2. P/C RATEMAKING AND LOSS RESERVING R.L. Brown & L. R. Gottlieb CHAPTER 4: RATEMAKING I. Introduction; II. Objectives; III. Data for Ratemaking; IV. Premium Data; V. Exposure Units; VI. The Expected Effective Period; VII. Ingredients; VIII. Rate Changes

3. 2 Introduction & Objectives A. Essential Objectives 1.Cover expected losses and expenses a.include investment income b.no inter-cohort subsidies c.no subsidies across risk classes d.must cover to last dollar paid out (several years later) e.selling at a loss is O.K., but subsidy should come from I.C. owner’s equity I. Introduction: Methods are applicable outside of P & C (e.g. health) II. Objectives:

4. 3 2.Produce rates that make adequate provision for contingencies a.The 100-year flood will happen! b.Tough to push rates up as industry is competitive and ph can often self-insure c.Inadequate rates endanger I.C. solvency A. Essential Objectives

5. 4 3. Encourage loss control a.get ph to reduce claim freq., severity or both b.methods used 1.good driver discounts 2.discounts for sprinklers, burglar alarms 3.discounts for accident prevention and rehab in W.C. c.leads to lower rates d.good for society A. Essential Objectives

6. 5 4. Satisfy rate regulators a.rates must be adequate, not excessive and not unfairly discriminatory unfairly discriminatory Too big to ignore: the impact of obesity on mortality trends Too big to ignore: the impact of obesity on mortality trends Does it matter where you live? ICCB Case#12ICCB Case#12 b.some methods that are actuarially acceptable may be refused (e.g., use of age, gender and marital status as rating variables)gender A. Essential Objectives

7. 6 B. Non-Essential but Desirable Objectives 1. Produce rates that are reasonably stable a.rate changes must be prospectively justifiable b.reinsurance can help stability B. Non-Essential Objectives

8. 7 2.Produce rates that are reasonably responsive to change a.real changes (vs. one-time aberration) should be reflected (e.g. change in speed limit) b.stability and responsive are in conflict (reflecting change only to the extent it is statistically credible helps) B. Non-Essential Objectives

9. 8 3.Be simple and easy to understand a.ph must understand connection to loss control to make it happen b.agents and brokers must be capable of using rate manual c.computer systems are cheaper if less complex system d.must be able to “sell” system to management and regulators (who are often not actuaries) B. Non-Essential Objectives

10. 9 III. Data for Ratemaking Data on claims, loss payments, and premiums may be collected and tabulated in any of the following three formats: accident year, policy year, and calendar year. (A fourth basis, called the reported year, which is used for claims- made policies will not be discussed here.)

11. 10 A. Accident Year 1.The most common method for compiling actuarial data. 2.All action of any loss occurring in accident year Z is held in the “accident year Z” file regardless of the date of action (e.g. payment). 3.You do not know the ultimate claim until the last dollar is paid. 4.All immature data then depend on reserve estimates. 中大生校內捱車撞 明報 2008-10-25 沙田中文大學校園內昨午 1 時許發生 交通意外。中大一年級女生楊 × 慧 （ 19 歲） 在大學營修樓對開橫過馬 路時，遭一輛沿大學道西行的私家 車撞倒，楊女頭部及兩腳膝部受輕 傷，救護員接報到場替她包紮，楊 女由兩名同學陪同送院 …… 33 歲私家車司機姚 × 志接受酒精測 試後，證實並無飲酒，警方正調查 意外原因。

12. 11 B. Policy Year 1.Any claim action arising on a policy that became effective in policy year Z is accounted for in the “policy year Z” file. 2.If policy year Z goes from Jan. 1, Z to Dec. 31, Z, and if all policies are effective for one year, the exposure period for policy year Z goes from Jan. 1, Z to Dec. 31, Z+1 (i.e. 24 months). 3.If all issue dates and accidents are uniformly distributed, the midpoint of PY(Z) is Dec. 31, Z or Jan. 1, Z+1. 4.Thus, you do not have complete PY(Z) data until Dec. 31, Z+1 (and then it is still immature). 5.However, pricing is done for policy years, so having policy year data is a consistent basis.

13. 12 C. Calendar Year 1.Any accounting action on a file in calendar year Z gets recorded in the “calendar year” Z file. 2.Calendar year Z is complete and mature as at Dec. 31, Z. 3.Incurred losses (Z) = Paid losses (Z) +  Reserve (Z) = Paid losses (Z) + Loss Reserve 12/31/Z - Loss Reserve 12/31/Z-1. 4.This system is not widely used as one cannot allocate “calendar year” action to any defined exposure period. Note: Again, The REPORT YEAR, used for claims-made liability policies, is not discussed in the text.

