Transforming Private Pensions: An Actuarial Model to Face Long-Term Costs



Ci a ion: De La Peña, J.I.; Fe nández-
Ramos, M.C.; Ga aye a, A.; Ma ín,
I.D. T ans o ming P i a e Pensions:
An Ac ua ial Model o Face Long-
Te m Cos s. Ma hema ics 2022,10,
1082. h ps://doi.o g/10.3390/
ma h10071082
Academic Edi o : Ana oliy
Swishchuk
Recei ed: 15 Feb ua y 2022
Accep ed: 22 Ma ch 2022
Published: 28 Ma ch 2022
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A ibu ion (CC BY) license (h ps://
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ma hema ics
A icle
T ans o ming P i a e Pensions: An Ac ua ial Model o Face
Long-Te m Cos s
J. Iñaki De La Peña 1,* , M. C is ina Fe nández-Ramos 2, Asie Ga aye a 1and I a xe D. Ma ín1
1Financial Economics I Depa men , Facul y o Business and Economics, Uni e si y o he Basque Coun y,
48015 Bilbao, Spain; asie [email p o ec ed] (A.G.); [email p o ec ed] (I.D.M.)
2Depa men o Educa ion, Jun a Cas illa-León, 47011 Valladolid, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +34-946-013-876
Abs ac :
A common esponse in public pension sys ems o popula ion ageing is o link pensions o
obse ed longe i y. This c ea es an au oma ic s abilise ha a ises om he alua ion o a p i a e
ac ua ially unded sys em. Howe e , no p i a e pension plan mechanism has been a icula ed o
adap o his ageing in ela ion o he inc eased cos s i en ails. P i a e pension plans ocus on
sa ing o e i emen ; capi al is accumula ed o pay o i . Howe e , pe cep ions o heal h s a us
change o e ime and, as e i emen age app oaches, conce ns abou long- e m ca e (LTC) inc ease.
Mo eo e , he e is no enough ime o plan o i su icien ly in ad ance. This pape p oposes o
inco po a e a mechanism o add an allowance o he inancial pension ( e i emen , disabili y, o a ion)
o co e LTC wi hin a p i a e de ined bene i pension plan, in he case o a pensione becoming
dependen . Depending on a pensione ’s heal h s a us, bo h he expec ed numbe o paymen s and
hei in ensi y a e ans o med. Fo his pu pose, a mechanism is de ined ( h ough Ma ko chains) o
adap he amoun o LTC suppo o a bene icia y’s heal h- ela ed li e expec ancy. The s udy’s main
con ibu ion is ha i es ablishes a p i a e pension plan model ha o e s o inco po a e dependency
aid h ough his mechanism in o he economic pensions wi hou inc easing he o al cos o he plan.
I adap s o li e expec ancy acco ding o a pe son’s s a e (heal hy, disabled, dependen ).
Keywo ds: ageing; dependency; long- e m ca e; p i a e pensions
MSC: 91G05
1. In oduc ion
Long- e m ca e (LTC) is de ined as hose expendi u es de o ed o he ca e o olde
people o e a pe iod o ime [
1
]. This ca e is he di ec suppo o ac i i ies o daily li ing
(ba hing, d essing, ea ing, e c.) o h ough suppo o ins umen al ac i i ies (p epa ing
meals, cleaning, managing money, e c.). LTC mainly a ises om he loss o au onomy
caused by old age [
2
,
3
]. The g owing numbe o olde people who need heal hca e hus
inc eases he inancial p essu e on heal hca e sys ems [
4
–
6
]. A he same ime, demand
o be e access o highe quali y se ices is g owing [
7
]. To ealise he impo ance,
acco ding o [
8
], o al LTC spending in he Uni ed S a es in 2012 was 8.7% o o al heal h
ca e expendi u e (USD 220 billion). Be ween one- hi d and one-hal o Uni ed S a es e i ees
needed nu sing and ca e se ices, and be ween 10% and 20% o hese equi ed i o mo e
han i e yea s [
9
]. I is es ima ed ha 43% o he Eu opean Union’s popula ion will be
o e 65 by 2025, eaching 129.8 million inhabi an s; his will inc ease heal h and ca e
expendi u e [10]. This is now an a ea o special ele ance in he scien i ic li e a u e [11].
Heal h ca e sys ems di e widely be ween coun ies and aken e o ms ha e changed
he app oach o LTC. As a esul , some au ho s [
12
,
13
] ha e classi ied ca e sys ems based
on in eg a ed dependency on a wel a e s a e and he ela ionship wi h o he ins i u ions.
