CAS Exam 7 Study Notes
Overview
I Estimation of Policy Liabilities
1
Loss Development Using Credibility - E. Brosius
1.1
Method Assumptions
1.1.1
Caveat with Traditional Method
1.1.2
Least Squares Pros & Cons
1.1.3
Practical Considerations for the Least Square Method
1.2
Best Estimate based on Bayes (with Theoretical Distribution)
1.2.1
Comparing Loss Development Methods
1.3
Bayesian Credibility
1.3.1
Practical Application (LS Development)
1.3.2
Credibility Form of the Dev’ Formula
1.3.3
Caseload Effect
1.4
Conclusions
1.5
Past Exam Questions
1.5.1
Question Highlights
2
Credible Claims Reserve: The Benktander Method - T. Mack
2.1
Gunnar Benktander Method (GB)
2.1.1
Method Comparison
2.1.2
MSE
2.2
Notation
2.3
Past Exam Questions
2.3.1
Question Highlights
3
Credible Claims Reserve: Benktander, Neuhaus and Mack - W. Hurlimann
3.1
Loss Ratio Claims Reserve Definition
3.1.1
Loss Ratio Claims Reserve Summary
3.1.2
Optimal Credibility Weights
3.1.3
MSE
3.2
Remark 6.1
3.3
Notation
3.4
Past Exam Questions
4
Measuring the Variability of Chain Ladder Reserve Estimate - T. Mack
4.1
Chainladder Underlying Assumptions
4.1.1
Implicit Assumptions
4.1.2
Proof for Assumption 3
4.1.3
LDF Selections Assumptions
4.1.4
Violation of Assumptions
4.1.5
Strength/Weakness of Chainladder
4.2
Mean Squared Error
4.2.1
Applying the MSE Formula
4.2.2
MSE Calculation Flow
4.3
Confidence Intervals
4.3.1
CI for All Years Reserves
4.3.2
CI Application & Range
4.4
Chain Ladder Assumptions Test
4.4.1
Intercept
4.4.2
Residuals
4.4.3
Calendar Year Test
4.4.4
Correlation of Adjacent LDFs
4.5
Past Exam Questions
4.5.1
Question Highlights
5
Testing the Assumptions of Age-to-Age Factors - G. Venter
5.1
Definitions and Assumptions
5.1.1
Mack’s Chainladder Assumptions
5.1.2
Variance Assumptions (for Chainladder)
5.2
6 Testable Implications
5.2.1
Goodness of Fit Measurement
5.3
Implication 1: Significance of Factors
5.4
Implication 2: Superiority of Alternative Emergence Patterns
5.4.1
Parameters: Alternative Emergence Pattern
5.4.2
Variance Assumptions: Alternative Emergence Pattern
5.4.3
Iteration Process
5.4.4
Counting
n
and
p
for SSE
5.5
Implication 3: Linearity
5.6
Implication 4: Stability
5.7
Implication 5: No Correlation on Columns (Test for independence)
5.8
Implication 6: No High of Low Diagonals (Test for independence)
5.8.1
Diagonal Trend as Inflation
5.9
Past Exam Questions
5.9.1
Question Highlights
6
LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach - D. Clark
6.1
Expected Loss Emergence
6.1.1
Expected Ultimate Loss Methods
6.2
Distribution of Actual Loss Emergence and Maximum Likelihood
6.2.1
Process Variance
6.2.2
MLE for Best Parameters
6.2.3
Parameter Variance
6.2.4
Variance of the Reserves
6.3
3 Key Assumptions of Model
6.4
LDF Method (Method 2)
6.4.1
Residual Review
6.4.2
Reserve Estimate
6.5
Cape Cod Method (Method 1)
6.5.1
Reserve Estimate
6.6
Other Use of Model
6.6.1
Variance of Prospective Losses
6.6.2
Calendar Year Development
6.6.3
Variability in the Discounted Reserves
6.7
Comments and Conclusion
6.8
Average Age of Claims
6.9
Past Exam Questions
6.9.1
Question Highlights
7
A Model for Reserving Workers Compensation High Deductibles - J. Siewert
7.1
Introduction
7.2
Method 1) Loss Ratio Method
7.3
Method 2) Implied Development
7.4
Method 3) Direct Development
7.5
Method 4) Credibility Weighting Techniques / Bornhuetter-Ferguson
7.6
Method 5) Development Method
7.6.1
Severity Trend
7.6.2
Claim Count Development
7.6.3
Severity Development
7.7
Method 6) Distribution Model
7.8
Aggregate Limits
7.8.1
Table M
7.9
Past Exam Questions
7.9.1
Question Highlights
8
Claims Development by Layer - R. Sahasrabuddhe
