A4. Mahler

Question 1

3 simplification are found in Mahler’s baseball data but not in insurance

Answer 1

1. a constant set of risks
2. data is readily available, accurate, and does not develop
   3/ each team is equal size and play roughly same number of games

Question 2

2 methods to test whether risk parameters change over time

Answer 2

1. Chi- squared test
   - Test statistics = summation of (Actual i - expected i )^2 / expected i
   df = number of data - 1
2. Correlation Test
   - Group data by pairs based on time lag
   - calculate the correlation for each pair
   - calculate the average correlation by time lag
   - if the correlation decreases as time lag increases, then risk parameter shift over time

Question 3

3 criterias to compare the performance of credibility models

Answer 3

1. Least Squared Error
   - determine the optimal credibility with the combinations of Z that results in smallest MSE.
   - minimize the squared error between the actual and predicted results
   SSE= summation (X est - X actual )^2
   MSE = SSE / (# of teams x # of years)
2. Limited Fluctuation (AKA small change of large errors)
   - Minimize the likelihood that any one actual observation will be a certain percent different from the predicted results.
   - Optimal credibility will minimize Pr( abs(X est - X act) / X est >K%)
3. Meyers/Dorweiler
   - minimize the correlation between the ratio of actual/predicted and predicted/avg actual
   -Instead of minimizing the prediction error, this focuses on the pattern of errors
   - calculate the correlation between Vector 1 (actual loss % / predicted loss %) and Vector 2 (predicted % / overall average actual loss %)

Question 4

6 options of credibility weighting

Answer 4

1. Xest = mu - grand mean
2. Xest = y1 (give last year 100% credibility)
3. Xest = Zy1 + (1-Z)mu (credibility weight of 1 and 2)
4. Xest = Z/n summation(yi) + (1-Z)mu (same as 3 but use more years)
5. Xest = Zy1 +(1-Z)Xest,i (exponential smoothing, give credibility to prior estimate)
6. Xest = summation (Zi*yi) + (1-summation(Zi))mu - more general formula