Disse a ion p esen ed as pa ial equi emen o ob aining he
mas e ’s deg ee in S a is ics and In o ma ion Managemen Risk,
wi h a specializa ion in Analysis and Risk Managemen
Es ima ion o Longe i y Risk and Mo ali y Modelling
TABI ROSY CHRISTY ATEMNKENG
NOVA In o ma ion Managemen School
Ins i u o Supe io de Es a ís ica e Ges ão de In o mação
Uni e sidade No a de Lisboa
Es ima ion o Longe i y Risk and Mo ali y Modelling
by
TABI ROSY CHRISTY ATEMNKENG
Disse a ion p esen ed as pa ial equi emen o ob aining he mas e ’s deg ee in
S a is ics and In o ma ion Managemen , wi h a specializa ion in Analysis and Risk
Managemen .
SUPERVISOR: P o . D . Jo ge Miguel Ven u a B a o
No embe 2021
Acknowledgemen
Fi s , I hank God o gi ing me knowledge and pa ience o ca y ou his esea ch.
My Family, Rela i e and Lec u e s, you suppo is app ecia ed.
A Special hanks o my Supe iso P o . D . Jo ge Miguel Ven u a B a o o you
guidance and assis ance h ough he en i e p ojec .
Abs ac
P e ious mo ali y models ailed o accoun o imp o emen s in human mo ali y a es
hus in gene al, human li e expec ancy was unde es ima e. Declining mo ali y and
inc easing li e expec ancy (longe i y) p o oundly al e he popula ion age dis ibu ion.
This demog aphic ansi ion has ecei ed conside able a en ion on pension and annui y
p o ide s.
Conce ns ha e been exp essed abou he implica ions o inc eased li e expec ancy o
go e nmen spending on old-age suppo . The goal o his pape is o lay ou a amewo k
o measu ing, unde s anding, and analyzing longe i y isk, wi h a ocus on de ined
pension plans. Lee-Ca e p oposed a widely used mo ali y o ecas ing model in 1992.
The s udy looks a how well he Lee-Ca e model pe o med o emale and male
popula ions in he selec ed coun y (F ance) om 1816 o 2018. The Singula Value
Decomposi ion (SVD) me hod is used o es ima e he pa ame e s o he LC model.
The mo ali y able hen assesses u u e imp o emen s in mo ali y and li e expec ancy,
aking in o accoun mo ali y assump ions, o see i pension unds and annui y p o ide s
a e exposed o longe i y isk. Mo ali y assump ions a e p edic ed dea h a es based on a
mo ali y able. The wo ypes o mo ali y a e mo ali y a bi h and mo ali y in old age.
Longe i y isk mus be e ec i ely managed by pension and annui y p o ide s. To
mi iga e his isk, pension p o ide s mus ac o in u u e imp o emen s in mo ali y and
li e expec ancy, as mo ali y a es end o dec ease o e ime.
The indings show ha ailing o accoun o u u e imp o emen s in mo ali y esul s in
an expec ed p o ision sho all. P o ec ion mechanisms and policy ecommenda ions o
manage longe i y isk can help o mi iga e he inancial impac o an unexpec ed inc ease
in longe i y.
Keywo ds: Lee-Ca e (LC) model, Mo ali y modeling, Fo ecas ing, Li e expec ancy,
Singula alue decomposi ion (SVD),
INDEX
Table o Con en s
Acknowledgemen ............................................................................................................ 3
Abs ac ............................................................................................................................. 4
LIST OF FIGURES .......................................................................................................... 7
LIST OF TABLES ........................................................................................................... 7
1. INTRODUCTION ..................................................................................................... 8
1.1. Backg ound o he s udy .................................................................................... 8
1.2. S a emen o he P oblem ................................................................................. 11
1.3. Objec i e .......................................................................................................... 12
1.3.1. Speci ic Objec i e..................................................................................... 12
1.4. Jus i ica ion o he S udy ................................................................................. 13
2. LITERATURE REVIEW ........................................................................................ 14
2.1. Theo e ical Backg ound: Longe i y Risk and Mo ali y Risk ......................... 14
2.1.1. P oduc s Wi h Longe i y Risk Exposu e ............................................................ 16
2.2. Decomposi ion o Mo ali y Risk .................................................................... 18
2.3. Managemen And Quan i ica ion o Longe i y Risk....................................... 19
3. METHODOLOGY .................................................................................................. 23
3.1. Basic Mo ali y Func ions ................................................................................ 23
3.2. Modelling S uc u e and Speci ica ion ............................................................ 26
3.3. Risk Model Classi ica ion ................................................................................ 28
3.4. Iden i ica ion and S uc u e o he APC S ochas ic Mo ali y Model ............. 29
3.5. S ochas ics Mo ali y Models .......................................................................... 30
4. EMPIRICAL ANALYSIS ...................................................................................... 41
4.1. Unce ain y Abou Mo ali y and Li e Expec ancy ......................................... 41
4.1.1. The ela ionship be ween mo ali y and li e expec ancy: Li e Table ....... 41
4.1.2. The unce ain y su ounding he imp o emen in mo ali y ..................... 42
4.1.3. Mo ali y and li e expec ancy o ecas ing me hods .................................. 43
4.2. Measu ing mo ali y and longe i y imp o emen unce ain y ........................ 44
4.2.1. Lee Ca e Model Measu emen ............................................................... 44
4.2.2. Fi ing and Es ima ing he Pa ame e s o he Lee-Ca e Model .............. 47
4.3. The impac o longe i y isk on de ined-bene i p i a e pension plans .......... 49
4.3.1. How does longe i y isk a ec DB p i a e pension plans?...................... 49

4.3.2. How p i a e pension unds accoun o u u e imp o emen s in mo ali y
and/o li e expec ancy?............................................................................................ 50
5. CONCLUSIONS ..................................................................................................... 52
5.1. Policy issues ..................................................................................................... 52
5.2. A eas in which addi ional esea ch is equi ed ................................................ 53
6. REFERENCES ........................................................................................................ 55
APPENDIX .................................................................................................................... 58
TABLES AND FIGURES .............................................................................................. 58
LIST OF FIGURES
Figu e 1: Male Dea h Ra e, F ance 1816 -2018. ............................................................ 44
Figu e 2: Female Dea h Ra e, F ance 1816-2018. .......................................................... 45
Figu e 3: 𝑎𝑥 and 𝑏𝑥 o F ance popula ion based on li e ables (1989 o 2018) .......... 45
Figu e 4: Pa e n o age acco ding o dea h a es o F ance Popula ion ....................... 46
Figu e 5: Pa e n o Dea h a e based on Yea (1918-2018)........................................... 46
Figu e 6: Es ima ed pa ame e o 𝑎𝑥,𝑏𝑥𝑘𝑡 .................................................................... 47
Figu e 7: P ojec ed alue o 𝑘𝑡 o 100 yea s................................................................. 48
Figu e 8: Pa e n o Pas and P ojec ed a es o people aged 65 ................................... 48
Figu e 9: Li e Expec ancy a age 65 in 2050 .................................................................. 63
LIST OF TABLES
Table 1: Mo ali y ables and imp o emen equi ed by egula ion and used in p ac ice
........................................................................................................................................ 38
Table 2:Mo ali y P ojec ion Scale AA compiled by he Socie y o Ac ua ies.............. 39
Table 3: Compa ing Li e Expec ancy a selec ed age g oups, F ance 1985-2018 ......... 43
Table 4: Li e able, F ance 2018 Males .......................................................................... 58
Table 5: An inc ease in he annui y paymen s' ne p esen alue ................................... 58
Table 6: Age g oup speci ic cen al dea h a es emale popula ion, F ance 1998-2018 59
Table 7: Na u al loga i hm o dea h a es o emale, F ance 1989-2018 ...................... 60
Table 8: 𝑎𝑥 and 𝑏𝑥 Es ima e, F ance 1989 o 2018 ...................................................... 61
Table 9: Es ima e o 𝑘𝑡, F ance 1989 o 2018 (Male and Female) ................................ 61
1. INTRODUCTION
1.1. Backg ound o he s udy
The con inuous imp o emen s in longe i y b ing new p oblems and challenges a
di e en le els o poli ical, social, economic, and egula o y. Howe e , one o he mos
obse ed e ec s o his imp o emen s in longe i y is on pensions. In ecen decades, mos
high-income coun ies ha e esponded o con inuous li e expec ancy inc eases, below
eplacemen -le el e ili y, an upwa d end in old-age dependency a ios, low
p oduc i i y gains and economic g ow h, a apidly shi ing labo ma ke and declining
inancial ma ke e u ns wi h sys emic (e.g., he swi ch owa ds a Non-Financial De ined
Con ibu ion (NDC) scheme in Sweden, I aly, Poland, La ia and No way; pension
inancializa ion, i.e., he expansion o p i a e complemen a y occupa ional and pe sonal
p e- unded de ined-con ibu ion (DC) pensions) and/o g adual pa ame ic e o ms in
na ional public pension schemes (e.g., upda es in he ea ly and no mal e i emen ages,
modi ica ions in he de ined bene i (DB) pension o mula) as pa o hei e o s o
educe o elimina e sho - e m and long- e m imbalances be ween e enues and
expendi u es, alle ia ing he p essu e on public inances, oge he wi h e o s o p ese e
minimum pension adequacy (OECD, 2019; B a o & He ce, 2020).
Fo na ional public pension schemes, a common denomina o o mos e o ms has been
o in oduce au oma ic adjus men o s abiliza ion mechanisms speci ically designed o
co ec o he inancial imbalance o he pension sys em, mechanically upda ing he
scheme’s pa ame e s o demog aphic and/o economic de elopmen s. A common
denomina o in mos pension e o ms adop ed in de eloped coun ies has been o
au oma ically link pension bene i s o li e expec ancy de elopmen s obse ed a
e i emen ages. The link has been es ablished and ein o ced in mul iple ways (Ayuso,
B a o & Holzmann, 2021b; B a o & Ayuso, 2020, 2021): i) by indexing no mal and
ea ly e i emen ages o li e expec ancy (e.g., Denma k, The Ne he lands, Po ugal, UK);
(ii) by linking en y pensions o sus ainabili y ac o s (e.g., Finland, Po ugal), (iii) by
indexing he eligibili y equi emen s o he con ibu ion leng h (e.g., F ance); (i ) by
condi ioning he annual pension indexa ion (e.g., The Ne he lands, Luxembou g); ( ) by
in oducing longe i y-linked isk-sha ing li e annui ies in public and p i a e pension
schemes (B a o & El Mekkaoui, 2018; B a o, 2019, 2020, 2021a).