8.1
Introduction
8.1.1
Claim Size Model
8.2
Part 1) Setup Base Layer Triangle
8.2.1
Setup the trend triangle
8.2.2
Calculate
unlimited
mean
8.2.3
Calculate LEV
8.2.4
Calculate the Base Layer Triangle and LDF
8.3
Part 2) Convert LDFs from Base Layer
8.3.1
LDFs for Layers from Basic Limits to Policy Limits
8.4
Other Practical Uses
8.5
Issues with Assumptions
8.6
Past Exam Questions
8.6.1
Question Highlights
9
Using the ODP Bootstrap Model: A Practitioner’s Guide - Shapland
9.1
Introduction
9.1.1
Stochastic vs Static Model
9.2
Notations
9.3
Bootstrap Model
9.3.1
GLM Parameters
9.3.2
Fitted Triangle
9.3.3
Residuals
9.3.4
Dispersion Factor
9.4
Bootstrap Simulation Procedure
9.5
Variations on the ODP Bootstrap
9.5.1
Bootstrapping the Incurred Loss Triangle
9.5.2
Bootstrapping the BF and Cape Cod Method
9.6
GLM Bootstrap Model
9.6.1
GLM Variation 1: Reduce Row Parameters
9.6.2
GLM Variation 2: Reduce Column Parameters
9.6.3
GLM Variation 3: Reduce Row and Column Parameters
9.6.4
GLM Variation 4: Calendar Year Parameter
9.6.5
GLM Variation 5: One Parameter for Each Dimension
9.7
ODP vs GLM Bootstrap Summary
9.8
Practical Issues
9.8.1
Negative Incremental Values
9.8.2
Non-Zero Sum of Residuals
9.8.3
Using L-year Weighted Average
9.8.4
Missing Value
9.8.5
Outliers
9.8.6
Heteroskedasticity
9.8.7
Heteroecthesious Data
9.8.8
Exposure Adjustment
9.8.9
Parametric Bootstrapping
9.8.10
Fitting a Distribution to ODP Bootstrap Residuals
9.9
Diagnostics
9.9.1
Residual Graphs
9.9.2
Normality Test
9.9.3
Outlier
9.9.4
Parameter Adjustments
9.9.5
Review Model Results
9.10
Using Multiple Models
9.10.1
Additional Useful Output
9.10.2
Estimated Cash Flow Results
9.10.3
Estimated Ultimate Loss Ratio Results
9.10.4
Estimated Unpaid Claims Runoff Results
9.10.5
Distribution Graphs
9.10.6
Correlation
9.11
Model Testing
9.11.1
Future Research
9.12
Past Exam Questions
9.12.1
Question Highlights
10
Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinions - R. Verrall
10.1
Introduction
10.1.1
Notation
10.2
Stochastic Models for the Chainladder
10.2.1
Mack-1993 (Non-parametric)
10.2.2
Over-dispersed Poisson and Negative Binomial
10.3
Incorporating Expert Opinion about the Development Factors
10.3.1
Reproduce the Chainladder
10.3.2
Intervention in a development factor in particular rows
10.3.3
Intervention in using L-years average
10.4
Bayesian Model for the Bornhuetter-Ferguson Method
10.4.1
Calculation Example
10.5
Stochastic Column Parameters for Bayesian BF
10.5.1
Calculate Gamma
10.6
Implementation
10.7
Past Exam Questions
10.7.1
Question Highlights
11
Stochastic Loss Reserving Using Bayesian MCMC Models - G. Meyer
11.1
Introduction & Synopsis
11.2
Data Set
11.3
Testing Procedure
11.3.1
Kolmogorov-Smirnov Test
11.3.2
p-p
Plot
11.3.3
Percentile Histogram
11.4
Models Overview
11.5
Non-Bayesian Model
11.5.1
Mack Model
11.5.2
ODP Bootstrap
11.6
Bayesian Models (Cumulative)
11.6.1
Leveled Chain Ladder
11.6.2
Correlated Chain-Ladder
11.6.3
Changing Settlement Rate
11.7
Skewed Distribution
11.7.1
Skewed Normal Distribution
11.7.2
Mixed Lognormal-Normal
11.8
Bayesian Models (Incremental)
11.8.1
Correlated Incremental Trend
11.8.2
Leveled Incremental Trend
11.9
Process, Parameter, and Model Risk
11.10
Conclusion
11.10.1
Results Summary
11.10.2
Final Comments
11.11
Appendix B: Intro to Bayesian MCMC Models
11.11.1
How Bayesian MCMC works in practice
11.11.2
Metropolis-Hastings Algorithm
11.11.3
Usecase Example for Actuaries
11.11.4
Bayesian interence Using Gibbs Sampling (BUGS)
11.12
Past Exam Questions
11.12.1
Question Highlights
12
A Framework for Assessing Risk Margins - K. Marshall et al.
