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THE ESTIMATION OF LOSS DEVELOPMENT TAILFACTORS: A SUMMARY REPORTCAS Tail Factor Working PartySteven C. Herman, co-chairMark R. Shapland, co-chairMohammed Q. AshabJoseph A. BoorAnthony R. BustilloDavid A. ClarkRobert J. FoskeyReviewed by:Aaron HalpertSejal HariaBertram A. HorowitzGloria A. HubermanRichard KollmarJoshua MerckRasa V. McKeanMichael R. MurrayBernard A. PelletierSusan R. PinoAnthony J. PipiaF. Douglas RyanScott G. SobelNancy Arico and Ryan Royce for the CAS Committee on ReservesABSTRACTMotivation. Tail factors are used by actuaries to estimate the additional development that willoccur after the eldest maturity in a given loss development triangle, or after the eldest credible link ratio.Over the years, many valuable contributions have been made to the CAS literature that describesvarious methods for calculating tail factors. The CAS Tail Factor Working Party prepared this paper onthe methods currently used by actuaries to estimate loss development ‘tail’ or ‘completion’ factors.Standard terminology for discussing aspects of link ratios and tail development is communicated withinthe paper. Descriptions of the advantages and disadvantages of each method are included as wellgeneral indications of what types of entities (companies, rating bureaus, or consulting firms) typicallyuse each method.Method. An extensive survey of existing CAS literature was performed, along with surveys ofmethods currently in use by various rating bureaus, insurers, and consulting organizations. The methodsidentified by the Working Party are grouped into six basic categories: (1) “Bondy Methods”; (2)algebraic methods that focus on relationships between paid and incurred loss; (3) methods based on useof benchmark data; (4) curve-fitting methods; (5) methods based on remaining open counts; (6)methods based on peculiarities of the remaining open claims; and (7) the remaining unclassifiedmethods.Results. Comparisons of the results of several key tail factor methodologies to the actual post-tenyear development for a number of long-tail lines using multiple realistic data sets are included, alongwith the advantages and vulnerabilities of each method.Availability. A copy of the Working Party’s paper and companion Excel template can be found onthe CAS website at s. Tail Factors; Completion Factors; Link Ratios; Age-to-Age Factors; DevelopmentFactors; Loss Reserving; Curve Fitting; Bondy Method; Benckmark; Loss Development.Casualty Actuarial Society E-Forum, Fall 20131

The Estimation of Loss Development Tail Factors: A Summary ReportTable of ContentsPage1. INTRODUCTION . 52. BONDY-TYPE METHODS . 92.1 Introduction and Description of Bondy-Type Methods .92.2 Bondy’s Original Method .92.3 Modified Bondy Method . 102.4 Generalized Bondy Method . 102.5 Fully Generalized Bondy Method . 103. ALGEBRAIC METHODS . 113.1 Introduction to Algebraic Methods . 113.2 Equalizing Paid and Incurred Development Ultimate Losses . 123.3 Sherman-Boor Method . 143.4 NCCI Method . 173.5 Summary of Algebraic Methods. 274. BENCHMARK-BASED METHODS . 274.1 Introduction to Benchmark-Based Methods. 274.2 Directly Using Tail Factors from Benchmark Data . 274.3 Use of Benchmark Tail Factors Adjusted to Match Pre-Tail Link Ratios . 294.4 Benchmark Average Ultimate Severity Method. 314.5 Use of Industry-Booked Tail Factors . 334.6 Benchmark Tail Factors Adjusted for Company-Specific Case Reserving . 354.7 Summary of Benchmark-Based Methods . 37CURVE-FITTING METHODS . 375.1 Introduction to Curve-Fitting Methods . 375.2 Exponential Decay Method . 375.3 McClenahan’s Method . 405.4 Skurnick’s Method. 465.5 Sherman’s Method . 515.6 Pipia’s Method . 535.7 England-Verrall Method . 555.8 Summary of Curve-Fitting Methods. 59Casualty Actuarial Society E-Forum, Fall 20132

