Course 3 - Actuarial Models
Course 4 - Actuarial Modeling

Background

The Joint CAS/SoA Working Group on Courses 3 and 4 was established under the direction of the Society of Actuaries Design Team (Jeffrey Beckley, Chair) and the Casualty Actuarial Society Vice President - Admissions (Kevin Thompson).

The Working Group was charged with developing a syllabus, learning objectives and sample examinations from the report of the Joint Ad Hoc Task Force on Examinations 3 and 4 (Harry Panjer, Chair). The Ad Hoc Task Force was established under the direction of the Presidents of both Societies (Mavis Walters, CAS and Anna Rappaport, SoA).

The Ad Hoc Task Force combined the work to date of the existing SoA Working Group on Courses 3 and 4 and the CAS Syllabus Committee. The Ad Hoc Task Force applied the principles set forth by the SoA Board Task Force on Education in its August 1996 Report to the Membership and the principles of the CAS Task Force on Education. The report of the Ad Hoc Task Force was approved in March 1998 by the SoA Board and in May 1998 by the CAS Board as a workable agreement for the joint administration of Courses 3 and 4.

The Ad Hoc Task Force reviewed all the topics on Courses 3 and 4 in total. Two approaches to dividing the material between the courses were considered: 1) models on Course 3 and modeling on Course 4; and 2) grouping the models into two groups. The Ad Hoc Task Force chose the first approach. This approach eliminates the overlap on some topics. This approach also eliminates the characterization of one course as the life course and one course as the non-life course. Courses 3 and 4 should not be viewed as a convenient grouping of topics, but instead as a course on the modeling process divided in two for practical reasons.

The approach chosen by the Ad Hoc Task Force means that the candidate will be expected to be familiar with material on Course 3 prior to taking the Course 4 examination. The expectation of the Working Group is that the examinations for both courses will be offered twice a year.

Learning Objectives

The learning objectives for Courses 3 and 4 are two-dimensional: understanding and applying. Detailed learning objectives are shown later in this report. A summary objective is shown below for each course.

Course 3: The candidate is expected to understand certain models and techniques and to be able to apply the models to solve problems set in a business context. The effects of regulations, laws, accounting practices and competition on the results produced by the models are not considered in this course.

Models presented: contingent payment; survival; frequency; severity; compound distribution; stochastic process; and ruin.

Course 4: The candidate is expected to understand the modeling process and to apply statistical methods to sample data to calibrate and evaluate the models presented on Course 3. The candidate should be able to carry out the steps of the modeling process in solving problems set in a business context. The effects of regulations, laws, accounting practices and competition on the results produced by the models are not considered in this course.

Readings

The approach of the courses in integrating several topics that have traditionally been presented separately or grouped differently in actuarial education eliminated the use of an exclusive textbook for each course. Some textbooks were selected from outside the stream of actuarial literature to expose the candidate to a wider range of techniques, as well as examples, outside of insurance and pensions. The Working Group anticipates that other textbooks may be presented in the future that are superior to those selected, and the Working Group encourages their consideration.

The texts and articles selected for Courses 3 and 4 are shown below. Detailed citations of chapters and sections for each topic are shown later in this report.

Bowers et al. Actuarial Mathematics (Second Edition). Chicago: Society of Actuaries, 1997. (Course 3)

Jones, B.L. "Stochastic Models for Continuing Care Retirement Communities," North American Actuarial Journal, Vol. 1, No. 1, pp. 50-64. (Course 3)

Klein, J.P. and Moeschberger, M.L. Survival Analysis. New York: Springer-Verlag, 1997. (Courses 3 and 4)

Klugman, S.A., Panjer, H.H. and Willmot, G.E. Loss Models: From Data to Decisions. New York: John Wiley and Sons, 1998. (Courses 3 and 4)

Pindyck, R.S. and Rubinfeld, D.L. Econometric Models and Economic Forecasts (Fourth Edition). New York: McGraw-Hill, 1997. (Course 4)

Ross, S.M. Introduction to Probability Models (Sixth Edition). San Diego: Academic Press, 1997. (Course 3)

Ross, S.M. Simulation (Second Edition). San Diego: Academic Press, 1997. (Courses 3 and 4)

_________. Models and Modeling Study Note. (Courses 3 and 4)

_________. Contingent Payment Models Applications Study Note. (Course 3)

 The candidate is expected to have a thorough knowledge of statistics. The Working Group recommends that a list of textbooks be provided to the candidates as a guide to the level of knowledge that will be assumed with the statement, "No examination questions will be based directly on these readings."