In 2009, mos companies in de eloping coun ies closed he de ined bene i e i emen
plans (such as 401(K) plans in he Uni ed S a es) o e o hei employees. The pension
plans p o ided o he employe can ei he be de ined con ibu ion o de ined bene i .
These plans ensu e employees ecei e a ce ain amoun a e i emen . In addi ion, de ined
bene i pension plans ha e been eplaced by de ined con ibu ion plans.
A de ined-bene i p og am is a p omise o li e ime e i emen bene i s and he mos i al
isk o e i emen esul ing om longe i y. Longe i y isk is he isk ha insu ance
companies o pension unds aced when assump ions abou li e expec ancies and
mo ali y a es a e inaccu a e. Mo ali y a es and longe i y end isk a e he main
indica o conside ed when a emp ing o ans e longe i y isk.
The insu ance sec o s aced he isk a ising om inc eased longe i y i.e., he end o
longe i y imp o emen will signi ican ly change in he u u e. To ace his long- e m isk,
mo e capi al mus be se asides. Hence i has become mo e impo an o li e o ice
(insu ance companies, pension unds) o ind e icien and sui able me hod o ans e pa
o longe i y isk o capi al o inancial ma ke . Howe e , longe i y isk canno be
ans e ed so easily since i is di icul o unde s and and manage due o i s long- e m
na u e, p ecisely p ojec ions o longe i y a e sensi i e and he modeling o in eg a ed
in e es a e isk emains challenging. Two main ac o s when ans e ing he longe i y
isk o a pa icula pension plan o insu e mus be conside ed. The i s is he cu en
mo ali y le els, which can be obse ed bu a y conside ably be ween socio-economic
and heal h ca ego ies. The second is he isk o longe i y, which is he isk ajec o y o
he ageing popula ion and is sys emic. Sys ema ic mo ali y end isk can be o se
di ec ly by keeping exposu e o inc eased mo ali y. One eason o ceding he isk is he
unce ain y conce ning he longe i y isk, especially because o he sys ema ic na u e, in
a pension plan o an insu ance company.
To manage he isk o longe i y be e , Indi iduals and li e o ices needs o ully
unde s and longe i y isk, o conside i s implica ions, when hey come o plan hei
e i emen income. Th ee causes o his a e unce ain y, unde es ima ion, and
complexi y. To help us be e unde s and hese e ms, (Yinglu Deng e al., 2012a) asse
ha
“Longe i y isk desc ibes he isk ha an indi idual o g oup will li e longe
li e han expec ed hus hei mo ali y a e will be lowe han expec ed, while
mo ali y isk desc ibes he isk ha an indi idual o g oup will li e a sho e
li e han expec ed hus hei mo ali y a e will be highe han expec ed.”, pp.
697,2012.
Longe i y isk o pensione s e e s o “ he possibili y ha hey will li e o such an
ad anced age ha hey will deple e hei e i emen sa ings and ha e o ely solely on
Social Secu i y and Medica e o hei expenses”.
Longe i y isk in e i emen planning can be de ined as “ he isk ha membe s o some
e e ence popula ion migh li e longe on a e age han an icipa ed” (S amp du y and
land ax o non- esiden owne s o Aus alian p ope y, n.d.). Longe i y isk o
indi iduals wi h DC pension sa ings can ha e signi ican implica ions when hey e i e.
The isk o people ou li ing hei e i emen sa ings o he isk o people unde spending
hei sa ings leads o lowe pension incomes. Fo pension plans, longe i y isk e e s o
he inc ease in e i emen pension du ies because o longe li espans. Fo indi iduals,
longe i y isks mean a pe son's possibili y o ou li ing on hei pension asse s. In he i s
place, he longe i y isk is due o he ac ha people a e now li ing longe due o a ious
ac o s like medical and heal h. This means ha one can easonably expec o add ano he
om he alse ce ain y o a single p ojec ion, and a s ep owa d explici ecogni ion o
he unce ain y su ounding he pa h o u u e imp o emen s.
2.1.1. P oduc s Wi h Longe i y Risk Exposu e
Longe i y isk exis s in any p oduc in which he issue is exposed o inancial losses i
policyholde s li e longe han expec ed. This is common when paymen s om he issue
a e con ingen on he policyholde 's su i al. T adi ionally, hese p oduc s ha e been
issued by insu ance companies and used o hedge agains an indi idual ou li ing hei
asse s. In ecen yea s, he numbe and a ie y o p oduc s exposed o longe i y isk has
g own. This can happen e en i ans e ing longe i y isk is no he p ima y goal o he
ansac ion. We examine some o he p oduc s on he ma ke ha a e ulne able o
longe i y isk. We also ake in o accoun he o he isks ha hese p oduc s ace, such as
inancial isk, p icing isk, and egula o y isk. Con e sely, longe i y isk is gene ally
de ined as he exposu e o a company o lowe - han-expec ed mo ali y(Owusu e al.,
2016).
i. Immedia e annui ies
An immedia e annui y is a p oduc ha usually p o ides paymen s o li e in exchange
o a lump sum. The equency and paymen amoun may a y o e he cou se o he
con ac . They can be designed o p o ide a ixed le el paymen , a s eam o paymen s
ha inc ease a a p ede e mined a e, o a s eam o paymen s ha is linked o an
unde lying equi y index.
Immedia e annui ies can be pu chased as ei he single li e o join -and-su i o policies.
In he la e case, annui y paymen s con inue as long as one o he wo li es is ali e,
hough he size o he annui y paymen may dec ease i he p ima y insu ed dies.
Immedia e annui ies a e also subjec o p icing isk. Companies ha se p ices o hei
p oduc s ha a e inconsis en wi h bes es ima e assump ions ace a g ea e isk ha he
ac ual expe ience will di e om wha was expec ed. Because annui y a es a e simple
o unde s and and compa e o insu e s, p icing o longe i y isk is compe i i e.
ii. Enhanced and impai ed li e annui ies
Impai ed o enhanced annui ies p o ide highe annui y paymen s o people who can
demons a e ha hey a e in poo heal h o a e e minally ill. Fo he insu e , he e is a
g ea e isk o medical b eak h oughs in a single condi ion ex ending an indi idual's li e,
which necessi a es ha enhanced p oduc s be p iced a a highe ma gin han s anda d
annui ies. This also has implica ions o es ima ing u u e mo ali y imp o emen s.
The isks associa ed wi h enhanced and impai ed li e annui ies a e simila o hose
associa ed wi h s anda d immedia e annui ies. Howe e , gi en he highe expec ed
mo ali y a es assumed o hese policies, he longe i y isk may be exace ba ed, as he e
is likely o be less da a on he mo ali y expe ience o subg oups o he popula ion.

iii. De e ed annui ies T adi ional
De e ed annui ies a e p ima ily used o accumula e ax-de e ed sa ings, which can hen
be dis ibu ed as an immedia e annui y o as a lump sum paymen . Fixed, a iable, and
equi y-indexed annui ies a e he h ee ypes o de e ed annui ies a ailable in he Uni ed
S a es. As a esul , hey a e less ulne able o he isks associa ed wi h aging. The addi ion
o gua an ees o p oduc o e ings has in oduced longe i y isk as he ma ke has
de eloped and become mo e compe i i e.
When a de e ed annui y is annui ized a ma u i y, i is subjec o a numbe o isks ha
a e no p esen when i is dis ibu ed in lump sum. These p oduc s ha e been in place o
a long ime, and i is di icul o p o ec he cash lows due o a sca ci y o asse s wi h he
app op ia e du a ion. As a esul , de e ed annui ies a e subjec o ein es men isk.
i . Ad anced Li e Delayed Annui ies
ALDAs (ad anced-li e delayed annui ies) a e a ype o longe i y insu ance. ALDAs a e
in la ion-linked annui ies sold o people in hei ea ly wen ies ha begin paying ou a
he age o 80, 85, o 90. The e is no cash alue, and no mo ali y insu ance bene i s ha
can be epaid a any ime. ALDAs a e designed o mimic a de ined bene i pension bene i
a ad anced ages o people who do no ha e access o his ype o p o ec ion. T adi ional
de e ed annui ies may be be e sui ed o p o ec ing agains ca as ophic longe i y
(Owusu e al., 2016).
. Co po a e pensions
The e a e wo ypes o co po a e pension plans: de ined bene i (DB) and de ined
con ibu ion (DC). The employee ecei es a ixed income s eam based on his o he
sala y, yea s o se ice, e i emen age, and o he ac o s unde a DB plan. Typically, he
bene i s eam is se . Con ibu ions a e made in o indi idual accoun s by each employee
unde a DC plan, and he employe may make a ma ching con ibu ion. When you e i e,
you can ake a lump sum equal o he alue o you cu en accoun . The lump sum can
be used o supplemen e i emen income.
i. S uc u ed se lemen s
S uc u ed se lemen s a e paymen s made as he esul o a gene al insu ance liabili y
in ol ing human li e (e.g., se ious inju y, medical negligence, o occupa ional inju y).
Paymen s a e some imes made in he o m o a lump sum o he inju ed pa y's los
ea nings and/o he cos o ca e i hey a e se iously inju ed. Annui ies payable o li e,
on he o he hand, ha e ecen ly been used as a ype o se lemen .
ii. Li e se lemen s
Pu chase s o li e se lemen s ace longe i y isk because lowe mo ali y means hey
mus pay insu ance p emiums o a longe pe iod o ime and ecei e he dea h bene i
la e han expec ed. Mos buye s o his ype o con ac a e no in he business o p o i ing
om mo ali y. Li e Se lemen s a e a way o an in es men bank o hedge und o
di e si y isk while po en ially achie ing a high a e o e u n, as has his o ically been he
case wi h hese po olios.