The Estimation of Loss Development Tail Factors: A Summary Report6. METHODS BASED ON REMAINING OPEN COUNTS . 596.1 Introduction to Open-Count Based Methods . 596.2. Static Mortality Method. 606.3 Trended Mortality Method . 626.4 Summary of Future Remaining Open Claims Methods . 637. METHODS BASED ON PECULIARITIES OF THE REMAININGOPEN CLAIMS . 647.1 Introduction . 647.2 The Maximum Possible Loss Method . 647.3 Judgment Estimate Method. 667.4 Summary of Methods Based on Peculiarities of the Remaining Open Claims . 698. OTHER METHODS . 698.1 Introduction . 698.2 Restate Historical Experience Method . 698.3 Mueller Incremental Tail Method . 738.4 Corro’s Method . 778.5 Sherman-Diss Method . 799. COMPARISON OF SELECTED RESULTS . 859.1 Discussion . 859.2 Future Research. 8610. CONCLUSIONS . 86REFERENCES . 88APPENDIX A – Alternative Organization of the Methods . 93APPENDIX B – Examples . 98B.1 Introduction . 98B.1.1 Paid Loss. 98B.1.2 Incurred Loss . 99B.1.3 Case Reserves . 99B.2 Bondy-Type Methods . 99B.3 Algebraic Methods . 101B.3.1 Sherman-Boor Method. 101Casualty Actuarial Society E-Forum, Fall 20133

The Estimation of Loss Development Tail Factors: A Summary ReportB.4 Curve-Fitting Methods. 103B.4.1 Exponential Method . 103B.4.2 McClenahan’s Method . 104B.4.3 Sherman’s Method . 106B.4.4 Pipia’s Method . 107B.4.5 England-Verrall . 108Casualty Actuarial Society E-Forum, Fall 20134

The Estimation of Loss Development Tail Factors: A Summary Report1. INTRODUCTION1.1 Importance of Loss Development Tail FactorsThe loss development tail factors (sometimes referred to as completion factors) are animportant part of any reserve analysis. They have a highly leveraged impact since they form aportion of the loss development applied to each of the accident years being analyzed.However, the discussions of tail factor estimation methods used, when they are contained inthe CAS literature at all, are generally just as adjuncts to the main topics of papers. Further,some methods are used in practice that are not described in the CAS literature at all.Therefore, the CAS Committee on Reserving sponsored a Tail Factor Working Party toundertake an exhaustive survey of the tail factor estimation methods in use and describe andcomment on each method.1.2 Research ContextAs stated above, tail factors have a highly leveraged impact on loss development since theyform a portion of the loss development of all accident years analyzed. Further, tail lossdevelopment reflects development occurring after the last development period in thereserving data triangle and is therefore somewhat more difficult to estimate than the variouslink ratios developed from the data triangle. For both those reasons, the Tail Factor WorkingParty believes it is helpful to provide information concerning tail factor estimation methodsto practitioners.1.3 ObjectiveThis paper is designed to be as exhaustive a listing of methods used to estimate tail lossdevelopment as is reasonably possible at the time of its writing. The Tail Factor WorkingParty hopes this will expose the various approaches to a wider audience, and help actuarieschoose the best method for each reserving circumstance from a larger toolkit. Further, thispaper lists at least some of the advantages and disadvantages of each method, which couldhelp the practitioner decide which method to use in a given circumstance.1.4 DisclaimerWhile this paper is the product of a CAS Working Party, its findings do not represent theofficial view of the Casualty Actuarial Society. Moreover, while we believe the approaches wedescribe are very good examples of how to estimate tail development in reserving, ratemakingand selecting the best method for a given circumstance, we do not claim they are the onlyacceptable ones or that we have ultimately addressed all of the issues that must be consideredin selecting a tail factor or tail factor methodology.Casualty Actuarial Society E-Forum, Fall 20135

The Estimation of Loss Development Tail Factors: A Summary Report1.5 Section References to MethodsThe classes of methods presented are discussed in the next sections. Within each class ofmethod, an introduction to the class of method, a summary of the methods, any particularfindings, and conclusions are presented.1.6 Alternate Grouping of Methods Included in the PaperWhile organizing this paper, working party members noted that the groupings of methodswere not inherently absolute and that the methods could be grouped in alternate ways. Thecommentary and listing in Appendix A represents an alternate but still logical view of how thevarious methods relate to each other.1.7 NotationThis paper describes many tail factor methods identified in the actuarial literature andelsewhere. For the sake of uniform notation, where appropriate we have adopted (andexpanded) the notation used by the CAS Working Party on Quantifying Variability in ReserveEstimates. In the paper produced by that Working Party, some models visualize loss statisticsas a two-dimensional triangle array. In the notation, the row dimension is the period 1 bywhich the loss information is subtotaled, most commonly an accident period. 2 For eachaccident period w , development age d the ( w, d ) element of the array is the total of the lossinformation as of development age d. 30F1F2FFor this discussion, we assume that the loss information available is an upper lefttriangular subset of the two-dimensional array for rows w 1,2, , n . For each row w , theinformation is available for development ages 1 through n w 1 . If we think of period n asthe latest accounting period for which loss information is available, the triangle represents theloss information as of accounting dates 1 through n . The diagonal for which w d equals aconstant k represents the loss information for each accident period w as of accountingMost commonly the periods are annual (years), but as most methods can accommodate periods other thanannual we will use the more generic term “period” to represent year, half-year, quarter, month, etc. unless notedotherwise.2Other exposure period types, such as policy period and report period, also utilize tail factor methods. Forease of description, we will use the generic term “accident” period to mean all types of exposure periods, unlessotherwise noted.3Depending on the context, the ( w, d ) cell can represent the cumulative loss statistic as of development age1d or the incremental amount occurring during the d th development period.Casualty Actuarial Society E-Forum, Fall 20136