This allows LTC o be included g adually in o bo h public and p i a e sys ems.
Ma hema ics 2022,10, 1082. h ps://doi.o g/10.3390/ma h10071082 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 1082 2 o 17
Al hough dependency cos s can be conside ed as a na u al ex ension o heal h insu -
ance, dependency is a con ingency ha should be conside ed as impo an as e i emen [
14
]
so ha he insu ed can be p o ec ed om he isk o ou li ing hei esou ces a e e i e-
men [
15
]. In his sense, he li e a u e p oposes linking dependency co e age o pensions
o ex end i s e ec [16–18].
LTC co e age should be in eg a ed in o an indi idual’s pa icula pension s a egy,
such as e i emen [
1
], a he han being conside ed an addi ional heal h se ice [
14
].
Financial suppo o LTC is he e o e a logical ex ension o he pu pose o pension plans:
o p o ide an adequa e complemen o mee e i emen needs, no ma e he indi idual’s
heal h s a us. Pension plans should he e o e inance LTC needs [
19
]: on he one hand hey
should p o ide an income o make up o he lack o a sala y and, on he o he , should
supplemen he pension income. In his way, as he indi idual ages, he needs change; when
he baby boom gene a ion eaches an age whe e LTC is needed, he e will be esou ces o
mee hem [20].
Re e ence [
21
] claims ha dependency co e age is al eady in o e i emen planning so
ha he p obabili y o becoming dependen is al eady included in he planning i sel . O he
au ho s [
22
,
23
] a gue ha dependency and mo ali y a e nega i ely co ela ed, hus c ea ing
a na u al demand o each p oduc . Two a ec ing ac o s a e undoub edly age and heal h
s a us, wi h a deg ee o unce ain y as o when he insu ed will become dependen [
24
],
al hough he e is e idence o ad e se isk selec ion o a ou able hedging: indi iduals
who a e mo e isk a e se and ake be e ca e, ypically li e longe and a e mo e likely o
unde w i e LTC p oduc s [25,26].
LTC co e age and e i emen should be planned well in ad ance [
27
,
28
] o a oid he
e ec s o demog aphic and social changes. Thus, in mos coun ies, he elde ly popula ion
will inc ease and amily ca egi ing will dec ease [
29
–
34
]. I will he e o e be necessa y o
p o ide esou ces, p oduc s and se ices ha a e adap ed o people´s needs [35].
The p oblem lies in he inancing o dependency. As a solu ion, some au ho s [
36
,
37
]
p opose o inance LTC co e age by paying he p emium om he e i emen bene i . In
con as o his app oach, o he au ho s [
38
,
39
] p oposed ha LTC should be in eg a ed
wi hin he co e age i sel , which adap s he pension o he dependen si ua ion h ough
an ac ua ial ac o ha ela es he pension o a pensione s’ li e expec ancy [
40
,
41
]. Social
secu i y in coun ies such as Sweden, No way, I aly, Poland and La ia employ a mecha-
nism in which indi iduals ecei e bene i s based on bo h es ima ed li e expec ancy and
con ibu ions. This educes p essu e on public esou ces while ailo ing he pension o
each gene a ion’s cha ac e is ics [
42
,
43
]. The inclusion o his esou ce in oduces ac ua ial
a ionali y o he sys em [44].
This s udy is no el because, as he e is unce ain y as o when a pe son will become
se e ely o highly dependen , i p oposes ha bene icia ies p io i ise hei esou ces o pay
o LTC. The e o e, his pape aims o es ablish a inancial–ac ua ial model ha ans o ms
p i a e economic esou ces (pensions) by including a supplemen o help pay o LTC a
a bene icia y’s eques . The inclusion o an ac ua ial model esul s in a social con ibu-
ion by adap ing he p i a e plan o a bene icia y’s needs, whe he hey a e a e i ed o
disabled pensione .
To mee his objec i e, he ollowing sec ion p esen s an ac ua ial model o aluing
pensions. This model, unde di e en ial mo ali y a es acco ding o a bene icia y’s s a us,
de e mines he ac ua ial pension co ec ion ac o o be applied, which is based on whe he
a bene icia y is e i ed o disabled. The choice o co ec ion ac o depends on a bene icia y’s
li e expec ancy (heal hy, disabled, se e ely dependen ). The hi d sec ion shows his ac o ’s
esul s when applied o he Spanish mo ali y expe ience o independen pe sons as well
as disabled and dependen pe sons. The inal sec ion discusses bo h he esul s and he
p oposed ac ua ial model, and p oposes u u e esea ch ecommenda ions.
