“Ratemaking” by Charles L. McClenahans3.amazonaws.com/zanran_storage/ “Ratemaking”, Chapter 3 of Foundations of Casualty Actuarial Science I. INTRODUCTION 1. The Concept of

Mahler’s Guide to

“Ratemaking” by Charles L. McClenahan

See CAS Learning Objectives: B2, D1-D6.

Prepared by Howard C. Mahler. [email protected] some questions prepared by J. Eric Brosius.

Copyright © 2008 by Howard C. Mahler

My Questions are in Study Guide 1B. Past Exam Questions are in Study Guide 1C.

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McClenahan “Ratemaking”, Chapter 3 of Foundations of Casualty Actuarial Science

I. INTRODUCTION1. The Concept of Manual Ratemaking

II. BASIC TERMINOLOGY1. Exposure2. Claim3. Frequency4. Losses and Loss Adjustment Expenses; Figure 3.15. Severity6. Pure Premium7. Expense, Profit and Contingencies8. Premium9. Loss Ratio10. The Goal of the Manual Ratemaking Process11. Structure of the Rating Plan

III. THE RATEMAKING PROCESS1. Basic Manual Ratemaking Methods2. Pure Premium Method3. Loss Ratio Method4. Relationship Between Pure Premium and Loss Ratio Methods5. Selection of Appropriate Method6. Need for Common Basis7. Selection of Experience Period8. Reinsurance9. Differences in Coverage10. Treatment of Increased Limits11. On-Level Premium—Adjusting for Prior Rate Changes; Figures 3.2-3.4

IV. TRENDED, PROJECTED ULTIMATE LOSSES1. Inclusion of Loss Adjustment Expenses2. Projection to Ultimate—the Loss Development Method3. Identification of Trends4. Reflection of Trends; Figures 3.5-3.6 5. Effects of Limits on Severity Trend; Figures 3.7-3.9 6. Trend Based Upon External Data7. Trend and Loss Development—The “Overlap Fallacy”; Figure 3.10 8. Trended Projected Ultimate Losses

V. EXPENSE PROVISIONS

HCMSA-S08-5-1A, McClenahan, Ratemaking, HCM 11/7/07, Page 1

VI. PROFIT AND CONTINGENCIES1. Sources of Insurance Profit2. Profit Provisions in Manual Rates3. Risk Elements

VII. OVERALL RATE INDICATIONS1. Credibility Considerations

VIII. CLASSIFICATION RATES1. Base Rates2. Indicated Classification Relativities3. Correction for Off-Balance4. Limitation of Rate Changes

IX. INCREASED LIMITS1. Trending Individual Losses2. Loss Development by Layer3. Fitted Size-of-Loss Distribution

X. SUMMARYXI. REFERENCESXII. QUESTIONS FOR DISCUSSIONXIII. APPENDIX; Exhibit 3.1-3.161

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 - 1993, m = 150.77, and b = 1455.13.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: 163.99/124.54 = 1.3167 rather than .9494.3 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.


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, 2002. 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, 2003. 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 2003.

Policy Year: All premiums and losses related to policies with effective dates during a given year.

For example, an accident occurs on March 15, 2003. 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.


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, 2003. 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, and5. 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 207-208 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.


Paid Losses and Types of Loss Reserves:6

PaidLosses

CaseReserves

Paid + Case Reserves

CaseReserveDevelopment

Reserve for UnreportedClaims

Bulk Reserves + Pure IBNR

TotalIncurred

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.


Calendar Year Data:

2006 Calendar Year Written Premium:Premium on policies with effective dates from 1/1/06 to 12/31/06.7

2006 Calendar Year Earned Premium: Premiums earned during 2006.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)/2.

2006 Calendar Year Paid Losses: Losses paid from 1/1/06 to 12/31/06.

2006 Calendar Year (Case) Incurred Losses:Losses paid during 2006 + (Case Reserves @ 12/31/06) - (Case Reserves @ 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.

C Y Contribution to Case Incurred Losses2004 $10002005 $1500 - $1000 = $5002006 $1200 - $1500 = -$300

Total $1000 + $500 - $300 = $1200.

7 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.


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.

2006 Accident Year Paid Losses:Losses paid on accidents that occurred from 1/1/06 to 12/31/06.

2006 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 2006 Losses:

@12/31/06 @12/31/07 @12/31/08Paid: 400 550 650

Incurred: 500 600 650

Policy Year Data:

2006 Policy Year Written Premium:Premium on policies with effective dates from 1/1/06 to 12/31/06.

2006 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.


For example, we might have for PY 2006 Premiums:

@12/31/06 @12/31/07Written: 500 500

Earned: 250 500

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)/2.

2006 Policy Year Paid Losses:Losses paid on accidents covered by policies with effective dates from 1/1/06 to 12/31/06.

2006 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 2006 Losses:

@12/31/06 @12/31/07 @12/31/08Paid: 200 450 700

Incurred: 300 650 700


Loss Development:8

Chuck is driving his car alone down a country road on December 29, 2003.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, 2004.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, 99-102, and Exhibit 3.3 in McClenahan.


For example, Chuck’s accident is part of Accident Year 2003. 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: $1400 - $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, 2004. 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 incurred12 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 2002. 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.


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) $10,625,0006/30/05 (42 months) $11,250,00012/31/05 (48 months) $11,875,0006/30/06 (54 months) $12,250,00012/31/06 (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 2002. 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 2005.

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: (1.053)(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, andwe estimate Policy Year 2005 losses at ultimate to be: (1.053)($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 1.053. (1.25)(1.053)($10 million) = $14.47 million.

