a/s/m Study Manual for Exam C/Exam 4: Construction and Evaluation of Actuarial Models

I Severity, Frequency, and Aggregate Loss
	1 Basic Probability
		1.1 Functions and moments
		1.2 Percentiles
		1.3 Conditional probability and expectation
		1.4 Moment and probability generating functions
		1.5 The empirical distribution
		Exercises
		Solutions
	2 Parametric Distributions
		2.1 Scaling
		2.2 Transformations
		2.3 Common parametric distributions
			2.3.1 Uniform
			2.3.2 Beta
			2.3.3 Exponential
			2.3.4 Weibull
			2.3.5 Gamma
			2.3.6 Pareto
			2.3.7 Single-parameter Pareto
			2.3.8 Lognormal
		2.4 The linear exponential family
		2.5 Limiting distributions
		Exercises
		Solutions
	3 Variance
		3.1 Additivity
		3.2 Normal approximation
		3.3 Bernoulli shortcut
		Exercises
		Solutions
	4 Mixtures and Splices
		4.1 Mixtures
			4.1.1 Discrete mixtures
			4.1.2 Continuous mixtures
			4.1.3 Frailty models
		4.2 Conditional Variance
		4.3 Splices
		Exercises
		Solutions
	5 Policy Limits
		Exercises
		Solutions
	6 Deductibles
		6.1 Ordinary and franchise deductibles
		6.2 Payment per loss with deductible
		6.3 Payment per payment with deductible
		Exercises
		Solutions
	7 Loss Elimination Ratio
		Exercises
		Solutions
	8 Risk Measures and Tail Weight
		8.1 Coherent risk measures
		8.2 Value-at-Risk (VaR)
		8.3 Tail-Value-at-Risk (TVaR)
		8.4 Tail Weight
		8.5 Extreme value distributions
		Exercises
		Solutions
	9 Other Topics in Severity Coverage Modifications
		Exercises
		Solutions
	10 Bonuses
		Exercises
		Solutions
	11 Discrete Distributions
		11.1 The (a,b,0) class
		11.2 The (a,b,1) class
		Exercises
		Solutions
	12 Poisson/Gamma
		Exercises
		Solutions
	13 Frequency— Exposure & Coverage Modifications
		13.1 Exposure modifications
		13.2 Coverage modifications
		Exercises
		Solutions
	14 Aggregate Loss Models: Compound Variance
		14.1 Introduction
		14.2 Compound variance
		Exercises
		Solutions
	15 Aggregate Loss Models: Approximating Distribution
		Exercises
		Solutions
	16 Aggregate Losses: Severity Modifications
		Exercises
		Solutions
	17 Aggregate Loss Models: The Recursive Formula
		Exercises
		Solutions
	18 Aggregate Losses—Aggregate Deductible
		Exercises
		Solutions
	19 Aggregate Losses: Miscellaneous Topics
		19.1 Exact Calculation of Aggregate Loss Distribution
			19.1.1 Normal distribution
			19.1.2 Exponential and gamma distributions
			19.1.3 Compound Poisson models
		19.2 Discretizing
			19.2.1 Method of rounding
			19.2.2 Method of local moment matching
		Exercises
		Solutions
	20 Supplementary Questions: Severity, Frequency, and Aggregate Loss
		Solutions
II Empirical Models
	21 Review of Mathematical Statistics
		21.1 Estimator quality
			21.1.1 Bias
			21.1.2 Consistency
			21.1.3 Variance and mean square error
		21.2 Hypothesis testing
		21.3 Confidence intervals
		Exercises
		Solutions
	22 The Empirical Distribution for Complete Data
		22.1 Individual data
		22.2 Grouped data
		Exercises
		Solutions
	23 Variance of Empirical Estimators with Complete Data
		23.1 Individual data
		23.2 Grouped data
		Exercises
		Solutions
	24 Kaplan-Meier and Nelson-Åalen Estimators
		24.1 Kaplan-Meier Product Limit Estimator
		24.2 Nelson-Åalen Estimator
		Exercises
		Solutions
	25 Estimation of Related Quantities
		25.1 Moments
			25.1.1 Complete individual data
			25.1.2 Grouped data
			25.1.3 Incomplete data
		25.2 Range probabilities
		25.3 Deductibles and limits
		25.4 Inflation
		Exercises
		Solutions
	26 Variance of Kaplan-Meier and Nelson-Åalen Estimators
