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NOTES

I have updated the manual originally written by Peter J. Murzda, Jr., FCAS, ASA and updated by Dean A. Westpfahl, FCAS, MAAA to reflect the 2014 syllabus material and 2013 CAS Exam 8. Questions and parts of some solutions have been taken from material copyrighted by the Casualty Actuarial Society. They are reproduced in this study manual with the permission of the CAS solely to aid students studying for CAS exams. Students may also request past exams directly from the CAS or find them on the CAS website. I am very grateful to the CAS for its cooperation and permission to use this material. The CAS is not responsible for the structure or accuracy of this manual. In some cases, questions and answers have been edited or altered to be more accurate, reflect syllabus changes, or provide a better organized manual. Students should keep in mind that there may be more than one correct way to answer a question even if only one is shown. Exam questions are identified by numbers in parentheses at the end of each question. Questions have four numbers separated by hyphens: the year of the exam, the number of the exam, the number of the question, and the points assigned. There are some questions labelled 14-8-MTS-(point value). These are questions I added in 2014 on the new syllabus material. I have made a conscientious effort to eliminate mistakes and incorrect answers, but a few may remain. I am grateful to students who previously pointed out errors and encourage those who find others to bring them to my attention. Please check the ACTEX website for corrections subsequent to publication.

Margaret Tiller Sherwood FCAS, ASA, MAAA, FCA, CPCU, ARM, ERMP, CERA

July 2014

AAA 1

© 2014 ACTEX Publications, Inc. Classification Ratemaking CAS Exam 8 – Sherwood

American Academy of Actuaries, Committee on Risk Classification, “Risk Classification Statement of Principles,” 1980.

OUTLINE

I. SUMMARY A. Introduction: Risk Classification 1. Insurance substitutes certainty for uncertainty

2. Risk classification – grouping of risks with similar risk characteristics used to determine equitable and average prices that is fundamental to a private, voluntary insurance system

3. Classification does not a. Predict the experience for an individual risk in a class b. Intend to identify good or bad risks or to penalize

4. Problem of determining similarity as cannot measure all characteristics, but only those that bear significantly on cost

B. Purposes of Risk Classification 1. Protect the insurance system's financial soundness 2. Achieve fairness

3. Allow the operation of economic incentives and the promotion of availability C. Principles Underlying Classification 1. Reflect expected cost differences 2. Distinguish risks based on cost-related factors 3. Apply objectively 4. Be practical and cost-effective 5. Be acceptable to the public D. The Context

Need to consider classification in relation to the other factors of the insurance system, e.g., marketing, underwriting, and administration

II. ECONOMIC SECURITY AND INSURANCE A. Hazard Avoidance and Reduction 1. Examples: not flying and sprinklers

2. Practical application is limited so there is a need for programs that transfer financial uncertainty

2 AAA


B. Transfer of Financial Uncertainty 1. Great variety of programs

2. Can distinguish those whose benefits are based on need (government assistance programs) and those based on defined contractual rights (private insurance)

C. Public and Private Programs 1. Both involve transfer and pooling that assures predictability 2. Differences a. Defined by law vs. by contract b. Usually compulsory vs. often voluntary

c. Little role for competition vs. a significant role d. Benefit need not have a long-term relationship to the amount paid vs. the

requirement of such a relationship 3. Government programs often devised because of a lack of private coverage 4. Difficult to generalize about private insurance as it is highly diverse III. THE NEED FOR RISK CLASSIFICATION A. Rationale for Risk Classification

1. Insurance system must find a way of establishing a fair price for assuming risk 2. Methods for estimating price

a. Wisdom, insight, and judgment – may be the only possible method for a new coverage, even though not the best

b. Observance of a risk's actual losses over an extended period of time – difficult or impossible in some cases, e.g., life insurance

c. Observance of the actual losses of risks with similar characteristics perhaps over a shorter period and use the average experience to apply to individuals in that class

3. Determination of relevant similar characteristics a. In perfectly competitive markets, would expect an optimal set to emerge b. But in the absence of such markets, characteristics based on fact and judgment

AAA 3


B. Three Primary Purposes 1. Protection of a program's financial soundness

a. Solvency threatened by adverse selection, a dramatic movement of buyers to or from different sellers that results in the sale of the product at less than the expected cost

b. Example: selling life insurance without reflecting age c. Classification reduces the potential for adverse selection d. Other methods include mandatory participation and choice restrictions

2. Enhanced fairness a. Differences in prices among classes should reflect differences in expected costs b. Also, individuals within a class should have expected costs similar to those of

the class 3. Economic incentive

a. To ensure profitable growth, insurers need a classification system that will permit them to offer insurance to as many as possible at adequate prices

b. Incentive for risk classifications to become as refined as possible in matching insureds with their expected costs, thereby achieving competitive success

c. Practical limit exists as the additional expense of obtaining more refinement should not be greater than the reduction in expected costs for the lower-cost risks