12.1
Introduction
12.2
Three Sources of Uncertainty
12.2.1
Quantitative vs Qualitative Analysis
12.3
Independent Risk
12.3.1
CoV for Independent Risk
12.4
Internal Systemic Risk
12.4.1
CoV for Internal Systemic Risk
12.5
External Systemic Risk
12.5.1
CoV for External Systemic Risk
12.6
Correlation (Aggregating the CoV)
12.6.1
Independent Risk Cov Correlation
12.6.2
Internal Systemic Risk Correlation
12.6.3
External Systemic Risk Correlation
12.7
Risk Margin
12.8
Additional Analysis
12.9
Documentation and Regularity
12.10
Past Exam Questions
12.10.1
Question Highlights
13
Reinsurance Loss Reserving - Patrik
13.1
7 Problems with Reinsurance Reserving
13.2
6 Components to a Reinsurance Loss Reserve
13.3
Reinsurance Reserving Procedure
13.3.1
Step (1): Portfolio Partition
13.3.2
Step (2) & (3): Analyze Historical Data and Projection
13.3.3
Step (4): Monitoring and Testing Predictions
13.4
Past Exam Questions
13.4.1
Question Highlights
14
Estimating the Premium Asset on Retrospectively Rate Policies - M. Teng and M. Perkins
14.1
Introduction
14.1.1
Discussion of Berry and Fitzgibbon Method
14.2
Teng Perkin Methods Overview
14.3
Formula Approach for PDLD
14.3.1
Retro Premium Formulas
14.3.2
Calculating PDLD
14.4
Empirical Approach for PDLD
14.4.1
PDLD Ratios Selection
14.4.2
Cumulative PDLD Ratios
14.4.3
Application
14.4.4
Further Issues
14.5
Feldblum’s Discussion
14.5.1
Improvements from Teng & Perkins
14.5.2
Teng & Perkins Assumptions
14.5.3
Enhancement
14.6
Past Exam Questions
14.6.1
Question Highlights
II Insurance Company Valuation
15
P&C Insurance Company Valuation - R. Goldfarb
15.1
Assumptions
15.1.1
Risk Adjusted Discount Rate
15.1.2
Growth Rate
15.2
Dividend Discount Model (DDM)
15.3
Free Cash Flow to Equity (FCFE)
15.4
Abnormal Earnings (AE)
15.5
Relative Multiples
15.5.1
Price to Earnings
15.5.2
Price to Book
15.5.3
Transaction Multiples
15.5.4
Relative Valuation for Multi-line firms
15.6
Option Pricing Models
15.6.1
Equity as Call Option
15.6.2
Real Options Valuation
15.7
Past Exam Questions
15.7.1
Question Highlights
III Enterprise Risk Management
16
ERA 1.1 - 1.3 Introduction - Brehm, et al.