The main esul s show ha expendi u e can be adap ed o eali y. E en while main-
aining he o iginal pension, an addi ional supplemen o co e LTC is achie ed. As a inal
esul , a dependen pe son’s quali y o li e imp o es when LTC expenses a e pa ly co e ed.
Ma hema ics 2022,10, 1082 3 o 17
As [
45
] highligh ed, a dependen pe son may li e longe by imp o ing hei unc ional
en i onmen .
2. Ma e ials and Me hods
2.1. The Ac ua ial Model
This pape ex ends he model ini ially p oposed by [46–48] ob aining complemen ed
model o pay LTC needs o highe deg ees o dependency. The ini ial assump ion akes a
de ined bene i (DB) plan and he indi idual lacks in o ma ion abou his/he u u e heal h
s a us. The e o e, he indi idual’s con ibu ion his o y is independen o he u u e heal h
s a us: he plan depends solely on he labou ca ee . Mo eo e , any addi ional in o ma ion
abou hei ue heal h s a us ha eme ges o e ime does no eally a ec he pension a
e i emen . In con as , i he plan we e a de ined con ibu ion (DC) plan, an indi idual
would p og essi ely acqui e in o ma ion abou his/he heal h s a us and could he e o e
unde w i e a co e age (insu ance, annui y, e e se mo gage, e c.) sui ed o hei LTC
needs [49–51].
The classi ica ion o deg ees o dependency depends on ins i u ional ac o s in each
coun y. In any case, as he deg ee o dependency inc eases, mo e LTC is needed. In
ac , milde deg ees o dependency may esul in disabili y pensions as hey do no allow
a wo ke o pe o m hei usual wo k. As de i ed om he s udy o popula ions in
he Uni ed Kingdom [
52
,
53
] and No way [
54
], i is also app ecia ed ha mo ali y o
dependen pe sons is p opo ional o hei age and he le el o ca e needed. I is he e o e
necessa y o ocus on he calcula ion o he p obabili y o a dependen pe son’s dea h ha
limi s he du a ion o LTC paymen s.
The e o e, le Xbe he andom a iable “age o dea h o a new bo n”. F ep esen s he
dea h dis ibu ion unc ion
F(x)=P(X≤x)(1)
whe e x≥0 and F(0)=0.
The complemen a y is he su i al unc ion. Fo each age x, i gi es he p obabili y
ha a new bo n will each ha age ali e. Tha is ∀x≥0.
s(x)=P(X>x)=1−F(x)(2)
The de i a i e unc ion (x)o he dea h unc ion F(x) esul s in
(x)=dF(x)
d x =−ds(x)
d(x)=−s0(x)(3)
being (µx) he ins an aneous mo ali y a e.
µx= (x)
1−F(x)(4)
As µx≥0 and (x)=−s0(x), hen
µx=−s0(x)
s(x)=−d Ln (s(x))
dx (5)
The p obabili y ha a pe son o age xwill li e yea s o mo e can he e o e be de ined
as
px=e−Rx+
xµxdz (6)
Likewise,
T
is he inancial discoun ac o om he - h ins an o he o igin o ze o
momen , whe e he inancial discoun unc ion is de ined by he discoun ing p ocess a he
ins an aneous a e o in e es δ( ).
T=e−RT
0δ( )d (7)
Ma hema ics 2022,10, 1082 4 o 17
The p esen alue o he compensa ion a ime - h will be
ZT=bT· T(8)
and bo h magni udes bTand Tdepend on he ime- o-dea h.
I he su i al unc ion (
s(x)
), he payo unc ion (
bT
) and he inancial unc ion (
T
)
a e known, i is possible o es ima e he p esen alue o u u e pension bene i s (assuming
a du a ion om, e.g., e i emen age o maximum li e expec ancy age w) o ac ua ial alue
a age (PVFB ) as
PVFB =E(ZT)=E(bT· T)=Zw
b ·e−Rw
µ d ·e−Rw
δ( )d ·d (9)
This s udy assumes ha when a bene icia y becomes se e ely o highly dependen
a an in e media e age x, be ween age and age w, he pension au oma ically inc eases
by he ac o
λd
x
. As a esul , he new amoun helps o pay o hei LTC. This ac o ,
which inc eases he pension, is coun e balanced by a dependen pe son’s inhe en mo ali y
di e ence. A each age he e is a di e en ac o ha is a ec ed only by he di e en
mo ali y p obabili ies: gene al mo ali y and dependen ´s mo ali y.