4. Close and reinforcing relationship among the three primary purposes exists IV. CONSIDERATIONS IN DESIGNING A RISK CLASSIFICATION SYSTEM A. Underwriting 1. Classification system developed independent of the actual risks 2. Determination of the acceptability of a risk based on its own merits

3. Design must recognize that underwriting controls the impact of the classification system B. Marketing 1. Marketing program has an important influence on the mix of business

2. Restrictions on or adjustments to the classification system may produce unintended changes in the mix of business

4 AAA


C. Program Design 1. Degree of choice available to the buyer a. Compulsory programs generally have broad classifications

b. Voluntary programs need refined classifications to prevent adverse selection 2. Experience-based pricing a. Some programs adjust prices to reflect emerging loss experience

b. May involve only risk's own experience or may include that of other similar individuals or groups

c. Rating classes may differ from dividend (a.k.a. equity or experience) classes d. If such adjustments based on experience occur, less-refined initial classification

systems are needed 3. Premium payer (different from insured) a. Nature of classification indifferent to the insured

b. Broader classifications may be appropriate as adverse selection has been reduced

D. Statistical Considerations 1. Homogeneity

a. Within a class there should be no clearly identifiable subclasses with significantly different loss potential

b. As concept is based on expected losses, actual losses should not be equal, but rather should be statistically distributed around the average experience for the class

c. Similarly, an overlap exists as the actual experience of some in one class will be the same as that of some in another class

2. Credibility a. Desirable that classes be large enough to allow credible statistical predictions b. Each class need not stand on its own, but can be weighted with the experience

of broader groupings of correlative classes 3. Predictive stability

a. Predictive capability must respond to changes in the nature of insurance losses b. Yet also need to avoid unwarranted abrupt changes in resulting prices c. Can use statistical tools to measure historical predictive stability, but need

judgment to evaluate noninsurance trends 4. Conclusions

a. Considerations may conflict as improving homogeneity may reduce credibility b. Relative weight to be given to the various considerations influenced by the

nature of the risks, management philosophy, and the judgment of the designer

AAA 5


E. Operational Considerations 1. Expense

a. Should be as low as possible, yet permit the system to minimize adverse selection and maximize equity

b. Cost of using a variable should be reasonable in relationship to the benefits achieved

2. Constancy a. Characteristic of the risk should remain the same for the policy period

b. Not preclude possible reclassification c. Lack of constancy increases expense and decreases utility 3. Availability of coverage

a. If the system reflects expected costs, availability should be enhanced b. In some cases, however, refinement that produces accurate expected costs may

make insurance unaffordable for some c. In other cases, risks may be uninsurable, though may get limited coverage

d. Overall, refinement should promote availability 4. Avoidance of extreme discontinuities

a. Number of classes should be large enough to present a reasonable continuum, especially at the extremes

b. But few enough so that differences are significant 5. Absence of ambiguity a. Clear and objective definition b. Collectively exhaustive and mutually exclusive 6. Manipulation – minimize the ability of an insured to manipulate or to misrepresent 7. Measurability a. Reliable measurement – age and sex b. Not easily measured – moral character F. Hazard Reduction Incentives 1. Examples: premium reduction for sprinklers and nonsmokers

2. Desirable, but not necessary, as there are limits to the extent that a classification can solve social problems

6 AAA


G. Public Acceptability 1. Difficulty with this principle as values a. Hard to determine b. Vary among social segments c. Change over time 2. Public acceptability considerations regarding classifications a. Should not discriminate unfairly b. Should be based on relevant data c. Should respect personal privacy

d. Should be structured so risks identify naturally with their classification 3. Constraints on classifications a. Laws b. Regulations c. Public opinion H. Causality

1. More acceptable to the public if a demonstrable cause-and-effect relationship exists 2. But often impossible to prove causality so this is not a necessary requirement 3. Often use causality to describe a plausible relationship, e.g., living in a valley and flood

claims I. Controllability

1. In many cases, desirable as closely associated with an effort to reduce hazards 2. But can also be associated with undesirable qualities, e.g., manipulation, impracticality,

and irrelevance 3. Must evaluate both positive and negative aspects

AAA 7


PAST CAS EXAMINATION QUESTIONS A. Introduction, Economic Security and Insurance, and the Need for Classification A1. For each of the following environments, describe how adverse selection can be controlled according to

the American Academy of Actuaries' “Risk Classification Statement of Principles.” a. A voluntary, competitive environment b. A government-controlled, mandatory environment. (83–9–15–2) A2. The Supreme Court's decision in the Norris case eliminated the use of sex as a rating variable in

pensions. Discuss the potential implications of this decision on automobile insurance classification in the context of the five basic principles of a sound risk classification system as outlined by the American Academy of Actuaries. (84–9–9–3)

