Ratemaking by Charles L. McClenahan

Transcription

1 Mahler s Guide to Ratemaking by Charles L. McClenahan See CAS Learning Objectives: B2, D1-D6. My Questions are in Study Guide 1B. Past Exam Questions are in Study Guide 1C. Prepared by Howard C. Mahler. hmahler@mac.com Including some questions prepared by J. Eric Brosius. Copyright 2009 by Howard C. Mahler New England Actuarial Seminars FAX: POB 315 Sharon, MA, Study Aid S09-5-1A Howard Mahler hmahler@mac.com

2 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 1 McClenahan Ratemaking, Chapter 3 of Foundations of Casualty Actuarial Science I. INTRODUCTION 1. The Concept of Manual Ratemaking II. BASIC TERMINOLOGY 1. Exposure 2. Claim 3. Frequency 4. Losses and Loss Adjustment Expenses; Figure Severity 6. Pure Premium 7. Expense, Profit and Contingencies 8. Premium 9. Loss Ratio 10. The Goal of the Manual Ratemaking Process 11. Structure of the Rating Plan III. THE RATEMAKING PROCESS 1. Basic Manual Ratemaking Methods 2. Pure Premium Method 3. Loss Ratio Method 4. Relationship Between Pure Premium and Loss Ratio Methods 5. Selection of Appropriate Method 6. Need for Common Basis 7. Selection of Experience Period 8. Reinsurance 9. Differences in Coverage 10. Treatment of Increased Limits 11. On-Level Premium Adjusting for Prior Rate Changes; Figures IV. TRENDED, PROJECTED ULTIMATE LOSSES 1. Inclusion of Loss Adjustment Expenses 2. Projection to Ultimate the Loss Development Method 3. Identification of Trends 4. Reflection of Trends; Figures Effects of Limits on Severity Trend; Figures Trend Based Upon External Data 7. Trend and Loss Development The Overlap Fallacy ; Figure Trended Projected Ultimate Losses V. EXPENSE PROVISIONS

3 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 2 VI. PROFIT AND CONTINGENCIES 1. Sources of Insurance Profit 2. Profit Provisions in Manual Rates 3. Risk Elements VII. OVERALL RATE INDICATIONS 1. Credibility Considerations VIII. CLASSIFICATION RATES 1. Base Rates 2. Indicated Classification Relativities 3. Correction for Off-Balance 4. Limitation of Rate Changes IX. INCREASED LIMITS 1. Trending Individual Losses 2. Loss Development by Layer 3. Fitted Size-of-Loss Distribution X. SUMMARY XI. REFERENCES XII. QUESTIONS FOR DISCUSSION XIII. APPENDIX; Exhibit While everything in McClenahan is very important, items in bold are especially important. Errata in McClenahan: 2 Page 95, Figure 3.2 should be On-Level Premium, Page 108, table at the bottom, third line, should be U/(1 + T) rather than U/(1 - T). Page 109, Figure 3.8, there is no label on the y-axis and all of the values are zero on the y-axis. I believe that the y-axis was intended to be number of claims, with 250 at the top. Page 129, General Expenses should be $45,000 (in order to match the solution.) In Exhibit 3.5 Note [1] is missing. I believe Note [1] was intended to read: y = mx + b, where x = Accident Year , m = , and b = The references in Exhibits 3.9 to 3.14 to other exhibit numbers are wrong. Exhibit 3.10, column 4, for Territories 2 and 3, the relativities to Class 1 are not correct. For example, for Territory 2, Class 2: / = rather than Page 146, Exhibit 3.14, territory and class are reversed in the whole exhibit. Page 147, [3].5 should be a power as should 1.5 in [4]. 1 See my notes on McClenahan s exhibits, later in this study aid. 2 I believe these are mistakes. Check the CAS website for any official errata they have posted. 3 Column 5 of Exhibit 3.10 is correct. The correct relativities to Class 1 are shown in Exhibit 3.11.

4 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 3 Basic Data Definitions: Calendar Year: All premiums and losses related to a given calendar year. Written premiums: those dollars of premiums on policies written during the period in question. Premiums are earned as coverage is provided. Normally, premium is earned at a constant rate over the policy effective period. Exercise: An annual policy is written with effective date October 1, The premium is $400. What are the contributions to the Calendar Year 2002 and 2003 written and earned premiums? [Solution: All of the $400 contributes to CY2002 written premium; none contributes to CY2003 written premium. One quarter of the $400, or $100 contributes to CY2002 earned premium; three quarters of the $400, or $300 contributes to CY2003 earned premium.] Similarly one can have written and earned exposures. If an annual policy covering one car is written with effective date March 1, 2002, then it contributes: 1 written car year to CY02 and no written car years to CY03, 5/6 earned car year to CY02 and 1/6 earned car year to CY03. Calendar Year Paid losses: those dollars of loss paid by an insurer during a given year. Calendar Year Incurred losses: those dollars of loss paid plus the change in loss reserves during a given year. Accident date: the date of the occurrence which gave rise to the claim. Accident Year: All the losses with accident dates during a given year. For example, an accident occurs on March 15, All payments related to claims resulting from this accident, are part of Accident Year 2003, regardless of when those payments are made. Calendar/Accident Year 2003 would consist of premiums for Calendar Year 2003 and losses for Accident Year Policy Year: All premiums and losses related to policies with effective dates during a given year. For example, an accident occurs on March 15, If the policy providing coverage was written effective November 1, 2002, then all payments related to claims resulting from this accident, are part of Policy Year 2002, regardless of when those payments are made.