11 One can work with either pure losses or loss plus alae.


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 110-111, based on “Trend and Loss Development Factors,” by C. Cook, PCAS 1970. 13 See Exhibit 3.3 in McClenahan.14 See Exhibit 3.5 in McClenahan.15 One could multiply by a trend factor of unity.


One could put our example into the form of a diagram:16

Experience Period

Exposure Period

Experience Loss Development

occurrence settlement

occurrence settlement

Exposure Loss Development

Trend

02 03 04 05 06 07 08 09

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.


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 3.3-3.6.19 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:

A Y 12 24 36 48 60 7294 1173 1440 1493 1562 1595 162695 1197 1483 1647 1639 170096 1305 1700 1827 181397 1443 1779 192298 1545 193399 1677

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 3.4.19 McClenahan uses the latest 6 Accident Years to compute trends, which is one reasonable choice.


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, 1677.

€

^β = {NΣXiYi - ΣXiΣYi }/ {NΣXi2 - (ΣXi)2} = {(6)(37,880) - (21)(10671)}/{(6)(91) - 212} =

3189/105 = 30.37.

€

α̂ =

€

Y -

€

^β

€

X = (10671/6) - (30.37)(3.5) = 1672.21.

Year Average Square of Avg. Size FittedSize Year times Year Severity

1 1626 1 1,626 1702.582 1700 4 3,400 1732.953 1813 9 5,439 1763.324 1922 16 7,688 1793.695 1933 25 9,665 1824.066 1677 36 10,062 1854.43

Sum 10,671 91 37,880

Fitted severity = 1672.21 + (30.37)(AY - 1993). Annual Severity Trend Factor (AY99/AY98 Least-Squares) = 1854.43/1824.06 = 1.017.]

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.


The Effects of Inflation20

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 2000.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 1999 2000 Amount 2000Probability of Loss Insurer Payment of Loss Insurer Payment

0.20 500 500 525 5250.30 1,000 1,000 1,050 1,0500.25 5,000 5,000 5,250 5,2500.15 10,000 10,000 10,500 10,0000.10 25,000 10,000 26,250 10,000

Average 5650.00 4150.00 5932.50 4232.504232.50 / 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 5932.5.

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.21

20 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.


Effect of a Deductible:



• An insurer pays all losses after applying a $1000 deductible to each loss.• Inflation of 5% impacts all losses uniformly from 1999 to 2000.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:


0.20 500 0 525 00.30 1,000 0 1,050 500.25 5,000 4,000 5,250 4,2500.15 10,000 9,000 10,500 9,5000.10 25,000 24,000 26,250 25,250

Average 5650.00 4750.00 5932.50 5027.505027.5 / 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 5932.5. 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.


The Loss Elimination Ratio in 1999 is: (5650 - 4750)/5650 = 15.9%. The Loss Elimination Ratio in 2000 is: (5932.5 -5027.5)/5932.5 = 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 107-109 of McClenahan and pages 168-169 of Lange.


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

10000 20000 30000 40000 50000Limit

2

4

6

8

10InflationH% L

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.


If instead there were a fixed deductible, then the increase in losses paid excess of the deductible as a function of the deductible is:

2000 4000 6000 8000 10000Deductible10

12

14

1618

20

2224

InflationH% L

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:



• 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 2000.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?


[Solution: One computes the average amount paid by the insurer per loss in each year:


0.20 500 0 525 00.30 1,000 0 1,050 500.25 5,000 4,000 5,250 4,2500.15 10,000 9,000 10,500 9,0000.10 25,000 9,000 26,250 9,000

Average 5650.00 3250.00 5932.50 3327.503327.5 / 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.


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.



• Inflation of 5% impacts all loss uniformly from 1999 to 2000.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 $12,000.

1999 Amount 1999 2000 Amount 2000Probability of Loss 10000 xs. 2000 of Loss 10000 xs. 2000

0.20 500 0 525 00.30 1,000 0 1,050 00.25 5,000 3,000 5,250 3,2500.15 10,000 8,000 10,500 8,5000.10 25,000 10,000 26,250 10,000

Average 5650.00 2950.00 5932.50 3087.503087.50/2950 = 1.047. 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 $15,000.

1999 Amount 1999 2000 Amount 2000Probability of Loss 5000 xs. 10000 of Loss 5000 xs. 10000

0.20 500 0 525 00.30 1,000 0 1,050 00.25 5,000 0 5,250 00.15 10,000 0 10,500 5000.10 25,000 5,000 26,250 5,000

Average 5650.00 500.00 5932.50 575.00575/500 = 1.150. 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.


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%) = $112.90.

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.26

At pages 88 - 91, the Pure Premium and Loss Ratio Methods are discussed.

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 factor27 = (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.28 At pages 88 - 91 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.

26 See Exhibit 1 in Schofield.27 In other words, the rate change factor.28 See page 90 of McClenahan.


At pages 112-114, 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:29 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% + 2.25% + 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%/1.0642 = 66.11%.

Unlike in Schofield, none of the general expense, other acquisition expense, or miscellaneous taxes are considered fixed here in McClenahan.30

Trying to understand expenses in terms of McClenahan's notation is not the real point!31

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.29 Exhibit 3.7 and page 117 in McClenahan use the same numbers and method as pages 112-114 in McClenahan.30 Compare Exhibit 1 in Schofield with page 114 in McClenahan.31 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.


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.

Loss Ratios, Loss plus ALAE Ratios, and Loss plus LAE Ratios:

For example, we are given the following expenses as percent of premium:32 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: 15 + 12 + 3 + 6 + 7 + 5 = 48, leaving: 100 - 48 = 52 to pay for Losses.