		Exercises
		Solutions
	27 Kernel Smoothing
		27.1 Density and distribution
			27.1.1 Uniform kernel
			27.1.2 Triangular kernel
			27.1.3 Other symmetric kernels
			27.1.4 Kernels using two-parameter distributions
		27.2 Moments of kernel-smoothed distributions
		Exercises
		Solutions
	28 Mortality Table Construction
		28.1 Individual data based methods
			28.1.1 Variance of estimators
		28.2 Interval-based methods
		Exercises
		Solutions
	29 Supplementary Questions: Empirical Models
		Solutions
III Parametric Models
	30 Method of Moments
		30.1 Introductory remarks
		30.2 The method of moments for various distributions
			30.2.1 Exponential
			30.2.2 Gamma
			30.2.3 Pareto
			30.2.4 Lognormal
			30.2.5 Uniform
			30.2.6 Other distributions
		30.3 Fitting other moments, and incomplete data
		Exercises
		Solutions
	31 Percentile Matching
		31.1 Smoothed empirical percentile
		31.2 Percentile matching for various distributions
			31.2.1 Exponential
			31.2.2 Weibull
			31.2.3 Lognormal
			31.2.4 Other distributions
		31.3 Percentile matching with incomplete data
		31.4 Matching a percentile and a moment
		Exercises
		Solutions
	32 Maximum Likelihood Estimators
		32.1 Defining the likelihood
			32.1.1 Individual data
			32.1.2 Grouped data
			32.1.3 Censoring
			32.1.4 Truncation
			32.1.5 Combination of censoring and truncation
		Exercises
		Solutions
	33 Maximum Likelihood Estimators—Special Techniques
		33.1 Cases for which the Maximum Likelihood Estimator equals the Method of Moments Estimator
			33.1.1 Exponential distribution
		33.2 Parametrization and Shifting
			33.2.1 Parametrization
			33.2.2 Shifting
		33.3 Transformations
			33.3.1 Lognormal distribution
			33.3.2 Inverse exponential distribution
			33.3.3 Weibull distribution
		33.4 Special distributions
			33.4.1 Uniform distribution
			33.4.2 Pareto distribution
			33.4.3 Beta distribution
		33.5 Bernoulli technique
		33.6 Estimating qx
		Exercises
		Solutions
	34 Variance Of Maximum Likelihood Estimators
		34.1 Information matrix
			34.1.1 Calculating variance using the information matrix
			34.1.2 Asymptotic variance of MLE for common distributions
			34.1.3 True information and observed information
		34.2 The delta method
		34.3 Confidence Intervals
			34.3.1 Normal Confidence Intervals
			34.3.2 Non-Normal Confidence Intervals
		34.4 Variance of Exact Exposure Estimate of j
		Exercises
		Solutions
	35 Fitting Discrete Distributions
		35.1 Poisson distribution
		35.2 Negative binomial
		35.3 Binomial
		35.4 Fitting (a,b,1) class distributions
		35.5 Adjusting for exposure
		35.6 Choosing between distributions in the (a,b,0) class
		Exercises
		Solutions
	36 Hypothesis Tests: Graphic Comparison
		36.1 D(x) plots
		36.2 p-p plots
		Exercises
		Solutions
	37 Hypothesis Tests: Kolmogorov-Smirnov
		37.1 Individual data
		37.2 Grouped data
		Exercises
		Solutions
	38 Hypothesis Tests: Anderson-Darling
		Exercises
		Solutions
	39 Hypothesis Tests: Chi-square
		39.1 Introduction
		39.2 Definition of chi-square statistic
		39.3 Degrees of freedom
		39.4 Other requirements for the chi-square test
		39.5 Data from several periods
		Exercises
		Solutions
	40 Likelihood Ratio Test and Algorithm, Schwarz Bayesian Criterion
		40.1 Likelihood Ratio Test and Algorithm
		40.2 Schwarz Bayesian Criterion
		Exercises
		Solutions
	41 Supplementary Questions: Parametric Models
		Solutions
IV Credibility
	42 Limited Fluctuation Credibility: Poisson Frequency
		Exercises
		Solutions
