TIA’s Exam 4/C Blog

Since I’m not doing a live seminar for the October sitting, I have temporarily added a copy of the live seminar notes and problems to the online seminar. I cover the material slightly differently in the live seminar, with the most obvious difference being that I break what is Section A in the online seminar into two halves, one of which is at the start of the notes (Loss Models pt I) and the other is at the end of the notes (Loss Models pt II). The second piece is stuff that is more self-contained and not really needed for any of the other topics (and some of it is material that is unlikely to be tested, such as discretization).

A less obvious difference is that some of the explanations are somewhat different (better?) in the live seminar notes. In particular, there are some timeline pictures in the discussion of risk sets for B.2 in the live notes that might be helpful to look at. Otherwise, the live seminar notes tend to be shorter and might therefore be useful when reviewing.

dave Uncategorized

Another common question that people have is how to figure out what to use for n when figuring out the credibility factor Z with Bühlmann (or more often either empirical or Bühlmann-Straub questions as that is where it gets trickier). There are a couple of things that you can keep in mind. One is that the value for n that you use in finding Z is the same as the denominator that you use in finding X-bar for the insured whose credibility you care about. I mention that since typically people are okay in finding X-bar (or X_A – bar in the empirical case) and it is the same logic as for n.

The other thing that you can look for is that n is the number of exposure units, or if you prefer, units of data, that you have. For example, in D.2 # 17 we are told that the number of claims per month for one insured has a Poisson distribution, so one unit of data is what happens to one person in one month. In the first month of our data, we have 100 insureds, so that is 100 data points for that month, etc., and that is why our n is 100 + 150 + 200.

For comparison, in D.2 #14, we are given that the claim size X has a certain distribution, and want the expected value of the next claim. That means we are looking at one unit of data = one claim, and our n = # of claims.

In order for n to be the number of months in #17, we would need to either have our data be for one single insured, so one month = one month for one insured = one data point, or we would need to change the description of the number of claims to only refer to the number of claims in a month. For example, we could say “The number of claims incurred in a month by a policyholder is … ” and then have our data be “The observed data for one policyholder is … ”

One hint that you can use in the problems is that they will almost always use the phrasing Bühlmann credibility when n is the number of years (or months) of data, or the number of claims. They will usually call it Bühlmann-Straub when we are in a case like #17 in which the number of data points per month varies (usually by varying either the number of insureds, as done here, or by varying the group size), and in that case typically your n will be the sum over all the months (or years) of the number of insureds (or number of people in the group).

Taking a peak at an empirical example, in D.2 #32, we want the estimate of the claim frequency per vehicle for Insured A in year 4. Since it says “per vehicle” then one exposure unit = one vehicle for one year. We are making an estimate for Insured A, so we only care about that insured for our n, and have 2 vehicles (=2 exposures) in the first year, 2 in the second, and 1 in the third for an n of 2 + 2 + 1 = 5.

dave Credibility

One of the standard questions that people have is when to use the empirical variance [1/n]*sum[(X_i - X-bar)^2] (aka biased estimate of the variance) versus the unbiased estimate [1/(n-1)]* sum[ (X_i - X-bar)^2].

Essentially the idea is that we want to use the empirical (biased) variance whenever the empirical distribution is the key to our definitions. That happens when we are using the method of moments estimate or when we are using the bootstrap method, as both are defined in terms of the empirical distribution. It also happens if we are in a case in which the MLE = MoM estimate of the variance, which happens for a normal distribution. Those are the main cases in which we want the empirical variance.

We want to use the unbiased estimate when we are doing anything with credibility theory, or simulation material other than the bootstrap method.

Those are the main situations in which we are estimating the variance on the exam. Other cases are less clearly spelled out in the text, so when they do appear on the exam it tends not to matter which you use. For example, that has been the case in old questions in which you are asked to choose between (a, b, 0) distributions by comparing the mean and variance. Still, when in doubt it I would tend to use the unbiased estimate.

dave Variance

As the live seminar is fast approaching, I briefly wanted to let you know what to expect and bring to the live seminar. Also, I will be out of the country for the week prior to the live seminar, so if you have questions you need to either get them to me by 3/18 or wait until the seminar itself. You can also find a handout (Live_Seminar_Packet_4C) with much of this information.

You should bring:

• A printed copy of the live seminar notes
• A copy of the exam tables
• One copy of the seminar problems. These are posted in two formats: 1 problem per page, or a more condensed form. My recommendation is to bring the 1 problem per page version so you can work the problems on that page, but you can certainly bring the condensed form and a notebook instead
• A calculator, which really should be the TI-3oXS MultiView

To download the handouts, first login, navigate to the live seminar page, and then expand the “Handouts to Bring to Seminar” section. Clicking the green arrows will let you download the handouts.

If you think you will have the energy to do work at night, you can also bring what is labeled as “Problem Set 1″ in the live seminar section. These are the end of section problems from the online seminar, so they are sorted by topic and you can work on them in the evening. Not everyone chooses to do this and many people prefer to relax in the evening instead.

We will meet from 8-5 M-Th, with a 1 hour lunch break, and from 8-11:30 on Friday. There will be a short break roughly every hour. We have to be out of the room by 5 each afternoon, so I will try to end around 4:45 so I can be available for questions afterwards.

dave Uncategorized

I’ve posted typed solutions to the new review problems (209-240). A couple of them are on the terse side and might be expanded later.

