Mahler.Credbility

Reading: Mahler, H. C., "An Example of Credibility and Shifting Risk Parameters"

Synopsis: Tested on the paper but not the appendices.

The paper deals with optimally combining different years of historical data when experience rating. The usual application of credibility, namely Data * Credibility + Prior Estimate * Complement of Credibility is used. The prior estimate of credibility is normally the average experience for a class of insureds but could also be the relative loss potential of the insured to the class average.

The goal is to understand how changes in the risk classification parameters over time results in older years of data adding less credibility to experience rating than expected.

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Estimated study time: x mins, or y hrs, or n₁-n₂ days, or 1 week,... (not including subsequent review time)

BattleTable

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

• fact A...
• fact B...

reference	part (a)	part (b)	part (c)	part (d)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)

In Plain English!

The efficiency of an experience rating plan is the reduction in expected squared error due to the use of the plan. The lower the expected squared error, the higher the efficiency.

Mahler uses data on baseball games to illustrate his points. He uses elementary statistics (binomial distribution, normal approximation) to conclude baseball teams do have significant differences between them over the years. Since they have differences between them, this means experience rating should predict future performance with some accuracy.

Next, Mahler asks if the differences in the win/loss record for a fixed team over time can be explained by random fluctuations from the same underlying distribution. He concludes the observed results cannot be explained this way, so the parameters of the distribution which describes the number of losses for a team in a year are changing over time.

One method for testing if parameters shift over time is the standard chi-squared test. Mahler groups the data into 5-year periods (other lengths could also be used) by team and uses the chi-squared test to measure if they could have the same underlying mean over the entire dataset. If a team didn't change its losing percentage over time, then its losing percentage should be normally distributed around its average. The chi-squared statistic is defined as [math]\displaystyle\sum_{i=1}^n\frac{\left(\mbox{Actual Loss}_i - \mbox{Expected Loss}_i\right)^2}{\mbox{Expected Loss}_i}[/math]. For a fixed team with n games, we have [math]n-1[/math] degrees of freedom if we have to estimate the expected loss.

Another method is to compute the average correlation between risks. #