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By Timothy Paris, October 21, 2009 8:54 am

Credibility theory often utilizes the following basic formula to calculate a credibility-weighted estimate:

Estimate = Z * [Mean of current observation] + (1-Z) * [Prior mean],

where 0<=Z<=1.

Z is the credibility ascribed to the mean of the current observation.  The prior mean may be based on prior observed data or some type of benchmark.  The question is – how do we determine Z?

Three approaches are commonly used:

• Under the Limited Fluctuation (LF) approach, also referred to as classical credibility or American credibility, it is assumed that the current observation of data includes independent trials and that the Central Limit Theorem holds.  In order to limit the effect that random fluctuations may have on the estimate, a confidence interval approach is used to determine how much data is required in the current observation in order to assign full credibility (i.e. Z=1), and how partial credibility (i.e. Z<1) should be calculated with smaller amounts of data.

• With the Greatest Accuracy approach, also referred to as Bühlmann’s approach or least squares credibility, the objective is to minimize the square of the error between the estimate and the true expected value of the quantity being estimated.  The credibility Z increases asymptotically toward 1 as the amount of data in the current observation increases, but also depends on an analysis of variance – how is the total variance comprised between the expected value of the underlying process variance and the variance of the hypothetical means?  As the former increases, Z decreases, but as the latter increases, Z increases.

• Bayesian Analysis utilizes Bayes’ Theorem, conditional probabilities, and distribution hypotheses to develop the estimate.  It can be shown that estimates using the Greatest Accuracy approach provide the best “least squares fit” to Bayesian estimates.

Each credibility approach includes subjective elements, and the actuary is often free to choose the approach that he or she deems appropriate for the situation and data.  Key considerations include the existence or reliability of the prior mean, the reliability of distribution hypotheses, the desired accuracy of the estimate both in overall terms and with respect to sub-classifications, regulatory or industry conventions, computational convenience, and ease of communication to technical and non-technical audiences.

In our recent work with VA industry studies, we have found that the LF approach provides a reasonable balance of technical sophistication, computational convenience, and ease of communication.  Importantly, we have also found that with appropriate choice of parameters, the LF approach produces results that are reasonably consistent with our intuition and that of our clients.  We believe that these attributes are particularly important in this type of situation, where we are trying to use the current observation to develop highly credible industry benchmarks where none existed before.

Hence, we view the LF approach as a natural starting point, and we will likely continue to use it in our ongoing VA industry studies, but this does not preclude further refinement or utilization of other credibility approaches as circumstances warrant.  In the next article, we will share details about how we have applied the LF approach to our VA industry studies.

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By Timothy Paris, October 20, 2009 12:18 pm

As actuaries, we are quite familiar with the rigorous use of double dots and bars in mathematical symbols and equations, and we often use software packages that facilitate their rendering in electronic documents.  However, it is sometimes tempting to ignore such diacritical marks when using foreign words or names in English.  This is particularly true for electronic documents — I see the tilde and circumflex on the keyboard, but how do I put it over the letter?
Of course, computer keyboard settings can be adjusted to accommodate non-English characters, and certain font settings also display as some of these characters.  But as a shortcut, it is useful to know that many of these characters can be produced by simply using the ALT key and a four digit code on the number keypad in the course of your normal typing.  For example, you need ALT-0252 to properly type “the Bühlmann credibility factor approaches one asymptotically”, and ALT-0244 for “Itô’s Lemma is used in the derivation of the Black-Scholes option pricing formula”.
You can find reference materials and other codes with an internet search on “diacriticals” or “international symbols”.  But be warned — those with a certain musical affinity and/or blatant disregard for the appropriate use of diacriticals may find some combinations difficult.  For example, in spite of our best efforts, we have been unable to code the dotless i or umlaut n as parodied in the movie “This is Spinal Tap”, which is troubling.  Then again, in the immortal words of the great David St. Hubbins, “It’s such a fine line between stupid and clever”.

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By Ruth Ann Woodley, October 20, 2009 8:30 am

Several Ruark employees will be at the Society of Actuaries’ Annual Meeting next week in Boston.  Peter Gourley, Mike Loftus, Sally Osit, Tim Paris, Rich Tucker and Ruth Ann Woodley will all be attending.  Ruth Ann will also be moderating the session “Project Management Success Strategies,” presented by Doris Orr at 10:45 Wednesday morning.  Be sure to let us know if you will be there too, or seek us out in Boston!

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By Timothy Paris, October 7, 2009 11:45 am

Credibility Theory is a branch of actuarial science that attempts to determine the extent to which data can be trusted to inform estimates of the future. While the concept of credibility is often associated with casualty actuaries, it is relevant to virtually all fields of actuarial endeavor. The academic literature on this subject is extensive, and includes several different approaches with a range of mathematical sophistication.

At RCL, we have been particularly mindful of credibility during the course of our VA industry studies, starting with our mortality study in 2007, our surrender study in 2008, and continuing with our benefit utilization study (in progress) in 2009. We believe that credibility theory has an important part to play in studies such as these, and in subsequent articles we will describe the considerations and approaches we used in applying credibility theory to our work.