2.2. Decomposi ion o Mo ali y Risk
Mo ali y isk is gene ally de ined as a company's exposu e o g ea e - han-expec ed
mo ali y. The In e na ional Ac ua ial Associa ion di ides mo ali y and longe i y isk
in o ou ca ego ies: le el, end, ola ili y, and ca as ophe. Risk can be classi ied in o
wo ypes: sys ema ic isk and speci ic isk. The e m "sys ema ic isk" e e s o inco ec
base assump ions (le el and end), whe eas "speci ic isk" e e s o ola ili y ha
su ounds he base assump ions ( ola ili y and ca as ophe). Speci ic isk is dec easing,
bu he sys ema ic isk canno be di e si ied as he numbe o li es co e ed inc eases.
The e a e conside able and inc easing cos s o sys ema ic isk o pension plans and
insu e s.
Mo ali y isk is a i al isk ac o o insu ance companies and mo ali y isk is b oken
up in o subca ego ies, sys emic isk, unsys ema ic isk, and ad e se emedies.
The isk o mo ali y e e s o he isk o a pe son li ing o a sho e li e han expec ed
and is, he e o e, highe han expec ed. The in e es o li e insu e s and pensione s in
longe i y isk o he design o a de ined bene i plan has inc eased (Ga ze & Weske ,
2014).
i. Unsys ema ic Mo ali y isk:
The isk o indi idual dea hs is a andom a iable wi h a ce ain p obabili y (see Bi iss,
Denui , and De olde , 2010). Thus, i may be di e si ied h ough na u al hedges, o
ans e s h ough mo ali y o he capi al ma ke , Con ingen bonds (MCBs).
ii. Sys ema ic Mo ali y isk:
The isk o sys ema ic mo ali y is he isk o sudden changes o unde lying popula ion
mo ali y, o example as a esul o common ac o s a ec ing dea hs o he en i e
popula ion ha igge li e dependencies and canno be di e si ied by b oadening he
po olio (see Wills and She is, 2010).
iii. Ad e se Selec ion:
This e e ed o he ac ha , o a ious popula ions o assu ed pe sons, o example, li e
insu e s and pensione s, he p obabili y dis ibu ion di e s in age le el and end (see
B ouhns, Denui and Ve mun , 2002a). In addi ion, ad e se selec ion is a majo sou ce o
isk when hedging longe i y isk ia MCB o o he capi al ma ke s ins umen s, because
o indi idual mo ali y he e ogenei y and in o ma ion asymme ies be ween he insu ance
company and Insu ed (see, e.g., Swee ing, 2007).
i . Basic Risk:
This occu s when hedge popula ion mo ali y does no coincide wi h he po olio hedge
mo ali y. This means ha he e is a base isk in longe i y hedges in he di e ences in
popula ion mo ali y and mo ali y o he insu ed pensione s caused by ad e se selec ion.
In his analysis, we explici ly conside he undamen al isk in hedges and models e e y
kind o mo ali y isk in o de o analyze i s impac on he isk si ua ion o he li e insu e .
2.3. Managemen And Quan i ica ion o Longe i y Risk
To ensu e ha insu e s' exposu e o longe i y isk is e ec i ely managed, ac ua ies mus
i s be awa e o he cu en me hods o quan i ying and managing his isk. Only hen
can hey ake an ac i e pa in iden i ying and building addi ional isk managemen
echniques ha a e mo e e ec i e in add essing longe i y isks.
Companies a e equi ed o main ain a ce ain pe cen age o hei ne isk o ese es o
co e he isk ha dea h is di e en om expec ed. As a esul , mos companies con inue
o quan i y he isk o longe i y wi h ela i ely undamen al me hodologies. Because he
isk o long li e o insu e s is inc easing, majo annui y au ho s and einsu e s look o
ways o manage hei cos s e ec i ely. To da e, p oduc design, con ac ing, na u al
hedging, and einsu ance a e he con en ional me hods ha di ec au ho s use.
Fu he mo e, companies ha e s a ed o use hei longe i y isk exposu e solu ions o he
inancial ma ke s.
• Buy-ins, A pension scheme's liabili ies such as pensione s' in-paymen , a e co e ed by
buy-in. The policy pays an income equi alen o he membe s' bene i s, emo ing he
dange o insu icien asse s o und u u e commi men s.
• Bulk Annui y and einsu ance ansac ions o ans e en s be ween insu e s and
einsu e s. • These solu ions a e also insu ance.
• Longe i y bonds which ans e a long- e m isk o ano he pa y in he o m o a
secu i y om a pension plan o annui y po olio. These a e solu ions o he capi al
ma ke s.
• Longe i y swaps o ans e longe i y only o ano he pa y om a pension scheme o
annui y po olio. These can ei he be insu ance solu ions o solu ions o he capi al
ma ke s.
• Mo ali y ca as ophe and swapping, ans e ing om li e insu e o einsu e o o he
pa ies, he isk o de as a ing (ca as ophe) inc eases in mo ali y due, e.g., o a pandemic
o na u al disas e . These a e solu ions o he capi al ma ke s.
• Li e secu i iza ions ha ans e isks ela ed o a speci ic block o insu ance
unde akings, as a secu i y, o capi al ma ke s. These a e solu ions o he capi al ma ke s.
• US li e se lemen s ansac ions ans e ing o in es o s small po olios o U.S. li e
insu ance policies. These a e solu ions o he capi al ma ke s.
• Pensions buy-ou s ha ans e pension obliga ions and all associa ed isks and
obliga ions o insu e s (also known as pension plan e minals). These a e he solu ions o
insu ance.
The hedging ins umen is he hi d ea u e o isk ansac ions wi h a pu e longe i y. The
longe i y swap o su i o s has p e iously been he mos common s uc u e.
Mo ali y o wa d (q- o wa d)
A o wa d mo ali y con ac is o en known as a o wa d, as he le e 'q' s ands o
ac ua ial mo ali y a e symbols. I is he simples ype o longe i y (and mo ali y) isk
ans e ins umen (Coughlan e al. 2007b) and was he i s ype o capi al ma ke s ha
we e used o longe i y hedges. This was an ag eemen be ween UK Lucida and J.P.
Mo gan pension insu e s and is desc ibed in he nex sec ion. The impo ance o q-
o wa ds is ha hey o m undamen al blocks om which o he li e- ela ed de i a i es
can be buil .
A q- o wa ds po olio can be used, i app op ia ely designed, o eplica e and sa egua d
a li e ime exposu e o o p o ec a li e insu ance book o a pension liabili y. A q- o wa ds
shall be de ined as an ag eemen be ween wo pa ies in which a sum p opo ional o he
ac ual mo ali y a es pe o med o a gi en popula ion (o subpopula ion) is exchanged
in exchange o he sum p opo ional o a ixed dea h a e ag eed upon a he ou se o be
payable in he u u e ( he ma u i y o he con ac ). I he e is a ai p ice o he q- o wa d,
he e is no change in paymen hands a he s a o he ade, bu a ma u i y one o he
wo coun e pa ies makes a ne paymen (unless he ixed and ac ual mo ali y a es
happen o be he same). The ma u i y paymen is based on he ne amoun payable and is
p opo ional o he di e ence be ween he ixed mo ali y a e ( he o wa d a e
ansac ed) and he e e ence a e ealized. I in he e e ence yea he a e is lowe han
he ixed a e ( ha is, a lowe dea h a e), he se lemen is posi i e, and he se lemen
paymen is ecei ed by he pension plan o make up o he inc ease in i s liabili y alue.
Whe e, on he o he hand, he e e ence a e is highe han he ixed a e (ie. highe
mo ali y), he epaymen is nega i e, and he pension plan pays he hedge p o ide he
se lemen paymen , which is o se by he decline in he alue o he paymen . The ne
liabili y alue is he e o e locked wi h ega ds o he mo ali y a es. The scheme is
p o ec ed agains unexpec ed mo ali y a e changes.
Su i o o wa d (S- o wa d)
A su i o o wa d, also known as a “S- o wa d,” is simila o a q- o wa d in concep bu
uses su i al a es a he han mo ali y a es. I is an ag eemen be ween wo pa ies o
exchange an amoun p opo ional o he ac ual, ealized su i al a e o a gi en popula ion
(o subpopula ion) in exchange o an amoun p opo ional o a ixed su i al a e ha
has been mu ually ag eed upon a he con ac 's incep ion o be payable a he con ac 's
ma u i y. As such, i en ails exchanging a no ional amoun mul iplied by a p e-ag eed-
upon ixed su i al a e o he same no ional amoun mul iplied by he ealized su i al
a e o a speci ied coho o e a speci ied ime pe iod (Coughlan e al., 2008b; Dawson
e al., 2010). I he con ac has a one-yea ma u i y, a su i o o wa d is he in e se o
a mo ali y o wa d. Howe e , i he con ac ma u i y exceeds a yea , his simple
ela ionship no longe exis s because su i al a es o e longe ime pe iods a e non-
linea unc ions o annual mo ali y a es. Because i is a unc ion o se e al mo ali y
a es a di e en ages and imes, a su i o o wa d is mo e complex han a q- o wa d. In
some si ua ions, i can ne e heless be a use ul building block.
Longe i y swaps
A longe i y swap can be classi ied as ei he a capi al ma ke s de i a i e o an insu ance
con ac . In ei he case, i is a inancial ins umen ha in ol es exchanging ac ual pension
paymen s o a se ies o p e-ag eed-upon ixed paymen s (Dowd e al., 2006; B a o &
Nunes, 2021). Each paymen is based on an amoun weigh ed su i al a e. In any
longe i y swap, he hedge o longe i y isk ( o example, a pension plan) ecei es he
ac ual paymen s i mus make o pensione s om he longe i y swap p o ide and, in
exchange, makes a se ies o ixed paymen s o he hedge p o ide . As a esul , i e i ees
li e longe han expec ed, he highe pension amoun s ha he pension plan mus pay a e
o se by he highe paymen s ecei ed om he longe i y swap p o ide . As a esul , he
swap o e s he pension plan a long ma u i y, cus omized cash low hedge o i s longe i y
isk. The July 2008 Canada Li e-J.P. Mo gan ansac ion (T ading Risk 2008; Li e &
Pensions 2008).