The provision to pay for loss and lae is: 52 + 6 + 7 = 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: 52 + 6 = 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%/1.1207 = 58%. We have backed the ULAE out of the target loss and lae ratio, by dividing by 1.1207. 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%/1.1154 = 52%. We have backed the ALAE out of the target loss and alae ratio, by dividing by 1.1154.33 This target pure loss ratio could be compared to an observed ratio that included no lae.

32 Values were taken from 5, 5/03, Q.28.33 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%.


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.34 Sometimes “loss ratio” will be used to refer to a loss and alae ratio.35 In some cases, the reason you are confused is because the paper or exam question is confusing!

34 In the case of essay exam questions, if you are uncertain what the question means, clearly state your assumptions.35 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.


Dividing an Indicated Rate Into Its Components:

In Exhibit 3.7: 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% + 2.25% + 5.6% + 6.8% = 29.65%.G = ULAE/(Loss + ALAE) = 6.42%.Q = Profit and Contingencies = 0.Target Loss and ALAE Ratio = (1 - V - Q)/(1 + G) = 70.35%/1.0642 = 66.11%.

In Exhibit 3.8: Trended On-Level Loss and ALAE Ratio is 72.82%.Indicated Rate Change is: 72.82%/66.11% - 1 = 10.14%, subject to rounding.36

In Exhibit 3.2:Earned Exposures (1997-1999): 52,267.On-Level Earned Premium (1997-1999): $11,403,572.Therefore, the current average rate is: $11,403,572/52,267 = $218.18.Therefore, the indicated average rate is: ($218.18)(1.1014) = $240.30.37

One can break this indicated average rate into components as follows:Commissions: (15%)($240.30) = $36.05. Taxes, Licenses and Fees: (2.25%)($240.30) = $5.41.Other Acquisition Expenses: (5.6%)($240.30) = $13.46. General Expenses: (6.8%)($240.30) = $16.34.Profit and Contingencies = (0)($240.30) = $0.38 Loss and ALAE = (66.11%)($240.30) = $158.86.39 ULAE = (6.42%)($158.86) = $10.20.

$36.05 + $5.41 + $13.46 + $16.34 + $0 + $158.86 + $10.20 = $240.32, which equals the indicated rate subject to rounding.36 Alternately, the target loss plus lae ratio is 70.35%. Projected Loss plus lae is: (1.0642)($23,163,751) = $24,650,864. Projected loss plus lae ratio is: $24,650,864/$41,811,448 = 77.49%. Indicated Statewide Rate Level Change is: 77.49%/70.35% - 1 = 10.14%.37 However, as shown in the bottom portion of Exhibit 3.14, due to rounding base rates to the nearest dollar, the proposed average rate change is 10.35%, rather than 10.14%.The proposed average rate is: $12,583,797/52,267 = $240.76. One could perform a similar breakdown of this $240.76, as I have done with the $240.30. 38 In most exam questions and practical applications, the profit and contingencies provision is not zero.39 Since in McClenahan’s Exhibits the loss and alae are shown combined, in this example there is no way to divide loss plus alae between losses and allocated loss adjustment expense.


Estimating Class Relativities:

McClenahan estimates relativities in two different places.

First, on pages 119-121, McClenahan has a very simple illustrative example.There are three classes and no territories.The loss ratios of the three classes are compared, without the use of credibility.

[1] [2] [3] [4] [5] [6] [7]Class Current On-Level Class 1 Experience Loss Indicated

Relativity Earned On-Level Loss & ALAE & ALAE Relativityto Class 1 Premium Earned Ratio to Class 1

=[3]/[2] =[5]/[4]

1 1.0000 $14,370,968 $14,370,968 $11,003,868 0.7657 1.00002 1.4500 $9,438,017 $6,508,977 $6,541,840 1.0050 1.31263 1.8000 $8,002,463 $4,445,813 $5,618,043 1.2637 1.6503

Total $31,811,448 $25,325,758 $23,163,751

Loss ratios have been calculated, but pretending that we had charged each class the rate used for the base class 1. If on this basis a class had a higher loss ratio than class 1, then it would have an indicated rate higher than class 1. For example, for class 3, the loss ratio on this basis is 1.2637 versus 0.7657 for class 1. Thus class 3 has an indicated relativity compared to the base class of1.2637/0.7657 = 1.6503.

Note that the output is indicated relativities, rather than changes to the relativities. These indicated relativities would not balance to either no overall change or a desired overall change. An off-balance factor (to be multiplied by) is then calculated, in order to balance to the desired overall rate change.

[1] [2] [3] [4] [5] [6]Class Current Indicated Indicated On-Level Indicated

Relativity Relativity Change Earned Premiumto Class 1 to Class 1 =[3]/[2] Premium =[4]x[5]

1 1.0000 1.0000 1.0000 $14,370,968 $14,370,9682 1.4500 1.3126 0.9052 $9,438,017 $8,543,6843 1.8000 1.6503 0.9168 $8,002,463 $7,336,925

Total $31,811,448 $30,251,576

Off-balance Factor = Premium for no overall change / Indicated Premium = $31,811,448/$30,251,576 = 1.0516.40

Assume we desire an overall increase of 10.14% and the current base rate for class 1 is $160.40 On page 120, McClenahan arrives at this same answer via a slightly different but equivalent calculation.


Then the changes by class and the new indicated rates by class are:41

[1] [2] [3] [4] [5] [6] [7] [8]Class Current Current Indicated Off-Balance Overall Indicated Indicated

Relativity Rate Change in Factor Rate Change Rateto Class 1 =[2]x$160 Relativity Change = [4]x[5]x[6] = [3]x[7]

1 1.0000 $160.00 1.0000 1.0516 1.1014 1.1582 $185.322 1.4500 $232.00 0.9052 1.0516 1.1014 1.0484 $243.243 1.8000 $288.00 0.9168 1.0516 1.1014 1.0619 $305.82

In each case, the indicated new class rate is: (current class rate)(indic. change in rel. prior to off-balance)(off-balance factor)(overall change factor).

