“The Effect of Changing Exposure Levels on Calendar Year Loss Trends” by Chris Styrsky, FCAS, MAAA

1. “The Effect of Changing Exposure Levels on Calendar Year Loss Trends”by Chris Styrsky, FCAS, MAAA Ratemaking Seminar March 10, 2005

2. Why are loss trends important? Loss trends are used to project the historical data to the future experience period so accurate loss costs will be reflected in the rates charged.

3. How should data be organized for loss trends? • Accident Year/Policy Year Benefit • Best matching of risk with exposure Drawback • Most recent years requires loss development • Calendar Year Benefit • Ease of use Drawback • Mismatching risk with exposure

4. Calendar Year Loss Trends Example Assumptions: • All policies are written on January 1st and are 12 month policies • The ultimate claim frequency for every risk in existence is 0.20 • 50% of the ultimate claims are paid within 12 months of the date the policy was written, 30% between 12 and 24 months, and 20% between 24 and 36 months (no claims paid past 36 months)

5. Calendar Year Loss Trends Example Assumptions (cont.): • The claim payment pattern does not change over time • During calendar year X+2, claims that were paid within 12 months of the date the policy was written were settled for $100, $200 for claims between 12 to 24 months, and $400 for claims between 24 to 36 months • Annual inflation is 5% for all claims

6. Calendar Year Loss Trends Example Assumptions (cont.):

7. Calendar Year Paid Frequency CYX Paid Frequency = (C0,12,X + C12,24,X + …) / EX Where: • CYX = Calendar year X • CT,T + 12,X = # of claims paid during CYX that were paid between T and T + 12 months after the claim occurred • EX = Earned Exposures from calendar year X

8. Calendar Year Paid Frequency Year X + 2 = (100,000 * 0.2 * 0.5 + 100,000 * 0.2 * 0.3 + 100,000 * 0.2 * 0.2) / 100,000 = 0.2 Year X + 6 = (48,575 * 0.2 * 0.5 + 63,475 * 0.2 * 0.3 + 78,500 * 0.2 * 0.2) / 48,575 = 0.243

9. Calendar Year Paid Frequency Trend

10. Why was there a trend??? There was a mismatch between the claims and exposures! For example: Calendar Year X + 6 paid claims come from Accident Years X + 4, X + 5, and X + 6 but are matched to Calendar Year X + 6 earned exposures

11. Will there always be an impact to paid frequency trends? There are two factors that need to occur to see a distortion: • Changing exposure levels • Significant amount of time between accident date and settlement date

12. CY Paid Pure Premium Trend Since CY paid frequency trend is 5% and inflation is 5% we would expect the CY paid severity to about 10%. Let’s take a look at CY paid pure premiums….

13. Calendar Year Paid Pure Premium CYX Paid Pure Premium= (L0,12,X + L12,24,X + …) / EX Where: • LT,T + 12,X = losses paid during CYX that were paid between T and T + 12 months after the claim occurred

14. Calendar Year Paid Pure Premium Year X + 2 = (100,000 * 0.2 * 0.5 *100 + 100,000 * 0.2 * 0.3 * 200 + 100,000 * 0.2 * 0.2 * 400) / 100,000 = $38.00 Year X + 6 = (48,575 * 0.2 * 0.5 * 100 * 1.05 4 + 63,475 * 0.2 * 0.3 * 200 * 1.05 4 + 78,500 * 0.2 * 0.2 * 400 * 1.05 4 ) / 48,575 = $62.42

15. Calendar Year Paid Pure Premium Trend

16. CY Paid Severity Trend In this example we know that inflation is 5%, so we want a measure that will produce a 5% severity trend Let’s take a look at CY paid severity….

17. Calendar Year Paid Severity CYX Paid Severity= (S0,12,X *C0,12,X + S12,24,X * C12, 24,X + …) / (C0,12,X + C12,24,X + …) Where: • ST,T + 12,X = losses paid during CYX that were paid between T and T + 12 months after the claim occurred

18. Calendar Year Paid Severity Year X + 2 = (100,000 * 0.2 * 0.5 *100 + 100,000 * 0.2 * 0.3 * 200 + 100,000 * 0.2 * 0.2 * 400) / (100,000 * 0.2 * 0.5 + 100,000 * 0.2 * 0.3 + 100,000 * 0.2 * 0.2) = $190.00 Year X + 6 = (48,575 * 0.2 * 0.5 * 100 * 1.05 4 + 63,475 * 0.2 * 0.3 * 200 * 1.05 4 + 78,500 * 0.2 * 0.2 * 400 * 1.05 4 ) / (48,575 * 0.2 * 0.5 + 63,475 * 0.2 * 0.3 + 78,500 * 0.2 * 0.2) = $257.75

19. Calendar Year Paid Severity Trend

20. Calendar Year Paid Severity Calendar Year Paid Severity represents a weighted average of the severities from the different settlement periods where the weights are the percentage of total paid claims from that specific settlement period

21. What Happened??? This example assumes uniform inflation of 5% annually, but the paid severity varies depending on how long it takes to settle the claim. With the declining exposures, the percentage paid claims from the early settlement times decreases with respects to total paid claims.