Va ian s on longe i y swaps
The ansac ion ca ied ou by Aegon and Deu sche Bank in Janua y 2012 is one a ian
o he s anda d longe i y swap. This was an “ou -o - he-money” longe i y swap because
i only ans e ed he longe i y isk associa ed wi h a signi ican inc ease in li e
expec ancy (o equi alen ly, a e y la ge and sus ained all in mo ali y a es). Aegon, he
hedge , ecei es no inc emen al paymen o modes inc eases in li e expec ancy un il a
ce ain h eshold, o "a achmen poin ," is c ossed. Aegon will hen be paid o which he
li e span inc eases un il a ce ain maximum le el o p o ec ion is a ained when li e
expec ancy ises o a e y ex eme le el. This swap is indeed a s anda d long-li e swap,
excep ha i has loa ing caps and loo s. The swap in capi al ma ke s was based on
indexes o e 20 yea s and he index ma ched he na ional popula ion da a o he
Ne he lands. This swap also included, like he A i a-RBS ansac ion, a swap paymen
a ma u i y o p o ec he longe i y o any esponsibili y cash low ha exceeds he
ma u i y da e.
Longe i y bonds
Since he s a o his ma ke , longe i y bonds ha e been widely spoken o p e en he
isks o longe i y. A longe i y bond (o a su i o bond as i was o iginally called) is a
bond ha pays coupons ha p opo ionally co espond o he numbe o su i o s s ill
li ing on he coupon paymen da e in he popula ion coho speci ied. (Wol , 2001; Blake
e al., 2006a, 2006a; Dowd, 2003). The cash lows o a single longe i y anilla bond a e
he same as hose o a longe i y swap loa ed bea ing. Howe e , longe i y bonds wi h

di e en s uc u es ha e ecen ly been p oposed. The cash lows o he bond a e indexed
o he mo ali y expe ienced in he Uni ed Kingdom by 65-yea -old men. The e is a 10-
yea de e men pe iod be o e he s a o paymen and a e minal swi ching paymen a
105 yea s is made o co e he isk o a long li e a e 105 yea s. I mo e people su i e
a each age, hen he bond pays mo e; i ewe people su i e, hen he bond pays less
(simila o he loa ing leg o he RBS-A i a longe i y swap).
3. METHODOLOGY
Li e expec ancy is he mos common s a is ical indica o o he a e age emaining
li espan an indi idual is expec ed o li e (Ayuso e al., 2021).
3.1. Basic Mo ali y Func ions
Le (𝑥) deno e a li e ha su i es o he age 𝑥. The li e (𝑥) is called a li e-age-𝑥.
Le 𝐷𝑥𝑡 be a andom a iable, in a popula ion who die a aged (𝑥) las bi hday
du ing a calenda yea .
𝑑𝑥𝑡 deno e he obse ed numbe o pe sons who die be ween ages (𝑥) and
(𝑥+𝑡)
𝑙𝑥 deno e numbe o pe sons who a ain age x acco ding o he mo ali y able.
𝑞𝑥 deno e he p obabili y ha (𝑥) will die wi hin 1 yea
𝑝𝑥 deno e he p obabili y ha (𝑥) will li e 1 yea .
𝑑𝑥 deno e
3.1.1. Ini ial Mo ali y Ra e
𝑞𝑥 is called he mo ali y a e a age 𝑥, in ac ua ial e minology 𝑞𝑥 is he p obabili y ha
(x) dies be o e age (𝑥 + 1). We can also subsc ibe a (𝑡) o ge 𝑞𝑥
𝑡 which is he
p obabili y ha (x) dies be o e age 𝑥 + 𝑡,
𝑞𝑥= 𝑑𝑥
𝑙𝑥
(1)
3.1.2. P obabili y o Su i al
The su i al unc ion o 𝑇𝑥 is deno ed by 𝑝𝑥
𝑡. I is he p obabili y ha a li e aged
𝑥 su i es 𝑡 mo e yea s o is he p obabili y ha an age (𝑥) su i es o a leas age (𝑥 +
𝑡). In simplici y emo ing (𝑡), we ge .
𝑝𝑥= 𝑙𝑥+1
𝑙𝑥
(2)
3.1.3. Cen al Dea h Ra e
The numbe o people who died du ing he yea di ided by he o al
numbe o people who we e ali e du ing he yea . The Cen al dea h Ra e (𝑚𝑥)
deno es as he cen al dea h a e o he yea o age (𝑥) o (𝑥+1).
𝑚𝑥= 𝑑𝑥
𝑙𝑥
(3)
In he ac ua ial modeling li e a u e, we use he ollowing s anda d de ini ions (Dickson
e al. (2013; 2009); Pi acco e al. (1998)). Le 𝑇𝑥 deno e he emaining li e expec ancy o
an indi idual o age 𝑥. The cumula i e unc ion o dis ibu ion and su i al o 𝑇𝑥 is
w i en as 𝜏 𝑞𝑥 = 𝑃(𝑇𝑥≤ 𝜏 ) and τ𝑝𝑥 = 𝑃(𝑇𝑥 > 𝜏 ) espec i ely. Fo an indi idual aged
𝑥, he o ce o mo ali y a age 𝑥 + 𝜏 is de ined as
𝜇𝑥+𝜏 ∶= lim
ℎ→01
ℎ𝑃(𝑇𝑥<𝜏+ℎ|𝑇𝑥>𝜏)=− 𝑑
𝑑𝜏ln𝜏𝜌𝑥
Le 𝑓𝑥 (𝑡) be he densi y unc ion o 𝑇𝑥, hen om (1) we ha e.
𝜏𝑞𝑥=∫ 𝑓𝑥(𝑠)𝑑𝑠
𝜏
0=∫ 𝑠𝜌𝑥
𝜏
0𝜇𝑥+𝑠 𝑑𝑠
The cen al dea h a e o 𝑥-yea -old, whe e 𝑥 𝜖 ℕ, is de ined as
𝑚𝑥:= 𝑞𝑥
∫𝑠𝑝𝑥𝑑𝑠
1
0=∫𝑠𝑃𝑥𝜇𝑥+𝑠 𝑑𝑠
1
0∫𝑠𝑃𝑥𝑑𝑠
1
0
which is a weigh ed a e age o mo ali y o ce (𝑞𝑥∶= 𝑞𝑥
1). Taking accoun , he so-
called cons an o ce o mo ali y assump ion, µ𝑥+𝑠 = µ𝑥 whe e 0 ≤𝑠 <1 and 𝑥 ∈ ℕ,
om (2), we ha e 𝑚𝑥= µ𝑥.
I a Poisson assump ion is deno ing o he ac ual numbe o dea hs, hen he maximum
likelihood es ima es o he o ce o mo ali y µ𝑥 is gi en by µ𝑥= 𝐷𝑥𝐸𝑥
⁄= 𝑚𝑥 whe e
𝐷𝑥 deno es he eco de numbe o dea hs a age 𝑥 las bi hday and exposu e o isk 𝐸𝑥
is he a e age numbe o indi iduals in he obse a ion yea who we e 𝑥 yea s old on
hei las bi hday. No ice ha 𝐸𝑥 is based on a popula ion es ima e o people who we e
𝑥 yea s old on hei las bi hday in he middle o he obse a ion yea .
𝐸𝑥𝑡
𝑐 ep esen he cen al exposed o isk a age 𝑥 in yea 𝑡, and 𝐸𝑥
𝑜 deno es he
ini ial exposed o isk o all a ays o 𝑥-age and 𝑡-yea comp ising ages (on he ows)
𝑥 = 𝑥1,𝑥2,𝑥3 ..., 𝑥𝑘, and calenda yea s (on he columns) 𝑡 = 𝑡1,𝑡2,𝑡3 ...,𝑡𝑛,
3.1.4. The o ce o mo ali y (𝝁𝒙,𝒕)
𝜇𝑥,𝑡 ep esen s he haza d a e o mo ali y o an indi idual a exac ly age x and dies a
he exac yea s.
The o ce o mo ali y ela ed o he dea h p obabili y as
𝜇𝑥,𝑡 = lim
𝑑𝑥→0+𝑃𝑟[𝑇0≤𝑥+𝑑𝑥|𝑇0>𝑥]
𝑑𝑥
𝜇𝑥𝑑𝑥≈ lim
𝑑𝑥→0+𝑃𝑟[𝑇0≤𝑥+𝑑𝑥|𝑇0>𝑥]
𝜇𝑥=−𝑑
𝑑𝑥𝑆0(𝑥)
𝑆0(𝑥)
(4)
𝐺𝑜𝑚𝑝𝑒𝑟𝑡𝑧: 𝜇𝑥=𝐵𝐶𝑥,0<𝐵<1,𝑐>0
3.1.5. Li e expec ancy (𝒆𝒙,𝒕)
𝑒𝑥,𝑡 means ha an indi idual o he gi en age 𝑥 can expec o li e wi h ime 𝑡 an
addi ional numbe o yea s on a e age. Li e expec ancy, which is equi alen o he o al
li e span, is mos common a bi h.
𝑒𝑥=𝑇𝑥
𝐿𝑥
(5)
Bu li e expec ancy o a gi en age in which he age plus li e expec ancy is equal o he
o al li e expec ancy.