5 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 4 Report Date: the date the insurer receives notice of the claim. Report Year: All the losses on claims for which the insurer first receives notice during a given year. For example, an accident occurs on March 15, If on June 5, 2004 the insurer first receives notice of a claim resulting from this accident, then all payments related to this claim, are part of Report Year 2004, regardless of when those payments are made. Some Loss Reserve Definitions: 4 A total loss reserve consists of 5 elements: 1. Case reserves assigned to specific claims, 2. A provision for future development on known claims, 3. A provision for claims that reopen after they have been closed, 5 4. A provision for claims that have occurred but have not yet been reported to the insurer, and 5. A provision for claims that have been reported to the insurer but have not yet been recorded. A loss reserve can be divided into two categories: known claims vs. unknown claims. The reserve for known claims represents the amount that will be required for future payments of claims that have already been reported to the insurer (the sum of (1), (2), (3)). This amount includes the case reserves, the aggregate of the individual estimates made by the adjusters. Some case reserves may be set by formula. The reserve for unknown claims is the amount for claims that have been incurred but not reported to the insurer. The reserve for unknown claims is commonly called an IBNR reserve. Often, it is not possible to distinguish between claims in the last two categories. These pure IBNR claims (4) and the claims in transit (5) are frequently combined together and called IBNR. This is the strict definition of IBNR, but in practice, future development on known claims, a provision for reopened claims, unreported claims and unrecorded claims are often combined together and called IBNR. Over time, losses develop and the IBNR claims emerge. One of the reasons loss reserves need to be estimated is due to the delay between when a loss occurs, when it is reported to an insurer, and when it is finally settled. Loss data used in ratemaking or individual risk rating will usually either consist of: paid losses, or paid losses plus case reserves (case incurred). Sometimes one will use paid losses plus all reserves (total incurred). Which type of data to use for a given application is a very important decision! 4 Quoted from pages of Loss Reserving, by Ronald F. Wiser, revised and updated by Jo Ellen Cockley and Andrea Gardner, Chapter 5 of Foundations of Casualty Actuarial Science, on the Exam 6 Syllabus. 5 For example, an injured worker with a strained back may go back to work and his Workers Compensation claim is closed, but the worker may later restrain his back and have to stop working for several weeks.

6 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 5 Paid Losses and Types of Loss Reserves: 6 Paid Losses Paid + Case Reserves Case Reserves Total Incurred Case Reserve Development Bulk Reserves + Pure IBNR Reserve for Unreported Claims Unfortunately, different terminologies are used by different authors. 6 Adapted from Individual Risk Rating by Margaret Wilkinson Tiller, Edition 1 of Foundations of Casualty Actuarial Science, not included in the current syllabus reading.

7 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 6 Calendar Year Data: 2006 Calendar Year Written Premium: Premium on policies with effective dates from 1/1/06 to 12/31/ Calendar Year Earned Premium: Premiums earned during Includes for example 1/4 of the premium for an annual policy with effective date 4/1/05, and 1/2 of the premium for an annual policy with effective date 7/1/06. CY Written and Earned Premiums are usually not equal to each other. For any policy, the average date of earning is the midpoint of the period for which the policy provides coverage: the date of writing plus (policy term)/2. For CY 2006 written premiums, the average date of writing is 7/1/06. For CY 2006 written premiums, the average date of earning is: 7/1/06 + (policy term)/2. For CY 2006 earned premiums, the average date of earning is 7/1/06. For CY 2006 earned premiums, the average date of writing is: 7/1/06 - (policy term)/ Calendar Year Paid Losses: Losses paid from 1/1/06 to 12/31/ Calendar Year (Case) Incurred Losses: Losses paid during (Case 12/31/06) - (Case 12/13/05). Assume a claim is opened on 10/1/04 with a case reserve of $1000. On 8/1/05, the case reserve is changed to $1500. On 2/1/06, $1200 is paid and the claim is closed. CY Contribution to Case Incurred Losses 2004 $ $ $1000 = $ $ $1500 = -$300 Total $ $500 - $300 = $ This would differ somewhat for lines of insurance with audited premiums, such as Workers Compensation or General Liability. The audit premium for a policy written 1/1/06 would be booked in 2007.

8 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 7 Loss Ratios: Loss Ratios commonly used in ratemaking: Paid Losses / Written Premium Incurred Losses / Earned Premium Calendar/Accident Year Data: There is no such thing as Accident Year Premiums. Calendar/Accident Year 2006 consists of Calendar Year 2006 premiums and Accident Year 2006 losses Accident Year Paid Losses: Losses paid on accidents that occurred from 1/1/06 to 12/31/ Accident Year (Case) Incurred Losses: Losses paid plus case reserves on accidents that occurred from 1/1/06 to 12/31/06. Both Accident Year Paid and Incurred Losses develop as they become more mature. At ultimate they are equal. For example, we might have for AY @12/31/08 Paid: Incurred: Policy Year Data: 2006 Policy Year Written Premium: Premium on policies with effective dates from 1/1/06 to 12/31/ Policy Year Earned Premium: Premiums earned on policies with effective dates from 1/1/06 to 12/31/06. As of 12/31/06, only 3/4 of the premium for an annual policy with effective date 4/1/06 has been earned. Policy Year Earned Premiums develop as they become more mature. At ultimate Policy Year Earned Premiums are equal to Policy Year Written Premiums.

