A4. An example of credibility and shifting risk parameters

Question 1

Simplifications in baseball data compared to insurance data

Answer 1

1. constant set of risks (no change in mix of business => baseball teams do not change over time)
2. loss data readily available (no development before loss is at ultimate => baseball scores are fixed after each game)
3. each risk is of equal size* (no change in exposures => baseball teams each play the same number of games per year)

Question 2

Mahler also observes that a team that has been worse than average over one period of time is likely to continue to be worse than average over another period of time, which implies there is value in using past experience of the team to predict future experience of that team.

2 tests to see if parameters shift over time of wining %

Answer 2

1. Chi-squared test

2. Correlation test

Question 3

3 criteria to evaluate the performance of credibility methods

Answer 3

1. Least square error
   - minimizes the mean squared error between predicted vs observed
   - seek to eliminate large error
   - Buhlmann and Bayesian credibilities use this criteria
   - Advantages: solvable equation, mitigate against large individual error
   - best use for experience rating
2. Limited fluctuation (small chance of large errors)
   - minimizes the likelihood that any observed will be a given % different from the predicted
   - smaller probability => better method
   - Classical credibility uses this criteria
   - seek to eliminate large error
   - Disadvantage: not easily solvable equation
3. Meyers/Dorweiler (correlation)
   - minimizes the pattern correlation between observed/predicted vs predicted/(average observed)
   - not concerned about large errors
   - assumes large predictions do not lead to large error (and small predictions do not lead to small error)
   - corr closer to 0 => better method
   - Advantage: mitigatepattern of error in prediction
   - Disadvantage: does not reduce the magnitude of large error

Question 4

Interpretation of the exponential smoothing credibility method

Answer 4

leads to exponentially decreasing creadibilities assigned to the older year’s experience

Question 5

Conclusions of paper

Answer 5

1. Optimal credibility:
   - experience should usually be given 60%-80% weight
2. if there are shifting parameters:
   - the credibility estimate gets worse as more years of data are used
   - older years are less relevant in predicting futur (and recent years are more relevant)
   - older years should have less credibility (and recent years should have more)
3. if there are delays in getting the historical data:
   - the credibility estimate is less accurate
   - SSE of estimates will significantly increase
   - optimal credibility significantly decreases

Question 6

is there an inherent difference between baseball teams.

Answer 6

calculates the average and standard deviation of the losing percentage of each team

1. Binomial test
   -Test if all teams win the same amount
   n is the number of games played
   p is the probability of losing (assumed to be 50%). Since there are many teams outside of 2 standard
   deviations of the mean losing percentage (divide the standard deviation by n to put it in percentage
   terms), Mahler concludes that there are inherent differences between the teams.