Conside he mo ali y model ha ep esen s he model which examines he
s uc u e o p obabili y o dea h o cen al mo ali y a es ac oss ages and o yea s.
ln[𝑚𝑥(𝑡)−𝛼×]≈𝜌1𝑈𝑥,𝑖𝑉𝑖,𝑡
(14)
Thus, es ima es o 𝛽𝑥 and 𝑘𝑡 can be ob ained:
𝛽𝑥=𝑈𝑥,𝑖
∑𝑈𝑥,𝑖𝑥 ,
(15)
𝑘𝑡=𝜌1𝑉1,𝑡∑ 𝑈𝑥,1
𝑥
(16)
The i ing e ec o he singula alue decomposi ion depends on he e iciency o
ex ac ing om he ln𝑚𝑥(𝑡)−𝛼𝑥 ma ix. I is gene ally conside ed ha he me hod can
explain mo e han 90% o he sum o squa es o de ia ions. A e ob aining he es ima ed
alues o he pa ame e s, i can be ound ha 𝛼𝑥 𝑎𝑛𝑑 𝛽𝑥 a e ixed o e ime, and he
a ie y o mo ali y o e ime is mainly e lec ed by (𝑘𝑡). The p edic ion alue o u u e
mo ali y can be ob ained by ex apola ing (𝑘𝑡). I is belie ed ha (𝑘𝑡) is a andom walk
wi h d i o ARIMA p ocess. Acco ding o he BIC in o ma ion c i e ion, (𝑘𝑡). Should
be he AR IMA (0,1,1) Model wi h d i e m.
Ad an ages o he Lee-Ca e model
I p o ides a good i o he his o ic da a. The 𝛼𝑥 aging unc ion makes i possible o a
model o be employed a all ages, e en young ages, when he li e able shape can be e y
complex, while he k e m ep esen s he p e alen endency in mo ali y e olu ion.
I is simple o i wi h ela i ely ew pa ame e s, pa icula ly compa ed o o he
complica ed models and bo h he o iginal decomposi ion o he single alue and B ouhns
e al (2002) a e well unde s ood and easy o implemen Poisson Likelihood i ing model.
The p ojec is easy. Because o he common linea end o mos da ase s in 𝜅𝑡's, he
andom walk-in d i ime se ies is used ex ensi ely o es ima ing he u u e cen al
mo ali y a es.
I is a simple concep o g asp. Bo h 𝛼𝑥 and 𝜅𝑡 a e easily unde s ood as he shape o
mo ali y ac oss ages and he le el o mo ali y each yea , which is use ul when epo ing
esul s o a la ge audience.
Disad an ages o Lee-Ca e Model
I has only one-pe iod e m 𝜅𝑡, which indica es ha he change in all he cen al mo ali y
a es in each yea o he p ojec ion is pe ec ly ied o he un ealis ic p oblem and o he
isk o liabili ies and secu i ies, based on he cen al mo ali y a e.
𝛽𝑥 does no ha e uni e sal in e p e a ion and can make unp edic able p ojec ions? The
shape o a 𝛽𝑥 becomes impo an when he cen al mo ali y a e is p ojec ed because a

model i ed in o a long ange o his o ical da a will con inue o show high a es o
imp o emen a he younge age and, a highe age a es, which migh be unlikely.
The e is no p o ision o “coho ” impac s based on a pe son's bi h yea . Renshaw and
Habe man we e among he i s o p opose models based on he Lee-Ca e model bu
in eg a ing coho e ec s (2006).
3.5.2. O he Models
II. The Cai ns-Blake-Dowd model
To add ess pe cei ed p oblems wi h he Lee-Ca e model and o e come p oblems wi h
p ojec ed dea h a es in single age/pe iod e m models, Cai ns e al. in oduced one o he
mos popula compe ing models o he LC model, he Cai ns-Blake-Dowd model (2006).
The Cai ns-Blake-Dowd model p esumes ha dea h p obabili ies can be modeled as
𝑙𝑜𝑔𝑖𝑡(𝑞𝑥,𝑡)= 𝜅𝑡
(1)+(𝑥−𝑥)𝜅𝑡
(2)
(17)
The logi o dea h p obabili ies is a linea age unc ion, which is easonable o high age
(abou 50 yea s old) bu is no ue o he younge age. I is assumed. The 𝜅𝑡
(1)pa ame e
de e mines dea h le els o e all yea s o a ce ain yea in he Cai ns-Blake-Dowd model.
The 𝜅 𝑡
(2)pa ame e de e mines he 'aging a e' o each yea , i.e., an inc ease in mo ali y
be ween one age and he ollowing age.
Cai ns e al. p esen ed a p edic o s uc u e wi h wo age-pe iod e ms (𝑁 = 2), age-
modula ing pa ame e s 𝛽𝑥
(1) = 1 and 𝛽𝑥
(2)= 𝑥 – 𝑥,, no s a ic age unc ion, and no coho
e ec (2006). The CBD model p edic o is p o ided by:
𝜂𝑥𝑡 = 𝜅𝑡
(1) + (𝑥 – 𝑥)𝜅𝑡
(2)
(18)
Whe e: 𝑥 ep esen he a e age age.
Ad an age o he Cai ns-Blake-Dowd model
The Cai ns-Blake-Dowd model is a commonly used mo ali y model, pa icula ly among
p ac i ione s conce ned wi h he iskiness o liabili ies ied o high- isk dea h p obabili y,
such as annui ies.
In compa ison o he Lee-Ca e model, i p o ides o a mo e sophis ica ed co ela ion
s uc u e be ween dis inc dea h p obabili y. This is especially signi ican when assessing
he possible iskiness o liabili ies, such as o insu ance sol ency conside a ions.
I is simple o pu oge he . Because he e a e no age unc ions in he model, i can be
i ed using leas squa es o likelihood maximiza ion app oaches o p oduce a sa is ac o y
i o he his o ical da a when u ilized o e long pe iods o ime.
I p o ides smoo h es ima es o dea h p obabili ies o e e y gi en yea . This is
p e e able i i is belie ed ha he basic p ocesses de e mining mo ali y should no
change as people age.
I is simple o p ojec . Fo p ojec ing he 𝜅𝑡 pa ame e s ac oss a numbe o coun ies, he
bi a ia e andom walk wi h d i has p o en o be a eliable and obus model.
In addi ion o agg ega e measu es o longe i y such as pe iod li e expec ancy, i p o ides
s ochas ic o ecas s wi h con idence anges o indi idual 𝑞𝑥,𝑡's ha a e deemed o be
ealis ic in con as o p e ious e idence.
Disad an ages o Cains. Blake-Dowd model
Models based on he Cai ns-Blake-Dowd model ha include coho e ec s ha e ecen ly
been p esen ed, mos no ably in Cai ns e al (2009) and Pla s e al (2009).
I does no i da a well ac oss he boa d. The assump ion o linea i y in 𝑙𝑜𝑔𝑖𝑡(𝑞𝑥,𝑡) is no
longe easonable below he age o 50, and i may no be easonable e en a highly
ad anced ages (abo e 90). The e ha e been a emp s o accommoda e his by in oducing
an age unc ion 𝛼𝑥, simila o ha ound in he Lee-Ca e model, o example in Pla
(2009).
III. The P-splines model
Cu ie e al. (2004) p oposed he P-splines model as a mechanism o eliably smoo hing
and p edic ing cen al mo ali y a es. I is ounded on Eile s' and Ma x's use o penalized
B-splines (1996). A "spline" is a piecewise polynomial unc ion de ined ac oss a ange o
alues.
A amily o splines known as a basis o splines (also known as B-splines) is la ge enough
o co e he comple e ange o an in e es . The linea sum o he B-splines can hen be
used o smoo h any discon inuous unc ion o e his ange. The numbe o splines
employed and whe e he kno s a e placed ha e a signi ican impac on he smoo hing
accomplished by his me hod.
This P-spline was used by Cu ie e al (2004) on wo-dimensional mo ali y da a o
smoo h he c ude es ima es o cen al dea h a es o e ages and yea s. They also p edic ed
cen al mo ali y a es in o he u u e by using missing alues in he model o u u e
yea s. The P-splines model implies ha he o ce o mo ali y may be ep esen ed as a
linea combina ion o smoo h unc ions o e ime and space, i.e.
log𝑚(𝑥,𝑡)=∑𝜃𝑖𝑗𝛽𝑖𝑗(𝑡,𝑥)
𝑖𝑗
(19)
Whe e: 𝛽𝑖𝑗(𝑡,𝑥) is he p ede e mined ounda ion unc ion wi h egula ly sp ead kno s,
and he 𝛽𝑖𝑗 is he age and coho pa ame e s o be calcula ed. I is commonly ecognized
ha he use o splines can esul in o e - i ed unc ions, esul ing in unnecessa ily lumpy
i ed mo ali y su aces.
Ad an ages o P-splines model
The P-splines me hod has become widely used o smoo hing his o ical da a, mos
no ably by he Con inuous Mo ali y Ins i u e o p oducing de e minis ic mo ali y
p ojec ions – o example, in CMI (2002) and CMI (2004). (2009b).
I gi es alues ha a e smoo h ac oss age and ime o cen al mo ali y a es and is hus
excellen in emo ing he e ec o andom noise om he c ude da a.
I 's ela i ely unpleasan . The smoo hing p ocedu e educes he o al numbe o model
pa ame e s and educes he e ec i e numbe o ee pa ame e s u he wi h he penal y
unc ion.
I p o ides p ojec ions o allow o changes in he cen al mo ali y a es o a ious ages
on he basis o he obse a ions.
Disad an ages o P-splines model
I s explana ion and implemen a ion a e complex. The e is no in ui i e meaning o
pa ame e s, and he i ing p ocedu e used by Cu ie e al (2004) and Cu iee al (2006)
in ol es manipula ing e y la ge ma ices ha educe he i ing speed and can cause
compu e memo y alloca ion p oblems.
The su aces a e i ed ha can be conside ed oo smoo h. The P-spline me hod i sel ies
o educe he impac o shocks on he da a o alle ia e po en ially alid cha ac e is ics
such as a one-o inc ease in he cen al dea h a e due o an epidemic.
The e a e no s ochas ic p ojec ions a ailable. Ins ead o allowing u u e a es o be
gene a ed by a s ochas ic p ocess, he P-splines model i s a de e minis ic su ace o he
da a and ex ends i in o he u u e. Cu ie (2006) a emp s o p o ide “con idence
in e als” o u u e p ojec ions, bu hese a e dependen on e o s in es ima ing he
unde lying pa ame e s a he han being uly s ochas ic.