One can verify that the indicated changes by class, produce the desired 10.14% increase.{(.1582)(14,370,968) + (.0484)(9,438,017) + (.0619)(8,002,463)}/31,811,448 = 3,225,640/31,811,448 = 10.14%.

One could have instead compared actual loss ratios for each class to those for the base class 1, in order to get indicated changes to the current relativities:

[1] [2] [3] [4] [7]Class On-Level Experience Loss Indicated

Earned Loss & ALAE & ALAE Change inPremium Ratio Relativity

=[3]/[2]

1 $14,370,968 $11,003,868 0.7657 1.00002 $9,438,017 $6,541,840 0.6931 0.90523 $8,002,463 $5,618,043 0.7020 0.9169

Total $31,811,448 $23,163,751

For example, .6931/.7657 = .9052.These match, subject to rounding, the indicated changes to relativities obtained previously.As before, one would have to calculate the off-balance factor and include a factor for the desired overall rate change, in order to calculate the indicated new rates by class.

41 On pages 120 and 121, McClenahan arrives at the same numbers using a somewhat different format.


Estimating Class and Territory Relativities:

In Exhibits 3.9 to 3.14, McClenahan presents a more complex example of calculating relativities.42 Unlike on pages 119-121, here there are both classes and territories, the method relies on pure premiums rather than loss ratios, and credibility is used.

Three separate accident years of data by class and territory are used.The first step is to trend and develop loss and alae by year, class, and territory.43 Then the developed and trended loss and alae are divided by exposures, to get pure premiums.44

PPtcy = (Dev. Factor)y(Sev. Trend Factor)y(Freq. Trend Factor)y(Loss & Lae)tcy/Expostcy.

The next step is to calculate for each territory/class/year cell, relativities to the base class and the base territory.45

ClassReltcy = PPtcy/PPt1y, for the relativity to the base class 1.For example, ClassRel1,3,99 = $279.29/$175.95 = 1.5873.

TerrReltcy = PPtcy/PP2cy, for the relativity to the base territory 2.For example, TerrRel1,3,99 = $279.29/$207.96 = 1.3430

Then for each class, the pure premium relativities by territory and year are averaged using exposures.46

ClassRelc = Σ Expostcy ClassReltcy / Σ Expostcy

t,y t,y

Then each relativity indicated by the pure premiums is credibility weighted with the current relativity.47

CredWghtClassRelc = Z ClassRelc + (1 - Z)CurrClassRelc.48

Z = Exposures/(Exposures + 25000).

42 All of McClenahan’s Exhibits are discussed subsequently.Finger discusses two somewhat different techniques of calculating relativities, which will be compared and contrasted with McClenahan’s in “Mahler’s Guide to Finger.”43 Exhibit 3.9.44 See Exhibit 3.10.45 See Exhibit 3.10.46 See Exhibit 3.11.47 See Exhibit 3.11.48 McClenahan then selects a relativity that is a round number close to the credibility weighted relativity.


Similarly, for each territory, the pure premium relativities by class and year are averaged using exposures.49

TerrRelt = Σ Expostcy TerrReltcy / Σ Expostcy

c,y c,y

Then each relativity indicated by the pure premiums is credibility weighted with the current relativity.50

CredWghtTerrRelt = Z TerrRelt + (1 - Z)CurrTerrRelt.51

The next step is to determine the indicated new base rate, for class 1 and territory 2.52 This is equivalent to determining the off-balance factor so that the desired overall rate change of 10.14% is achieved. One uses the latest year of on-level earned premium, AY99.One calculates the premium level effect of the proposed new relativities.

Effect of the relativity change on class c and territory t:53 Effecttc = {(CredWghtClassRelc CredWghtTerrRelt)}/ {(CurClassRelc CurTerrRelt)} - 1.

Σ Premtc99 Effecttc / Σ Premtc99 = -$398,873/$11,403,572 = -3.50%.

Thus if the base rate were kept the same, the overall average rate would decrease by 3.50%.In order to get the desired 10.14% overall average increase, we need to multiply the current base rate by a factor of: 1.1014/.9650 = 1.1413.54

New Base Rate = (Current Base Rate)(Desired Rate Increase Factor)/(Effect of Change in Rels.)

Since the current base rate is $160, the proposed new base rate is: (1.1413)(160) ≅ $183.

49 See Exhibit 3.12.50 See Exhibit 3.12.51 McClenahan then selects a relativity that is a round number close to the credibility weighted relativity.52 See Exhibit 3.13.53 See column [8] of Exhibit 3.13.The class and territory relativities multiply.54 In the third to last row of Exhibit 3.13, it mistakenly has 1.014/.9650, but has the correct answer.


Diagram of the Exhibits in the Appendix of McClenahan:

Exhibit 1Current Rate Manual

Exhibit 3Loss Devel.

Exhibit 4Claim Count Development

Exhibit 5Severity Trend

Exhibit 6Frequency Trend

Exhibit 8Overall Rate Change

Exhibit 9LossesDeveloped& TrendedClass/Terr.

Exhibit 10Pure Premium Relativities

Exhibit 11Class Rels.

Exhibit 12Terr. Rels.