A3. According to the American Academy of Actuaries' “Risk Classification Statement of Principles,” in a

perfectly competitive market, the optimal set of risk classification characteristics tends to emerge over time through the competitive mechanism. (85–9–1g–.5)

A4. According to the “Risk Classification Statement of Principles” by the American Academy of Actuaries

Committee on Risk Classification, what three primary purposes should be served by the process of risk classification? (86–9–6–2)

A5. Under California's Proposition 103, the use of territories in automobile insurance may be greatly

curtailed or even eliminated.

a. List the three primary purposes of risk classification as stated in the AAA's “Risk Classification Statement of Principles.”

b. Indicate whether each of the three purposes would be affected by the curtailment of territorial rating under California's Proposition 103. Briefly explain your answer. (89–9–8–1.5/1.5)

A6. According to the “Risk Classification Statement of Principles” by the American Academy of Actuaries,

in insurance programs that are largely or entirely compulsory with broad classifications and no voluntary choice among competing institutions, adverse selection will likely occur. (91–9–3–MC)

A7. According to the “Risk Classification Statement of Principles” by the American Academy of Actuaries, risk classification reduces adverse selection by balancing the economic forces governing buyers and sellers. (91–9–3–MC)

A8. Based on the American Academy of Actuaries' paper, “Risk Classification Statement of Principles,” the

presence of strong competition decreases the need for an insurer to have a refined classification system. (92–9–1–MC)

A9. According to the American Academy of Actuaries' “Risk Classification Statement of Principles,” which

of the following are considered primary purposes of risk classification? 1. To protect the insurance program's financial soundness 2. To enhance fairness 3. To permit economic incentives to operate A. 2 B. 1,2 C. 1,3 D. 2,3 E. 1,2,3 (94–9–5–1)

8 AAA


A1. a. It can be controlled by a risk classification system of sufficient refinement. b. It can be controlled by a restriction of the buyer's freedom, p. 8. A2. The five principles outlined by the American Academy are: 1) “The system should reflect expected cost differences.”

2) “The system should distinguish among risks on the basis of relevant cost-related factors.” 3) “The system should be applied objectively.” 4) “The system should be practical and cost-effective.” 5) “The system should be acceptable to the public.” If the Norris case were applied to automobile insurance, its effect on principles 3 and 4 would be

undetectable. Generally, public opinion would seem to favor its application (principle 5). It is also clear that principles 1 and 2 would be violated. The system would not reflect expected costs as males would be undercharged and females overcharged as one relevant cost-related factor would be disregarded, pp. 2–3.

A3. T, p. 6. A4. Risk classification should “protect the insurance system's financial soundness; be fair; [and] permit

economic incentives to operate and thus encourage widespread availability of coverage,” p. 2. A5. a. See A4.

b. The elimination of territorial rating may have significant effects on the financial soundness and

fairness of the insurance system and on the existence of economic incentives. Elimination of a factor that is related to expected loss costs may result in adverse selection. In such a situation, rates would be based on the experience of all territories, but drivers in low-cost territories would not purchase insurance, making the rates inadequate for the drivers from high-cost territories who remain in the system. Depending on the definition of fairness used, fairness may be increased or decreased. Social equity is increased because insurance becomes more affordable for the urban poor. On the other hand, actuarial equity, which underlies the AAA's definition of fairness, is decreased because the relationship between rates and expected costs is undermined. Finally, availability of insurance should decrease as there is no economic incentive for companies to insure most drivers from high-cost territories, pp. 7–10.

A6. F, p. 8 – Restrictions on a buyer's freedom reduce the possibility of adverse selection. A7. T, p. 8. A8. F, p. 9 – Substitute “increases” for “decreases.”

A9. E, p. 2.

AAA 19


B11. In the American Academy of Actuaries' monograph, “Risk Classification Statement of Principles,” several operational considerations in designing a successful classification system are cited. List four of these considerations and briefly explain how each contributes to the success of a classification system. (Only the first four considerations listed will be graded.) (94–9–26–2)

B12. According to the American Academy of Actuaries' “Risk Classification Statement of Principles,” which

of the following are true?

1. In contrast to the assignment of a risk to a class based on individual and possibly unique characteristics of each risk, the underwriting process involves the evaluation of the risk based on general criteria.

2. To the extent that prices are adjusted based on a risk's emerging actual experience after the insurance and its initial price have been established, less refined initial risk classification systems are needed.

3. As the proportion of the total premium paid by the insured increases, the use of a broader classification system becomes more appropriate.

A. 1 B. 2 C. 3 D. 2,3 E. 1,2,3 (95–9–4–1) B13. According to the American Academy of Actuaries' “Risk Classification Statement of Principles,” which

of the following are true?

1. Operational expenses for a risk classification system include those expenses associated with determining a price for each class.

2. Particular attention often is required in defining classes at the extreme ends of the expected claim cost range to reduce large differences in anticipated average claim costs between the extreme class and the adjacent class.