9 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 8 For example, we might have for PY Written: Earned: For any policy, the average date of earning is the midpoint of the period for which the policy provides coverage: the date of writing plus (policy term)/2. For PY 2006 premiums, the average date of earning is: 7/1/06 + (policy term)/ Policy Year Paid Losses: Losses paid on accidents covered by policies with effective dates from 1/1/06 to 12/31/ Policy Year (Case) Incurred Losses: Losses paid plus case reserves on accidents covered by policies with effective dates from 1/1/06 to 12/31/06. Both Policy Year Paid and Incurred Losses develop as they become more mature. At ultimate they are equal. For example, we might have for PY @12/31/08 Paid: Incurred:

10 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 9 Loss Development: 8 Chuck is driving his car alone down a country road on December 29, The bad news is a deer suddenly dashes onto the road in front of Chuck s car. The good news is Chuck is able to swerve and avoid the deer. The bad news is Chuck s car skids off the icy road and hits a tree. The good news is Chuck is not hurt. The bad news is the front end of Chuck s car is badly damaged. The good news is Chuck s car still barely runs. The bad news is it takes Chuck over an hour to very slowly drive his car to Mac s garage. The good news is Mac says that he and his son Nugget can fix Chuck s car in no time. The bad news is no time turns out to be 6 hours. The good news is Chuck s car looks and runs almost as it did before the accident. The bad news is Chuck has to pay Mac $1200 for repairing his car. The good news is Chuck has a collision policy with Gecko Insurance. The bad news is Chuck has to submit a claim, before Gecko Insurance will pay him. The good news is Chuck s brother-in-law Woody is a claims handler for Gecko. The bad news is Woody is off from work until January 5, The good news is on January 5, 2004, Chuck visits Woody s office and his claim is settled. The bad news is Chuck has a $500 deductible on his collision policy. The good news is Woody hands Chuck a check for $700 from Gecko on January 5, 2004! In this case, Chuck s collision insurance claim was closed very quickly. The insured was reimbursed within a week of the accident and the case was closed. As of January 5, 2004 this claim payment of $700 should be in Gecko s computer system and the claim would be marked closed. However, this is the exception rather than the norm. Even for collision insurance, it will often take a month or two for a claim to be settled, and could take 2 years for all payments to be made and all claims to be closed. For liability insurance and workers compensation insurance, it may take up to several years for a claim to be reported to the insurer, and many more years or even decades for all payments to be made and a claim closed. This delay between the occurrence of an incident and the knowledge of the final cost of the incident to an insurer, has very important effects on the data available to actuaries. 8 See pages 80, , and Exhibit 3.3 in McClenahan.

11 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 10 For example, Chuck s accident is part of Accident Year However, as of 12/31/03 nothing had been paid on it. Thus Accident Year 2003 paid losses as of 12/31/03 would include nothing for this accident. However, Accident Year 2003 paid losses evaluated as of 12/31/04 would include $700 for this accident. Eventually, all the dollars on collision incidents during 2003 will have been paid by Gecko Insurance. At that point, Accident Year 2003 collision losses would be at ultimate. Along the way, the Accident Year Losses develop towards their ultimate value. 9 Assume Chuck had contacted Gecko Insurance on December 29, and told them he had been in an accident, and that the expected cost to repair his car was $1400. On December 30, 2003, Gecko would set up a case reserve of: $ $500 = $900. This case reserve would have been part of the Accident Year 2003 paid + case incurred losses as of 12/31/03. Since it only turned out to cost $1200 to repair Chuck s car, he was paid $700 on January 5, At that point the case reserves was removed and $700 in payment was put in its place. Chuck s accident contributes only $700 to Accident Year 2003 paid + case incurred losses as of 12/31/04. So again there is development of the losses. Whenever the amount paid is different than the case reserve (estimate of what will be paid), there is loss development. The combination of the impacts of different things happening on different incidents results in the overall development of an Accident Year of losses. Policy Year losses develop for the same reason. 10 Patterns of development vary from year to year due to random fluctuations as well as changing conditions. However, actuaries can use recent historical patterns of loss development to predict future loss development, for a given line of insurance and situation. For example, assume that for a given policy year, historically the following percentages of ultimate losses have been reported by the following times from the the end of the policy year: Time from beginning of Policy Year % of ultimate reported as paid + case incurred 12 months 30% 18 months 55% 24 months 70% 30 months 80% 36 months 85% 42 months 90% 48 months 95% 54 months 98% 60 months 100% Policy Year 2002 paid plus case incurred losses as of 6/30/04 are $10,000,000. 6/30/04 is 30 months after 1/1/02, the start of Policy Year Thus we expect that $10,000,000 is 80% of the ultimate losses. 9 See McClenahan s Exhibit 3.3. I discuss this exhibit as well as the others in McClenahan subsequently. 10 Calendar Year losses are an accounting concept and do not develop.