	43 Limited Fluctuation Credibility: Non-Poisson Frequency
		Exercises
		Solutions
	44 Limited Fluctuation Credibility: Partial Credibility
		Exercises
		Solutions
	45 Bayesian Methods—Discrete Prior
		Exercises
		Solutions
	46 Bayesian Methods—Continuous Prior
		46.1 Calculating posterior and predictive distributions
		46.2 Recognizing the posterior distribution
		46.3 Loss functions
		46.4 Interval estimation
		46.5 The linear exponential family and conjugate priors
		Exercises
		Solutions
	47 Bayesian Credibility: Poisson/Gamma
		Exercises
		Solutions
	48 Bayesian Credibility: Normal/Normal
		Exercises
		Solutions
	49 Bayesian Credibility: Bernoulli/Beta
		49.1 Bernoulli/beta
		49.2 Negative binomial/beta
		Exercises
		Solutions
	50 Bayesian Credibility: Exponential/Inverse Gamma
		Exercises
		Solutions
	51 Bühlmann Credibility: Basics
		Exercises
		Solutions
	52 Bühlmann Credibility: Discrete Prior
		Exercises
		Solutions
	53 Bühlmann Credibility: Continuous Prior
		Exercises
		Solutions
	54 Bühlmann-Straub Credibility
		54.1 Bühlmann-Straub model: Varying exposure
		54.2 Hewitt model: Generalized variance of observations
		Exercises
		Solutions
	55 Exact Credibility
		Exercises
		Solutions
	56 Bühlmann As Least Squares Estimate of Bayes
		56.1 Regression
		56.2 Graphic questions
		56.3 Cov(Xi,Xj)
		Exercises
		Solutions
	57 Empirical Bayes Non-Parametric Methods
		57.1 Uniform exposures
		57.2 Non-uniform exposures
			57.2.1 No manual premium
			57.2.2 Manual premium
		Exercises
		Solutions
	58 Empirical Bayes Semi-Parametric Methods
		58.1 Poisson model
		58.2 Non-Poisson models
		58.3 Which Bühlmann method should be used?
		Exercises
		Solutions
	59 Supplementary Questions: Credibility
		Solutions
V Simulation
	60 Simulation—Inversion Method
		Exercises
		Solutions
	61 Simulation—Special Techniques
		61.1 Mixtures
		61.2 Multiple decrements
		61.3 Simulating (a,b,0) distributions
		61.4 Normal random variables: the polar method
		Exercises
		Solutions
	62 Number of Data Values to Generate
		Exercises
		Solutions
	63 Simulation—Applications
		63.1 Actuarial applications
		63.2 Statistical analysis
		63.3 Risk measures
		Exercises
		Solutions
	64 Bootstrap Approximation
		Exercises
		Solutions
	65 Supplementary Questions: Simulation
		Solutions
VI Practice Exams
	1 Practice Exam 1
	2 Practice Exam 2
	3 Practice Exam 3
	4 Practice Exam 4
	5 Practice Exam 5
	6 Practice Exam 6
	7 Practice Exam 7
	8 Practice Exam 8
	9 Practice Exam 9
	10 Practice Exam 10
	11 Practice Exam 11
	12 Practice Exam 12
	13 Practice Exam 13
Appendices
	A Solutions to the Practice Exams
		Solutions for Practice Exam 1
		Solutions for Practice Exam 2
		Solutions for Practice Exam 3
		Solutions for Practice Exam 4
		Solutions for Practice Exam 5
		Solutions for Practice Exam 6
		Solutions for Practice Exam 7
		Solutions for Practice Exam 8
		Solutions for Practice Exam 9
		Solutions for Practice Exam 10
		Solutions for Practice Exam 11
		Solutions for Practice Exam 12
		Solutions for Practice Exam 13
	B Solutions to Old Exams
		B.1 Solutions to CAS Exam 3, Spring 2005
		B.2 Solutions to SOA Exam M, Spring 2005
		B.3 Solutions to CAS Exam 3, Fall 2005
		B.4 Solutions to SOA Exam M, Fall 2005
		B.5 Solutions to Exam C/4, Fall 2005
		B.6 Solutions to CAS Exam 3, Spring 2006
		B.7 Solutions to CAS Exam 3, Fall 2006
		B.8 Solutions to SOA Exam M, Fall 2006
		B.9 Solutions to Exam C/4, Fall 2006
		B.10 Solutions to Exam C/4, Spring 2007
	C Cross Reference from Loss Models
	D Exam Question Index