The Estimation of Loss Development Tail Factors: A Summary Reportperiod k . 43FIn general, the two-dimensional array will also extend to columns d 1, 2, , n . Forpurposes of calculating tail factors, we are interested in understanding the developmentbeyond the observed data for periods d n 1, n 2, , u , where u is the ultimate timeperiod for which any claim activity occurs – i.e., u is the period in which all claims are finaland paid in full.The paper uses the following notation for certain important loss statistics:c( w, d ) :cumulative paid or incurred loss from accident period w as ofdevelopment ages d . ( w and d may be thought of as representing“when” and “delay,” respectively.) In the context of this and othernotation, c Paid ( w, d ) denotes cumulative paid loss and c Inc ( w, d ) denotescumulative case incurred loss.q( w, d ) :incremental paid or incurred loss on accident period w during thedevelopment age from d 1 to d . Also denoted as qPaid (w, d ) orqInc (w, d ) .s( w, d ) :case reserves at end of development age d for accident period w .c(w, u) U (w) :total loss from accident period w when at the end of ultimatedevelopment.R( w) :future development after age d n w 1 for accident period w , i.e., U (w) c(w, n w 1) .S (d ) :estimated ratio of unpaid costs to case reserves at the end of the triangledata d .S:estimated ratio of unpaid costs to case reserves as of the end of thetriangle data.f (d ) 1 v(d ) : factor applied to c( w, d ) to estimate c(w, d 1) or more generally anyfactor relating to age d . This is commonly referred to as a link ratio. v(d )is referred to as the ‘development portion’ of the link ratio, which is usedto estimate q(w, d 1) . The other portion, the number one, is referred to4For a more complete explanation of this two-dimensional view of the loss information see the Foundations ofCasualty Actuarial Science [5], Chapter 5, particularly pages 210-226.Casualty Actuarial Society E-Forum, Fall 20137

The Estimation of Loss Development Tail Factors: A Summary Reportas the ‘unity portion’ of the link ratio.fˆ (d ) 1 vˆ(d ) : an estimate of the link ratio for development age to development aged 1.F (d ) 1 V (d ) : ultimate development factor relating to development age d . The factorapplied to c( w, d ) to estimate c( w, u ) or more generally any cumulativedevelopment factor relating to development age d . The capital indicatesthat the factor produces the ultimate loss level. As with link ratios, V (d )denotes the ‘development portion’ of the loss development factor, thenumber one is the ‘unity portion’ of the loss development factor. G(d ) isused interchangeably with F (d ) and by convention, G may also be usedto denote the ultimate loss development factor needed for period w whenwritten as G(w) .T T (n) :tail factor at end of triangle data.Tˆ :estimate of the tail factor.h(w d ) :factor relating to the diagonal k along which w d is constant.e( w, d ) :a mean zero random fluctuation that occurs at the w , d cell.r (k ) :annual rate of loss cost inflation, in this case related to payment period,although in cases where r is either constant or estimated as a constant, ris the cumulative impact over k years (1 r) k .r̂ :an estimate of the rate of annual loss cost inflation.m:development or delay time in months.D(m) :rate of loss cost inflation per month, when D is constant over m , theimpact over m months is (1 D) m .D̂ :an estimate of the rate of monthly loss cost inflation.l:lag until payouts start. Used in McClenahan and Sherman methods.B(d ) 1 b(d ) : notation for a benchmark link ratio and the ‘development portion’ of thebenchmark. Note that BT 1 bT represents the benchmark tail factor.i:a specific accident month, similar to w .pi :the month-to-month decay rate of the pre-inflation loss payouts for agiven accident month, also used as a constant over all months, p .qi 1 pi :the complement of p , also used as a constant over all months, q .Casualty Actuarial Society E-Forum, Fall 20138