Exhibit 13Off Balance

Exhibit 14Revised Basic LimitRates

Exhibit 15Increased LimitsFactors

Exhibit 16Proposed Rate Manual

Exhibit 2On Level Premiums

Exhibit 7Expenses


Notes On the Exhibits in the Appendix of McClenahan:

McClenahan in an appendix has 16 exhibits. Since McClenahan is a basic and important reading, study all of these exhibits carefully. Then go back and go through these exhibits again in about a month. The appendix contains a complete, though simplified, example of a manual rate analysis of private passenger automobile bodily injury.55

Exhibit 3.1 is meant to represent the existing rate manual, effective 7/1/1998, for the coverage under review. The manual contains basic limits rates for each of three classifications within each of three territories,56 along with a single increased limits factorto adjust the rates for basic limits of $20,000 per person, $40,000 per occurrence (20/40) upward to limits of $100,000 per person, $300,000 per occurrence (100/300).57 Territorial and classification rates are keyed to a base rate of $160 for Territory 2, Class 1.

Current Rate LevelTerritory Class 1 Class 2 Class 3

1 $224 $325 $4032 $160 $232 $2883 $136 $197 $245

Pay more in urban Central City (territory 1) and Youth Owner or Principal Operator (Class 3).Pay less in remainder of the state (territory 3) and with no Youth Operators (Class 1).

Note the use of a base class and base territory. In this case, the base class 1 and base territory 2 have the most exposures, although this need not be the case.58

Dividing by the $160 rate for Class 1 and Territory 2:Current Relativity, with respect to Class 1 and Territory 2

Territory Class 1 Class 2 Class 31 1.4 2.03 2.522 1.0 1.45 1.803 .85 1.23 1.53

These are the product of territory relativities of: 1, 1.45 and 1.80, and class relativities of: 1.4, 1 and .85. For example, (1.45)(.85) = 1.23.

55 There would some differences for other lines of insurance.56 There would usually be more than three classes and more than three territories.57 We will make rates for basic limits, 20/40, and then apply an increased limits factor to get the rate for 100/300. 20/40 is presumably the minimum limits of liability required by law in this state.58 The choice of a base class and territory is just a matter of convenience. Usually an insurer will use the same base class in every state.


Currently an insured pays 30% more to buy increased limits of 100/300, than if he bought basic limits of 20/40, regardless of class or territory.59

Exhibit 3.2 demonstrates the computation of the on-level earned premium based upon the extension of exposures technique.60 The experience period is the three years 1997–1999 and the earned exposures, by class and territory, for each of those years are multiplied by the appropriate current rate to yield the on-level earned premium.

For example, (7807)($224) = $1,748,768.The average rate is: $11,403,572 / 52,267 = $218.18.

Calendar Year Exposures.An excellent example of the use of the extension of exposures technique!

Note the use of a two dimensional grid of exposures by class and territory.61 The distribution of classes differs by territory. For example, in 1999, Class 3 is: 9575/52,267 = 18.32% overall, but only 1870/15,787 = 11.85% of Territory 1.62

Exhibit 3.3 shows the projection of ultimate loss and allocated loss adjustment expense for accident years 1994–1999, using the case-incurred loss development method.

Accident Year Losses Plus ALAE:63 paid + case reserves.64 Good example of the standard chain ladder technique of getting age-to-age link ratios! Very basic and important!

59 The increased limits factor of 1.30 is the same regardless of class or territory. We would normally have more than one increased limit available. For example, 25/50 would cost less than 100/300, while 250/500 would cost more than 100/300. 60 We do not use on-level factors as per McClenahan pages 91-97. We make no use of premiums actually collected.61 It is actually a three dimensional array by year, class, and territory.62 In such situations it is better to work directly with the individual cells, as McClenahan did here, rather than with the marginal distributions as in Schofield’s Exhibit 2.63 One could analyze the development of losses and alae separately.64 One could do a similar calculation with paid losses. The loss development factors would be larger, and the projected ultimates would differ, being in some cases larger and in some cases smaller.


Here is McClenahan’s Exhibit 3.3:65 Accident Cumulative Basic Limits Case-Incurred Losses and ALAECumulative Basic Limits Case-Incurred Losses and ALAECumulative Basic Limits Case-Incurred Losses and ALAECumulative Basic Limits Case-Incurred Losses and ALAE

Year Age 12 Age 24 Age 36 Age 48 Age 60 Age 72

1994 $2,116,135 $3,128,695 $3,543,445 $3,707,375 $3,854,220 $3,928,8051995 $2,315,920 $3,527,197 $3,992,805 $4,182,133 $4,338,7651996 $2,743,657 $4,051,950 $4,593,472 $4,797,1941997 $3,130,262 $4,589,430 $5,230,4371998 $3,625,418 $5,380,6171999 $3,919,522

Accident Accident Incremental Loss and Allocated Development FactorsAccident Incremental Loss and Allocated Development FactorsAccident Incremental Loss and Allocated Development FactorsAccident Incremental Loss and Allocated Development FactorsYear 12 to 24 24 to 36 36 to 48 48 to 60 60 to 72

1994 1.4785 1.1326 1.0463 1.0396 1.01941995 1.5230 1.1320 1.0474 1.03751996 1.4768 1.1336 1.04441997 1.4661 1.13971998 1.4841

Selected 1.4800 1.1350 1.0450 1.0385 1.0200 1.0000UltimateFactor 1.8594 1.2564 1.1069 1.0593 1.0200 1.0000

Accident Loss & ALAE Ultimate ProjectedYear @ 12/31/99 Factor Ultimate

1994 $3,928,805 1.0000 $3,928,8051995 $4,338,765 1.0200 $4,425,5401996 $4,797,194 1.0593 $5,081,5241997 $5,230,437 1.1069 $5,789,7651998 $5,380,617 1.2564 $6,760,0661999 $3,919,522 1.8594 $7,288,089

For example, $3,128,695/$2,116,135 = 1.4785.