12 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 11 Therefore, we would estimate ultimate Policy Year 2002 paid plus case incurred losses as: $10,000,000/.8 = $12,500,000. Another way to say the same thing, is that we would apply a loss development factor of 1/.8 = 1.25 to the Policy Year 2002 paid plus case incurred losses as of 6/30/04 (30 months), in order to predict the ultimate losses for that Policy Year. One could predict the Policy Year 2002 losses at various evaluation dates: 11 6/30/04 (30 months) $10,000,000 (observed) 12/31/04 (36 months) $10,625,000 6/30/05 (42 months) $11,250,000 12/31/05 (48 months) $11,875,000 6/30/06 (54 months) $12,250,000 12/31/06 (60 months) $12,500,000 (ultimate) The actual loss development would vary somewhat from this estimate. Nevertheless, we are assuming $12.5 million is the best estimate, as of 6/30/04, of the ultimate losses on Policy Year This is one of the costs that will have to be paid out of Policy Year 2002 premiums. When the rates for Policy Year 2002 were determined, hopefully a good estimate was included of the ultimate losses for that Policy Year. However, let us assume we are now making rates for Policy Year Trend: The average cost of claims usually changes over time due to inflation. For example, let us assume that losses will increase on average 5% per year. Then since as of 6/30/04 (30 months) Policy Year 2002 losses are $10 million, we estimate Policy Year 2005 losses as of 6/30/07 (30 months) as: ( )(10 million) = $11.58 million. Exercise: Estimate Policy Year 2005 losses at ultimate. [Solution: We estimate Policy Year 2002 losses to ultimately be $12.5 million, and we estimate Policy Year 2005 losses at ultimate to be: ( )($12.5 million) = $14.47 million. Alternately, we estimate Policy Year 2005 losses as of 6/30/07 (18 months) as $11.58 million, and we estimate PY 2005 losses at ultimate to be: (1.25)($11.58 million) = $14.47 million.] So in order to estimate Policy Year 2005 losses at ultimate, we need to multiply the Policy Year 2002 losses as of 6/30/04 (30 months) of $10 million by two factors: a loss development factor of 1.25 and a trend factor of (1.25)( )($10 million) = $14.47 million. 11 One can work with either pure losses or loss plus alae.

13 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 12 The Overlap Fallacy: 12 We apply loss development factors because it takes time from when an accident occurs to when all of the money is paid. 13 We apply trend factors because our data is for older years than the year in which the new rates will apply. 14 It is appropriate to both trend and develop losses; there is no overlap. If we are using Policy Year 2002 data (as of 6/30/04) in order to make rates for Policy Year 2005, then we need to both develop the Policy Year 2002 losses to ultimate and trend the Policy Year 2002 losses to the Policy Year 2005 level. The trend factor would adjust for inflation, while the development factor would take into account the fact that Policy Year 2002 losses would take some more time to reach their ultimate level. The two factors adjust for different phenomena For example, if all losses were paid on the same day they occurred, then Policy Year 2002 data evaluated as of 6/30/04 would be at ultimate. In this unrealistic example, there would be no need to apply any loss development factor. However, we would still apply 3 years of inflation, in order to adjust to the Policy Year 2005 cost level. If instead there is no change expected in the average size of loss for this line of insurance, in other words we expect no inflation, then there would be no need to apply any trend factor. 15 However, we would still apply a loss development factor, in order to adjust the Policy Year 2002 losses at early reports to ultimate. 12 See McClenahan pages , based on Trend and Loss Development Factors, by C. Cook, PCAS See Exhibit 3.3 in McClenahan. 14 See Exhibit 3.5 in McClenahan. 15 One could multiply by a trend factor of unity.

14 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 13 One could put our example into the form of a diagram: 16 Trend Exposure Loss Development occurrence settlement Experience Loss Development occurrence settlement Exposure Period Experience Period The Experience Period is Policy Year 2002, which includes some accidents from 1/1/02 to 12/31/03. Focus on an accident that occurred in the middle of the experience period, 1/1/03, and is settled (closed) on 7/1/05, two and a half years after it occurred. 17 The Exposure Period (when we assume the rates will be in effect) is Policy Year 2005, which includes some accidents from 1/1/05 to 12/31/06. There is a three year long trend period, from 1/1/03, the average date of accident in the experience period, to 1/1/06, the average date of accident in the exposure period. Focus on an accident that occurs in the middle of the exposure period, 1/1/06, and is settled (closed) on 7/1/08, two and a half years after it has occurred. 16 See Figure 3.10 at page 111 of McClenahan, which deals with an accident year of data. Some candidates find these diagrams helpful, while others do not. 17 Other accidents contributing to the Policy Year losses would occur at other times and take various amounts of time to be closed.

15 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 14 Maturity of Trend Data: When one puts together a time series of data to trend, one needs to have data at the same maturity. 18 For example, a series of severities based on accident years each at first report would be valid. For example, one could use: AY 2001 as of 12/31/01, AY 2002 as of 12/31/02, AY 2003 as of 12/31/03, AY 2004 as of 12/31/04, and AY 2005 as of 12/31/05. In this case, one is ignoring any additional information on AY 2001 available after 12/31/01. Instead one could use second reports: AY 2000 as of 12/31/01, AY 2001 as of 12/31/02, AY 2002 as of 12/31/03, AY 2003 as of 12/31/04, and AY 2004 as of 12/31/05. In this case, in order to get a series of 5 data points, we need to use older accident years. The usual approach is to use each accident year at the latest maturity currently available, and then develop them to some common maturity. This is what is done in McClenahan s Exhibits While McClenahan develops losses plus alae and number of claims to ultimate, the key idea is to take each year at the latest available report and to develop it to the same level of maturity. For example, dividing McClenahan s Exhibit 3.3 (reported losses and alae) by Exhibit 3.4 (reported number of claims), we get the following triangle of reported severities: AY For a given maturity, the severities tend to increase with Accident Year. For a given Accident Year, the severities tend to increase with maturity. We are trying to estimate a trend factor to adjust the most recent Accident Year losses and alae at ultimate to the level of the losses and alae at ultimate that will underlie the proposed rates. If we were to use severities at different maturity levels, since severities change with maturity in a systematic way, the future trend would be misestimated. 18 See Exhibits 3.5 and 3.6 in McClenahan, which use data from Exhibits 3.3 and McClenahan uses the latest 6 Accident Years to compute trends, which is one reasonable choice.

