Bailey.Simon

Reading: Bailey, R. A. & Simon, L. J.: "An Actuarial Note on the Credibility of Experience of a Single Private Passenger Car", plus discussion paper by Hazam, W. J.

Synopsis: You should read this article with the source paper open so you can refer to the tables as you go along. The article introduces the Canadian merit rating plan used during the 1950's and 1960's. It explains a way of measuring the credibility of a single car year of experience. The note by Hazam goes into more depth about the assumptions which underpin the credibility calculations. On the exam you could be expected to calculate the credibility or test the assumptions or both.

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Study Tips

This is a short paper which is somewhat hard to extract the key points from the source first time round. Read the wiki article first and focus on understanding the old exam questions. There are only two or three ways the CAS can test this material but unfortunately every couple of years they try a fourth way on the exam...

Estimated study time: 1 day (not including subsequent review time)

BattleTable

Based on past exams, the main things you need to know (in rough order of importance) are:

• How to calculate the credibility for a group with at least one year of claims free experience.
• Be able to justify a choice of exposure base.
• How to calculate the experience mod using relative claim frequency.
• How to calculate the credibility for a group with claims in the past year.

Questions are held out from Fall 2019 exam. (Use these to have a fresh exam to practice on later. For links to these questions see Exam Summaries.)

reference	part (a)	part (b)	part (c)	part (d)
E (2018.Fall #3)	Experience Mod - apply formula	Merit Rating Factor - calculate	Mal-distribution - relation to premiums
E (2017.Fall #3)	Exposure base - recommend	Relative Credibility - calculate
E (2016.Fall #1)	Relative Credibility - calculate
E (2015.Fall #1)	Exposure Base - explain choice	Credibility - calculate	Credibility - calculate
E (2014.Fall #5)	Exposure Base - discuss assumptions	Credibility - calculate	Premium - calculate
E (2012.Fall #6)	Exposure Base - Recommend & Justify

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In Plain English!

Bailey & Simon's Paper

Canadian merit ratings depend on the number of full years since the insured's most recent accident or, if they have had no accidents, the number of full years since the insured became licensed.

Merit ratings of A, X, Y, and B are available and these correspond to three or more years, two years, one year, and no years since the most recent accident or since licensing respectively. Merit rating groups may be aggregated; for instance, grouping A and X together is denoted by A+X and this gives the experience for two or more accident free years.

Earned premiums in the study are on-levelled to account for prior rate changes and also modified to a common basis to account for differences in premiums between the merit ratings. That is, the merit rating factor is backed out of the premiums and the result is known as the adjusted earned premium.

Relative claim frequency is calculated using premiums rather than earned car years to avoid distortions due to higher frequency territories producing more X, Y, and B risks and consequently higher territorial premiums. The papers refer to this as correcting for maldistribution.

The data used in the paper is split into five classes labelled 1 – 5. Class 1 is broadly defined to include vehicles used for pleasure with no male operators under 25 years old. Classes 2-5 are defined much more specifically.

Each class contains two policy years of data for merit plan ratings A, X, Y, and B. The claim frequency per $1,000 of premium is calculated for each class and merit rating. These are normalized so the overall class has a relativity of 1.000.

The paper uses the following experience modification formula [math]\mathrm{Mod}=ZR+(1-Z)\cdot1[/math], where Z is the credibility and R is the ratio of actual losses to expected losses for the past policy term. The complement of credibility is the ratio of 1, i.e. actual losses equal expected losses. Since risks with at least one year accident free by definition have no losses in their previous term, we know R = 0 for risks in groups A, X, Y, A+X, and A+X+Y.

It's helpful to recall that experience rating attempts to measure the deviation of an individual risk from the average risk. Whereas class ratemaking is the process of finding the average. An increase in the volume of experience increases the reliability of the indication in proportion to the square root of the volume.

By setting the modification equal to the rebased claim frequency and setting R equal to the ratio of the merit rating group experience to the experience for the whole class, the experience modification formula can be used to solve for credibility, Z.

We can compute the credibility for merit rating plan combinations A, A+X, A+X+Y, which are claim free for 3+, 2+ and 1+ years respectively. Since [math]R=0[/math] in each of these situations the experience modification formula reduces to [math]\mathrm{Mod}=1-Z[/math].

Now go try parts (b) and (c) of 2014 Q5.

2014 Q5 Example & Practice Problem

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The paper looks at the claim frequency per earned car year as well as the ratio of 3-year credibility (merit rating plan group A) to claim frequency. If the variation in the individual insured's likelihood of an accident is the same within each of Classes 1 — 5, the experience rating credibility should be proportional to the average claim frequency. The results (see Table 2 in the paper) show the ratio of 3-year credibility to claim frequency is much lower for Classes 2 — 5 than Class 1. Hence, the credibility of the experience rating depends on the volume of data in the experience period and the amount of variation present in the individual hazards in the class.

The paper looks at normalizing the 1, 2, and 3-year credibility by class using 1-year as the base. The authors note that if the individual insured's accident likelihood remains constant from year to year and the risks in the classes remained constant, then the credibility should be proportional to the number of years. However, this isn't observed. Probable reasons include:

1. Risks are entering and leaving each class. For instance, people get older and age out of the under 25 class.
2. An insured's likelihood of an accident occurring changes during the year and from one year to the next. Alternatively, the risk distribution of insureds within a class may have material skewness that reflects different levels of accident likelihood.