I does no conside “coho ” impac s. " I desi ed, he P-splines model can be changed
om an age/pe iod o an age/coho model, as desc ibed by CMI (2006), al hough his
emo es he pe iod e ec s, which a e equen ly domina ing and cause p oblems because
some coho s ha e limi ed obse a ions.
IV. The CMI Model
The Con inuous Mo ali y In es iga ion (CMI) de eloped he CMI mo ali y p ojec ion
model (2009). I is a model o mo ali y imp o emen a es a he han mo ali y a es
hemsel es, as he p e ious models o mo ali y we e. The mo ali y imp o emen a es
a e de ined as
𝑟𝑥𝑡 = 1− 𝑞𝑥𝑡
𝑞𝑥,𝑡−1
(20)
To de i e he pa e n o mo ali y imp o emen s, he s uc u e o mo ali y a es in a
popula ion is analyzed o e age, ime, and yea s. The age/pe iod and coho componen s
disco e ed a e hen assumed o pe sis o se e al yea s be o e blending in o a use -
speci ied “long- e m a e o imp o emen .”
Ad an age o CMI Model
Based on a single and ela i ely simple inpu om he use , i can quickly gene a e a
cen al p ojec ion o mo ali y a es. This is ex emely bene icial o ac ua ial consul an s
who wo k p ima ily in de e minis ic en i onmen s ( o ins ance, alua ion o pension
schemes o ese ing o li e assu ance). In his con ex , i can also se e as a "common
cu ency" o ansla ing he pa e n o imp o emen s in mo ali y a es o li e expec ancy
obse ed in ano he model ( o example, he Lee-Ca e model) in o a oughly equi alen
long- e m a e o imp o emen .
Disad an age o CMI Model
The CMI model's inabili y o gene a e s ochas ic p ojec ions o mo ali y a es means ha
i is unsui able o measu ing he isk inhe en in any p ojec ion, excep when compa ing
compe ing scena ios. I is also a e y complex model when compa ed o he o he models
used, hough his complexi y is la gely hidden om he in ended end use and is only
isible he e because he me hodology mus be applied o di e en da ase s.
3.6. Regula o y F amewo k o Mo ali y Assump ion
Mo ali y assump ions used in he alua ion o pension and annui y liabili ies a e ypically
p esen ed in he o m o a able, wi h he p obabili y o dea h o e he nex yea s, 𝑞𝑥,
gi en o each indi idual age 𝑥. Usually, di e en assump ions a e used o males and
emales, howe e ce ain dis ic s' egula ions necessi a e he use o unisex a es.
Tables o mo ali y can be one-dimensional, accoun ing jus o di e ences in dea h by
age, o wo-dimensional, accoun ing o mo ali y e olu ion h ough ime. One-
dimensional ables, o en known as s a ic ables, ha e only one dea h a e o each age
g oup.
As many yea s o su icien mo ali y expe ience a e equi ed, es ablishing assump ions
o p edic ed mo ali y imp o emen needs subs an ially mo e da a and is hus mo e
di icul o se . As a esul , mo ali y imp o emen assump ions a e equen ly based on
gene al popula ion mo ali y.
A e he mo ali y assump ions ha e been de e mined, hey can be applied o he ini ial
mo ali y le el o es ablish a gene a ional able gi ing he mo ali y assump ion a any
u u e poin in ime. They a e commonly used in he ollowing ways, whe e 2000 is he
yea in which he ini ial le el o mo ali y was de e mine and 𝑟 is he annualized a e o
mo ali y imp o emen o age 𝑥:
𝑞𝑥,2000+𝑡 =𝑞𝑥,2000((1−𝑟𝑥)𝑡
(21)
In p ac ice, 𝑟 may a y o e ime, bu i usually jus a ies by age and gende .
3.6.1. Mo ali y Assump ions in P ac ice and Regula ion
The egula o y amewo k may demand he use o specialized mo ali y ables. These
ables indica e minimal mo ali y assump ions and may o may no accoun o u u e
imp o emen s in mo ali y and li e expec ancy. Howe e , when minimum ables a e
necessa y, pension unds and annui y p o ide s a e o en allowed o employ mo ali y
ables ha a e mo e conse a i e han hose equi ed in o de o accoun o and p epa e
o signi ican u u e imp o emen s in mo ali y and li e expec ancy i deemed sui able.
Whe e he legisla i e amewo k does no c ea e speci ic mo ali y ables, pension unds
and annui y p o ide s may use hei own ables, o he ables mos commonly used by he
indus y.
The ex en o which mo ali y assump ions a e egula ed a ies g ea ly be ween coun ies
and is no always uni o m be ween pension unds and annui y p o ide s wi hin he same
coun y. Table 1 illus a es whe he he egula ion manda es minimum mo ali y
assump ions o whe he he egula ion equi es ha u u e imp o emen s in mo ali y be
accoun ed o in he assessmen o pension and annui y liabili ies, while he speci ic
assump ions o be used a e no equi ed. The analysis e alua es whe he i is s anda d
ma ke p ac ice o accoun o u u e mo ali y imp o emen in he p icing o liabili ies,

e en i egula ion does no demand i . In hal o he coun ies, nei he pension unds no
annui y p o ide s a e equi ed o accoun o u u e mo ali y imp o emen . Despi e he
lack o a legisla i e obliga ion, he majo i y o coun ies do so in p ac ice, wi h annui y
p o ide s doing so mo e equen ly han pension unds.
Table 1: Mo ali y ables and imp o emen equi ed by egula ion and used in
p ac ice
Coun y
Minimum able equi ed
by Regula ions
Mo ali y
Imp o emen equi ed
by Regula ions
Mo ali y
Imp o emen s used
in P ac ice
Annui y
P o ide s
Pension
Plans
Annui y
P o ide s
Pension
Plans
Annui y
P o ide s
Pension
Plans
B azil
No
Yes
No
No
No
No
Canada
No
Yes
Yes
Yes
Yes
Yes
Chile
Yes
Yes
Yes
Yes
Yes
Yes
China
Yes
Yes
No
No
No
No
F ance
Yes
Yes
Yes
Yes
Yes
Yes
Ge many
Yes
Yes/No
Yes
Yes
Yes
Yes
Is ael
Yes
Yes
Yes
Yes
Yes
Yes
Japan
No
Yes
No
No
Yes
No
Ko ea
No
No
No
No
No
No
Mexico
Yes
No
Yes
No
Yes
No
Ne he land
No
No
Yes
Yes
Yes
Yes
Pe u
Yes
Yes
No
No
Some
Some
Spain
No
No
Yes
Yes
Yes
Yes
Swi ze land
No
No
No
No
Yes
Some
Uni ed
Kingdom
No
No
Yes
Yes
Yes
Yes
Uni ed
S a es
Yes
Yes
No
Yes
Yes
Yes
Sou ce: OECD
No es: The s a is ical da a o Is ael a e supplied by and unde he esponsibili y o he
ele an Is ael au ho i ies.
The use o such da a by he OECD is wi hou p ejudice o he s a us o he Golan
Heigh s, Was Je usalem and Is aeli se lemen s in he Wes Bank unde he e ms o
in e na ional law.
1. Fo non/ egula ed Pensionskassen and insu ance o ien ed Pensions onds.
2. Fo egula ed Pensionskassen and non/insu ance o ien ed Pensions onds.
Despi e he lack o a legal obliga ion o p o ision o mo ali y imp o emen , he majo i y
o coun ies do so in p ac ice, wi h annui y p o ide s doing so mo e equen ly han
pension unds. In p ac ice, hi een o he six een na ions' annui y p o ide s use mo ali y
imp o emen assump ions, whe eas only ele en o he six een coun ies' pension unds
do.
3.6.2. S anda d Mo ali y Table
The analysis is based on he si ua ion in which con en ional mo ali y ables a e used by
pension unds and annui y p o ide s. Mo ali y a es o mos plans will be based on
s anda d ables c ea ed and published by he Socie y o Ac ua ies o go e nmen al
o ganiza ion. Tables a e o en i led based on (1) he ypes and cha ac e is ics o da a
unde lying he able and (2) because mo ali y a es gene ally change o e ime, he
calenda yea o expe ience ha he mo ali y a es a e assumed o ep esen . In mos
cases, de ailed in o ma ion abou he da a's sou ce is included in he epo ha is
published alongside he able. In addi ion, a b eakdown o able a es o subg oups may
be p o ided.
3.6.3. Mo ali y Imp o emen
Cu en mo ali y ables, which ha e been speci ically cons uc ed o he e i emen a ea,
ypically ha e no oom o u u e mo ali y imp o emen . Mos Socie y o Ac ua ies
mo ali y ables used in he e i emen a ea, howe e , include p ojec ion scales o use in
es ima ing u u e mo ali y imp o emen . These scales a e ypically di e en ia ed by age
and gende .
Table 2:Mo ali y P ojec ion Scale AA compiled by he Socie y o Ac ua ies
Age
Male
Female
60
.016
.005
61
.015
.005
62
.015
.005
63
.014
.005
64
.014
.005
65
.014
.005
66
.013
.005
67
.013
.005
68
.014
.005
69
.014
.005
70
.015
.005
Sou ce: Mo ali y P ojec ion Scale AA compiled by he Socie y o Ac ua ies
Fo ull scale see Table 7-3 in RP-2000 Mo ali y Table,
h ps://www.soa.o g/globalasse s/asse s/Files/Resea ch/Exp-
S udy/ p00_mo ali y ables.pd
These scales a e used o educe he likelihood o dea h in he ollowing way:
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑡ℎ 𝑤𝑖𝑡ℎ 𝑛 𝑦𝑒𝑎𝑟𝑠 𝑜𝑓 𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡
= (𝑚𝑜𝑟𝑡𝑎𝑙𝑖𝑡𝑦 𝑟𝑎𝑡𝑒 𝑎𝑡 𝑎𝑔𝑒 𝑥) (1
− 𝑝𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑠𝑐𝑎𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑎𝑡 𝑎𝑔𝑒 𝑥)𝑛
3.6.4. Gene a ional Mo ali y Imp o emen
I i is assumed ha he o ces leading o mo ali y imp o emen will con inue in he
u u e, hen mo ali y a es will a y by bo h age and he calenda yea o a ainmen o
age, because hose a aining he age la e will be exposed o he o ces leading o mo ali y
imp o emen o a longe ime pe iod. Thus, he p obabili y o dying a 60 would be
highe o a pe son u ning 60 in 2012 han o a pe son becoming 60 in 2016. Ano he
way o look a i is ha a ious gene a ions ( hose bo n in 1952 e sus hose bo n in 1956)
will ha e di e en mo ali y a es a he age o 60.