22. Calendar Year Paid Severity Distribution by Settlement Period

23. What can you do to measure the correct paid frequency? Calendar Year Paid Frequency was distorted by the mismatch between paid claims and exposures, why not match the paid claims to the exposures that produced them?

24. Adjusted Paid Frequency Adjusted Paid Frequency (APF) = C0,12,X / EX + C12,24,X / EX-1 + C24,36,X / EX-2 + … This formula can be thought of as adding the incremental frequencies

25. Adjusted Paid Frequency Year X + 2 = 100,000 * 0.2 * 0.5 / 100,000 + 100,000 * 0.2 * 0.3 /100,000 + 100,000 * 0.2 * 0.2 / 100,000 = 0.2 Year X + 6 = 48,575 * 0.2 * 0.5 /48,575 + 63,475 * 0.2 * 0.3 / 63,475 + 78,500 * 0.2 * 0.2 / 78,500 = 0.2

26. Adjusted Paid Frequency Trend

27. What about paid pure premium? Calendar Year Paid Pure Premium is also distorted by the mismatch between paid claims and exposures, so a similar adjustment would seem warranted.

28. Adjusted Paid Pure Premium Adjusted Paid Pure Premium (APPP) = L0,12,X / EX + L12,24,X / EX-1 + L24,36,X / EX-2 + … This formula can be thought of as adding the incremental pure premiums

29. Adjusted Paid Pure Premium Year X + 2 = 100,000 * 0.2 * 0.5 * 100 / 100,000 + 100,000 * 0.2 * 0.3 * 200 /100,000 + 100,000 * 0.2 * 0.2 * 400 / 100,000 = $38.00 Year X + 6 = 48,575 * 0.2 * 0.5 * 100 * 1.05 4/48,575 + 63,475 * 0.2 * 0.3 *200 * 1.05 4/ 63,475 + 78,500 * 0.2 * 0.2 * 400 * 1.05 4/ 78,500 = $46.19

30. Adjusted Paid Pure Premium Trend

31. What about paid severity? Since we have formulas for adjusted paid frequency and adjusted paid pure premium, the formula for paid severity can be backed into using the relationship of: Frequency * Severity = Pure Premium

32. Adjusted Paid Severity Adjusted Paid Severity (APS)= (L0,12,X / EX + L12,24,X / EX-1 + L24,36,X / EX-2 + … )/(APF) = (L0,12,X / EX)/APF + (L12,24,X / EX-1)/APF + … = ((L0,12,X / C0,12,X ) * (C0,12,X / EX))/APF + ((L12,24,X / C12,24,X ) * (C12,24,X / EX-1))/APF + … = (S0,12,X * (C0,12,X / EX))/APF + (S12,24,X * (C12,24,X / EX-1))/APF + …

33. Adjusted Paid Severity Adjusted Paid Severity represents a weighted average of the severities from the different settlement periods where the weights are the percentage of total paid frequency from that specific settlement period

34. Adjusted Paid Severity You have a formula to derive adjusted paid severity, but you can use the same relationship used to derive that formula and just divide the Adjusted Paid Pure Premium by the Adjusted Paid Frequency.

35. Adjusted Paid Severity Trend

36. Benefits of using Adjusted Loss Trends • Adjusted loss trends remove the implicit assumption with CY loss trends that exposure levels are constant • If exposure levels are constant, CY loss trends are equal to adjusted loss trends • No development needed (issue w/ AY) • No issues with seasonality of reporting patterns, plus adjustment is made for severity issues (issue w/ reported frequency)

37. Pitfalls or Issues to Watch for if using this method #1 • How many years to match claims/losses with exposures? • Claims can be paid many years after the accident occurred • Not practical to match every accident year within a calendar year’s paid claim • Recommend matching enough years where a “significant” portion of claims/losses have been paid (in PPA 8 years should be sufficient)

38. Pitfalls or Issues to Watch for if using this method (cont.) #2 • What to do with the claims/losses from the years not match? • Recommend creating an “all others” accident year category where all of the paid claims/losses are summed • These “all others” paid claims/losses should then be matched to the calendar year exposures from the most recent year that falls in the “all others” group since this should be most representative of the exposure level of the claims/losses

39. Pitfalls or Issues to Watch for if using this method (cont.) #3 • Some older CY earned exposures could be very small if company is relatively new, potentially causing unusual results • Ex. There might be 1 paid claim matched to 2 earned exposures causing frequencies to look extremely high • Could remove incremental frequency that is distorted • Could match back to years w/ at least X exposures • Actuarial judgment should be used as to what the appropriate action should be