Mahler - shifting risk parameters

Question 1

Mahler uses baseball team win/loss records to

Answer 1

illustrate the potentially significant impact of shifting risk parameters on credibilities and experience rating

Question 2

-Mahler notes 3 simplifications that are found in baseball set that are not found in insurance data

Answer 2

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

Question 3

standard formula used for credibility weighting

Answer 3

new estimate = data*credibility+(1-credibility)*prior estimate

-often prior estimate is class average or previous estimate for particular insured/group

Question 4

to answer whether there is an inherent difference between baseball teams

Answer 4

-Mahler calculates average and std dev of losing percentage of each team

std dev = sqrt(np(1-p))

p=losing %, assumed to be 505

-since there are many teams outside of 2 std dev of mean losing percentage (divide std dev by n to put in % terms), concludes that there are inherent differences between teams

Question 5

observes that a team that has been worse than average over 1 period of time is likely to

Answer 5

continue to be worse than average over another period of time which implies there is value in using past experience of team to predict future experience of that team

Question 6

2 methods to test: are the observed team loss percentages over time random fluctuations?

Answer 6

• essentially asking whether risk parameters for team are changing over time
  1. chi squared
  2. compare correlation between years aka lagged correlations test

Question 7

chi squared test

Answer 7

Null hypothesis: risk parameters do no shift over time

• group data into appropriate intervals, calculate overall expected value
• calculate χ2=(A-E)^2/E
• sum up for all interval
• if there are n time period groups, χ2 table value to compare against n-1 DoF
• if test statistic > tabular value, then reject null and conclude risk parameters have shifted over time

Question 8

Compare correlation between years

aka

lagged correlations test

Answer 8

• group data by pairs based on time lag (separation in time)
• calculate correlation between for each pair
• calculate average correlation for each time lag
• if correlation decreases as time lag increases, risk parameters shift over time

*implies risk parameters are shifting over time and recent years can help predict future

Question 9

using both tests, concluded

Answer 9

risk parameters of teams are changing over time -> they are not random fluctuations in the same distribution

Question 10

3 criteria to evaluate the quality of solutions (different credibility weighted options)

Answer 10

1. Least Squared Error
2. Limited Fluctuation
3. Meyers/Dorweiler

Question 11

Least Squared Error

Answer 11

• calculate the mean squared error of prediction compared with actual observed result
• Buhlmann/Bayesian credibility methods attempt to minimize this criteria

*minimize squared error between actual and predicted result

Question 12

final points

Answer 12

1. Mahler notes that when there are shifting risk parameters over time, older years of data will be less relevant predicting the future and so should be given less or no credibility compared with recent years
2. Related to prior item, Mahler looks at the impact of having a delay in getting historical data as is the case with experience rating; notes that not having the most recent year of historical data significantly increases the squared error of the estimate

Question 13

Limited Fluctuation

Answer 13

• AKA Small Chance of Large Errors
• measures the probability that observed result differs by more than a certain percent from predicted result
• classical credibility method targets this criteria

*minimize likelihood that any one actual observation will be certain percent different from predicted result

Question 14

Meyers/Dorweiler

Answer 14

• calculates correlation between ratio of actual to expected losses and ratio of predicted losses to overall losses
• 2 vectors used: V1=actual team t losing %/estimated team t losing %

V2=estimated team t losing %/mean losing %

-Kendall tau statistic is used to compute the correlation between 2 vectors

*minimize correlation between ratio of actual/predicted and predicted/average actual

Question 15

Meyers/Dorweiler criteria confirms

Answer 15

that there is no evidence that large predictions lead to large errors and small predictions lead to small errors

Question 16

Evaluate whether claim freq is shifting over time using Chi2 test

Answer 16

-calculate overall parameter for all years -> frequency

λ=sum(exposures*avg freq)/sum(exposures)

-calculate expected for each grouping (exposures*λ; λ from prior step)

X^2=sum(actual-expected)^2/expected

-compare test statistic to critical value for relevant chi2 dist

critical value = X^2 (n-1,α%)

n=#groups

n-1=DoF

• null hypothesis=parameter is not shifting over time
• if statistic > critical value, reject null hypothesis and conclude that frequency is shifting over time

Question 17

when parameters are shifting over time, should

Answer 17

place less credibility on older data and higher credibility on more recent data & important to minimize delays in data used for predictions and ratemaking

Question 18

Using Correlations between LRs at different lags, test if LR is shifting over time

Answer 18

-set up tables of historical LRs @ increasing lag periods

t | t-1 t| t-2 etc.

• calculate correlation of parameter @ increasing lags
• if correlation decreases as lag increases, LRs are shifting over time (higher correlations for years closer together indicate parameters are shifting)

Question 19

Use least squared error to determine which credibility weight produces better LR estimates

Answer 19

• calculate estimated LRs for each credibility scenario and set up table of estimated and actual values for each scenario
• calculate mean squared error for each credibility scenario
• select credibility with smallest MSE

*could find optimal Z by minimizing the Sum Squared Error (SSE) and not dividing by n

Question 20

what criterion (used to compare performance of credibility methods) is different on conceptual level

Answer 20

Meyers/Dorweiler

focues on pattern of errors instead of minimizing prediction errors

Question 21

risk parameters will have had less time to shift between years that are

Answer 21

closer together

so correlation should be higher for years that are closer together

Question 22

Meyers/Dorweiler criterion would choose the plan with

Answer 22

no correlation between the modification factors and the errors

Question 23

What would you conclude about your ability to estimate future losses for these risks?

Answer 23

Since losses for small separations in years are highly correlated, past experience will likely be helpful in predicting future experience. Since the correlations drop off over time, more recent years should get more weight (credibility) in estimating future losses.

Question 24

compare 3 criteria that can be used to select optimal estimates

Answer 24

Least Squared Error: Minimizes the squared error between the predicted results and the actual results. I believe this is the best choice for experience rating since it leads to mathematically solvable equations and because it mitigates against large individual errors.
LimitedFluctuation: Measures the probability that the predicted result doesn’t differ from the actual result by more than a certain percentage. This criteria will yield similar results to the least squared error method, but is less practical to implement given that it doesn’t lead to easily solvable equations.

Meyers/Dorweiler: Measures the correlation between the ratio of actual to predicted losses, and the ratio of predicted losses to overall losses. While this criteria helps mitigate patterns of errors in the loss predictions, it doesn’t help reduce the magnitude of the errors. As such, it would be an inferior criteria to use for experience rating.