . . . . . . . . . . . . . 2
A Review of Probability Related Results
5
2 A Brief Review of Probability . . . . . . . . . . . . . . . . . . . . 6
3 A Review of Random Variables . . . . . . . . . . . . . . . . . . . 18
4 Raw and Central Moments . . . . . . . . . . . . . . . . . . . . . 34
5 Empirical Models, Excess and Limited Loss variables . . . . . . . 45
6 Median, Mode, Percentiles, and Quantiles . . . . . . . . . . . . . 62
7 Sum of Random Variables and the Central Limit Theorem . . . . 69
8 Moment Generating Functions and Probability Generating Func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Tail Weight of a Distribution
87
9 Tail Weight Measures: Moments and the Speed of Decay of S(x) 88
10 Tail Weight Measures: Hazard Rate Function and Mean Excess
Loss Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
11 Equilibrium Distributions and Tail Weight . . . . . . . . . . . . 102
Risk Measures
109
12 Coherent Risk Measurement . . . . . . . . . . . . . . . . . . . . 110
13 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
14 Tail-Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Characteristics of Actuarial Models
125
15 Parametric and Scale Distributions . . . . . . . . . . . . . . . . 126
16 Discrete Mixture Distributions . . . . . . . . . . . . . . . . . . . 131
17 Data-dependent Distributions . . . . . . . . . . . . . . . . . . . 138
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