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**Actuarial Models**

The examination for this material consists of four hours of multiple-choice questions. This material 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. Knowledge of risk management at the level of Exam P 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 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.

A variety of tables will be provided to the candidate in the study note package and at the examination. These include values for the standard normal distribution, illustrative life tables, and abridged inventories of discrete and continuous probability distributions. These tables are also available on the SOA Web site. Since they will be included with the examination, candidates will not be allowed to bring copies of the tables into the examination room.

## LEARNING OUTCOMES

- Survival and severity models.
- Define survival-time random variables
- for one life, both in the single- and multiple-decrement models;
- for two lives, where the lives are independent or dependent (including the common shock model); whether they be left-truncated, right-censored, both, or neither.

- Assuming a uniform distribution of deaths, define the continuous survival-time random variable that arises from the discrete survival-time random variable.
- Define severity random variables
- with or without a deductible;
- with or without a limit;
- with or without coinsurance.

- For any survival-time or severity random variable defined above, with single or mixed distributions, calculate
- expected values;
- variances;
- probabilities;
- percentiles.

- Define non-homogeneous and homogeneous discrete-time Markov Chain models and calculate the probabilities of
- being in a particular state;
- transitioning between particular states.

- Define survival-time random variables
- Frequency models.
- Define and calculate expected values, variances and probabilities for frequency random variables
- under the Poisson distribution;
- under the Binomial distribution;
- under the Negative Binomial distribution;
- under the Geometric distribution;
- under any mixture of the above.

- Define and calculate expected values, variances and probabilities for Poisson processes,
- using increments in the homogeneous case;
- using interevent times in the homogeneous case;
- using increments in the non-homogeneous case;
- resulting from special types of events in the Poisson process;
- resulting from sums of independent Poisson processes.

- Define and calculate expected values, variances and probabilities for frequency random variables
- Compound (aggregate) models.
- Define compound random variables, combining severity distributions with frequency distributions and Poisson processes.
- Calculate, for the compound random variables defined above,
- expected values, including recursion for aggregate deductibles (stop-loss insurance);
- variances;
- probabilities.

- Life contingencies
- Define present-value-of-benefit random variables for life insurances defined on survival-time random variables
- for one life, both in the single- and multiple-decrement models;
- for two lives, where the lives are independent or dependent (including the common shock model).

- Define present-value-of-benefit random variables for annuities defined on survival-time random variables
- for one life, in the single-and multiple-decrement models;
- for two lives, where the lives are independent or dependent (including the common shock model).

- Calculate the expected values, variances and probabilities for present-value-of-benefit random variables for the life insurances and annuities described above.
- Define and calculate the expected values, variances and probabilities for the present-value-of-lossat- issue random variables, as a function of the considerations (premiums), for the life insurances and annuities described above.
- Calculate considerations (premiums) for life insurances and annuities,
- using the Equivalence Principle;
- using percentiles.

- Define and calculate the expected values, variances and probabilities for the present-value-of-futureloss random variables for life insurances and annuities.
- Calculate liabilities, analyzing the present-value-of-future-loss random variables for life insurances and annuities,
- using the prospective method;
- using the retrospective method;
- using special formulas.

- Using recursion, calculate expected values (reserves) and variances of present-value-of-future-loss random variables for general fully-discrete life insurances written on a single life.
- For the life insurances and annuities described above, calculate
- gross considerations (expense-loaded premiums);
- expense-loaded liabilities (reserves);
- asset shares.

- Extending present-value-of-benefit, present-value-of-loss-at-issue, present-value-of-future-loss random variables and liabilities to discrete-time Markov Chain models, calculate
- actuarial present values of cash flows at transitions between states;
- actuarial present values of cash flows while in a state;
- considerations (premiums) using the Equivalence Principle;
- liabilities (reserves) using the prospective method.

- Define present-value-of-benefit random variables for life insurances defined on survival-time random variables

Note: Concepts, principles and techniques needed for Exam M are covered in the references listed below. Candidates and professional educators may use other references, but candidates should be very familiar with the notation and terminology used in the listed references. Texts

### Suggested Texts

- Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J., Chapter 3 , Sections 3.1–3.3, 3.5, 3.6 (excluding constant force and hyperbolic assumptions), 3.7 and 3.8, Chapter 4, Sections 4.1–-4.4, Chapter 5, Sections 5.1–5.4, Chapter 6, Sections 6.1(excluding utility-theory approach), 6.2–6.4, Chapter 7, Sections 7.1(excluding utilitytheory approach), 7.2–7.6, Chapter 8, Sections 8.1–8.2, 8.3 (only the recursion in Equation 8.3.9 and its equivalent variants), 8.4 (only Equation 8.4.6 for UDD and its equivalent variants), Chapter 9, Sections 9.1–9.5, 9.6.1, 9.7, Chapter 10, Sections 10.1–10.3, 10.5–10.5.1, 10.5.4, 10.6, Chapter 11, Sections 11.1–11.3 and Chapter 15, Sections 15.1–15.2.1, 15.4, 15.6–15.6.1.
- Introduction to Probability Models (Eighth Edition), 2003, by Ross, S.M., Chapter 5, Sections 5.3.1, 5.3.2 (through Definition 5.1), 5.3.3, 5.3.4 (through Example 5.14 but excluding Example 5.13), Proposition 5.3 and the preceding paragraph, Example 5.18, 5.4.1(up to example 5.23), 5.4.2 (excluding Example 5.25), 5.4.3, and Exercise 40.
- Loss Models: From Data to Decisions, (Second Edition) 2004, by Klugman, S.A., Panjer, H.H., and Willmot, G.E., Chapter 2 (background only), Chapter 3 (background only), Chapter 4, Sections 4.1–4.4 (excluding data-dependent distributions), 4.6.1–4.6.5, 4.6.7 through Theorem 4.51 (excluding zero-modified distributions, in particular Example 4.46 and Theorem 4.49), 4.6.9–4.6.11, Chapter 5, Sections 5.1–5.6, Chapter 6, Sections 6.1–6.3, 6.7 (excluding discretization), Chapter 8, Section 8.1.1.

### Study Notes

SNs for the Preliminary Education examinations and Course 6 are available on the SOA Web site under Education and Jobs/Candidate and Exam Information/Spring Exam Session/Spring 2005 Basic Education Catalog – Study Notes Information. Hard copies may be purchased by using the Study Note and Published Reference order form in the back of the printed catalog or by downloading the form from the Spring Exam Session Web page.

Code Title

- M-05-05# M Introductory Study Note
- -09-05# M Sample Exam Questions and Solutions
- M-24-05# Multi-State Transition Models with Actuarial Applications
- M-25-05# Section 8.5 from the second printing of Actuarial Mathematics, Second Edition (only the
- variance recursion given by Equation 8.5.16 with i =1)