McClenahan selects round values for the age-to-age factors, somewhere in the historical range. For example, for 36 to 48 months the observed factors are: 1.0463, 1.0474, and 1.0444; McClenahan selects 1.0450.

Here we assume that 72 months (from the start of the accident year) is ultimate.66 The 48th to ultimate factor is: 1.0385)(1.0200) = 1.0593.Projected ultimate for Accident Year 1996 is: (1.0593)($4,797,194) = $5,081,524.

Even though the incurred loss plus alae for AY96 as of 12/31/99 includes paid plus case reserves, they are only an estimate of the ultimate loss plus alae paid on accidents that occurred during 1996.65 Some values differ slightly from those shown in McClenahan due to rounding.66 For other lines of insurance, such as medical malpractice, it would take many more years in order to reach ultimate.


Based on the observed past pattern, we expect that if we come back in several years, when all of the AY96 claims have been closed, we will find that the total paid on AY96 will be about 6% higher than the incurred as of 12/31/99.

Thus rather than the $4.8 million at latest report, we project an ultimate for AY96 of $5.1 million.It is important to note that this is just an estimate based on the information available as of 12/31/99.The ultimate losses for AY 96 could be either higher or lower than $5.1 million.

One should also note that for some lines of insurance or insurers the average incremental development at some reports may be downwards. For example, an insurer may set case reserves that on average are more than is eventually paid, although the reverse is more common. On automobile physical damage coverages, at later reports development tends to be downwards due to salvage and subrogation.67

Exhibit 3.4 contains the projected ultimate claim counts for accident years 1994–1999 based upon the reported count development method.

Parallel to Exhibit 3.3, but it deals with claim counts rather than dollars of loss. Claim counts generally develop less than dollars of loss.68

Exhibit 3.5 details the calculation of the severity trend factor based upon the projected incurred losses and ultimate claims for accident years 1994–1999.69 The trend factor is based upon a linear least-squares fit.70

The severities by Accident Year are the ratio of projected ultimate losses and alae from Exhibit 3.3 and the projected ultimate claim counts from Exhibit 3.4.

€

^β = {NΣXiYi - ΣXiΣYi }/ {NΣXi2 - (ΣXi)2} = {(6)(44,280) - (21)(11897)}/{(6)(91) - 212} =

15843 /105 = 150.89.

€

α̂ =

€

Y -

€

^β

€

X = (11897/6) - (150.89)(3.5) = 1454.72.

67 For example, if the insurer pays its insured to replace a damaged car, the insurer can sell the damaged car for whatever it is worth. Also in some of these situations there may be an at-fault driver other than its own insured. In that case, the insurer may be able to recover some money from the at-fault driver and/or his insurer. See page 245 of Insurance Operations by Webb, Harrison, and Markham.68 Even after all the claims have been reported, they have not all been settled.69 One could use a different number of years than six.70 One could instead use an exponential regression. See “Mahler’s Guide to Marker & Mohl,” for a brief review of regression.


Year Ultimate Ultimate Average Square of Avg. Size FittedLoss & ALAE Claims Size Year times Year Severity

1 3,928,805 2,416 1626.2 1 1,626 1605.612 4,425,540 2,552 1734.1 4 3,468 1756.503 5,081,668 2,646 1920.5 9 5,762 1907.394 5,790,094 2,844 2035.9 16 8,144 2058.285 6,760,207 3,068 2203.5 25 11,017 2209.176 7,288,351 3,066 2377.2 36 14,263 2360.06

Sum 11,897 91 44,280

Fitted severity = 1454.72 + (150.89)(AY - 1993).71 Annual Severity Trend Factor (AY99/AY98 Least-Squares) = 2360.06/2209.17 = 1.0683.

Exhibit 3.6 addresses the frequency trend factor based upon the earned exposures and projected ultimate claims for accident years 1994–1999, based upon an exponential least-squares fit.72

€

^β = {NΣXilnYi - ΣXiΣlnYi }/ {NΣXi2 - (ΣXi)2} = {(6)(-58.44) - (21)(-16.63)}/{(6)(271) - 212} =

-1.41/105 = -.0134.

€

α̂ = ΣlnYi/N -

€

^β

€

X = (-16.63/6) - ( -.0134)(3.5) = -2.725.

Year Ultimate Earned Projected Ln(freq) Square of Ln(freq) FittedClaims Exposures Frequency Year times Year Frequency

1 2,416 37,846 0.0638 -2.751 1 -2.75 0.06472 2,552 39,771 0.0642 -2.746 4 -5.49 0.06383 2,646 42,135 0.0628 -2.768 9 -8.30 0.06304 2,844 45,231 0.0629 -2.767 16 -11.07 0.06215 3,068 48,583 0.0631 -2.762 25 -13.81 0.06136 3,066 52,267 0.0587 -2.836 36 -17.02 0.0605

Sum -16.630 91 -58.44

Fitted Frequency = exp[-2.725 + (-.0134)(AY - 1993)] = .0656 exp[(-.0134)(AY - 1993)].73 Annual Frequency Trend Factor (AY99/AY98 Least-Squares) = 0.0605/0.0613 ≅ .9868.

71 Differs from Exhibit 3.5 due to rounding.72 One could instead use a linear regression.73 Differs from Exhibit 3.6 due to rounding.


Exhibit 3.7 contains the calculation of the target loss and allocated loss expense ratio. Note that there is no specific provision for profit and contingencies in this example, the assumption being that the investment profits will be sufficient.

Basic and important!

Standard calculation as per McClenahan pages 113-114.74 Here we calculate a target loss and alae ratio, rather than a target loss ratio.

T = (1 - V - Q)/(1 + G) = (1 - 29.65% - 0%)/(1 + 6.42%) = 66.11%.

