Selecting Mortality Tables: A Credibility Approach Gavin Benjamin October 2008

Transcription

1 Selecting Mortality Tales: A Creiility Approach Gavin Benjain Octoer 8 Historically, pension actuaries have tene to focus their attention on the estalishent of appropriate econoic assuptions ecause changes in assuptions such as the liaility iscount rate can have a significant effect on the easureent of a plan s liailities an costs. The selection of eographic assuptions has often receive less attention since changes to these assuptions ten to have a less aterial effect. However, the use of inappropriate eographic assuptions for valuation purposes can, over the long ter, have averse consequences if it results in a aterial isstateent of a pension plan s liailities or costs. In toay s environent, the assuptions selecte y pension actuaries are coing uner increase scrutiny. It is ecoing ore an ore coon for actuaries to e aske to provie justification for all of their assuptions, incluing the eographic assuptions, to various stakeholers such as regulators. As part of the ongoing valuation an plan anageent process, an actuary shoul conuct perioic reviews of the eographic eperience of a pension plan an, ase on these reviews an future epectations, ake appropriate ajustents to the eographic assuptions. The use of creiility theory can, an perhaps shoul, e a key aspect in the analysis that leas the actuary to the selection of a ortality tale for a pension plan valuation. While use of creiility theory to ajust ortality rates can e very useful, the following liitations shoul e kept in in with respect to the iscussion: This article oes not attept to reflect any rules on the selection of ortality tales that ay e ipose y applicale legislation or regulators. Also, it is not the intent of this article to take a position on the appropriateness of the threshols at which a regulator ay have chosen to perit a plan s ortality eperience to e reflecte in the selection of a ortality tale. The focus is on the selection of the ase ortality tale, which reflects the ortality rates of plan eers at the tie of a ortality eperience stuy. While assuptions regaring future ortality iproveent after the stuy ate are very iportant, the selection of a ortality projection assuption is eyon the scope of this article.

2 Definitions For purposes of our analysis, the following efinitions for a given pension plan are require:, = α toτ,is the nuer of eers an forer eers in a ortality eperience stuy age anτ is the aiu age to which a plan eer can live. It has een assue that, for eperience stuy purposes, integral ages are use. (For eaple, a eer' s actual age at the tie of the stuy ay e roune to the closest integral age.) q, = α toτ,is the actual unerlying ortality rate for eers an forer eers age. q, = α toτ,is the ortality rate at age,ase on the ortality tale currently in use for valuation purposes. q, = α toτ,is the estiate ortality rate at age, which reflects the results of the eperience stuy. j, = α toτ, j = to, is the accrue pension for the th j eer or forer eer who is age. j, = α to τ, j = to, is equal toif the ies uring the year,an is zero otherwise. th j eer or forer eer who is age Z p is the p th percentile of the stanar noral istriution. Mortality perience Stuies: Aounts versus Lives Pulishe ortality tales such as GAR-94, UP-94, an RP- were evelope using aounts rather than lives (i.e., they were eterine y iviing total accrue pension aounts for those who ie y total accrue pension aounts for all epose at a given age). This approach is intene to e a proy for weighting ortality rates y liaility. Since stuies have consistently shown that aountweighte ortality rates are aterially lower than lives-weighte rates, ortality eperience stuies shoul e conucte using aounts. In other wors, the use of lives-weighte rates for valuation purposes will ten to result in an unerstateent of pension liailities.

3 When Shoul You Consier Changing a Valuation Mortality Tale? When assessing whether to change a valuation ortality tale ase on the results of an eperience stuy, egin y coparing the oserve eaths weighte y enefit aount to the epecte eaths weighte y enefit aount. For eaple, Chart elow shows the ratios of aounts of actual eaths ( j j ) to the aounts of epecte eaths ( ) for a pension plan with 4, active an jq retire ales within the age an of = 5 to 97. pecte eaths in Chart are ase on the UP-94 static tale projecte to 5 using ortality iproveent Scale AA. Chart : Ratio of Actual to pecte Deaths (using aounts) Actual to pecte Deaths Age If the current ortality tale is a goo representation of the actual unerlying ortality rates of plan eers an forer eers, the ratios of actual to epecte eaths shoul closely track (which is shown as the lack line in Chart ). In Chart, actual eaths are generally greater than epecte eaths efore age 75, an are usually less than epecte eaths after age 85. However, after age 85 there are very little ata, so the creiility of the ratios after age 85 is questionale. Between ages 75 an 85, actual eaths closely track epecte eaths. Overall, the ifferences etween actual an epecte eaths efore age 75 see to inicate that a revision to the current tale is warrante. In the aove eaple, it is fairly clear that a change in the ortality tale shoul e consiere. In ost situations, however, the ifferences etween actual an epecte eaths will not e so etree. The actuary ust always eercise jugent

4 when eciing whether to revise a ortality tale. One of the consierations coul e the characteristics of plan eers an forer eers (e.g., whether they are lue or white collar workers) an the potential effect on future ortality eperience. Pension plan liailities are usually not very sensitive to the ortality assuption at younger ages. This is ecause ortality rates at younger ages are relatively sall an any pension plans provie a eath enefit to the eneficiary of a eer who ies while in active eployent. Therefore, as in the aove eaple, eperience stuies often focus on ages 5 to 55 an aove. Builing a Tale fro Scratch Full Creiility If a ecision is ae to change the ortality assuption, one approach is to create copletely new tales using the ata fro the eperience stuy. In this case, for each age, the estiate of q (i.e., q ) woul e eterine as: q = j j j An ovious question is How creile is y estiate of q?. One way of efining full creiility is to select a sall constant r an large proaility p, an say that q is fully creile if: Pr[ q q rq ] p Assuing that eaths are inepenent an using the Central Liit Theore, it can e shown that full creiility is achieve when the epecte nuer of eaths is greater than or equal to: j j, where Z = ( p) r /