16 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 15 Exercise: Use the above severities at latest report to compute a linear trend as per Exhibit 3.5 in McClenahan. Note: Using severities at latest report is not the appropriate way to estimate a trend. [Solution: Take X = 1, 2, 3, 4, 5, 6. Y = 1626, 1700, 1813, 1922, 1933, ^ β = {NΣX i Y i - ΣX i ΣY i }/ {NΣX i 2 - (ΣX i ) 2 } = {(6)(37,880) - (21)(10671)}/{(6)(91) } = 3189/105 = ^ α = Y - ^β X = (10671/6) - (30.37)(3.5) = Year Average Square of Avg. Size Fitted Size Year times Year Severity , , , , , , Sum 10, ,880 Fitted severity = (30.37)(AY ). Annual Severity Trend Factor (AY99/AY98 Least-Squares) = / = ] The 1.7% trend computed using severities at latest report differs very significantly from the 6.8% trend computed in McClenahan s Exhibit 3.5 based on data developed to ultimate. The 6.8% is a valid estimate of the annual trend expected in the future, while the 1.7% is not.

17 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 16 The Effects of Inflation 20 Inflation is a very important consideration when pricing Health Insurance and Property/Casualty Insurance. Important ideas include the effect of inflation when there is a limit and/or deductible. Effect of a Limit Exercise: You are given the following: For 1999 the amount of a single loss has the following distribution: Amount Probability $500 20% $1,000 30% $5,000 25% $10,000 15% $25,000 10% An insurer pays all losses after applying a $10,000 limit to each loss. Inflation of 5% impacts all losses uniformly from 1999 to Assuming no change in the deductible, what is the inflationary impact on dollars paid by the insurer in the year 2000 as compared to the dollars the insurer paid in 1999? [Solution: One computes the average amount paid by the insurer per loss in each year: 1999 Amount Amount 2000 Probability of Loss Insurer Payment of Loss Insurer Payment ,000 1,000 1,050 1, ,000 5,000 5,250 5, ,000 10,000 10,500 10, ,000 10,000 26,250 10,000 Average / 4150 = 1.020, therefore the insurer s payments increased 2.0%.] Inflation on the limited losses is 2%, less than that of the total losses. Prior to the application of the limit, the average loss increased by the overall inflation rate of 5%, from 5650 to In general, for a fixed limit, limited losses increase more slowly than the overall rate of inflation. As the limit increases, the rate of inflation of the limited losses increases, eventually approaching that of the unlimited losses A review of material on an earlier exam. See McClenahan, pages 106 to 110. Note that in the table at the bottom of page 108 of McClenahan, the third limit should be: U/(1 + T) < X U. 21 See for example, Figure 3.9 in McClenahan.

18 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 17 Effect of a Deductible: Exercise: You are given the following: For 1999 the amount of a single loss has the following distribution: Amount Probability $500 20% $1,000 30% $5,000 25% $10,000 15% $25,000 10% An insurer pays all losses after applying a $1000 deductible to each loss. Inflation of 5% impacts all losses uniformly from 1999 to Assuming no change in the deductible, what is the inflationary impact on dollars paid by the insurer in the year 2000 as compared to the dollars the insurer paid in 1999? [Solution: One computes the average amount paid by the insurer per loss in each year: 1999 Amount Amount 2000 Probability of Loss Insurer Payment of Loss Insurer Payment , , ,000 4,000 5,250 4, ,000 9,000 10,500 9, ,000 24,000 26,250 25,250 Average / 4750 = 1.058, therefore the insurer s payments increased 5.8%.] Inflation on the losses excess of the deductible is 5.8%, greater than that of the total losses. Prior to the application of the deductible, the average loss increased by the overall inflation rate of 5%, from 5650 to In general, for a fixed deductible, losses paid excess of the deductible increase more quickly than the overall rate of inflation. In general, there are two effects that cause the excess losses to increase more quickly. 22 First, for a loss that was greater than the deductible, the excess piece increases quicker than the overall rate of inflation. For the above example, the excess portion of the $25,000 loss goes from 24,000 to 25,250, an increase of 5.2%. However, the portion below the deductible did not increase. Second, a loss that was close to but did not exceed the deductible, after inflation will exceed the deductible. This increases the number of losses contributing to the excess layer. For the above example, the $10,000 loss did not lead to a payment before inflation, but does lead to a payment after inflation. 22 See 5, 6/05, Q.31.

19 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 18 The Loss Elimination Ratio in 1999 is: ( )/5650 = 15.9%. The Loss Elimination Ratio in 2000 is: ( )/ = 15.3%. In general, under uniform inflation for a fixed deductible amount the LER declines. The effect of a fixed deductible decreases over time. Similarly, under uniform inflation the Excess Ratio over a fixed amount increases. If a reinsurer were selling reinsurance excess of a fixed limit such as $1 million, then over time the losses paid by the reinsurer would be expected to increase faster than the overall rate of inflation, in some cases much faster. The same mathematics applies when we look at losses excess of a basic limit, rather than a deductible. Excess losses increase more quickly than basic limit losses, for a fixed limit, when there is positive inflation. 23 Limited Losses increase slower than the total losses. Excess Losses increase faster than total losses. Limited Losses plus Excess Losses = Total Losses. 23 See pages of McClenahan and pages of Lange.

20 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 19 Graphical Examples: Assume for example that losses follow a Pareto Distribution with α = 3 and θ = 5000 in the earlier year. 24 Assume that there is 10% inflation and the same limit in both years. Then the increase in limited losses as a function of the limit is: 25 InflationH% L Limit As the limit increases, so does the rate of inflation. For no limit the rate is 10%. 24 You should not be asked to work with the Pareto Distribution on Exam 5. F(x) = 1 - (θ/(θ + x)) α. 25 See Figure 3.9 in McClenahan.