Exam Length

The Working Group concluded that two examinations, each at least four hours in length, will be required to adequately test the topics. An examination length of up to five hours may be necessary. The Working Group reviewed the page counts for all recommended readings and decided that they were inconclusive for recommending the appropriate exam length. The Working Group believes that a better opinion on the optimal examination length could be formed after sample examination questions are developed. The expectation of the Working Group is that the questions will be set in a business context and integrated across topics. It is likely that the candidate will require more time per question than the historical standards indicate. For these reasons, the final decision on exam length should be deferred until after the sample examinations are created.

Sample Examinations

The Working Group was not able to construct sample examinations in the time frame allowed for the completion of this report. The expectation of the Working Group is that all questions will be framed in a business context and that a majority of the questions will integrate topics (i.e., require the candidate to draw upon knowledge from two or more topics to answer the question). These expectations will be difficult to achieve immediately and should initially be viewed as goals. The Working Group believes that these expectations can be achieved over a short period of time.

In developing sample examinations, the Working Group will consider the merits of both multiple choice and written answer questions. The Working Group will assess the advantages and disadvantages of both approaches in testing the learning objectives. Consideration will also be given to the additional resources required in the administration of written answer examinations.

Ken Bonvallet	Rich Hertling
Nancy Braithwaite, Co-Chair	Bruce Jones
Peggy Brinkman	Don Jones
Bob Campbell	Clive Keatinge
Frank Cerasoli	Jim Miles, Co-Chair
Nancy Davis	Gordon Willmot
Tom Gallagher	Judy Anderson, SoA Staff Liaison

 June 11, 1998

Course 3 - Actuarial Models

Course Description

This course develops the candidate’s knowledge of the theoretical basis of actuarial models and the application of those models to insurance and other financial risks. A thorough knowledge of calculus, probability and interest theory is assumed. A knowledge of risk management at the level of Course 1 is also assumed.

The candidate will be required to understand, in an actuarial context, what is meant by the word "model," how and why models are used, their advantages and their limitations. The following specific models will be introduced:

• Contingent Payment Models
• Survival Models
• Frequency and Severity Models
• Compound Distribution Models
• Stochastic Process Models
• Ruin Models

The candidate will be expected to understand what important results can be obtained from these models for the purpose of making business decisions, and what approaches can be used to determine these results. Simulation and recursion are two very useful methods that are introduced.

Learning Objectives

Understanding Actuarial Models

The candidate is expected to understand the models and techniques listed below and to be able to apply the models to solve problems set in a business context. The effects of regulations, laws, accounting practices and competition on the results produced by the models are not considered in this course.

1.  Explain what a mathematical model is and, in particular, what an actuarial model can be.

2. Discuss the value of building models for such purposes as: forecasting, estimating the impact of making changes to the modeled situation, estimating the impact of external changes on the modeled situation.

3. Identify the models and methods available, and understand the difference between the models and the methods.

4. Explain the difference between a stochastic and a deterministic model and identify the advantages/disadvantages of each.

5. Understand that all models presented (e.g., survival models, stochastic processes, aggregate loss models) have the same structure.

6. Formulate a model for the present value, with respect to an assumed interest rate structure, of a set of future contingent cash flows. The model may be stochastic or deterministic.

7. Determine the characteristics of the components and the effects of changes to the components of the model in 6. Components include:

• a deterministic interest rate structure;
• a scheme for the amounts of the cash flows;
• a probability distribution of the times of the cash flows; and
• the probability distribution of the present value of the set of cash flows.

8. Apply a principle to a present value model to associate a cost or pattern of (possibly contingent) costs with a set of future contingent cash flows.

• Principles include: equivalence, exponential, standard deviation, variance, and percentile.
• Models include: present value models based on 9-12 below.
• Applications include: insurance, health care, credit risk, environmental risk, consumer behavior (e.g., subscriptions), and warranties.