3. Hazard reduction incentives are desirable and necessary features of a risk classification system. A. 1 B. 3 C. 1,2 D. 2,3 E. 1,2,3 (95–9–5–1) B14. List three reasons given by the AAA, in “Risk Classification Statement of Principles,” on why public

acceptability is difficult to apply in practice. (96–9–47a–1.25)

B15. A property insurance company is considering adding a new classification rating variable to its homeowners insurance program based on an individual risk's actual loss experience over the past five year period as follows:

Class A – no claims Class B – one or two claims Class C – three or more claims

Evaluate this new classification rating variable based on the following considerations as described in the American Academy of Actuaries' “Risk Classification Statement of Principles”: a. Controllability b. Operational expense c. Hazard reduction incentives. (96–9–48a–1.5)

B16. According to the American Academy of Actuaries Committee on Risk Classification's “Risk Classification Statement of Principles,” which of the following are not operational considerations relating to classification plans?

A. Availability of coverage B. Avoidance of extreme discontinuities C. Absence of ambiguity D. Measurability E. All of these are operational considerations. (00–9–16–1)

20 AAA


B11. 1) Expenses of obtaining and maintaining data, of classifying risks, and of determining prices should be minimized to promote efficiency and competitiveness.

2) Risk characteristics should be constant over a period to reduce expense and increase their usefulness.

3) Proper matching of expected costs and prices should promote availability. 4) Extreme discontinuities should be avoided to discourage manipulation and customer

dissatisfaction. 5) Classes should be clear, objective, exhaustive and mutually exclusive to avoid ambiguities that

hamper efficient and correct classification. 6) The ability to manipulate classes should be minimized so that risks are properly classified.

7) Risk variables should be easily measurable so that risks are properly classified, pp. 16–19. B12. 1. F, p. 11 – Reverse “individual and possibly unique” and “general.”

2. T, p. 13 3. F, p. 13 – Substitute “decreases” for “increases.” Answer: B

B13. 1. T, p. 16 2. T, p. 18 3. F, p. 19 – Substitute “but not” for “and.” Answer: C B14. 1) Social values “are difficult to ascertain.”

2) Social values “vary among segments of the society.” 3) Social values “change over time,” p. 19.

B15. a. Not all ratable losses would be under the insured's control (e.g., many of those resulting from a hurricane), but some would be (e.g., those prevented by installing window shutters), p. 21.

b. Operational expenses would be minimal for those insureds who have been with the company for

the five-year experience period. For other insureds, the company would have the additional expenses of obtaining and maintaining claim histories, p. 16.

c. The system would promote hazard reduction as the possibility of lower future premiums would

encourage insureds to take steps to reduce losses, e.g., install window shutters, p. 19.

B16. A. F, p. 17 – It is an operational consideration.

B. F, p. 18 – It is an operational consideration. C. F, p. 18 – It is an operational consideration. D. F, p. 18 – It is an operational consideration. Answer: E

Bailey 27


Robert A. Bailey and LeRoy J. Simon, “An Actuarial Note on the Credibility of Experience of a Single Private Passenger Car”;

Discussion by W. J. Hazam, PCAS XLVI, 1959, pp. 159–64; XLVII, 1960, pp. 150–52

OUTLINE

I. INTRODUCTION A. The Problem

What credibility should be assigned to the accident experience of an automobile in determining its liability premium?

B. The Authors' Solution

Automobiles are divided by class, reflecting use and driver characteristics, and by subclass, reflecting claim experience over three years. Their claim frequency per $1,000 of premium for two subsequent years was then compiled and each class's frequency was compared to the average. Credibility equals the ratio of the future relative frequency less unity to the past relative frequency less unity.

C. Symbols 1. A - subclass with three or more accident-free years 2. B - subclass with no accident-free years 3. m - claim frequency of a class 4. N - radix, number of persons in the population 5. R - ratio of actual to expected losses 6. X - subclass with only two accident-free years 7. Y - subclass with only one accident-free years 8. Z - credibility

28 Bailey


II. THE AUTHORS' APPROACH A. Determination of Subclass Relative Frequency

1. Convert each subclass's earned premium to earned premium at rates for subclass B 2. Divide the earned premium by $1,000 3. Divide the number of claims by the premium in 2. 4. Take the ratio of subclass frequency to the average frequency for all classes

5. Compute the relative frequency for the following classes: a. A b. (A + X) c. (A + X + Y) B. Determination of Accident-Free Credibilities

1. One-year credibility equals unity minus the relative frequency for (A + X + Y) 2. Two-year credibility equals unity minus the relative frequency for (A + X) 3. Three-year credibility equals unity minus the relative frequency for A