5 Alternatively, full creiility is achieve when the epecte ollars of eath are greater than or equal to: j j For eaple, consier ales age 7 an assue that a ortality eperience stuy is ase on nuer of lives, in which case 7, j is set equal to for all j. With r =.5 an p =.9, full creiility is achieve when the epecte nuer of eaths is greater than or equal to: 7 7 7, j 7, j.645 = =.5 =,8 For creiility calculations, r =.5 an p =.9 are coon values use for those variales an are ase on achieving a typical esire level of confience. If the actual unerlying ortality tale is UP-94 static, projecte to 5 using Scale AA, then q =. 7 an ore than 49, eposures for ales age 7 are neee to epect the,8 eaths require for the estiate of q 7 to e fully creile. For an eperience stuy ase on aounts rather than lives, the nuer of eposures neee woul e even greater. trapolating fro the aove eaple for ales age 7 to other ages, it is clear that a large aount of ata is require to uil a fully creile ortality tale fro scratch. Only a few of the largest pension schees in the worl can uil a fully creile ortality tale fro scratch using their own eperience ata. In aition, creating a ortality tale fro scratch is not a trivial eercise. For eaple, once the raw ortality rates have een eterine, they ust then e soothe so that the rates progress in a reasonale pattern fro age to age. Grauation techniques are often use for purposes of soothing these rates. The process can e tie-consuing an challenging. Ajusting a Stanar Tale Full Creiility As can e seen aove, uiling a ortality tale fro scratch is a non-starter for ost pension plans. A ore practical approach is often to rate a version of a stanar tale, such as the UP-94 or RP-, up or own ase on the total eath aounts fro the eperience stuy. More precisely, efine f as the ratio of actual

6 to total epecte eath aounts for all ages, an then, assuing this value is fully creile, set the ortality rate at each age to f q For eaple, in the eperience stuy use to evelop Chart aove: 97 = 5 f = =., an q 97 7 woul e set equal to f q7 =.. =.484. q = 5 j j j For ages close to τ, the aiu age to which a plan eer is assue to live, ortality rates shoul e ajuste, if necessary, so that there is a reasonale progression fro the ortality rates close to τ to the ortality rate of at age τ. Chart shows in lue actual to epecte eaths using the ajuste ortality rates.. Actual to pecte Deaths Chart : Ratio of Actual to pecte Deaths (using aounts) Age Stanar tale Ajuste tale It is reasonale to ask whether the resulting ortality tale create y ajusting a variation of a stanar tale as escrie aove will e creile. If creiility is assue to e achieve when: Pr[ f f rf ] p, where f = a j = = = a j = j j q q an a an are the iniu an aiu ages, respectively, of the plan eers inclue in the eperience stuy,

7 then it can e shown that the tale is fully creile when the epecte nuer of eaths, N, is greater than or equal to: = a q q = a j = q = a j j Alternatively, full creiility is achieve when the epecte ollars of eath, D, are greater than or equal to: q = a j = q = a j = j j Since the actual ortality rates are unknown, it is reasonale to use estiate q. f q to Applying the previously-efine forula for N to the ata fro our eaple, N ecoes,635 while the actual nuer of stuy eaths is only 73. Therefore, the ajuste tale in this case woul not e fully creile. Ajusting a Stanar Tale Partial Creiility If there are insufficient ata for the ajustent to a stanar tale escrie in the previous section to e fully creile, a ore appropriate approach woul e to assign partial creiility to the results of the ortality stuy. In this case, q woul e estiate as: Z f q + ( Z) q, where < Z < is the creiility weighting assigne to the eperience stuy. Z can e eterine as: = a j = N j = a j =, or equivalently D j j. In our eaple, Z is equal to 73 =. 656 an,635

8 q7 woul e set equal to Z f q7 + ( Z) q7 = (.656). =.393 which is etween the stanar tale value of. an the full creiility ajuste value of.484 eterine aove. Chart 3 shows, in green, the actual to epecte eaths ase on ajuste ortality rates using the partial creiility approach. Actual to pecte Deaths Chart 3: Ratio of Actual to pecte Deaths (using aounts) Stanar tale Ajuste tale - full creiility Ajuste tale - partial creiility Age Conclusion As with all actuarial work, jugent ust e use when ajusting ortality assuptions. For eaple, the creiility approach outline aove assues that the shape of the stanar tale is appropriate for the plan eing value, an all that is require is a proportional ajustent (either up or own) to the stanar tale. If an actuary elieves strongly that the unerlying shape of the ortality curve for a plan iffers significantly fro all availale stanar tales, he or she ay choose to uil a tale fro scratch using eperience ata fro the plan, even if a creile aount of eperience ata is not availale. Nevertheless, as long as jugent is use, creiility theory can serve as a powerful tool for an actuary setting the ortality assuption for a pension plan. Gavin Benjain, FSA, FCIA, is a principal with Towers Perrin in Toronto, Ontario. He is Chair of the Pension Section Retireent Plans perience Coittee. He can e reache at gavin.enjain@towersperrin.co.