不合标准的结构性结算年金——最佳估计死亡率：支持改进的对数线性递减法

MILLIMANWHITE PAPER Substandard Structured Settlement Annuities–Best Estimate Mortality: Support for theModified Log-Linear Declining (LLD) Method Gordon Klein, FSA, MAAACameron Denker Executive Summary The mortality assumption forSubstandard Structured SettlementAnnuities(SSSA)presentsan interestingchallenge,as it mustbe inferredfrom actual ages and “rated ages,” which express the reduction in life expectancy due to animpairment.Three major basic methods have been usedinacademic studies: the Rated Age method, the ConstantExtra Deaths method, and the Log-Linear Declining method.These basic methods assign annual mortality ratesdifferently, buteachpreservesthe life expectancy assigned at issue.Unfortunately,these basic methods fail toaccurately predict future mortalityon actual blocks ofexperience study data. Actuaries havedevelopedmodifications of these methodsto correct for theirweaknesses.We examine three ofthem:Modified Rated Age, Modified Constant Extra Deaths, and Modified Log-Linear Declining.Wefiteach to actual datausing a“training period” of early data. We then test that model on experience from the later period, the “test period.” We foundthat the best methodfor developing a mortality assumption from SSSA experience study datais theModified Log-Linear Declining method.It had the closest fit of expected to actual at higher durations, whichisimportant for pricing and valuation of in-force blocks. More qualitatively,theparameters and modificationsofthismethodalsoseem to be the most intuitive,allowing foreasier interpretation and explanation as well as modificationfor actuarial judgment. Although we advocate for a commonmethodology, we do not advocate for a single commonassumption.Instead,theModified Log-Linear Declining methodshould be applied anew to each block, to reflect idiosyncrasies such as mix ofages, conditions, marketing focus, and sales era. I.SubstandardStructured Settlement Annuities For a more thorough understanding of Structured Settlement Annuities (SSA), please refer to [1]in the Referencessection below.ASubstandardStructured Settlement Annuity (SSSA)is one written onan annuitantwho has an“impairment” that makes their mortality higher than that of a “standard” life of the same age. This white paper is onlyconcerned withSSSAand the estimation oftheirmortality. Consider aprospective annuitant,age𝑥, who hasan impairment.Acasualty insurerhas become obligated topaythisindividualadefinedannuityin the future.Alifeinsurersellsthe casualty insurer an SSSA,agreeing to pay thosefuture payments to theprospective annuitantin return for a premium. The premium is based on “substandard(impaired)mortality rates” that reflect the conditionof this annuitant. These rates are based on an underwriter’sopinionas to the “rated age,” say𝑥+𝑟, of theindividual.The rated age ismeant to summarize the mortality effect ofthe impairment ina single number.The difference between the rated age and the actual age,𝑟>0, is referred to asthe rate-up.Using𝑖to represent “impaired” and𝑠to represent “standard,” we haveequation (1): where the sum on the right side is calculated from a given standard mortality table. II.ThechallengeofSSSAmortality The actuary must come up with a set of mortality rates that lead to the stated life expectancy,forwhichthere areinfinitely many solutions. How can we choose one and justify it? Several reasonably simple methods have been proposed. We willstudythree ofthembrieflyandseewhetheranyofthemareadequate for our purposes. If not,we willconsidermodificationsof these modelsand identify the best method. First, how can we evaluate these methods, so that we can choose the best one or ones? We want to use a method thatfits thehistoricalexperience dataand,moreimportantly,thatfitsthefutureexperience.Inorder to be confident that amethodwill fit in the future, we will comparethe performance of candidate methods acrossanumberofSSSAblocksbyfittingmethods to “training periods” and then evaluating over “test periods”as described inSection VIIbelow. III.Threebasic methodsforinferring mortality ratesfromlife expectancy As a basis of developing basic methods for estimating mortality ratesof impaired individuals, please refer to [2],whichcomparesbasicmortality models.Each of the threebasicmethodsbelowassign annual mortality ratesusingthe issue ageandrated age.Although they assign mortality rates differently,all three basic methodsmaintain thePrincipleofEquivalentLifeExpectancy,which is that the equation (1) holds.(Later,in Section VIbelow, thisprinciplewill no longer hold as thebasicmethods arecalibratedto fit the actual experience data.) A.RATED AGE(RA) The Rated Age method stipulates that the life expectancy for the impaired lifeage𝑥will be set equal to that of astandard life of the rated age𝑥+𝑟,provided by the underwriter. This is showninequation(2): Thismethodisa rathersimplemethodandiseasy tocompute, butthe link between issue age and rated age resultsin the undesirable property tha