9. Characterize discrete and continuous univariate probability distributions for failure time random variables in terms of the life table functions, l_x, q_x, p_x, _nq_x, _np_x, and _{m½ n}q_x, the cumulative distribution function, the survival function, the probability density function and the hazard function (force of mortality), as appropriate.

• Establish relations between the different functions.
• Develop expressions, including recursion relations, in terms of the functions for probabilities and moments associated with functions of failure time random variables, and calculate such quantities using simple failure time distributions.
• Express the impact of explanatory variables on a failure time distribution in terms of proportional hazards and accelerated failure time models.
  

10. Given the joint distribution of two failure times

• Calculate probabilities and moments associated with functions of these random variables.
• Characterize the distribution of the smaller failure time (the joint life status) and the larger failure time (the last survivor status) in terms of functions analogous to those in 9, as appropriate.
• Develop expressions, including recursion relations, for probabilities and moments of functions of the joint life status and the last survivor status, and express these in terms of the univariate functions in 9 in the case in which the two failure times are independent.
• Characterize the joint distribution of two failure times, the joint life status and the last survivor status using the common shock model and using copulas.
  
  

11. Characterize the joint distribution (pdf and cdf) of the time until failure and the cause of failure in the competing risk (multiple decrement) model, in terms of the functions l_x, _tq_x, _tp_x, _td_x, m _x(t).

• Establish relations between the functions.
• Given the joint distribution of the time of failure and the cause of failure, calculate probabilities and moments associated with functions of these random variables.
• Apply assumptions about the pattern of failures between integral ages to obtain the associated (discrete) single decrement models from a discrete multiple decrement model as well as the discrete multiple decrement model that results from two or more discrete single decrement models.

12.Generalize the models of 9, 10, and 11 to multiple state models characterized in terms of transition probability functions or transition intensity functions (forces of transition).

13. Define a counting distribution (frequency distribution).

• Characterize the following distributions in terms of their parameters and moments: Poisson, mixed Poisson, negative binomial, binomial, and the (a,b,1) class of distributions.
• Identify the applications for which these distributions are used and the reasons why they are used.
• Given the parameters of a distribution, apply the distribution to an application.
  
  

14. Define a loss distribution.

• Characterize the following families of distributions in terms of their parameters and moments: transformed beta, transformed gamma, inverse transformed gamma, lognormal and inverse Gaussian.
• Apply the following techniques for creating new families of distributions: multiplication by a constant, raising to a power, exponentiation, and mixing.
• Identify the applications in which these distributions are used and the reasons why they are used.
• Given the parameters of a distribution, apply the distribution to an application.
  

15. Define a compound distribution.

16. Calculate probabilities associated with a compound distribution when the compounding distribution is a member of the families in 13, and the compounded distribution is discrete or a discretization of a continuous distribution.

17. Adjust the calculation of 16 for the impact of policy modifications such as deductibles, policy limits and coinsurance.

18. Define a stochastic process and distinguish between discrete-time and continuous-time processes.

19. Characterize a discrete-time Markov chain in terms of the transition probability matrix.

• Use the Chapman-Kolmogorov equations to obtain probabilities associated with a discrete-time Markov chain.
• Classify the states of a discrete-time Markov chain.
• Calculate the limiting probabilities of a discrete-time Markov chain.
  

20. Define a counting process.

21. Characterize a Poisson process in terms of:

• the distribution of the waiting times between events,
• the distribution of the process increments,
• the behavior of the process over an infinitesimal time interval.
  

22. Define a nonhomogeneous Poisson process.

• Calculate probabilities associated with numbers of events and time periods of interest.

23.  Define a compound Poisson process.

• Calculate moments associated with the value of the process at a given time.
• Characterize the value of the process at a given time as a compound Poisson random variable.
  

24. Define a continuous-time Markov chain.

• Characterize such a process in terms of transition intensity functions and in terms of transition probability functions.
• Use the Chapman-Kolmogorov equations to calculate probabilities associated with the value of the process at a given time.
• Use the Kolmogorov differential equations to obtain transition probability functions from the transition intensity functions in special cases.
• Calculate the limiting distribution of the value of the process.

25. Define a Brownian motion process.

• Determine the distribution of the value of the process at any time.
• Determine the distribution of a hitting time.
• Calculate the probability that one hitting time will be smaller than another.
• Define a Brownian motion process with drift and a geometric Brownian motion process.

