Tag Archives: actuarial exams

Risk Theory (I)

31 Jan

The two chapters Risk Theory I and Risk Theory II are obviously related. Together, they cover the individual and collective risk models, but in addition to the mathematical treatment there is also a lot of general insurance general knowledge. Hence, Risk Models 1 starts with an explanation of the desirable features of a risk portfolio, from a non-numerical perspective.

This beginning material about Risk theory emphasises stochastic processes which model counts – in order to model a count of claims on an insurer, known as a Poisson process – and how they can be combined with other statistical distributions – which can model the size of the claims.
Three compound distributions are introduced to help model these processes – the compound Poisson distribution, compound binomial distribution and compound negatvie binomial distribution. Each is good for in different conditions

Poisson is a good all purpose count distribution

Binomial is good where the portfolio size sets an upper limit on the possible number of claims

Negative binomial is an alternative to the Poisson in that it doesn’t set an upper limit, but allows for a variance greater than the mean.

In each case we are trying to model an integer number of claims, which can be extended by considering the severity of a claim to model the total amount payable across all claims. Hence, for example, arrive at the compound Poisson distribution, where the number of claims is modelled by the Poisson distribution, and the severity is modelled independently and separately for each claim by another distribution. For example, you could potentially model a team’s score in an AFL football match by using a Poisson distribution to model the number of scoring shots, and a Bernoulli trial on each scoring shot to determine if it was a goal worth six points or a behind worth one point. This would obviously simplify to a Binomial distribution, with the ‘n’ variable itself a random variable with a Poisson distribution.

Chapter 5: Credibility Theory

6 Nov

This is likely to be a little bit shorter than usual, but  there will be more on this topic.

One of my goals in this blog is to identify cheap or free (most likely online) resources. Although it seems straight forward to find academics who have posted lecture notes covering Financial Mathematics (CT1) or the standard undergraduate probability and statistics covered in CT3, it is pretty hard to find stuff relating to the core of CT6 (as distinct from some of the mainstream statistic topics such as GLMs or Time Series).

Hence for the first bite at Credibility Theory, I want to briefly mention some of the free resources on this topic made available by the American Associations. The material in what is called Learning Outcome H for Exam 4/C for Casualty Actuaries Society can be studied either using the text by Willmot, Panjer and Klugman, which is also suggested reading for the Institute and Faculties CT6, or it can be covered from one of two study notes made available by the Society of Actuaries.

The first is the Credibility Theory chapter from the text called ‘Foundations of Casualty Actuarial Science’ (avaialbe from the CAS) and is available here: www.soa.org/files/pdf/C-21-01.pdf

The second is a study note called Topics in Credibility, which is here: www.soa.org/files/pdf/C-24-05.pdf

This one is specifically written for people studying Exam 4/C, partly in order to address some gaps in the first reference. However, the first is possibly more in tune with the CT6 notes in the sense that it more explicitly discusses the connection between credibility and Bayesian statistics. Both study notes in come with a number of exercises and answers.

A much briefer look at credibility for those who want to read a concise treatment before jumping into detail is:


I like the way this one discusses, to some extent, when to use the different approaches.


For a more statitiscal look at Credibility, that is not necessarily connected to the exam material, you could look at:


I like this stuff because it hierachical model and geospatial models are what I was doing at uni, but to me it also helps to build a bridge between what statisticians are doing, and what actuaries are doing. Also, it sheds some light on what analyses insurance companies are doing in real life, rather than what people study for the exams. This last is for interest, rather than for exam study, per se.

Chapter 4: Reinsurance (Equivalent CAS/SOA C/4 Section 29)

25 Oct

So far, I’ve had the inside track inasmuch as the topics were topics that were important to my earlier statistical studies. ‘Reinsurance’ however is not something that is required to be understood by all that many statisticians. (Apologies for pretending English is Latin, and trying to use the ablative absolute)

First, let us summarise the chapter:

We start with some terminology. Reinsurance is a method of risk transfer from one insurer to another. This chapter looks at three simple forms:

  1. Proportional reinsurance. Does what is says on the tin
  2. Excess of loss reinsurance. The insurer is responsible for paying the claim to a value M. Above M the reinsurer is responsible, until an excess limit reach, at which point the original insurer is again responsible
  3. Stop loss insurance. The reinsurers has the responsibility to pay the amount owing over a claim amount M with no upper limit.