To accoun o his di e ence, p ojec ion scales o he numbe o yea s be ween he
alua ion yea and he yea he indi idual eaches a ce ain age can be used. This is known
as he gene a ional app oach o p ojec ing mo ali y imp o emen .
Fo example, i a alua ion is being pe o med as o Janua y 1, 2014, using a mo ali y
able wi h mo ali y a es ep esen a i e o 2014, a p esen alue ac o a age x would
use he ollowing mo ali y a es, whe e he supe sc ip ep esen s he calenda yea in
which he indi idual a ains a gi en age.
𝑞2014𝑥,
𝑞2015𝑥+1 =𝑞2014𝑥+1(1−𝑠𝑐𝑎𝑙𝑒𝑥+1),
𝑞2016𝑥+2 =𝑞2014𝑥+2(1−𝑠𝑐𝑎𝑙𝑒𝑥+2)2,.....,
4. EMPIRICAL ANALYSIS
This sec ion examines how de ined bene i (DB) pension plans would be a ec ed by
unce ain y abou u u e mo ali y and li e expec ancy ou comes. In his ega d, he i s
s ep is o assess he unce ain y su ounding u u e changes in mo ali y and li e
expec ancy, also known as longe i y isk. Second, i conside s he impac o longe i y
isk on de ined bene i (DB) pension plans p o ided by employe s. The link be ween
mo ali y and li e expec ancy, as well as how li e ables a e cons uc ed om mo ali y
da a, is examined in o de o assess he unce ain y su ounding u u e mo ali y and li e
expec ancy ou comes.
Finally, a ocus on he mos p essing issue con on ing pension unds: o ecas ing he
u u e pa h o mo ali y and li e expec ancy in o de o de e mine hei u u e liabili ies.
As a esul , he sec ion p esen s a s ochas ic app oach o modeling mo ali y and li e
expec ancy unce ain y. I p o ides he esul s o es ima ing he Lee-Ca e model o he
selec ed coun y in his ega d. F ance da a was chosen o his s udy o es ima ion and
modeling. The da a o he es ima ion came om he Socie y o Ac ua ies Annui y
Mo ali y da abase and he human mo ali y da abase. The Au ho used R-p og amming
o analysis o he da a.
4.1. Unce ain y Abou Mo ali y and Li e Expec ancy
4.1.1. The ela ionship be ween mo ali y and li e expec ancy: Li e Table
Fo a gi en popula ion, li e ables p o ide a summa y o mo ali y, su i o ship, and li e
expec ancy. They can con ain da a o each and e e y yea o li e (comple e li e ables)
o by 5- o 10-yea in e als (ab idged li e ables). A li e able can be c ea ed in i s mos
basic o m by combining a se o age-speci ic dea h a es. Age-speci ic dea h a es a e
calcula ed as he a io o dea hs in a gi en yea o he popula ion size. They' e usually
exp essed in e ms o people pe 1,000. Mo ali y a es, on he o he hand, a e he chances
ha someone o a speci ic age will die du ing he ime pe iod unde conside a ion (i.e.,
he p obabili y o dying). The nume a o is he numbe o indi iduals om his gene a ion
who die be ween age n and age n+1, and he denomina o is he size o he gene a ion
who each age n du ing he yea in ques ion. The annual dea h a e is di e en om he
annual p obabili y o dying by age because he la e is he p opo ion o people o ha
age who die du ing he yea , whe eas he p obabili y o dying is he p opo ion o people
o ha age dying du ing he age in e al. The e o e, li e ables p o ide a link be ween
mo ali y and li e expec ancy. As a esul , li e ables es ablish a connec ion be ween
mo ali y and li e expec ancy. The mean numbe o yea s s ill o be li ed by a pe son who
has eached ha exac age (i.e., age-speci ic li e expec ancies) i subjec ed o he cu en
age-speci ic p obabili ies o dying o he es o his o he li e is he inal esul o a li e
able. Table 4 shows a li e able o males in F ance 2018. The i s column lis s he
Figu e 7: P ojec ed alue o 𝒌𝒕 o 100 yea s
Fo ecas ing is he main aim behind he s ochas ic modeling. One o he no ewo hy
p ope ies o he LC model is ha , once i is i ed (i.e., once alues o 𝑎𝑥, 𝑏
𝑥, and 𝑘
𝑡 a e
ound), only he mo ali y index (𝑘𝑡) o e ime needs o be o ecas ed o u u e ime
poin s. Lee and Ca e (1992) i ed au o eg essi e in eg a ed mo ing a e age (ARIMA)
(0,1,0) (i.e., andom walk wi h d i ) o modeling mo ali y index o F ench popula ion.
The igu e below shows he u u e p ojec ed alues o 𝑘𝑡𝑠 up o 60yea s.
Figu e 8: Pa e n o Pas and P ojec ed a es o people aged 65
Sou ce: Human Mo ali y Da abase (h p://www.mo ali y.o g/index.h ml).
No es: HMD F ance 5x1 (age by yea ), Au ho Calcula ions

Finally, he en i e a e pa e n is simple o deduce. In his ma ix, pas and p ojec ed a es
a e bo h blinded. We p esen he e a pa e n o pas and p ojec ed a es o people o e he
age o 65 based on di e en popula ions. Figu e 8 clea ly shows he expec ed
imp o emen . This could be a ibu ed o HIV/AIDS Pandemics, disease, and d ugs. We
obse ed ha in nex decades mo ali y is expec ed o decline o bo h emale and male
popula ion in F ance. This is due o dec easing na u e o 𝑘𝑡. We ha e o ecas ed alues
o age speci ic dea h a e, by using es ima ed pa ame e s 𝑎𝑥, 𝑏𝑥 and o ecas ed alues o
mo ali y index 𝑘𝑡.
4.3. The impac o longe i y isk on de ined-bene i p i a e pension plans
The impac o longe i y isk on employe -p o ided DB p i a e pension schemes is
examined in his sec ion. The p e ious sec ion demons a ed ha o ecas ing mo ali y
and li e expec ancy using a s ochas ic app oach allows you o assign p obabili ies o a
a ie y o possible p ojec ions and hence es ima e he unce ain y su ounding u u e
mo ali y and li e expec ancy ou comes. P i a e pension unds, on he o he hand, a e
conce ned abou he impac o his unce ain y on hei pension commi men s. This
sec ion assesses he changes in he ne p esen alue o annui y paymen s as mo ali y and
li e expec ancy e ol es, as his is he p incipal impac o longe i y isk on ne pension
obliga ions. These adjus men s a e assessed o membe s o pension unds o a ious
ages, as well as pension unds wi h a ious age membe ship s uc u es.
4.3.1. How does longe i y isk a ec DB p i a e pension plans?
Longe i y isk has he g ea es in luence on he ne pension liabili ies o employe -
p o ided DB p i a e pension plans because o annui y paymen s. An annui y is a con ac
in which one pe son o o ganiza ion ag ees o pay a s eam o se ies o paymen s o
ano he pe son o o ganiza ion ( he annui an ) (annui y paymen s). Annui ies a e designed
o gi e a cons an s eam o income o he annui an o e a pe iod o ime, which can
begin immedia ely o a any ime in he u u e. Capi al gains and in es men p o i s a e
usually ax-de e ed. The e a e nume ous ypes o annui ies. They can be classi ied in a
a ie y o ways, including: (1) by he unde lying in es men in o ixed o a iable; (2) by
he p ima y pu pose, i.e., accumula ion o pay-ou , in o de e ed o immedia e; (3) by he
na u e o he pay-ou commi men in o ixed pe iod, ixed amoun , o li e ime; and (4) by
he p emium paymen a angemen in o single o lexible p emium. In a ixed annui y,
he insu ance company o pension und gua an ees he p inciple as well as a minimum
a e o in e es , bu in a a iable annui y, he annui y paymen is based on he unde lying
po olio's in es men pe o mance. An immedia e annui y is in ended o pay a lump sum
o a se ies o paymen s immedia ely a e he annui y is pu chased, whe eas a de e ed
annui y pays he annui an a a la e da e. Fixed pe iod annui ies pay an income o a se
leng h o ime (e.g., 10 yea s), whe eas li e ime annui ies pay income o he es o he
annui an 's li e. A single p emium annui y is one ha is unded wi h a single paymen ,
whe eas a lexible p emium annui y is one ha is unded o e a se ies o paymen s. Only
de e ed annui ies a e lexible.
Because employe -p o ided DB p i a e pensions p omise hei membe s a gua an eed
u u e s eam o paymen s a e i emen o he es o hei li es, he esea ch concen a es
on he impac o longe i y isk on ixed, de e ed, li e ime, and lexible p emium
annui ies h oughou . Longe i y isk would ha e a g ea e impac on annui ies ha a e
ixed, de e ed, and o he annui an 's li e ime once e i emen age is achie ed. The
impac o longe i y isk on ixed pe iod annui ies, on he o he hand, is less ob ious.
Fu he mo e, he ex en o he impac o longe i y isk on annui y paymen s would be
de e mined no jus by he ype o annui y gua an ees, bu also by how pension unds
accoun o imp o emen s in mo ali y and li e expec ancy when calcula ing he ne
p esen alue o annui y paymen s.
4.3.2. How p i a e pension unds accoun o u u e imp o emen s in mo ali y
and/o li e expec ancy?