C. Determination of Credibility for Risks Having at Least One Accident in the Last Year 1. Calculate past relative frequency for these risks a. Assume Poisson applies with average frequency m

b. Determine the percentage of persons with at least one claim, unity minus the percentage of persons with no claims, i.e., (1 e-m)

c. Average frequency for those with at least one claim

AF = m

1 e-m

d. Substitute actual frequency for m e. Relative frequency

RF = 1

1 e-m

2. Determine the future relative frequency for this group of risks

3. Apply the following formula:

Z = Future Relative Frequency 1.0Past Relative Frequency 1.0

Bailey 29


III. PERSPECTIVE A. The Authors' Major Conclusions for Canadian Private Passenger Cars

1. One-year experience has significant and measurable credibility for experience rating, ranging from .038 to .071 for a subclass

2. In a highly refined private passenger rating system reflecting inherent hazard, the accuracy of a merit rating plan would be low, but if there is a wider range of hazard, credibility would be larger

3. Adding a second year's experience to one year's will increase credibility by two-fifths; adding a third year's experience to two years' will increase credibility by one-sixth

B. Other Comments

1. Class 1 (no male operator under 25) is the least homogeneous and thus its subclasses have the most credibility

2. Credibility also increases with size C. Reasons That Credibility Does Not Vary in Proportion to Time

1. An individual's accident propensity changes over time 2. The population of a class changes as individuals enter and leave 3. Individuals within a class have different accident propensities, which are markedly

skewed 4. In the credibility formula, Z is not exactly proportionate to n

D. Use of Premium as a Base Rather Than Car-Years

1. According to the authors, this avoids the maldistribution created by having territories with higher claim frequencies produce more X, Y, and B risks and higher territorial premiums

2. According to Hazam, this eliminates maldistribution only if both of the following two conditions are met:

a. High-frequency territories are also high-premium territories b. Territorial differentials are proper

E. Use of Losses Rather Than Claim Counts

1. Accident frequency used to reduce chance variations caused by claim size variations 2. But subclass B risks have a consistently higher-than-average severity and subclass A

risks have one that is lower than average 3. This consistency is the reason that using losses instead of counts produces an increase in

credibility

30 Bailey


F. Hazam's Conclusions

1. Credibility is measurable and significant 2. But not large enough to justify the credits now offered by many U.S. plans 3. May, however, reduce the gap by also taking into account conviction frequencies

G. Comparison with Dropkin 1. Use of accidents rather than violations 2. Emphasis on the results accomplished rather than on the limits of segregation 3. Their levels of data a. Possible levels of data 1) Class 2) Subdivision of class by violations or accidents 3) Individual drivers b. Bailey and Simon

1) Weight class and subclass data to get a better predictor of future experience

2) Subclass has some credibility and thus is somewhat homogeneous 3) Refer to subclass data as individual experience c. Dropkin 1) Separates class data into subclasses 2) Still finds subclasses heterogeneous and overlapping

3) Implies that further segregation needed so that subclass experience approximates that of the individual

Bailey 31


PAST CAS EXAMINATION QUESTIONS A. The Credibility Equation A1. Bailey and Simon in their paper, “An Actuarial Note on the Credibility of a Single Private Passenger

Car” computed credibilities for accident-free risks based upon the commonly used experience rating formula, Modification = Z(R) + (1 Z) where Z is the credibility factor and R is the ratio of actual losses to expected losses. Given the following information and using Bailey and Simon's technique, compute the credibilities for automobiles with at least one, two, and three years, respectively of accident-free driving.

Earned Premium Number Claim Frequency Relative Years Since Earned at Present of Claims per $1,000 Claim Last Accident Car Years Rates ($000) Incurred of Premiums Frequency 3 2,757 159,108 217,151 1.365 .920 2 131 7,910 13,792 1.744 1.175 1 164 9,862 19,346 1.962 1.322 None 274 17,226 37.730 2.190 1.476 Total 3,326 194,106 288,019 1.484 1.000 (75S–9a–2a–6) A2. You have been asked to develop a compensation experience rating plan for one-employee manufacturing

risks. For ease of administration, it has been decided that only one year's claim experience will be utilized. The table below gives the last calendar year's experience sorted by the time elapsed since the previous claim.

Years Since Earned Number of Latest Claim Employee Years Incurred Claims 0 25,000 7,500 1 75,000 7,500 2 or more 300,000 15,000 Total 400,000 30,000

Determine the amount of credibility that can be assigned to one year's claim experience. (77–9–13–5) A3. a. From the information below on a particular class of private passenger automobile business,

determine the experience rating credibility of the experience of one private passenger car for one year.

Years Since Most Recent Earned Car Number of Claim Accident Years Claims Frequency 0 50,000 9,000 .180 1 50,000 7,500 .150 2 50,000 6,500 .130 3 50,000 6,000 .120 ≥ 4 800,000 56,000 .070 Total 1,000,000 85,000 .085

b. Assuming the Poisson distribution represents the risk distribution and using the additional information and notations below, set up the final equation you would use to confirm your answer above for the credibility of one year's experience.