21 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 20 If instead there were a fixed deductible, then the increase in losses paid excess of the deductible as a function of the deductible is: InflationH% L Deductible For no deductible the rate of inflation is 10%. As the size of the deductible increases, the losses excess of the deductible becomes more excess, and the rate of inflation increases. One can also illustrate the effects of inflation using Lee Diagrams, as shown in Mahler s Guide to Lee Diagrams. Effect on Layers of Loss: Exercise: You are given the following: For 1999 the amount of a single loss has the following distribution: Amount Probability $500 20% $1,000 30% $5,000 25% $10,000 15% $25,000 10% An insurer pays all losses after applying a $10,000 limit to each loss and then a $1000 deductible to each loss. Inflation of 5% impacts all loss uniformly from 1999 to Assuming no change in the deductible or limit, what is the inflationary impact on dollars paid by the insurer in the year 2000 as compared to 1999?

22 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 21 [Solution: One computes the average amount paid by the insurer per loss in each year: 1999 Amount Amount 2000 Probability of Loss Insurer Payment of Loss Insurer Payment , , ,000 4,000 5,250 4, ,000 9,000 10,500 9, ,000 9,000 26,250 9,000 Average / 3250 = 1.024, therefore the insurer s payments increased 2.4%.] In this case, the layer of loss from 1000 to 10,000, in other words 9000 excess of 1000, increased more slowly than the overall rate of inflation. However, there were two competing effects. The deductible made the rate of increase larger, while the limit made the rate of increase smaller. Which effect dominates depends on the particulars of a given situation. For example, a loss of size 0.8 million would contribute nothing to the layer from 1 to 2 million prior to inflation, while after 50% inflation it would be of size 1.2 million, and would contribute 0.2 million. In addition, losses which were less than the top of the layer and more than the bottom of the layer, now contribute more dollars to the layer. For example, a loss of size 1.1 million would contribute 0.1 million to the layer from 1 to 2 million prior to inflation, while after 50% inflation it would be of size 1.65 million, and would contribute 0.65 million to this layer. On the other hand, a loss whose size was bigger than the top of a given layer, contributes no more to that layer no matter how much it grows. For example, a loss of size 3 million would contribute 1 million to the layer from 1 to 2 million prior to inflation, while after 50% inflation it would be of size 4.5 million, and would still contribute 1 million. A loss of size 3 million has already contributed the width of the layer, and that is all that any single loss can contribute to that layer. So for such losses there is no increase in the contribution to this layer. Thus for an empirical sample of losses, how inflation impacts a particular layer depends how the varying effects from the various sizes of losses contribute to the combined effect.

23 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 22 Comparing the Effect on Different Layers of Loss: As has been discussed, limited losses increase at a slower rate due to inflation, and the lower the limit, the lower the rate of increase. As has also been discussed, excess losses increase at a faster rate due to inflation, and the higher the deductible, the higher the rate of increase. However, the behavior of layers in general depends on the particular size of loss distribution. Exercise: You are given the following: For 1999 the amount of a single loss has the following distribution: Amount Probability $500 20% $1,000 30% $5,000 25% $10,000 15% $25,000 10% Inflation of 5% impacts all loss uniformly from 1999 to What is the rate of inflation on the layer $10,000 excess of $2,000? What is the rate of inflation on the layer $5000 excess of $10,000? [Solution: The layer $10,000 excess of $2,000 is the layer from $2000 to $12, Amount Amount 2000 Probability of Loss xs of Loss xs , , ,000 3,000 5,250 3, ,000 8,000 10,500 8, ,000 10,000 26,250 10,000 Average /2950 = Therefore the layer $10,000 excess of $2,000 increased by 4.7%. The layer $5,000 excess of $10,000 is the layer from $5000 to $15, Amount Amount 2000 Probability of Loss 5000 xs of Loss 5000 xs , , , , , , ,000 5,000 26,250 5,000 Average /500 = Therefore the layer $5,000 excess of $10,000 increased by 15%.] For this size of loss distribution, the rate of inflation on the first layer $10,000 excess of $2,000 is less than that on the second layer $5,000 excess of $10,000.

24 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 23 Loss Ratio Method of Ratemaking: 26 Experience Loss Ratio = (Trended and Developed Losses) / (Trended Premiums at Present Rates). 27 Exercise: Reported loss and alae is $10 million. The loss development factor to ultimate is 1.2. The loss trend is 5% per year over 3 years. The reported premium is $20 million. The on-level factor for premium is There is no premium trend. What is the experience loss and alae ratio? [Solution: (10)(1.2)( )/{(20)(1.04)} = 66.8%.] Target Loss Ratio = (1 - variable expenses - profit provision)/(1 + ratio of fixed expense to losses). 28 Exercise: Variable Expenses are 20% of premiums. The provision for profit and contingencies is 5% of premiums. ULAE is 7% of loss and lae. What is the target loss and alae ratio? [Solution: (1-20% - 5%)/1.07 = 70.1%.] Indicated Rate Change Factor = (Experience Loss Ratio)/(Target Loss Ratio). Exercise: Using the information from the two previous exercises, what is the indicated rate change? [66.8%/70.1% = A 4.7% decrease.] If the current average rate is $1000, then the indicated new average rate is: (0.953)($1000) = $ See pages of McClenahan. 27 Premiums are trended if there is an inflation sensitive exposure base such as sales. Premiums are either put on the current rate level via the parallelogram method, or premiums at present rates are calculated via the extension of exposures technique. The numerator may or may not include ALAE; the target and experience loss ratios have to be consistent in this regard. 28 The losses may or may not include ALAE; the target and experience loss ratios have to be consistent in this regard. Fixed expenses may include ULAE, part general expenses, etc.