16.1
ERA 1.1 Historical Context
16.2
ERA 1.2 Overview of Enterprise Risk Management
16.3
Key aspects of Enterprise Risk Management
16.4
Risk Taxonomy
16.5
Enterprise Risk Model Process
16.5.1
Diagnose
16.5.2
Analyze
16.5.3
Implement
16.5.4
Monitor
16.6
ERA 1.3 Enterprise Risk Modeling Overview
16.7
Quality of Models
16.8
Key Elements of Enterprise Risk Model
16.8.1
Underwriting Risk
16.8.2
Reserving Risk
16.8.3
Asset Risk
16.8.4
Dependencies
16.9
Setting Capital Requirements
16.9.1
Convert Probability to Loss
16.9.2
Risk Adjusted Performance
16.10
Past Exam Questions
16.10.1
Question Highlights
17
ERA 2.1 Corporate Decision Making Using an Enterprise Risk Model - Don Mango
17.1
Evoluation of Corporate Decision Making Under Uncertainty
17.1.1
Deterministic Project Analysis
17.1.2
Risk Analysis
17.1.3
Certainty Equivalent
17.2
Decision Making with an Internal Risk Model (IRM)
17.2.1
5a - Corporate Risk Tolerance
17.2.2
5b - Cost of Capital Allocated
17.2.3
5c - CBA for Mitigation
17.3
Conclusion
18
ERA 2.2 Risk Measures and Capital Allocation - G. Venter
18.1
Introduction
18.2
Risk Measures
18.2.1
Moment-Based Measures
18.2.2
Tail-Based Measures at
\(\alpha\)
Percentile
18.2.3
Probability Transforms
18.2.4
Generalized Moments
18.3
Required Capital
18.4
Capital Allocation
18.4.1
Proportional Method
18.4.2
Co-Measures
18.4.3
Having a Marginal Method
18.4.4
Marginal Impact
18.4.5
Using Decomposition
18.5
Allocating the Cost of Capital
18.6
Summary
19
ERA 2.3 Regulatory and Rating Agency Capital Adequacy Models - Witcraft
19.1
Introduction
19.2
Leverage Ratios
19.3
Risk-Based Capital Models
19.3.1
Credit Risk
19.3.2
Reserve Risk
19.3.3
Accumulation Risk
19.4
Scenario Testing
19.5
Evaluating Capital Structure Strategies
20
ERA 2.4 Asset-Liability Management - Brehm
20.1
Introduction
20.2
Optimal Porfolio Under 4 Scenarios
20.2.1
Additional Consideration: Tax
20.2.2
Additional Consideration: Equity
20.2.3
VFIC 2002
20.3
Asset-Liability Modeling Approach
20.3.1
Step 1: Model Asset Classes, Liabilities, and Current Business Operations
20.3.2
Setp 2: Define Risk Metrics
20.3.3
Step 3: Return Metrics
20.3.4
Step 4: Time Horizon
20.3.5
Step 5: Consider Relevant Constraints
20.3.6
Step 6: Simulation Model
20.3.7
Step 7: Efficient Frontier Graph
20.3.8
Step 8: Liabilities
20.3.9
Step 9: Review Results
20.4
Future Research
21
ERA 2.5 Measuring Value in Reinsurance - Venter, Gluck, Brehm
21.1
Introduction
21.2
Quantifying Stability and Its Value
21.2.1
Reinsurance Premium and Expected Recoveries
21.2.2
Amount of Protection
21.2.3
Premium Less Expected Loss
21.2.4
Distribution Based
21.2.5
Efficient Frontier Charts
21.3
Reinsurance as Capital
21.3.1
Change in Capital
21.3.2
Theorectical Model Example: Marginal ROE
21.3.3
Accumulation Risk
21.3.4
Capital Consumed
21.4
Reinsurance and Market Value
21.5
Conclusion
21.6
Past Exam Questions
21.6.1
Question Highlights
22
ERA 3.1 Considerations on Implementing Internal Risk Model - Mango
22.1
Introduction
22.2
Startup: Staffing and Scope
22.2.1
Recommendations
22.3
IRM Parameter Development
22.3.1
Recommendtaions
22.4
Implementation
22.4.1
Recommendations
22.5
Integration and Mainteneance
22.5.1
Recommendations
23
ERA 3.2 Modeling Parameter Uncertainty - Venter, Gluck
23.1
Introduction
23.2
Impact of Parameter Risk
23.3
Projection Risk
23.3.1
Simple Trend
23.3.2
Severity Trend and Inflation
23.3.3
Trend as a Time Series
23.4
Estimation Risk
23.5
Model Risk
23.6
Projection Models
23.7
Conclusion
24
ERA 3.3 Modeling and Dependency: Correlations and Copulas - G. Venter
24.1
Introduction
24.2
Pearson’s Correlation