We saw in the last chapter that given a set of claim data, it is relatively straight forward to model the size of claims – one can check various distributions for their goodness of fit. The meat of this chapter is how to convert the expressions for conditional probability from both the original insurer’s and the reinsurer’s point of view into tracatble algebraic expressions.

To provide the future actuary with the tools to apply these expressions, we are given  formulae for the moments of lognormal and normal, to simplify finding the moments of these distributions without needing to explicitly integrate the cdf, necessary we suppose due to the difficulty of integrating the Gaussian distribution.

There is also some salad in the form of discussions of the implications of inflation (if the size of claim is changed by inflation, but the retention limit is stationary, who does this effect the probability of exceeding the limit?), the difficulty in estimating the values of estimators if the data is censored by the retention limit e.g. claim amounts above the retention limit only show as the retention limit and a simple statement of fact that an excess paid by a consumer policyholder holding a general insurance is mathematically identical to the position of an insurer holding a reinsurance policy.


Chapter 1: Games and Decisions

7 Oct

This is my first real post. I tried to express in my introductory post that my intention in this blog was to challenge myself to look Institute and Faculty of Actuaries’ Core Technical material in a way not necessarily suggested by the material itself.

The first chapter of CT6 Statistical Methods is a brief look at Game Theory and Decision Theory.

In this chapter some of the essential terminology of these two related topics is introduced.To wit:

  1. Dominated strategies
  2. Maximin criterion
  3. Saddle point strategy
  4. The Bayes criterion

I think the authors’ of this part of the notes which is that the opportunity to use whimsy in the examples – one of the best opportunities throughout the Core Technical series, given it is hard to find spice in interest theory calculations. In this regard, I recommend Luce and Raiffa’s Games and Decisions (1957) available from Dover for no more than semi-drinkable bottle of wine (in Australian bottle shops, anyway. Depending on the country you may get anything from a completely undrinkable to really quite decent for that money). Note though that out of fourteen chapters, only two (chapter 4 and chapter 13) really correspond to the material in CT6, though.

While the above objection is only half serious, another way in which the Luce and Raiffa treatment is more interesting, is that for each classification of game it discusses, it gives an example game either of research interest (whether for theoretical or practical reasons). In particular the egg craking story ( borrowed from Savage) used to motivate statistical decision theory is a far better illustration that a statistician tossing a coin used in CT6. Although, as a married man, I am a little distracted by the questions thrown up by the opening sentence of this example – ‘Your wife has broken five eggs into a bowl when you…volunteer to finish making the omelet’. The problem is to decide whether or not to check if an egg is rotten before craking it into the bowl and either making a large omelet or ruining five eggs.  If it was not your wife, but you girlfriend, daughter, kitchen hand (to your chef) or housemate, how would your decision process change?

In the end there is a purpose to fitting game and decision theory to real life situations, so to redress the lack in the CT6 notes I offer an inversion of the problems suggested there based on TV’s The West Wing.

In the final series, a Republican and Democratic presidential candidate are in the race to become Bartlet’s successor. In the episode Duck and Cover a nuclear power plant whose approval was made possible by the lobbying of the Republican candidate goes into meltdown, and at least one of the repair crew dies after trying to fix it. The respective campaign managers must decide whether to put out a statement telling the world about the Republican candidate’s role. For the Republican candidate, putting out a statement first could minimise the political damage because their side of the story will get told first, but if there is no statement from the Democrats, it will reveal the existence of the connection. On the other hand, if the Democratic candidate points out the connection to the press, it could either be helpful to their campaign if t harmful depending on whether there is a Republican press statement available to neutralise it. Propose a pay-off matrix which describes this situations. Is there a spy proof strategy?

The example above operates under the rules of decision making under certainty. In the show, there is an element of uncertainty in the form of whether the press discovers the connection and when. Adding this element to the problem above makes for a more complicated example.