Let's now turn our attention to calculating the credibility of subclasses where at least one claim occurs during the past year, i.e. subclasses 1B, 2B, ... , 5B. A key assumption we'll make is the observed claim frequency for the class is representative of all subclasses. For instance, Class 1 has a claim frequency of 0.087 (see Table 2 in the paper and note this is calculated using the earned car years found in Table 1). We assume subclasses 1A, 1X, 1Y, and 1B all have this claim frequency.

We will also suppose the subclass claim frequency has a Poisson distribution with mean λ.

If we consider the subgroup of Class 1 insureds that have at least one claim in the past year (Class 1B) then (by assuming Class 1B has the same mean claim frequency as Class 1, i.e. [math]\lambda=0.087[/math]) the expected number of claims for a risk in Class 1B is [math]\frac{0.087}{1-e^{-0.087}}=1.044[/math].

This is because if there are N risks in the Class 1B then we expect Nλ claims in total. However, some of the risks in Class 1B will have 0 claims while others will have more than one claim as since having a claim in the past year doesn't guarantee you'll have one in the next year. The probability of a risk having no claims is [math]e^{-\lambda}[/math], so the probability of a risk having at least one claim is [math]1-e^{-\lambda}[/math]. Hence there are [math]N\cdot(1-e^{-\lambda})[/math] risks in Class 1B which have at least one claim. Therefore the ratio [math]\frac{N\lambda}{N\cdot(1-e^{-\lambda})} = \frac{\lambda}{1-e^{-\lambda}}[/math] is the expected number of claims for a risk in Class 1B.

Referring back to the formula for the experience modification factor, [math]\mathrm{Mod}=Z\cdot R +(1-Z)\cdot 1[/math], let [math]R=\frac{1.044}{0.087}=12[/math]. That is, let R be the expected number of claims for a risk in Class 1B divided by the expected number of claims for a risk in Class 1, λ.

Now use the relative claim frequency per $1,000 of premium as the modification factor for Class 1B. That is, the relative claim frequency for Class 1B is the Class 1B claim frequency per $1,000 divided by the Class 1 claim frequency per $1,000. Using the data in Table 1 this is [math]\mathrm{Mod}=\frac{2.190}{1.484} = 1.476[/math].

Plugging these figures into the experience modification formula, [math]\mathrm{Mod}=ZR+(1-Z)\cdot 1[/math], we can solve for the credibility, Z. We get [math]Z=0.043[/math] for Class 1B which is very close to the 1-year credibility [math]Z=0.046[/math] for the Class 1 (see Table 2).

When reading Table 2 the credibility is referring to the credibility of the number of years claims free. This means the credibility for 1-year is based on data for merit rating group A+X+Y for Class i, the 2-year credibility is based on data for merit rating group A+X, and the 3-year credibility is based on data for merit rating group A.

Since the credibility calculated for Class 1B is very close to the 1-year claims free credibility for Class 1 and both are materially non-zero, the authors deduce the credibility of the experience for a single car year is measurable and non-trivial.

Key Point: When performing the above calculation make sure you use earned car years to set the Poisson distribution parameter λ but then use claim frequency per $1,000 premium everywhere else.

The analysis is based on accident frequency in order to limit fluctuations due to claim severity. If you repeat the analysis on a set of data which is large enough to make claim severity reliable (such as Class 1) then you get very similar results using a loss ratio approach instead of claim frequency.

The paper concludes we can measure the credibility of one car for one year and it is non-trivial. However, this method isn't much use in a highly refined rate classification plan. Lastly, adding more years of experience improves the credibility for a single vehicle at a decreasing rate.

Hazam's Note

The key point of the Bailey & Simon paper is we can assign significant and measurable credibility to individual cars that have been claim free for one, two, or three or more years.

A weakness is the paper relies on premiums when developing frequencies. Using premiums to account for differences in frequency distribution is only going to work if high frequency groups are also high premium groups. Further, the differentials between groups must be accurate.

To see how you can apply this, try solving 2012 Q6.

2012 Q6 Example & Practice Problem

Another point made by Hazam is the relativities between the theoretical 1-, 2-, and 3-year credibilities do not vary in proportion to the number of years. That is, the relativities aren't 1x, 2x, and 3x respectively. Instead, they are less.

This is seen using the experience rating credibility formula [math]\frac{P}{P+K}[/math] where the one year credibility from the paper, 0.046, is used along with an assumption of 100 claims per year to derive the K value of 2074. By moving to 200 and 300 claims, we consider the 2-year and 3-year credibility under this fixed K value. By rebasing to use the 1-year result as the base, the relativities are 1.00, 1.91, and 2.74 respectively. These are higher than observed in the paper but are lower than if the credibility varied in proportion to the number of years.

One way they have addressed this material in past exams is seen in 2011 Q1. Try it now!

2011 Q1 Example & Practice Problem

The comment ends by remarking all of the conclusions were drawn using claim frequencies only. If information about conviction frequencies were added then maybe there would be greater support for the magnitude of experience rating credits being offered in the US in the 1950s-60s.

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