26. For a discrete-time surplus process:

• Calculate the probability of ruin within a finite time by a recursion relation.
• Analyze the probability of ultimate ruin via the adjustment coefficient and establish bounds.

27. For a continuous-time Poisson surplus process:

• Derive an expression for the probability of ruin assuming that the claim amounts are combinations of exponential random variables.
• Calculate the probability that the surplus falls below its initial level, determine the deficit at the time this first occurs, and characterize the maximal aggregate loss as a compound geometric random variable.
• Approximate the probability of ruin using the compound geometric recursion.
• Analyze the probability of ruin: analytically (e.g., adjustment coefficient); numerically; and by establishing bounds.
• Determine the characteristics of the distribution of the amount of surplus (deficit) at: first time below the initial level; and the lowest level (maximal aggregate loss).

28. Analyze the impact of reinsurance on the probability of ruin and the expected maximum aggregate loss of a surplus process.

29. Generate discrete random variables using basic simulation methods.

30. Generate continuous random variables using basic simulation methods.

31.Construct an algorithm to appropriately simulate outcomes under a wide variety of stochastic models.

Applications of Actuarial Models

The candidate is expected to be able to apply the models above to business applications. The candidate should be able to determine an appropriate model for a given business problem and be able to determine quantities that are important in making business decisions, given the values of the model parameters. Relevant business applications include, but are not limited to:

• Premium (rate) for life insurance and annuity contracts.
• Premium (rate) for accident and health insurance contracts.
• Premium (rate) for casualty (liability) insurance contracts.
• Premium (rate) for property insurance contracts.
• Rates for coverages under group benefit plans.
• Loss reserves for insurance contracts.
• Benefit reserves for insurance contracts.
• Resident fees for Continuing Care Retirement Communities (CCRC’s).
• Cost of a warranty for manufactured goods.
• Value of a financial instrument such as: a loan, a stock, an option, etc.
• Risk classification.
• Solvency (ruin).

Topics

Topic	Weighting
Classification of Models	0-5%
Contingent Payment Models	38-42%
Survival Models	13-17%
Frequency and Severity Models Compound Distribution Models	15-20%
Stochastic Process Models	13-17%
Ruin Models	3-7%
Simulation of Models	3-7%
Total	100%

Readings

• Classification of Models

A study note on Actuarial Models and Actuarial Modeling. The study note will present the modeling process, provide a definition of a model, and discuss the various types of models along with their uses, advantages and limitations. The study note will be approximately 25 pages in length, and may be computer based. It will be used on both Course 3 and Course 4.

• Contingent Payment Models

Bowers et al. Actuarial Mathematics (Second Edition). Chicago: Society of Actuaries, 1997. Chapter 4, Sections 5.1-5.4, 6.1-6.4, 7.1-7.6, Chapter 8, Sections 9.1-9.8, Chapter 10.

A study note, consisting of examples only, illustrating the concepts of contingent payment models applied to property-casualty insurance, finance, consumer behavior and other areas. The study note will be 30-50 pages in length. Until a suitable study note is available the Working Group believes two to three articles of modest length on non-life insurance applications of contingent payment models could serve as interim readings. The study note is not written but authors are being solicited.

• Survival Models

Bowers et al. Actuarial Mathematics (Second Edition). Chicago: Society of Actuaries, 1997. Chapter 3.

Klein, J.P. and Moeschberger, M.L. Survival Analysis. New York: Springer-Verlag, 1997. Chapters 2 and 3 (excluding 3.6).

• Frequency and Severity Models

Klugman, S.A., Panjer, H.H. and Willmot, G.E. Loss Models: From Data to Decisions. New York: John Wiley and Sons, 1998. Sections 1.3, 2.1, 2.7 (excluding example 2.51), 2.10 (excluding 2.10.1 and following), 3.1, 3.2.1-3.2.2, 3.3.1-3.3.2, 3.4.1, 3.5 (through paragraph ending on page 223), 3.6.1, 3.7 (excluding 3.7.1 and 3.7.2), Example 3.29.

• Compound Distribution Models

Klugman, S.A., Panjer, H.H. and Willmot, G.E. Loss Models: From Data to Decisions. New York: John Wiley and Sons, 1998. Sections 1.4, 4.1-4.3, 4.5-4.6, 4.9.1, 4.9.4 (page 336 only).