N - total number of cars insured x - claim frequency of class Ne-x - number of cars having no claim last year. (78–9–8–4/4)

32 Bailey


A1. 1) Calculate future absolute claim frequencies: FACF = (Number of Claims Incurred)/(Earned Premium at Present Rates)

FACF1 = 217,151 + 13,792 + 19,346

159,108 + 7,910 + 9,862 = 1.415

FACF2 = 217,151 + 13,792159,108 + 7,910 = 1.383 FACF3 =

217,151159,108 = 1.365

2) Calculate future relative claim frequencies: FRCF = FACF/FACFOverall FRCF1 = 1.415/1.484 = .953 FRCF2 = 1.383/1.484 = .932 FRCF3 = 1.365/1.484 = .920

3) Calculate credibilities: Z = 1 FRCF Z1 = 1 .953 = .047 Z2 = 1 .932 = .068 Z3 = 1 .920 = .080, pp. 159–60.

A2. 1) Calculate future absolute claim frequencies:

FACF1 = Number of Claims Incurred

Earned Employee Years = 7,500 + 15,000

75,000 + 300,000 = .060

FACFOverall = 30,000/400,000 = .075

2) Calculate the future relative claim frequency: FRCF1 = FACF1/FACFOverall = .060/.075 = .800 3) Calculate the credibility: Z = 1 FRCF1 = 1 .800 = .200, pp. 159–60.

A3. a. 1) Calculate the future absolute claim frequency:

FACF = (Number of Claims)/(Earned Car Years)

FACF1 = 7,500 + 6,500 + 6,000 + 56,000

50,000 + 50,000 + 50,000 + 800,000 = .080

2) Calculate the future relative claim frequency:

FRCF1 = FACF1/FACFOverall = .080/.085 = .941 3) Calculate the credibility:

Z1 = 1 FRCF1 = 1 .941 = .059, pp. 159–60. b. 1) Calculate the future relative claim frequency:

FRCF0 = FACF0/FACFOverall = .180/.085 = 2.118 2) Since the past relative claim frequency equals 1/(1 e-x) or 1/(1 e-.085), we get the

following equation:

FRCF0 = 2.118 = Z0

1 e-.085 + (1 Z0) Z0 = (1.118)(1 e-.085)

e-.085 = .099, pp. 159–160, 164.

Grossi 2 721

© 2014 ACTEX , Publications, Inc. Catastrophic and Reinsurance Pricing CAS Exam 8 – Sherwood

Patricia Grossi, Howard Kunreuther, and Don Windeler, Chapter 2: “An Introduction to Catastrophe Models and Insurance”

in Catastrophe Modeling: A New Approach to Managing Risk, edited by Patricia Grossi and Howard Kunreuther, pp. 23–42

with June 2, 2013 Clarification and Errata of Section 2.4

by the CAS Syllabus Committee

OUTLINE I. OVERVIEW OF CATASTROPHE MODELS

A. History of Catastrophe Models

1. Some origins in property insurance – crude mapping of insured structures ends in 1960s

2. Other origins in seismological and meteorological measurements a. Measurement of earthquakes begins with the first seismograph in 1800s b. By 1970s studies published on sources and frequency of earthquakes, floods,

and hurricanes c. Late 1980s – mapping and measurement components combined to produce

catastrophe modeling

3. Advances made in information technology and geographic information systems a. Combine information on hazards with properties at risk b. Hurricane Hugo (1989), Loma Prieta Earthquake (1989) and Hurricane Andrew

(1992) promote the use of modeling c. Hurricane Andrew results in nine insolvencies

4. Government recognition of the need for an accurate assessment of the impact of disasters leads to a FEMA report on earthquake estimation methodologies and the development of HAZUS, an open source model

B. Structure of Catastrophe Models – Four Components

1. Hazard, e.g., for earthquakes, the epicenter, moment magnitude, etc.

2. Inventory – portfolio of properties a. Geocoding assigns geographic coordinates

b. Other features included 1) Construction type, stories, age 2) Deductibles, coverage limits

3. Vulnerability – susceptibility to damage

722 Grossi 2

© 2014 ACTEX Publications, Inc. Catastrophic and Reinsurance Pricing CAS Exam 8 – Sherwood

4. Loss – quantification of the physical impact of natural hazard phenomenon a. Varies from model to model

b. Loss to inventory 1) Direct losses – “cost to repair and/or replace a structure” 2) Indirect losses – “business interruption impacts and relocation costs of

residents forced to evacuate their homes”

C. Uses of a Catastrophe Model for Risk Management

1. Model output quantified and presented to stakeholders 2. Assessment of alternative risk management strategies

a. Mitigation b. Insurance c. Reinsurance d. Catastrophe bonds

3. Examples of outputs a. GIS map

1) Spatial representation of the number of displaced households 2) Useful for emergency response officials

b. Exceedance probability curve 1) Exceedance probability curve – “graphical representation of the

probability that a certain level of loss will be surpassed in a given time period”

2) Enables insurer to assess whether risk of the current loss profile is acceptable

II. DERIVATION AND USE OF AN OCCURRENCE EXCEEDANCE PROBABILITY CURVE

A. Definitions

1. Occurrence Exceedance Probability (OEP) – Probability that at least one loss exceeds the specified amount a. This is the distribution of the largest loss in the period b. Based on the theory of order statistics.

