A.4. An Example of Credibility and Shifting Risk Parameters

Question 1

Simplifications in baseball data compared to insurance data

Answer 1

1. A constant set of risks (teams).
2. The baseball loss data is readily available, accurate, and not subject to development.
3. Each risk (team) is of equal size; they each play roughly the same # of games each year.

Question 2

Tests to determine if risk parameters are shifting over time

Answer 2

1. χ^2 test.

For each risk, calculate the χ^2 test statistic as follows:
χ^2 test statistic =∑i=1 [(ActualLossesi−ExpectedLossesi)^2]/ExpectedLossesi

If there are n time period groups, the χ^2 table value to
compare against will have n − 1 degrees of freedom. If the
test statistic exceeds the tabular value with the acceptable
percentage level, then conclude that the distributions are different, and thus risk parameters for that risk have shifted
over time.

2. Compare correlations between years.

Question 3

6 methods to estimate X

Answer 3

1. Xest = µ
2. Xest = Y1
3. Xest = ZY1 + (1 − Z)µ
4. Xest = Z/n∑i=1 Yi + (1 − Z)µ
5. Xest,i+1 = ZYi + (1 − Z)Xest,i
6. Xest =∑i=1 ZiYi + (1 −∑i=1 Zi)*µ

Question 4

3 criteria to evaluate possible estimates

Answer 4

1. Least Squared Error - Calculate the Mean Squared Error of the prediction compared with the actual observed result. The Bühlmann/Bayesian credibility methods attempt to minimize this criteria.
2. Limited Fluctuation - Also known as Small Chance of Large Errors. Measure the probability that the observed result differs by more than a certain percent from the predicted result. The classical credibility method targets this criteria.
3. Meyers/Dorweiler - Calculate the correlation between the ratio of actual to expected losses and the ratio of predicted losses to overall losses. This criteria confirms that there is no evidence that large predictions lead to large errors and small predictions lead to small errors.