The Estimation of Loss Development Tail Factors: A Summary ReportA(i ) :constant of proportionality reflecting total expected pre-inflation losses ina given accident month i .H (w) :a constant of proportionality used in curve-fitting. Often, for global curvefitting across an entire triangle, simply used as H .a and b :constant terms representing the multiplier and exponent of an inversepower curve, respectively.RE :the reinsurance retention applying to a given triangle. RE ( w) refers to theretention of a specific period w .E (x) :the expectation of the random variable x .Var (x) :the variance of the random variable x .U ( w) :ultimate loss amount in accident period w c(w,u).Also, for some methods, additional or slightly different notation is used.2. BONDY-TYPE METHODS2.1 Introduction and Description of Bondy-Type MethodsThis class of methods is discussed first due to its simplicity. Martin Bondy suggested thismethod of just repeating the last observed link ratio for use as the tail factor. Note, that at thetime Bondy developed his method in the 1960s, most lines of insurance were believed to be“short-tailed” in nature compared to assumptions assumed for many casualty lines ofinsurance today. Bondy’s Original Method (see section 2.2) may seriously understate theneeded tail factor for “long-tail” lines or for any case where substantial development occursin the tail. Several alternate versions of the Bondy approach have been developed in anattempt to mitigate the original method’s shortcomings.The formulas for the Bondy-Type methods are described in the sub-sections below.Starting with the original method, we move through modifications that lead to a fullygeneralized method.2.2 Bondy’s Original MethodBondy’s Original Method used the link ratio f (n 1) at the last observed developmentage, n , to develop losses to ultimate; that isF (n) f (n 1) .(2.1)The assumption for age-to-age development factors in the tail is thatCasualty Actuarial Society E-Forum, Fall 20139

The Estimation of Loss Development Tail Factors: A Summary Reportf (d ) f (d 1) .(2.2)2.3 Modified Bondy MethodIn these revisions of Bondy’s Original Method, some recognition is given to moreextended development patterns; the first approach is multiplicative, the second additive.The first approach consists of simply squaring the last link ratio, rather than just repeatingit:F (n) f (n 1)2 .(2.3)The second approach, utilized by some practitioners, is to merely double the developmentportion of the last link ratio:F (n) 1 [2 v(n 1)] .(2.4)2.4 Generalized Bondy MethodSubsequently, Weller [16] suggested a generalization by setting f (n) f (n 1) B , where Bis a number between 0 and 1. We call B the Bondy exponent. It follows thatF (n) f (n 1) B f (n 1) B f (n 1) B /(1 B ) .(2.5)1Thus, if B , we recover the original Bondy method.2Let f (d ) be the development ratio chosen for age d 1 to age d . In his paper, Wellerused the average of the latest three observed development ratios for f (d ) . (Fewer or moreobservations could be utilized.) Set ld log f (d ) , B̂ the estimated Bondy parameter, fˆ (i )the estimated development ratio for the earliest development period used to estimate theparameters, and lˆ log fˆ (i) . The parameters, fˆ (i ) and B̂ , are chosen to minimize2i lnd id lˆi Bˆ d 1 2.(2.6)The parameters, fˆ (i ) and B̂ , can be calculated easily using a readily available spreadsheetoptimization function such as the “Solver” function in Microsoft Excel.2.5 Fully Generalized Bondy MethodGile [6] devised a further generalization by letting the estimated development ratios varyby accident period, while using the same estimated Bondy parameter for each accident period.Two parameters, as well as the development ratios, are chosen for each accident period byminimizing the sum of squared differences using more than one development period for eachaccident period.Casualty Actuarial Society E-Forum, Fall 201310