2. Aggregate Exceedance Probability (AEP) – Probability that the sum of all losses during

a given period exceeds some amount

3. Conditional Exceedance Probability (CEP) – Probability that the amount of a single event exceeds a specified loss amount. The CEP is equal to 1-CDF of the severity curve used by actuaries in other contexts

Grossi 2 723


B. Generating an Occurrence Exceedance Probability Curve

1. Expected loss for a given event E[L] = piLi, where

pi – probability of the occurrence of event Ei

Li – loss associated with event Ei

2. Average Loss Event

ALE =

i piLi

3. Occurrence exceedance probability of event Ei if only one of each event occurs in a year

OEP(Li) = 1 – j=1

i (1 – pj)

4. Probable maximum loss (PML) – “subjective risk metric . . . associated with a given

probability of exceedance specified by the insurer” a. Loss amount corresponding to a selected acceptable probability level

b. Alternatively, loss amount corresponding to a return period c. Return period is the inverse of the annual probability of exceedance

C. Stakeholders and the Occurrence Exceedance Probability Curve

1. If liability split among several stakeholders, can develop OEP curves for each party 2. Various expected losses possess inherent uncertainty 3. CV on event loss generally decreases with the size of loss

III. INSURABILITY OF CATASTROPHE RISKS

A. Overview

1. Insurance a principal mechanism for managing risk and may be compulsory for

mortgages and car licensing

2. Insurance pricing can signal riskiness of an activity, e.g., higher prices for younger drivers and smokers

3. But regulation may prevent insurance prices from fully reflecting risks 4. Question of how to allocate catastrophe risks in a similar way to the allocation of

noncatastrophe risks a. Sufficient data allows use of actuarial-based models for automobile coverages b. But lack of data means catastrophe models must be used for natural disasters

724 Grossi 2


B. Conditions for Insurability of a Risk 1. Two conditions for the insurability of a risk

a. Able to estimate probability of an event’s occurrence and extent of likely losses 1) Base on past data or modeling and expert assessment of a risk 2) Use data to create an exceedance probability

b. Able to set premiums for each potential customer or class of customers 1) If considerable uncertainty about a risk, higher premium may result 2) If reduced capacity because of recent losses, industry may charge higher

prices, especially if demand increases 2. Meeting the conditions for insurability does not imply profitability – may not be able to

set a rate for which there is sufficient demand and cover costs and profit 3. Factors affecting the rate level decision

a. State regulation b. Competition c. Uncertainty of losses d. Correlation of losses e. Adverse selection – situation “when the insurer cannot distinguish (or does not

discriminate through price) between the expected losses for different categories of risk, while the insured, possessing information unknown to the insurer, selects a price/coverage option more favorable to the insured”

f. Moral hazard – increased expected loss caused by the policyholder’s behavior g. Adverse selection and moral hazard not major problems for natural hazard risks

C. Uncertainty of Losses

1. Natural disaster hazards involve uncertain and potentially high losses 2. For such hazards, median loss is low but maximum loss is very high

D. Highly Correlated Losses

1. Simultaneous occurrence of many losses from a single event 2. Insurance markets do best when they have many policies whose potential losses are

spatially and otherwise independent 3. Application of the law of large numbers

E Determining Whether to Provide Coverage

1. Stone’s two constraints

a. Survival constraint – insurers seek to maximize profits while meeting constraint b. Stability constraint – less applicable in the case of catastrophe risks

Grossi 2 725


2. Following 1989 disasters insurers focus on survival constraint a. FL – insurers only sell wind damage coverage because required to do so; state

insurance pools enable them to limit risk b. CA – California Earthquake Authority makes earthquake coverage available

3. Meeting the survival constraint

a. Select portfolio with insolvency probability less than a selected probability b. Number of policies needed to satisfy the constraint

Pr[Total Loss > (nz + A)] < p1, where

n - maximum number of policies that can be written z - insurance premium A – current surplus p1 – selected insolvency probability

c. Whether risk is insurable depends on whether cost of marketing and policy issuance is sufficiently low

d. Ways of meeting the constraint 1) Increase the premium amount – will increase number of possible

policies but will reduce the demand 2) Reduce the coverage 3) Transfer some of risk to others