Exhibit 3.8 presents the calculation of the indicated statewide rate level change, using the loss ratio method.

Very important and basic!

Note that we use the latest three Calendar/Accident Years of Data in order to calculate the overall rate increase.75 The experience loss and alae ratio is the sum of three years of losses and alae divided by three years of premiums. In this case, the loss ratios for individual years are not used.76

Accident Projected Midpoint Years Trend Factor to 7/0/01 Trend Factor to 7/0/01Year Loss & ALAE Experience to 7/1/01 Severity Frequency

Period 1.0683 0.9867

1997 $5,790,094 7/1/97 4.0 1.3025 0.94791998 $6,760,207 7/1/98 3.0 1.2192 0.96061999 $7,288,351 7/1/99 2.0 1.1413 0.9736

Accident Trended On-Level Trended Target IndicatedYear Loss & ALAE Earned On-Level On-Level Statewide

Premium Loss & ALAE Loss & ALAE Rate LevelRatio Ratio Change

1997 $7,148,239 $9,831,957 72.70%1998 $7,917,627 $10,575,919 74.86%1999 $8,098,153 $11,403,572 71.01%

Total $23,164,019 $31,811,448 72.82% 66.11% 10.14%

The midpoint of AY97 is 7/1/97. The average date of accident of the effective period is 7/1/01.77

74 We do not assume any fixed expenses, in contrast to what would be done in Schofield.75 Using more years would make the indication less responsive. Using fewer years would make the rate indication less stable. AY 1999 is less mature, requiring we apply larger loss development factors. AY 1997 is less recent, requiring that we apply a larger trend factor.76 In other cases, one might take a (weighted) average of the loss and alae ratios by year.77 Proposed rates are to be effective 7/1/2000, for one year.


Apply 4 years of trend to AY97. Severity: 1.06834 = 1.3025.78 Frequency: .98674 = .9479.

AY97 loss and alae developed and trended: (1.3025)(.9479)($5,790,094) = $7,148,680.

Trended On-Level Loss and ALAE Ratio = $23,163,751/$31,811,44879 = 72.82%.

Since the experience loss ratio & alae of 72.82% is higher than the target loss & alae ratio of 66.11%, basic limits rates are indicated to increase.72.82%/66.11% ≅ 1.1014. ⇔ 10.14% increase indicated overall for basic limits.80

Exhibit 3.9 contains projections of trended projected ultimate losses and allocated loss expenses by accident year, classification, and territory for accident years 1997–1999.

Losses and alae, by class/territory/year cell, a 3-dimensional array.Multiply each value by the appropriate loss development factor from Exhibit 3.3.81 Multiply each value by the appropriate trend factors from Exhibit 3.8.

For example, (1.1070)(1.3025)(0.9479)($986,617) = $1,348,455.

Exhibit 3.10 demonstrates the calculation of indicated classification and territorial pure premiums and pure premium relativities.

For each cell, divide the trended, developed loss & alae from Exhibit 3.9 by the earned exposures from Exhibit 3.2.Use these pure premiums to calculate relativities to the base Class 1.For example, for Territory 1, Class 2, AY 1997: $239.99 /$172.72 ≅ 1.3894.Use these pure premiums to calculate relativities to the base Territory 2.For example, for Territory 1, Class 2, AY 1998: $247.55 /$155.47 = 1.5923.

Exhibit 3.11 shows the calculation of credibility-weighted classification relativities and the selection of relativities to be used.

For each class, there are (3)(3) = 9 relativities from Exhibit 3.10.Then weight these 9 relativities together, using earned exposures.For example, for Class 3:{(1553)(1.6580) + ... + (3036)(1.8162)}/(1553 + ... + 3036) = 45,433.93/27,104 = 1.6763.78 Note that even though as discussed previously the annual severity trend was determined by fitting a straight line, the severity trend factor is determined here by taking this annual factor to the appropriate power.79 On-Level Earned Premium for 1997-1999, from Exhibit 3.2, by extension of exposures.80 The indicated overall change is given full credibility; it is not weighted with something else.81 The references in Exhibits 9-14 to other exhibit numbers are incorrect.


Then credibility weight this indicated relativity with the current relativity.Z = E/(E + 25,000).82

For Class 3, Z = 27,104/(27,104 + 25,000) = .5202.Then the Credibility Weighted Relativity is: (.5202)(1.6763) + (1 - .5202)(1.800) = 1.7357.McClenahan then selects a round number, in this case 1.74, for the Class 3 relativity.

Exhibit 3.12 shows the calculation of credibility-weighted territorial relativities and the selection of relativities to be used.

Parallel to the previous exhibit, but territory rather than class relativities.

Exhibit 3.13 details the correction for off-balance resulting from the selected classification and territorial relativities.

Based on the latest year, 1999, of premiums at current rates, from Exhibit 3.2.

For Class 3, Territory 3, the current relativity is: (1.8)(.85) = 1.53,83 and the proposed relativity is: (1.74)(.8) = 1.392.84 The effect of the change in relativities is: 1.392/1.53 - 1 = -9.0196%.Since there is $743,820 in on-level earned premium, this has an effect of:(-9.0196%)($743,820) = -$67,090.

Adding up the effects for each class/territory cell, one gets -$398,873.In other words, the proposed changes in relativities would result in a decrease in premiums, if not corrected via an off-balance factor.