The Estimation of Loss Development Tail Factors: A Summary Report2.6 ExamplesSee Appendix B, Section B.2.2.7 Advantages and Disadvantages of the Bondy MethodsThe method is easily implemented using standard spreadsheet functions. It only uses thedata in cumulative paid or incurred loss triangles. Finally, loss development is described interms of only one factor, the Bondy exponent.The fully generalized Bondy method is not always useful for incurred loss data because itmay produce Bondy exponents not in the range from 0 to 1. For this same reason, themethod fails to give meaningful answers when the pattern of development factors isincreasing. Since the Bondy method describes loss development in terms of only oneparameter, the method may also fail if the development pattern is complicated in some otherway.2.8 UsersThe Bondy-type methods (including the specific forms discussed above) are widelyaccepted and used in current practice.2.9 SummaryBondy methods give a simple solution to the problem of determining tail factors. They areeasy to explain and to implement. However, they describe loss development in terms of onlyone parameter so that complicated development patterns may not be accurately projected.3. ALGEBRAIC METHODS3.1 Introduction to Algebraic MethodsAlgebraic methods are methods that focus on the relationships between the paid andincurred loss triangles. They are based on relatively simple calculations in the sense thatcomplex mathematical formulae and curve fitting, etc. is not required. Additionally, ancillaryinformation beyond readily available paid and incurred data is not required for any of thesemethods.Casualty Actuarial Society E-Forum, Fall 201311

The Estimation of Loss Development Tail Factors: A Summary Report3.2 Equalizing Paid and Incurred Development Ultimate LossesThis method is one of the oldest tail factor methods used and also has perhaps thebroadest usage of all the methods. It was designed to provide an easy methodology fordetermining a paid loss tail factor when the incurred loss tail factor is available.3.2.1 Description 54FThis method is most useful when incurred loss development essentially stops after acertain stage (i.e., the link ratios are near to unity or are equal to unity). Then, due to theabsence of continuing development, the current case incurred (e.g., case incurred as of end ofmost recent accounting period, sometimes called reported) losses are a good predictor of theultimate losses for the older or oldest years without the need for additional tail factordevelopment. A tail factor suitable for paid loss development can then be computed as theratio of the case incurred for the oldest accident period in the triangle divided by the paidlosses to date for the same accident period. This results in a paid to ultimate developmentfactor estimate which when multiplied by the cumulative paid equals the ultimate (which arealso the current) incurred losses for that oldest accident year.This method relies on one axiomatic (meaning plainly true rather than an assumption assuch) assumption and two true assumptions. The axiomatic assumption is that the paid lossand incurred loss development estimates are estimating the same quantity, therefore theultimate loss estimates they produce should be equal. The second assumption (the first trueassumption) is that the incurred loss estimate of the ultimate losses for the oldest accidentperiod is accurate. The last assumption is that the other periods will show the samedevelopment in the tail as the oldest period. An appropriate way to test this assumption is toestimate the paid loss tail based on several accident periods.This method may also be generalized to the case where the current case incurred is stillshowing development near the tail. In this situation, the implied paid loss tail factor is, orcInc (1, u ).cPaid (1, n)5(3.1)Section 3.2.1 is reproduced from [1] with permission. Minor edits have been made for consistency with therest of this Report.Casualty Actuarial Society E-Forum, Fall 201312

The Estimation of Loss Development Tail Factors: A Summary ReportIn this instance, the incurred loss development estimate for the oldest accident period isusually the current case incurred losses for the oldest period multiplied by an incurred loss tailfactor developed using other methods.3.2.2 ExampleWe are given the following selected incurred loss development factors:12-24 ed losses for the oldest year in the triangle as of 120 months is 50,000,000 and thecorresponding paid loss is 40,000,000. The incurred estimated ultimate using the 1.004 tailfactor is 50,200,000. The paid loss tail factor to equalize the paid estimated ultimate to theincurred estimated ultimate would be 50,200,000 divided by 40,000,000 or 1.255.3.2.3 Advantages and DisadvantagesThis method has a substantial advantage in that it is based solely on the information in thetriangle itself. One of its weaknesses is that a reliable estimate of the ultimate loss for theoldest year is needed before it can be used. In addition, if the ultimate incurred lossdevelopment of the oldest accident year is estimated using a tail factor estimate, then thismethod also relies on the incurred loss tail factor. Lastly, there is an assumption that the ratioof the case incurred loss to the paid loss will be the same for less mature years once theyreach the level of maturity used initially to calculate the paid tail. This assumption can betested by looking at the stability of the paid to incurred ratio.3.2.4 UsersThis method is such a basic part of most loss development analyses that it is probablyunder-reported on surveys. For example, most users will attempt to at least compare theestimated ultimate paid and estimated ultimate incurred loss for the oldest years.Casualty Actuarial Society E-Forum, Fall 201313

The Estimation of Loss D

identified by the Working Party are grouped into six basic categories: (1) "Bondy Methods"; (2) algebraic methods that focus on relationships between paid and incurred loss; (3) methods based on use of benchmark data; (4) curve-fitting methods; (5) methods based on remaining open counts; (6) methods based on peculiarities of the remaining .

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