The Estimation Of Loss Development Tail Factors A Summary Report Cas .

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 .