• Stochastic Process Models

Ross, S.M. Introduction to Probability Models (Sixth Edition). San Diego: Academic Press, 1997. Sections 2.8, 4.1-4.4, 4.5.1, 4.6, 5.3-5.4, 6.1-6.5, 6.8, 10.1-10.4.

Jones, B.L. "Stochastic Models for Continuing Care Retirement Communities," North American Actuarial Journal, Vol. 1, No. 1, pp. 50-64.

• Ruin Models

Bowers et al. Actuarial Mathematics (Second Edition). Chicago: Society of Actuaries, 1997. Chapter 13 (excluding autoregressive discrete-time model and appendix), Section 14.5.

Klugman, S.A., Panjer, H.H. and Willmot, G.E. Loss Models: From Data to Decisions. New York: John Wiley and Sons, 1998. Sections 6.2.3, 6.3.1, 6.3.2.1.

• Simulation of Models

Ross, S.M. Simulation (Second Edition). San Diego: Academic Press, 1997. Sections 3.1, 4.1-4.3, Chapters 5 (excluding 5.3), 6.

 Supplemental Readings

A list of readings may be provided for the candidate who is seeking additional background material on selected topics. No examination questions would be based on these readings. These background readings would be added before the formal syllabus is published.

Course 4 - Actuarial Modeling

Course Description

This course provides an introduction to modeling and covers important actuarial and statistical methods that are useful in modeling. A thorough knowledge of calculus, linear algebra, probability and mathematical statistics is assumed.

The candidate will be required to understand the steps involved in the modeling process and how to carry out these steps in solving business problems. The candidate should be able: 1) to analyze data from an application in a business context, 2) to determine a suitable model including parameter values, and 3) to provide measures of confidence for decisions based upon the model. The candidate will be introduced to a variety of tools for the calibration and evaluation of the models on Course 3.

Learning Objectives

Understanding Actuarial Models

The candidate is expected to apply statistical methods to sample data to quantify and evaluate the models presented on Course 3 and to use the models to solve problems set in a business context. The effects of regulations, laws, accounting practices and competition on the results produced by the models are not considered in this course.

1. Identify the steps in the modeling process and discuss how they interrelate.

2. Identify the models and methods available, and understand the difference between the models and the methods.

3. Explain the difference between a stochastic and a deterministic model and identify the advantages/disadvantages of each.

4. Discuss the possible limitations imposed by the data available for input for constructing a model.

5. Understand that all models presented in Courses 3 and 4 have the same structure. Apply models from more than one family (e.g., regression, stochastic process, time series) to a particular business application.

6. Identify the underlying assumptions implicit in each family of models and recognize which set(s) of assumptions are applicable to a given business application.

7. Estimate the parameters of a tabular failure time or loss distribution when the data is complete, or when it is incomplete, using maximum likelihood, method of moments, and Bayesian estimation.

8. Obtain nonparametric estimates for a failure time or loss distribution using the empirical distribution, the Kaplan-Meier estimator and the Nelson-Aalen estimator.

9. Construct the likelihood model needed to estimate the parameters of a parametric failure time or loss distribution regression model.

10. Construct the partial likelihood model needed to estimate the regression coefficients in a semiparametric failure time or loss distribution regression model.

11. Adjust an estimation based on the presentation of the sample data: complete, incomplete, censored, truncated, grouped, shifted.

12. Apply statistical tests to determine the acceptability of a fitted model:

• Pearson’s chi-square statistic
• Likelihood ratio test
• Kolmogorov-Smirnov statistic

13. For estimators, define the terms: efficiency, bias, consistency, mean squared error.

14. Calculate the least squares estimates of the parameters used in single and multiple linear regression models, and use knowledge of their distributions for hypothesis testing and development of confidence intervals.

15. Test a given linear regression model’s fit to a given data set.

16. Assess the appropriateness of the linear regression model for a given data set by checking for such irregularities as heteroscedasticity, serial correlation, and multicollinearity.

17. Perform statistical tests to determine the presence of measurement error or specification error.

18. Develop deterministic forecasts from time series data, using simple extrapolation and moving average models, applying smoothing techniques and seasonal adjustment when appropriate.