IV. FRAMEWORK TO INTEGRATE RISK ASSESSMENT WITH RISK MANAGEMENT

A. Framework

1. Assessment of risk using catastrophe modeling involving four modules

2. Stakeholders’ decision processes produce risk management strategies

3. Goal is to maximize expected profits subject to meeting survival constraint

B. Risk Management Strategies 1. Risk reduction measures

a. Mitigation b. Well-enforced building codes c. Land-use regulations

2. Risk transfer

a. Insurance b. Reinsurance c. Catastrophe bonds

726 Grossi 2


3. Strategies for insurers to reduce risk a. Charge higher rates to reflect uncertainty b. Change portfolio c. Transfer risk

Grossi 2 727


PAST CAS EXAMINATION QUESTIONS

1. The following occurrence exceedance probability curve is available for an insurer's portfolio:

0.0, 0.75

0.5, 0.50

1.0, 0.30

2.0, 0.185.0, 0.10

10.0, 0.020.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 2.0 4.0 6.0 8.0 10.0

Occurrence Exceedan

ce

Probab

ility

Loss (in $M)

a. Briefly explain what an occurrence exceedance probability curve represents. b. The insurer wants to hold capital to support a 1 in 25 year Probable Maximum Loss (PML).

Determine the loss level associated with this PML implied by the occurrence exceedance probability curve above.

c. Briefly discuss three common uses for occurrence exceedance probability curves. (12–8–9–1.5 Modified)

728 Grossi 2


1. a. An occurrence exceedance probability curve shows all possible levels of annual loss and the annual probability that the given loss level is exceeded.

The probability for a loss to exceed a certain level given a period of time.

b. 1/25 = .04 => PML = 10M – [((0.04-‐0.02)/(0.1-‐0.02))*5M] = 8.75M

c. (Need three)

Calculate the PML for a given payout period Calculate if the portfolio meets solvency goal Decide how much proportion of the risks should be ceded Used to calculate average annual loss Set level of conservativeness like PML in 1/X chance Used to find a strategy to change the portfolio if it is currently above the level of

conservativeness Used by emergency response unit to determine where damage might be and build

strategies in times of catastrophes Used when running logic trees. Instead of using point estimates, each branch of tree can

have its own exceedance probability curve for the different outcomes. Then can combine. Emergency management services: to evaluate potential risk of some regions and plan

evacuation. Reinsurance Broker: to evaluate the level of risk of their portfolio and estimate the impact

of accepting new risks.

Grossi 2 729


2. The following Occurrence Exceedance Probability curve is available for an insurance company's portfolio:

Return Period

Occurrence Exceedance Probability

Loss10,000 0.0001 $200,000,000

500 0.0020 $50,000,000

200 0.0050 $20,000,000

100 0.0100 $12,000,000

50 0.0200 $7,000,000

33 0.0300 $3,500,000

25 0.0400 $1,500,000

20 0.0500 $500,000

a. The insurer specifies that its acceptable risk level is 1-in-250 year PML. Define PML and calculate the 1-in-250 year PML.

b. The insurer decides to buy property catastrophe reinsurance protection up to the 1-in-500 year

PML in the following treaties:

Quota share, where 30% is ceded up to a $40 million loss limit, which inures to the benefit of the following:

• 100% placed 1st layer property catastrophe excess of loss treaty $6 million xs $4 million

• 90% placed 2nd layer property catastrophe excess of loss treaty $10 million xs $10 million

• 75% placed 3rd layer property catastrophe excess of Loss treaty $30 million xs $20 million

During the treaty year, the insurer suffers a $45 million earthquake loss. Calculate the amount of loss ceded to each of the reinsurance treaties and the net retained loss by the primary insurer.

(13-8-24-1/1.25)

3. Define the following terms:

Occurrence Exceedance Probability Aggregate Exceedance Probability Conditional Exceedance Probability

(14-8-MTS-0.5/0.5/0.5)

730 Grossi 2


2. a. PML = Probable Maximum Loss – largest loss likely to occur for the insurer 1/250 = 0.004 Must interpolate between 0.002 and 0.005

(0.004 0.002)

(20 50) 20(0.005 0.002)

50,000,000 – 20,000,000 = 30,000,000 is 1/250 PML b. Ceded to Quota Share = 0.3*40M = 12M Retained from QS = 45 – 12 = 33M Ceded to first layer = 100% * 6M = 6M (retained 4M) Ceded to second layer = 90% * 10M = 9M (retained 1M) Ceded to third layer = 75% * (33-20)M = 9.75M (retained 3.25M)

Net retained = 4M + 1M + 3.25M = 8.25M

3. Occurrence Exceedance Probability – The probability that at least one loss exceeds the specified loss

amount. Aggregate Exceedance Probability – The probability that the sum of all losses during a given period

exceeds some amount. Conditional Exceedance Probability – The probability that the amount on a single event exceeds a

specified loss amount.

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