Calculates an off-balance factor to be divided by.premium effect / total premium = -398,873/11,403,522 = -3.50%off-balance factor = 1 - 3.50% = .9650.Overall increase factor: 1.1014.85 Factor to apply to base rate: 1.1014/.9650 = 1.1413.Proposed Base Rate = (1.1413)($160) = $183.86

82 Buhlmann Credibility, with the Buhlmann Credibility Parameter, K = 25,000. For 25,000 exposures, the indicated and current relativity would each be given 50% weight. The source of K is not explained or shown in McClenahan. Buhlmann Credibility is covered on Course 4, see for example “Credibility” by Mahler and Dean.83 See Exhibit 3.1.84 From Exhibits 3.11 and 3.12.85 Prior to off-balance. From Exhibit 3.8.86 Rounded to the nearest dollar. Current Base Rate for Class 1 and Territory 2 is $160 from Exhibit 3.1.


Exhibit 3.14 shows the development of the revised basic limits rates and the calculation of the resulting statewide rate level change.

The top part of the exhibit calculates the proposed (basic limits) rates as a product of the base rate, class relativity, and territory relativity.For example for Class 3 and Territory 1, (1.74)(1.40)($183) = $446.87 88

The bottom part of the exhibit calculates the resulting change in basic limits rates.Expanding by the exposures for the latest year, 1999, the premiums for the current rates is compared to that for the proposed rates: $12,583,797/$11,403,372 = 1.1035.Due to rounding, the 10.35% rate change proposed, is not exactly equal to the 10.14% indicated change from Exhibit 3.8.

Exhibit 3.15 describes the calculation of the revised 100/300 increased limits factor using the individual trended loss approach.

Review of increased limits factor, as described at pages 122-123 of McClenahan.

Data is individual unlimited losses89 from closed claims90 from the most recent 13 years,91 1987-1999.

Each individual loss is trended to 12/31/99, using an annual trend of 8.5%.For example, a loss of size $15,000 on 1/1/90 would become (1.08510)($15,000) = $33,915.

Applying no limit, these trended losses sum to $47,574,875.

Going through these trended losses, one then limits each one to the basic limits of $20,000 per person and $40,000 per accident. So for example, the $33,915 loss would be limited to $20,000. In addition, one would add up all the claims associated with a single accident, and limit the sum to $40,000.

87 Rates are rounded to the nearest dollar. This value for the proposed rate is shown in Exhibit 3.16.88 Class and territory are reversed throughout Exhibit 9. Since this mislabeling is consistent with respect to relativities, rates, and exposures, the final result of 10.35% is correct. 89 “Since insurers are frequently unaware of the unlimited loss amounts associated with closed claims, this method is often based upon special data surveys.” This is a major difficulty with McClenahan’s method!90 “Generally, closed claim data are used in order to avoid the problems associated with projecting loss development on individual claims.” McClenahan does not mention a serious problem with the use of closed claims. For a recent year such as 1999, a significantly larger proportion of large claims are still open. Thus the closed claim data from recent years would underestimate the proportion of losses in higher layers, and therefore produce erroneous estimates of increased limits factors.91 Using 13 years of data makes the questionable assumption that there have been no significant changes in the shape of the size of loss distribution over time. I would normally use about the latest five years of data.


For example, let us assume an accident had trended losses from 3 injured persons of: $33,915, $22,610, and $6783. Then we would limit each person to $20,000.$20,000 + $20,000 + $6783 = $46,783. We then limit the sum to $40,000.92

Applying basic limits of 20/40, these trended losses sum to $34,215,312. Applying instead limits of 100/300, these trended losses sum to $45,230,399.As expected, the insurer would pay more in total when there are higher limits of 100/300.

The indicated increased limits factor for 100/300 is: $45,230,399/$34,215,312 = 1.3219.93

However, the losses have only been trended to 12/31/1999, rather than the average date of accident for the proposed rates, 7/1/2001.

McClenahan includes this additional year and a half, by comparing the current indicated increased to limit factor to that indicated two years ago.94

100/300 Factor Indicated as of 12/31/1997: 1.2683.100/300 Factor Indicated as of 12/31/1999: 1.3219. Ratio over two years: 1.3219/1.2683 = 1.0423.Inferred increase per year: 1.04230.5 = 1.0209.Inferred increase from 12/31/1999 to 7/1/2001, 1.5 years: 1.02091.5 = 1.03151.100/300 Factor Indicated as of 7/1/2001: (1.03151)(1.3219) = 1.3636.

McClenahan then selects a round number, 1.3500, as the proposed 100/300 increased limits factor.

92 It may be difficult to get data that allows one to apply per accident limits.93 We are assuming all expenses vary in proportion to the increased losses that result from the increased limits. If we assumed some expenses were fixed regardless of the limit purchased, the increased limits factors would be closer to one.94 Presumably, the indication as of 12/31/97 was done in a similar manner. Rather than use McClenahan’s somewhat unusual technique, one could instead have trended the losses themselves to 7/1/2001, rather than 12/31/1999.


Exhibit 3.16 is the proposed rate manual to be effective 7/1/2000.

Parallel to Exhibit 3.1.

Shows the results of Exhibits 3.14 and 3.15.

Rate Change for those buying Basic Limits95 Territory Class 1 Class 2 Class 3

1 14.3% 8.0% 10.7%2 14.4% 8.2% 10.4%3 7.4% 2.0% 4.1%

Rate Change for those buying 100/300 Increased Limits96 Territory Class 1 Class 2 Class 3

1 18.7% 12.2% 14.9%2 18.8% 12.4% 14.7%3 11.5% 6.0% 8.1%

95 Comparing Exhibits 3.1 and 3.16. 96 Comparing Exhibits 3.1 and 3.16.


Documents

“Ratemaking” by Charles L. McClenahans3.amazonaws.com/zanran_storage/ “Ratemaking”, Chapter 3 of Foundations of Casualty Actuarial Science I. INTRODUCTION 1. The Concept of