I’ve also made some minor changes to the problems that probably aren’t worth re-printing: on #212, I changed the notation for the aggregate loss variable from X to S, on #229 I spelled out more explicitly that n=# of data points, and on #239 I clarified that we are using the MoM for the standard parameters for a zero-modified geometric (this is important as the method of moments estimate is affected by what parameterization you use, unlike the much nicer MLE, but the parameters that I am using are probably the ones you would have guessed to use).

dave Uncategorized

I’ve added one short lesson and about 40 problems to the seminar. The new lesson (A.2.8) is on frailty models, which is a relatively rarely tested topic that has started to come up recently on the exam. There are also a couple of new problems in A.2 related to frailty models. In addition, I’ve added 32 more review problems, as well as some problems to A.1, A.2, A.3, B2, C2, and C3. Some of the new end of section problems are on topics that are newish to the syllabus and also show up in the new review problems. For example, I’ve added a number of questions on (a,b,1) distributions (in particular, finding MLEs and MoM estimates), and there are some of those in both the new C.3 problems and the new review problems.

If you are taking the exam in February and are relatively low on study time, I would still recommend looking at the new review problems and the new lesson, but the other new problems are somewhat redundant with the new review problems and can be skipped.

In addition to the sections with new problems, all of the problem sets and solutions are flagged as updated. The reason for that is that I have made some cosmetic changes to all of the typed solutions, plus re-compiled the solutions to make sure that all of the typos that have been fixed on my computer are also fixed on the website. Other than the six sections with new problems (A1, A2, A3, B2, C2, C3), these changes are minor enough that I wouldn’t recommend printing new copies of the problem sets/solutions.

Edit: The new review problems also include essentially all of the SOA 289 problems that were previously missing from either the end of section problems or other review problems. There are still 1 or 2 problems from the SOA sample problems that I don’t include because they are old exam M problems that use life contingency notation that is not on the 4/C syllabus and I haven’t yet figured out a good way to rephrase them to make them relevant for the our exam.

dave Uncategorized

I’ve added 3 more bootstrap problems to the end of the E.1 problem set. Currently these problems just have typed solutions, but I’ll work on videos soon. I’ve also corrected some minor typos on the solutions of some other problem sets.

dave Uncategorized

Since Bayesian credibility is one of the harder topics on the exam, I wanted to address a couple of common questions so you don’t have to keep hunting back through the lessons.

First, the Bayesian credibility premium is often easily linked to the posterior mean. If N has a Poisson distribution with mean lambda, then E[N]=lambda if lambda is fixed. Applying double expectation, E[N] = E[E[N | lambda]]=E[lambda], so our unconditional mean of N = the prior mean of lambda, and the predictive mean of N = Bayesian credibility premium = posterior mean of lambda. If N is a binomial with parameters m=3 and q, then E[N]=mq=3q, and we get unconditional mean of N = 3*prior mean of q, and predictive mean of N=3*posterior mean of q. In other words, if we want to find the expected value of N given some data, we often first find the expected value of our parameter, given the same data (i.e., find the posterior mean) and then use whatever usual relationship we have between N and our parameter to find the Bayesian credibility estimate.

Second, Bayesian = Bühlmann whenever the Bayesian credibility premium is a linear function of X-bar. This happens for the easier conjugate priors (Poisson/Gamma, Beta/Binomial, Normal/Normal) but not some of the harder ones (Lognormal/Normal) and also tends not to hold if you don’t have a conjugate prior.

Finally, a lot of people are comfortable with using conjugate priors but not with finding the posterior distribution from first principles. One way to work on that is to work problems involving conjugate priors twice: first, do them the way that you would on the exam and use what you know about conjugate priors. Then redo the problem from first principles. That way you will get practice working the problem from first principles in a case in which the answer will work out relatively nicely and so you can use it to just practice the techniques. If you only use first principles in the problems that can’t be done with conjugate priors, then you will only be using first principles on the hardest questions without a good warmup.

dave Uncategorized

We have added a CBT format to the sample exams. This means that you now have the choice of printing out the practice exams and taking them as a traditional paper and pencil exam, or you can take it on your computer in a format that is very similar to what you will see on the exam day. There are a few differences between our CBT version and the SOA’s version, but we think it is close enough that it will be good practice. In particular, you can get used to using the online exam tables.

For a demo of how to use the sample exams, we have a sample videos that you can watch in either QuickTime or WMP format. When you take a CBT sample exam, the exam will end whenever time runs out or if you click the “End” button on the review screen. At that point, your exam will be automatically scored. Note that closing the exam window will exit the exam without saving your progress or grading anything.

One feature that you should experiment with is the ability to mark questions for review. At any point during the exam, you can click the “Review” button on the bottom of the screen and that will give you the option to review either all of the exam problems, only those problems that you have marked for review, or only those problems that you haven’t answered. For example, one possible exam strategy is to make a first pass through the exam working all of the problems that you can do quickly, marking problems that you think you can do but will take a long time, and skipping problems that you don’t know how to do. Then when you are done with your first pass, you can select “Review Marked” and do the longer problems that you know how to do. After finishing that, you can review the unanswered problems that you didn’t initially know how to do and either guess or take a second stab at them.

As the CBT format is a new feature, we welcome any comments and suggestions that you may have for it. In particular, if you have taken the official exam and can let us know about some format changes that we need to make we would really appreciate it. There will always be some differences since we have to deal with different browsers and screen sizes, but we would like to minimize them. Known differences are:

1. On the official CBT exam, if the problem is long enough that you need to scroll down to see all of the answer choices, you must scroll down before selecting your answer. We do not force you to do so.

dave Uncategorized

I’ve added two new study schedules for those of you taking the November exam. The one that most of you should use is a spreadsheet that generates a study schedule based on the actual date of the CBT exam that you sign up for (this has that advantage that it can also be used for other exam sittings, not just this one). The second schedule is a PDF that has the live seminar dates hard coded in. The live seminar is earlier than usual this year, so that schedule has more review time built into it but also starts earlier than the spreadsheet does.

dave Uncategorized