25 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 24 Pure Premium Method of Ratemaking: 29 loss & alae pure premium = (loss & alae)/exposures. 30 Exercise: Reported loss and alae is $10 million. The loss development factor to ultimate is 1.2. The loss trend is 5% per year over 3 years. The reported exposures are 20,800. There is no exposure trend. What is the indicated loss & alae pure premium? [Solution: (10,000,000)(1.2)( )/20,800 = $ ] Indicated Rate = (loss & alae pure premium + fixed expenses)/(1 - variable expenses - profit provision). 31 Exercise: Variable Expenses are 20% of premiums. The provision for profit and contingencies is 5% of premiums. ULAE is 7% of loss and ALAE. What is the indicated average rate? [Solution: ($667.86)(1.07)/(1-20% - 5%) = $953.] Loss Ratio Method versus Pure Premium Method of Ratemaking: 32 Given consistent inputs, the loss ratio and pure premium methods produce the same indicated rate. 33 In this example, both techniques produced the same indicated rate of $953. Here we have assumed a current rate of $1000. The reported premiums had a lower average rate of: $20 million/20,800 = $ This is consistent with the given on level factor of 1.040; $1000/$ = See page 89 of McClenahan. 30 As before, the loss and alae have to be trended and developed to ultimate. Premiums are trended if there is an inflation sensitive exposure base such as sales. 31 As before, the loss and alae have to be trended and developed to ultimate. 32 See pages of McClenahan. 33 See pages of McClenahan. See also 5, 5/00, Q.39 and 5, 5/02, Q.44.

26 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 25 Pure Premium Method Based on exposure Does not require existing rates Does not use on-level premium Produces indicated rates Loss Ratio Method Based on premium Requires existing rates Uses on-level premium Produces indicated rate changes Pure premium method requires well-defined, responsive exposures. Where the exposure unit is not available or is not reasonably consistent between risks, as in the case of commercial fire insurance, the pure premium method cannot be used. Loss ratio method cannot be used for a new line. Where manual rates are required for a new line of business, and assuming there are relevant loss statistics available, the pure premium method must be used. For example, in the 1960s when Homeowners Insurance was introduced, the pure premium method of ratemaking was used. The pure premiums were estimated from monoline experience: fire, theft, liability, etc. Pure premium method is preferable where on-level premium is difficult to calculate. Therefore, for commercial lines where individual risk rating adjustments are made to individual policies, it is more appropriate to use the pure premium method if possible. Expenses: McClenahan discusses expenses in a number of different places. Unfortunately, his different discussions are not a single consistent example! At page 83, F = $12.50 in fixed expenses per exposure and V = 17.5% variable expenses as a percent of premiums. With a loss and loss adjustment expense pure premium of P = $75, and Q = 5% provision for profit and contingencies, using the pure premium method: Indicated Rate = ($75 + $12.5)/(1-17.5% - 5%) = $ Indicated Rate = (P + F)/(1 - V - Q). Here P = $75 includes loss and loss adjustment expense. Therefore, F = $12.50, here must include some of the other expenses, perhaps most of the general expenses and some of the other acquisition expenses. 34 As mentioned previously, at pages 88-91, McClenahan discusses the Pure Premium and Loss Ratio Methods. 34 See Exhibit 1 in Schofield.

27 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 26 In the Loss Ratio Method, target loss ratio = T = (1 - V - Q)/(1 + G), where G = (non-premium related expenses)/losses. It is unclear what is included in non-premium related expenses at pages 88-91! They might include some or all of the following: unallocated loss adjustment expense, a portion of general expenses, a portion of other acquisition expense, and a portion of licenses and fees. Experience loss ratio = (experience losses)/(on-level premiums). A = adjustment factor 35 = (experience loss ratio)/(target loss ratio). The relationship between the factor to load non-premium related expenses in the loss ratio method and the fixed expense pure premium in the pure premium method is: G = F/P. 36 At pages there are no numbers, and it is unclear what is included in G and F. Using the numbers from page 83 of McClenahan, F = $12.50 in fixed expenses per exposure and a loss and lae pure premium of P = $75, one would get G = 12.5/75 = 16.67%. T = (1-17.5% - 5%)/(1.1667) = 66.4%. In this case, T is a target loss and loss adjustment expense ratio. In other cases, T could be a pure loss ratio, or a loss and alae ratio. At pages , G consists only of Unallocated Loss Adjustment Expense, and T = 66.11% is a target loss and alae ratio. G = ULAE/(Loss + ALAE) = 6.42%. V = 29.65%, here includes general expense, other acquisition expense, commissions, and taxes, licenses and fees. In Exhibit 3.7, this target loss and alae ratio of 66.11% is computed as follows: 37 Commissions: 15% of premium. Taxes, Licenses and Fees: 2.25% of premium. Other Acquisition Expenses: 5.6% of premium. General Expenses: 6.8% of premium. V = 15% % + 5.6% + 6.8% = 29.65%. Q = Profit and Contingencies = 0. Target Loss and LAE Ratio = 1 - V - Q = 70.35%. Target Loss and ALAE Ratio = (1 - V - Q)/(1 + G) = 70.35%/ = 66.11%. Unlike in Schofield, none of the general expense, other acquisition expense, or miscellaneous taxes are considered fixed here in McClenahan In other words, the rate change factor. 36 See page 90 of McClenahan. 37 Exhibit 3.7 and page 117 in McClenahan use the same numbers and method as pages in McClenahan. 38 Compare Exhibit 1 in Schofield with page 114 in McClenahan.