14. 13 a.If $120 received Oct. 15, Z, then written premium in Z is $120, but earned premium is $25 for Z. b.The remaining $95 is unearned premium in Z, and will only be earned in year Z+1. IV. The Premium Data Written Premium vs. Earned Premium: Premium data usually come as “written” premiums (cash accounting) or “earned” premiums (accrual accounting).

15. 14 The XYZ Insurance Company had an unearned premium reserve of $20 million at the end of 2015. During 2016 it wrote $25 million in annual premiums. At the end of 2016, its unearned premium reserve was $23 million. What were earned premiums during 2016? Question:

16. 15 V. The Exposure Unit 1.Premium = (rate) * (exposure units) (e.g. for W.C.  $100 of payroll) 2. A good exposure base should: a.be an accurate measure of quantitative exposure to risk b.be easy for I.C. to determine (at time of premium calculation) c.not to be subject to manipulation by ph d.be easy to record and admin

17. 16 3. You seldom get all criteria - must compromise (e.g. annual mileage is not used for car insurance) 2. A good exposure base should: e. be understood by ph f. not essential but desirable - automatically adjust with inflation (e.g. payroll) V. The Exposure Unit

18. 17 VI. The Expected Effective Period 1.There is a time lag from the most recent data to the expected effective period. 2.The effective period is a policy period, normally a policy year. 3.If rates take effect, Oct. 15, Z, and are to be left unchanged for exactly one year, on one- year policies, then the effective period starts Oct. 15, Z; ends Oct. 15, Z+2; and has a midpoint of Oct. 15, Z+1. (note: exposure starts at zero, peaks on Oct. 15, Z+l and ends at zero). 4.Claims data: actuary normally uses accident- year or policy-year data.

19. 18 VII. Ingredients of Ratemaking A.Loss-Development Factors (LDF) B.Trend Factors (TF) C.Expenses D.Loading for Profits and Contingencies E.Credibility Factors (CF) F.Investment Income

20. 19 A. Loss-Development Factors 1.These data come from the “Loss-Reserving” actuary. Your price must cover all claim costs for this cohort until the last $ is paid (many years out). 2.Early claims estimates may understate the ultimate value. a.reserve estimates are deficient or optimistic b.there exist claims incurred but not reported (IBNR) c.this leads to “+” loss-development (study Tables 4.1~4.3) 3.It is possible to have loss-development factors < 1 a.reserve estimates prove to be conservative (even with IBNR) b.some lines allow for salvage and subrogation (i.e. claim payment recoveries)

21. 20 1.Must adjust past experience period data to mid- point of future exposure period. 2.Reasons for cost adjustment a.economic inflation b.judicial decision c.change in mandated benefits d.technical advances e.legislative changes f.change in U/W criteria or definitions g.level of economic activity B. Trend Factors

22. 21 3.Note: not all of above can be modeled with trend factor (trend factor only projects past trends) 4.If data accident year, mid-pt of A.Y.(Z) is June 30, Z or July 1, Z. 5.If data policy year, mid-pt of P.Y.(Z) is Dec. 31, Z or January 1, Z+1. 6.If rates take effect at date t, Z for one year on one-year policies, mid-pt of future exposure period is t, Z+1. Trend Factors

23. 22 Trend Factors 7.Trend factor methods: a.linear regression/projection from past period b.log linear regression/projection (i.e. exponential trend) c.fit frequency and severity separately vs. fitting loss cost (= f * S) data d.find slope of data trend line; then extrapolate only last one or two data points using this slope (say weighted 70/30 for example) 8.No overlap between loss-development and trend (overlap fallacy) since trend is from average past date of claim to future average claim date and loss development goes from future average claim date to final dollar paid.

24. 23 Example 4.1: Given the loss costs shown in Table 4.5, estimate the expected loss cost for rates that take effect September 1, AY6 on one-year policies. Assume that rates will be in effect for one year. Table 4.5 Average Frequency, Average Severity, and Loss Cost Accident Average claim Average Loss Loss Cost per Year Frequency Severity Unit Exposure AY1.05141 2,323119.39 AY2.05335 2,502133.97 AY3.05311 2,445 129.89 AY4.05648 2,807 158.57 AY5.05765 3,274 188.72