\(\rho\)
24.3
Kendall’s
\(\tau\)
24.4
Motivations for Using Copulas
24.4.1
Joint Distribution Plots
24.4.2
Advantages of using copula to describe dependency with percentiles
24.5
How to Use a Copula
24.6
Summary of Copulas
24.6.1
Frank’s Copula
24.6.2
Gumbel Copula
24.6.3
Heavy Right Tail Copula
24.6.4
Normal Copula
24.6.5
Partial Perfect Correlation Copula
24.7
Tail Concentration Functions
24.7.1
Left Tail Concentration Function
24.7.2
Right Tail Concentration Function
24.7.3
LR Graph
24.8
How to Select Copula Given Dataset
24.8.1
Multivariate Copulas
24.9
Fitting Copulas to Data
24.9.1
\(J(z)\)
24.9.2
\(\chi(z)\)
24.10
Past Exam Questions
24.10.1
Question Highlights
25
ERA 4.1 & 4.2 Operational and Strategic Risk - D. Mango, G. Venter
25.1
ERA 4.1 Operational Risk Intro
25.2
Operational Risk to Insurers
25.3
Insurer Op Risk: Plan Loss Ratios
25.3.1
Three reasons why the planned LR did not work
25.4
Insurer Op Risk: Cycle Management
25.4.1
System Performance Perspective
25.4.2
Performance Improvement via Cycle Management
25.5
Miscellaneous
25.5.1
Agency Theory Perespective
25.5.2
Operational Risk Management in Banking and Manufacturing
25.5.3
Control Self-Assessment (CSA)
25.5.4
Key Risk Indications (KRIs)
25.5.5
Six Sigma
25.5.6
Operational Risk Modeling
25.6
ERA 4.2 Strategic Risk Intro
25.7
History and Definition
25.8
Strategic Risk Management Research to Date
25.8.1
Miller
25.8.2
Baird and Thomas
25.8.3
Slywotzky and Drzik
25.8.4
Hertz and Thomas
25.9
Scenario Planning
25.9.1
Key Steps in Scenario Planning Process
25.9.2
Insurance Example
25.10
Advanced Scenario Planning & ERM
25.10.1
Agent-Based Modeling
26
ERA 5.4 Approaches to Modeling the Underwriting Cycle - Major
26.1
Introduction
26.1.1
Underwriting cycle
26.1.2
Four Stages Insurance Business
26.2
Theories of the Cycle
26.2.1
Institutional Factors
26.2.2
Competition
26.2.3
Supply and Demand, Capital Constraints and Shocks
26.2.4
Economic Linkages
26.2.5
All of the Above
26.3
Approaches to Modeling the Cycle
26.3.1
Soft Approach
26.3.2
Technical Approach
26.3.3
Behavioral Modeling
26.4
Supply and Demand
26.4.1
Gron Supply Curve
26.5
Capital Flows
26.6
Assembling the Components
26.7
Conclusion
26.8
Past Exam Questions
26.8.1
Question Highlights
IV Overall Summary
27
Formulas
27.1
Loss Development Using Credibility - E. Brosius
27.2
Credible Claims Reserve: The Benktander Method - T. Mack
27.3
Credible Claims Reserve: Benktander, Neuhaus and Mack - W. Hurlimann
27.4
Measuring the Variability of Chain Ladder Reserve Estimate - T. Mack
27.5
Testing the Assumptions of Age-to-Age Factors - G. Venter
27.6
LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach - D. Clark
27.7
A Model for Reserving Workers Compensation High Deductibles - J. Siewert
27.8
Claims Development by Layer - R. Sahasrabuddhe
27.9
Using the ODP Bootstrap Model: A Practitioner’s Guide - Shapland
27.10
Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinions - R. Verrall
27.11
Stochastic Loss Reserving Using Bayesian MCMC Models - G. Meyer
27.12
A Framework for Assessing Risk Margins - K. Marshall et al.
27.13
Reinsurance Loss Reserving - Patrik
27.14
Estimating the Premium Asset on Retrospectively Rate Policies - M. Teng and M. Perkins
27.15
P&C Insurance Company Valuation - R. Goldfarb
27.16
ERA 2.2 Risk Measures and Capital Allocation - G. Venter
27.17
ERA 2.5 Measuring Value in Reinsurance - Venter, Gluck, Brehm
27.18
ERA 3.3 Modeling and Dependency: Correlations and Copulas - G. Venter
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CAS Exam 7 Study Notes
Chapter 27
Formulas
Links to all key formulas