28 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 27 Trying to understand expenses in terms of McClenahan's notation is not the real point! 39 One could think of expenses as: ALAE, ULAE, plus all of the items discussed in Schofield and Werner: General Expenses, Commissions, Other Acquisition Expense, Taxes, Licenses and Fees. This is merely a way to put into convenient categories dollars spent by the insurer. How expenses are treated for purposes of ratemaking, in other words predicting future needs, varies. Regardless of the particular method of determining the indicated rate, every dollar expected to be paid by the insurer, whether for losses, loss adjustment expense, or expenses, should be included, plus an appropriate amount to cover profit and contingencies. Depending on the situation, ALAE could be put together with losses, treated as varying proportional to losses, or as a fraction of premiums (part of G). Depending on the situation, ULAE could be put together with losses and ALAE, treated as varying proportional to losses, treated as varying proportional to losses plus ALAE, or as a fraction of premiums (part of G). Depending on the situation, General Expenses could be treated as being a percentage of premium (Variable), as being a certain amount per exposure (Fixed), or with a portion fixed and a portion variable, as discussed in the Schofield and Werner papers. Therefore, G in different contexts might include some or all of the following: allocated loss adjustment expense, unallocated loss adjustment expense, a portion of general expenses, a portion of other acquisition expense, and a portion of licenses and fees. 39 Personally, I do not memorize formulas for this, but rather I understand the concepts. A problem with memorizing formulas, is that the situations and exam questions vary so much.

29 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 28 Loss Ratios, Loss plus ALAE Ratios, and Loss plus LAE Ratios: For example, we are given the following expenses as percent of premium: 40 General Expense 15% Acquisition Expense 12% Premium Taxes 3% Allocated Loss Adjustment Expense 6% Unallocated Loss Adjustment Expense 7% Profit Load 5% Assuming for example 100 dollars in premium, then the provisions in the premiums are: 15 for General Expense, 12 for Acquisition Expense, 3 for Premium Taxes, 6 for Allocated Loss Adjustment Expense, 7 for Unallocated Loss Adjustment Expense, and 5 for Profits. This adds to: = 48, leaving: = 52 to pay for Losses. The provision to pay for loss and lae is: = 65. The target Loss plus LAE ratio for the proposed rates is: 1-15% - 12% - 3% - 5% = 65%. This target would be compared to an observed ratio that included all of the loss adjustment expense. The provision to pay for loss and allocated loss adjustment expense is: = 58. Therefore, the target Loss and ALAE ratio for the proposed rates is: 58%. We note that the ULAE is 7/58 = 12.07% of the loss and alae. Therefore, the target Loss and ALAE ratio for the proposed rates could be determined as: 65%/ = 58%. We have backed the ULAE out of the target loss and lae ratio, by dividing by This target loss and alae ratio could be compared to an observed ratio that included alae, but not ulae. Since the provision to pay losses is 52, the target loss ratio for the proposed rates would be 52%. We note that the ALAE is 6/52 = 11.54% of the loss. Therefore, the target loss ratio for the proposed rates could be determined as: 58%/ = 52%. We have backed the ALAE out of the target loss and alae ratio, by dividing by This target pure loss ratio could be compared to an observed ratio that included no lae. 40 Values were taken from 5, 5/03, Q LAE is 13/53 = 25% of losses. Therefore, the target loss ratio for the proposed rates could also be determined by backing the LAE out of the loss plus lae ratio: 65%/1.25 = 52%.

30 HCMSA-S09-5-1A, McClenahan, Ratemaking, HCM 11/3/08, Page 29 For example, let us assume that the premiums adjusted for trend, development, rate level changes, etc., are 10,000,000. Let us assume that losses adjusted for trend, development, etc., are 6,000,000. Then the observed pure loss ratio is: 6/10 = 60%. Comparing to the target pure loss ratio of 52%, the indicated rate change would be: 60%/52% - 1 = 15.4%. Assume in addition, that the allocated loss adjustment expenses adjusted for trend, development, etc., are 400,000. Then the observed loss and alae ratio is: 6.4/10 = 64%. Comparing to the target loss and alae ratio of 58%, the indicated rate change would be: 64%/58% - 1 = 10.3%. Depending on the various methods used to project the loss and alae, the results of the two methods of arriving at an indicated rate increase may differ, as is the case in this example. Assume in addition, that the unallocated loss adjustment expenses adjusted for trend, development, etc., are 800,000. Then the observed loss and lae ratio is: 7.2/10 = 72%. Comparing to the target loss and lae ratio of 65%, the indicated rate change would be: 72%/65% - 1 = 10.8%. Again, depending on the various methods used to project the loss, alae, and ulae, the results of the different methods of arriving at a rate increase may differ, as is the case in this example. Loss plus LAE Ratio > Loss plus ALAE Ratio > Pure Loss Ratio Pure loss ratios, loss and alae ratios, and loss and lae ratios, are each used in different exam questions, as well as different practical applications. The key idea is to compare apples to apples or oranges to oranges. If the target is a pure loss ratio, then so should the projected. If the target is a loss and alae ratio, then so should the projected. If the target is a loss and lae ratio, then so should the projected. Some of the readings and exam questions are not as clear as they could be as to which of these ratios they are using. 42 Sometimes loss ratio will be used to refer to a loss and alae ratio. 43 In some cases, the reason you are confused is because the paper or exam question is confusing! 42 In the case of essay exam questions, if you are uncertain what the question means, clearly state your assumptions. 43 For example, in 5/01, Q.37, what is called a target loss ratio in part a, is actually a target loss and alae ratio, which can be usefully compared to the projected loss and alae ratio in part c.