25. 24 1.Loss adjustment expenses (LAE) a.Allocated loss adjustment expenses (ALAE) (e.g. lawyers’ fees, claim adjusters’ fees, court costs) go into claim file and are part of total incurred losses. b.Unallocated loss adjustment expenses (ULAE) (e.g. head office claims dept., salaries) do not go into any claim file and are allocated en masse at year-end to line-of-business using an appropriate formula 2.Commissions, premium taxes, licenses and fees and misc. (usually set as % of Gross Premium). C. Expenses

26. 25 3.Expense ratio = all expenses (not LAE) as % of GP. 4.Permissable loss ratio (PLR) = 1 - expense ratio 5.. 6.If expenses (not LAE) are partly fixed (F) per unit and partly variable (V) then: 7.Book examples assume all non LAE expenses are variable unless indicated otherwise Expenses

27. 26 Expense Loadings as a Percentage of Premium

28. 27 1.Need profit to compensate capital providers and need to cover “expected” deviations (insolvency is possible) a.implicit approach - adopt conservative parameters and live off the margins b.explicit approach - adopt best-estimate parameters and add an explicit margin for profit and cont. c.ignore investment income and call it profit and contingencies d.regulators prefer (b) D. Loading for Profit and Contingencies

29. 28 Concept Check Assume that the discounted expected claim costs and administrative costs for hurricane and fire insurance are the same. Which type of coverage will have higher fair premium? Why?

30. 29 1.Sparse data will not be statistically credible. 2.Also, credibility strikes a balance between responsiveness and stability (only make a change to the extent it is credible). 3.You need more data for any set credibility for business that is more variable (e.g. auto liability vs. collision). E. Credibility Factors

31. 30 4.Properties of credibility (Z) a.. b. (more data, higher credibility) c. (additional units of exposure provide less than pro rata additional credibility) 5.Examples of credibility formulae: a., (E  measure of exposure) (K  measure of variability) b., (n  # of claims(e.g.)) (K  criteria for full credibility) Credibility Factors

32. 31 F. Investment Income 1.Should discount for time to payment 2.Covered more later in loss reserving Example Assume no administrative costs one year policies, premium received at beginning certain claim costs = $100 paid according to table below Fair Premium

33. 32 Example: Effect of Investment Income Assume no administrative costs one year policies, premium received at beginning certain claim costs = $100 paid according to table below Fair Premium

34. 33 Example: Effect of Investment Income Assume no administrative costs one year policies, premium received at beginning certain claim costs = $100 paid according to table below Fair Premium

35. 34 Example: Effect of Investment Income Assume no administrative costs one year policies, premium received at beginning certain claim costs = $100 paid according to table below Fair Premium

36. 35 Effect of Investment Income Varies Across Lines of Business

37. 36 P-C Industry Break-even Combined Ratio

38. 37 VIII. Rate Changes 1.Overall Average Rate Change Determine the average, or overall, rate change required. 2.Changing Risk Classification Differentials Decide on changes required in the differentials that apply to rate classification parameters. 3.Balancing back Adjust the results so that the overall change in premium income is actually desired.

39. 38 1.Loss Cost Method 2.Loss Ratio Method A. Overall Average Rate Change

40. 39 LR Method where Denominator = today’s earned premium at current rates Or adjust account dept. records of “Earned Premiums” to “Earned Premiums @ Current Rate Level” using parallelogram method of Example 4.2 A. Overall Average Rate Change

41. 40 Example 4.2: Consider the following data: Calendar YearEarned Premium CY33853 CY44600 CY55125 Date Rate Change July 1, CY1+12.5% November 15, CY3+10.0 October 1, CY4+8.0 Average all policies have a one-year term, the policy issues are uniformly distributed, and the following rate changes have occurred: Rates are currently at the level set on October 1, CY4. Calculate the earned premium at current rates for calendar years CY3, CY4, and CY5.

42. 41 Example 4.3: Calculate the new average gross rate given the following information: Expected Effective Incurred Losses (Trended and Developed)30,000,000 Earned Exposure Units 1,000,000 Earned Premium at Current Rates45,000,000 Present Average Manual Rate 45 Permissible Loss Ratio (1 – Expense Ratio).700

43. 42 3.If data are consistent, Loss Cost and Loss Ratio methods achieve same answer (see proof page 127/8 - this is important). 4.Credibility: if company data not fully credible (Z < 1) use: Cred-wtd Average Indicated Rate (or change) = Z (Company Indication) + (1-Z) Industry Indication A. Overall Average Rate Change