19. Use the concept of the autocorrelation function of a stochastic process to test the process for stationarity.

20. Generate a forecast using the general ARIMA model and develop confidence intervals for the forecast.

21. Test the hypothesis that a given stochastic process is a random walk.

22. For an ARIMA process (including simpler models as special cases), estimate the model parameters, and perform appropriate diagnostic checks of the model.

23. Apply limited fluctuation (classical) credibility including criteria for both full and partial credibility.

24. Perform Bayesian analysis using discrete and continuous examples.

25. Apply the Buhlmann-Straub credibility model to basic situations. Understand the relationship to the Bayesian model.

26. Apply the conjugate prior in Bayesian analysis and Buhlmann-Straub credibility, and, in particular, to the Poisson-gamma model.

27. Apply empirical Bayesian methods in the nonparametric and semiparametric cases.

28. Compare and contrast the assumptions underlying limited fluctuation credibility, Bayesian analysis, and the Buhlmann-Straub credibility model.

29. Determine an appropriate number of simulations to perform in order to estimate a quantity of interest.

30. Quantify the variability of an estimate in the context of simulation.

31. Determine the bootstrap estimates of the mean squared error of an estimator.

32. Use basic simulation methods to validate a model.

 

Applications of Actuarial Models

The candidate is expected to apply the models presented in Course 3 and the statistical methods presented on this course to business applications. As discussed above, the candidate should be able to take data from a given application and determine a suitable model, including parameter estimates, for use in making business decisions related to the application. The candidate should be able to assess the variability of the parameter estimates and the goodness of fit of the model, and therefore provide an opinion on the confidence that should be given to the model output in making decisions. Relevant business applications include, but are not limited to:

• Premium (rate) for life insurance and annuity contracts.
• Premium (rate) for accident and health insurance contracts.
• Premium (rate) for casualty (liability) insurance contracts.
• Premium (rate) for property insurance contracts.
• Rates for coverages under group benefit plans.
• Loss reserves for insurance contracts.
• Benefit reserves for insurance contracts.
• Resident fees for Continuing Care Retirement Communities (CCRC’s).
• Cost of a warranty for manufactured goods.
• Value of a financial instrument such as: a loan, a stock, an option, etc.
• Risk classification.

Topics

Topic	Weighting
The Modeling Process	3-7%
Estimation and Fitting of Models:
Survival	18-22%
Frequency and Severity	18-22%
Regression, Forecasting and Time Series	25-30%
Credibility Theory	20-25%
Simulation in Estimating and Fitting	3-7%
Total	100%

Readings

• The Modeling Process

A study note on Actuarial Models and Actuarial Modeling. The study note will present the modeling process, provide a definition of a model, and discuss the various types of models along with their uses, advantages and limitations. The study note will be approximately 25 pages in length, and may be computer based. It will be used on both Course 3 and Course 4.

• Estimation and Fitting of Models

Klein, J.P. and Moeschberger, M.L. Survival Analysis. New York: Springer-Verlag, 1997. Chapters 4, 5, 6, Sections 7.1-7.3, Chapter 8, Sections 12.1-12.4.

Klugman, S.A., Panjer, H.H. and Willmot, G.E. Loss Models: From Data to Decisions. New York: John Wiley and Sons, 1998. Sections 2.2-2.6, 2.8-2.9, 2.10.1-2.10.5, 3.2.3, 3.3.3-3.3.4, 3.4.2.

• Regression, Forecasting and Time Series

Pindyck, R.S. and Rubinfeld, D.L. Econometric Models and Economic Forecasts (Fourth Edition). New York: McGraw-Hill, 1997. Chapters 3-7, 15-18.

• Credibility Theory

Klugman, S.A., Panjer, H.H. and Willmot, G.E. Loss Models: From Data to Decisions. New York: John Wiley and Sons, 1998. Sections 1.5, 5.1-5.5 (excluding 5.4.6 and 5.5.3).

• Simulation in Estimation and Fitting

Ross, S.M. Simulation (Second Edition). San Diego: Academic Press, 1997. Chapters 7 and 9.

Supplemental Readings

A list of readings may be provided for the candidate who is seeking additional background material on selected topics. No examination questions would be based on these readings. These background readings would be added before the formal syllabus is published.