Pension unds do no appea o accoun ully o p ojec ed inc eases in mo ali y and li e
expec ancy. Recen s udy, pa icula ly ha o he Ac ua ial P o ession and Cass Business
School (2005), disco e ed ha cu en p ac ice di e s signi ican ly ac oss he EU.
Pension unds in ce ain coun ies accoun o p edic ed u u e imp o emen s in mo ali y,
whils o he s use ables based on mo ali y eco ded in he pas , wi hou accoun ing o
he possibili y ha li e expec ancy will con inue o ise (Belgium, Denma k, No way,
Sweden, and Swi ze land). O hose coun ies inco po a ing an allowance o u u e
imp o emen s in mo ali y, Aus ia, F ance, Ge many ( o only 25 yea s and using 1996
as he base yea ), I eland (imp o emen s inco po a ed only un il 2010), I aly, he
Ne he lands, Spain, and he Uni ed Kingdom use o ecas s; while Canada, Finland, and
he Uni ed S a es, despi e o ha ing mo ali y ables wi h buil in mechanisms o ake in o
accoun u u e changes in mo ali y, gene ally do no use hem.
Fu he mo e, he e is no s anda dized o consis en mechanism o accoun ing o u u e
inc eases in mo ali y and li e expec ancy. In his aspec , assessing longe i y isk is
challenging due o he lack o a consis en me hodology, which makes mo ali y
p ojec ions a bi a y and impossible o compa e among pension unds, le alone coun ies.
As a esul , he impac o he longe i y isk is ampli ied. The impac o u u e
imp o emen s in mo ali y and li e expec ancy (i.e., longe i y isk) on employe -p o ided
DB p i a e pension plans is compounded by he ac ha ew ac ua ies and pension
schemes accoun o u u e imp o emen s in mo ali y and li e expec ancy, and hose ha
do so only pa ially. Fu he mo e, e en wi h adjus men s o an icipa ed imp o emen s in
mo ali y, he base ables used o demog aphic assump ions a e nea ly en yea s old,
da ing om he ea ly o mid-1990s. Fu he mo e, he lack o s anda d me hods o o ecas
mo ali y and li e expec ancy, and he ac ha hese me hods a e gene ally a om being
ully s ochas ic complica e any compa a i e analysis and make he ask o examining he
impac o longe i y isk on pension und liabili ies uzzie .
Fu he mo e, he lack o s anda d me hods o o ecas ing mo ali y and li e expec ancy,
as well as he ac ha hese me hods a e a om being o ally s ochas ic, complica es
any compa ison s udy and makes he ask o analyzing he impac o longe i y isk on
pension und liabili ies e en mo e hazy.
5. CONCLUSIONS
Li e expec ancy o ecas s a e necessa y o es ima ing u u e heal hca e and pension
cos s. The Lee-Ca e (LC) model (1992), which o ecas s age-speci ic dea h a es log
bilinea ly, is a commonly used model o an icipa e mo ali y. The LC model is employed
because pa ame e es ima ion is simple, and i p o ides a good i o e a wide ange o
ages. The da a collec ion includes da a on F ance's popula ion mo ali y om 1816 o
2018. The pa ame e s o he LC model a e es ima ed using he Singula Value
Decomposi ion (SVD) me hod. The mo ali y alues a e o ecas ed using he Au o
Reg essi e In eg a ed Mo ing A e age (ARIMA) ime se ies model.
We o ecas ed he ime-index using a andom walk wi h d i , which is ypically ound o
be an app op ia e mode (Callo e al. 2016). The o e all pa e n o mo ali y ( 𝑎𝑥) o bo h
emale and male popula ions e ealed high in an mo ali y, an acciden al hump a ound
he age o 20, and a nea ly exponen ial inc ease a olde ages. The sensi i i y o mo ali y
( 𝑏
𝑥) has e ealed ha mo ali y declines a a highe a e o emales aged 25-34 yea s
and males aged 15-24 yea s han o o he ages. The Mo ali y index ( 𝑘
𝑡) has been
declining.
Female mo ali y imp o emen has ou paced male mo ali y as well as he se ies o he
gene al indices clea ly end o dec ease, al hough no mono onically o e ime. Fo he
i s hal o he pe iod, he e is a signi ican inc ease in emale mo ali y o e men, which
dec eases signi ican ly in he second hal o he pe iod. The sensi i i y o mo ali y has
shown ha mo ali y declines a a apid a e o people aged 20 o 25. Since Wo ld Wa I
and Wo ld Wa II, he mo ali y index has shown a dec easing end wi h wo spikes. The
p edic ed Lee Ca e model i s F ance popula ion da a well o e a wide age ange bu
pe o ms poo ly below he age o ou and a e he age o 55.
5.1. Policy issues
Longe i y isk, de ined as he unce ain y su ounding u u e de elopmen s in mo ali y
and li e expec ancy, has a non-negligible impac on he liabili ies o employe -p o ided
pension plans because li e ime annui y paymen s a e based on he leng h o ime people
a e expec ed o li e, acco ding o he pape . The impac o his on he ne p esen alue
o annui y paymen s o a " heo e ical pension und" was calcula ed in Table 5. I was
disco e ed ha he amoun o his in luence is de e mined by he pension und
membe ship's age s uc u e. As a esul , pension unds wi h a younge membe ship
s uc u e will be mo e a ec ed by longe i y isk since hey will be exposed o unce ain
changes in mo ali y and li e expec ancy o a longe pe iod o ime. Un o una ely, he
impac o longe i y isk is agg a a ed by he ac ha ew pension plans accoun o u u e
changes in mo ali y and li e expec ancy, and hose ha do only accoun o pa ial
imp o emen s. To make ma e s wo se, mos pension unds ely on mo ali y ables ha
a e almos a decade old. Fu he mo e, he lack o a consis en echnique o calcula ing
longe i y isk makes de e mining he op imal way o accoun o gains in mo ali y and
li e expec ancy di icul .
Using a common me hodology o p edic dea h a es and li e expec ancy has an ob ious
ad an age in his ega d. This esea ch a gues o he use o a s ochas ic model in his
case because i allows o he a achmen o p obabili ies and consequen ly he assessmen
o he deg ee o unce ain y a ound u u e mo ali y and li e expec ancy ou comes.
Un o una ely, many small and medium-sized pension unds may lack he inancial and
echnical capabili ies o c ea e o ecas s using a s anda dized echnique. Go e nmen
en i ies may be able o de elop hem i hey ha e he necessa y esou ces and echnical
expe ise. Howe e , assump ions abou o al popula ions a he han speci ic membe ship
g oups o p i a e pension plans may no be use ul. Go e nmen al en i ies migh c ea e
o ecas s o he o e all popula ion as well as o a ious subg oups based on gende , age,
weal h, and educa ional a ainmen . As a esul , sepa a e pension unds could use he sub-
popula ion ha mos closely e lec s hei cu en membe ship composi ion.
Using mo ali y ables ha di e en ia e based on socioeconomic posi ion and gende , on
he o he hand, has i s own se o issues because i may gi e ise o disc imina o y issues.
A gumen s in a o o dis inguishing ables include he ac ha adop ing an a e age li e
expec ancy index penalizes pe sons wi h g ea e li e expec ancy (e.g., women, well
educa ed, and well-o people) while ewa ding people wi h lowe li e expec ancy (e.g.,
men, low educa ed and low-income people). Fu he mo e, p i a e pension plans mus
hedge agains hei own longe i y isk, i.e., he isk associa ed wi h hei own membe ship
s uc u e, a he han an a e age longe i y isk.
Finally, in addi ion o inco po a ing mo ali y imp o emen s h ough he adop ion o a
s anda d me hodology and a e age o di e en ia ed mo ali y ables, he impac o
longe i y isk on employe -p o ided DB plans can be mi iga ed in pa by indexing
pension bene i s o li e expec ancy. Indexing bene i s o li e expec ancy, on he o he
hand, mo es some o he longe i y isk back o indi iduals, educing one o he main
easons people buy annui ies. Di e en ia ing be ween indi idual and agg ega e o coho
longe i y isk can be use ul in his ega d. Indi idual isk is unique o each pe son, bu i
can be easily mi iga ed by sha ing isks. As a esul , assuming i by pension unds would
be mo e e icien , as hey a e bes posi ioned o pool indi idual unique isks. On he o he
side, he agg ega e o coho isk is mo e di icul o add ess o mi iga e. As a esul , by
indexing bene i s o coho longe i y changes, his isk can be bo ne mo e easily by
pension unds and people.
5.2. A eas in which addi ional esea ch is equi ed
We made an e o o be ho ough by iden i ying and e iewing li e a u e on he subjec
o longe i y isk. Howe e , mo ali y isk is dynamic, and con inual s udy is equi ed o
ensu e ha he indus y is up o da e on he cu en ends. In ecen yea s, his has
included inc easingly ex ensi e analysis o cha ac e is ics such as sepa a ing li es in o
coho s, ocusing on speci ic causes o dea h as d i e s o mo ali y, and inc easing he

oughness o he isk a iables used in mo ali y in es iga ions. The e is s ill oppo uni y
o mo e complex s udy, which would only se e o be e unde s anding o mo ali y and
longe i y isk p o ile. On he opic o s ochas ic mo ali y models, he e is a lo o
li e a u e, p ima ily om academics. These a e usually conce ned wi h he shape o he
models and how well hey i his o ical da a. One a ea whe e he e is a less in o ma ion
is he discussion abou he p ac ical applica ion o such models.
I would be use ul o see some in-dep h analysis om a company s andpoin o he ela i e
cos s and bene i s o implemen ing s ochas ic mo ali y analysis in a ious s ages o he
p oduc cycle (p icing, ese ing, managing capi al, hedging longe i y isk, and so on)
and ac oss di e en p oduc ca ego ies (payou annui ies, li e se lemen s, e c.). This
could be because insu ance businesses specialize in his sec o , hus all p oduc
ad ancemen s will mos likely o igina e om wi hin he indus y. Insu e s, on he o he
hand, a e equen ly equi ed o sa is y a a ie y o s akeholde s. Sugges ions o new and
unique p oduc concep s could be ascina ing o see
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