44. 43 1.Production of Rate Manual a.One cell is called the base cell (usually the cell with the largest credibility) and its rate is the Base Rate(BR) b.Then each risk class variable will have a vector of differentials (d i ) or relativities with Base Cell entry always set = 1.000. c.So for 3 risk class variables x i (i=1, l), y j (j=1, m) and z k (k=1, n), you can produce l  m  n rates B. Changing Risk Classification Differentials

45. 44 2.Changing Differentials: Loss Ratio Method a.(lndicated Diff) i = (Existing Diff) I. Note: if experience period LR j = LR BASE then the existing differential need not be changed (understand?). i.e. the existing differential proved to be just right! 3.Changing Differentials: Loss Cost Method a.L.C. method takes you directly to the indicated diff. (one-step) b.. B. Changing Risk Classification Differentials

46. 45 Example 4.4: Determine new differentials for Class B and Class C, given the following information and assuming full credibility in all classes. ExistingExperience Period Loss Class Differential Ratio at Current Rates Loss Cost A 1.00.65129 B.85.71120 C 1.21.66157

47. 46 Example 4.5: You are the consulting actuary for a property/casualty insurer that proposes to begin selling automobile insurance in State Z which has three classification territories. The insurer wants advice on what territorial differentials to adopt. You have the following information showing the present average territorial differentials used by the top five insurers. Territory Differential 11.00 2.95 31.25 The following industry statistics are also available for the latest calendar year. Earned Premium Incurred Losses Territory Cars Insured(in thousands) (in thousands) 1 65,354 12,046 7,215 2 56,182 10,093 5,987 3 24,858 5,840 3,580 What territorial differentials would you recommend, and what comments would you include in your report?

48. 47 4.Loss Ratio Diff  Loss Cost Diff if: a.populations across cells are heterogeneous (e.g. more young male drivers in Terr 1 than in Terr 3). b.Since Loss Cost = $L/# exposure units, all exposure units are treated as equal. c.But they are not: e.g. if young male drivers pay twice the base rate then, in solving for the Territorial Diff young male drivers should count as 2 units of exposure (see Example 4.6) B. Changing Risk Classification Differentials

49. 48 d.The Loss Ratio method self adjusts for this problem to the extent that last year’s differentials were correct since 5.With appropriate adjustments, the Loss Cost and Loss Ratio methods achieve same answer (see Appendix A now). 6.Credibility: if cell credibility Z < 1, use (normally): New Diff = Z (Indicated Diff) + (1-Z)(Existing Dift) (that is, only change the differential to the extent that the need for a change is credible). B. Changing Risk Classification Differentials

50. 49 1.Because d BR =1.000 a.given an overall rate change indication of  R, then equation BR NEW = BR OLD (1 +  R ) b.And New Rate i,j,k = BR NEW. (new x i ). (new y j ). (new z k ) c.But this combination may not produce a premium income increase of  R as needed (see handout!) d.This is because the New Average Diff  Old Average Diff e.So: off-balance and f.balance-back factor is its reciprocal (see p.135) C. Balancing Back

51. 50 2.Under the Lost Cost method, the balance back is achieved in one step: Base rate 3.Review in detail example 4.7 (p.138-145) C. Balancing Back

52. 51 Example 4.6: Given the following information, calculate the proposed Class 1A rate for Territory 2. Class differentials will not be changed, and the province wide rate change is +5%. The base cell is Territory 1, Class 1A. EarnedExisting Average Existing ExposureAverage Loss Class Differentials Territory Units Rate Cost within Territory 1 2000 250 200 1.50 2 1000 500 300 1.25

53. 52 Example 4.7: Given the following information, and assuming the revised rates take effect July 1, 2007 for one year on one-year policies, determine new rates for each of Class 1 and Class 2, for each of Territory 1 and Territory 2. (Class ½ differentials will not change.) Use the loss ratio and loss cost methods, and base the overall average rate change on 2005 policy year data, assuming they are fully credible for that purpose. The permissible loss ratio is.600. Policy Year 2004 Losses As of March 31, 2006 As of March, 2007 PaidOutstandingPaid Outstanding 400,000100,000625,000 0 Trend Factors July 1, 2006 July 1, 2006 January 1, 2006January 1, 2006 to tototo July 1, 2007 July, 2008 July 1, 2007July 1, 2008 1.18 1.30 1.241.36

54. 53 Territory 1Territory 2 Present Rates Class 1 (Differential)100(1.00)200(2.00) Class 2 (Differential)300(3.00)600(6.00) Collected Earned Premium1,000,0001,000,000 Policy Year 2005 Incurred Losses as of March 31, 2007 360,000 240,000 Earned Exposure Units Class 1 5,000 2,000 Class 2 1,000 500