When an indi idual becomes dependen ,
bx·λd
x
eplaces
bx
( he pension due a age x).
Thus, he pension is au oma ically inc eased o p o ide a highe pension o help o pay o
LTC cos s.
2.2. The Ac ua ial Equi y Fac o
This ac o means ha he e is no addi ional unding. The e o e, a an age x> , i
a bene icia y becomes dependen and decides o ans o m his/he pension, he p esen
alue o u u e pension bene i s has o be equal o he p esen alue o u u e pension
bene i s combined wi h LTC assis ance o a dependen pe son.
PVFBx=PVFLTCx(10)
PVFBx
: Ac ua ial alue o u u e pensions o be ecei ed by an independen pensione
alued a age x, such ha x> .
PVFLTCx
: Ac ua ial alue o a u u e pension, including he new LTC aid a age x,
such ha x> .
The esul is he con e sion ac o a age x, such ha x > . The ac o depends on he
di e en ial o mo ali y ables (gene al e sus dependen ’s mo ali y) discoun ed o he
expec ed e u n o he pension und.
λd
x=Rw
xe−R +1
µ d ·e−R +1
δ( )d ·d
Rw
xe−R +1
µd
d ·e−R +1
δ( )d ·d
=am
x
dam
x
(11)
e−R +1
µd
d
: P obabili y ha a dependen pe son o age will li e o age + 1 as a
dependen pe son.
e−R +1
µ d
: P obabili y ha a pe son o age will li e o age + 1, based on a gene al
mo ali y able.
am
x
: Ac ua ial li e annui y o a heal hy pe son a age x. I can be a iable o cons an ,
depending on whe he i is indexed o an ex e nal benchma k.
dam
x
: Ac ua ial li e annui y o a dependen pe son a age x. I can be a iable o
cons an , depending on whe he i is indexed o an ex e nal benchma k.
The esul ing LTC complemen depends on:
•The age o decision making.
•A coho ’s expec ed mo ali y.
•A pension plan’s pe o mance.
Ma hema ics 2022,10, 1082 5 o 17
•A dependen pe son’s expec ed mo ali y.
•The amoun o pension ha a bene icia y ecei es.
Wi h he excep ion o a dependen pe son’s expec ed mo ali y, all o he pa ame e s
a e s anda d in he de elopmen o a p i a e pension plan.
2.3. The Mo ali y o Dependen Pe sons
In he li e a u e [
55
–
58
], he e is consensus ha a a gi en age, he disabled pe sons’
mo ali y a e (
iqm
x
) is highe han he gene al popula ion mo ali y a e (
qm
x
); he e is
unanimi y ha he dependen pe sons’ mo ali y a e (
dqm
x
) is di e en o and highe han
he gene al mo ali y a e, as shown in he s anda d mo ali y ables used by insu e s. I is,
o cou se, signi ican ly highe han he mo ali y a e o insu ed pe sons (aq(m)
x).
dqm
x>iqm
x>qm
x>aq(m)
x(12)
This pape ini ially s a s om a simpli ied ype o mul i-s a e ansi ion model based
on s ochas ic Ma ko p ocesses [
59
,
60
] desc ibing he p obabili ies be ween a ious s a es:
ac i e wo ke o e i ed (bo h independen and dependen ), ac i e wo ke o disabled
(independen o dependen ) and o deceased (Figu e 1). Fo an annual pe iod, i is a mul i-
s a e disc e e model, whe e i is assumed ha he e can be no mo e han one ansi ion pe
yea and he e a e no e u ns o p e ious s a es.
Ma hema ics2022,10,xFORPEERREVIEW5o 17


The esul ingLTCcomplemen dependson:
 Theageo decisionmaking.
 Acoho ’sexpec edmo ali y.
 Apensionplan’spe o mance.
 Adependen pe son’sexpec edmo ali y.
 Theamoun o pension ha abene icia y ecei es.
Wi h heexcep iono adependen pe son’sexpec edmo ali y,allo he pa ame e s
a es anda din hede elopmen o ap i a epensionplan.
2.3.TheMo ali yo Dependen Pe sons
In heli e a u e[55–58], he eisconsensus ha a agi enage, hedisabledpe sons’
mo ali y a e(

)ishighe  han hegene alpopula ionmo ali y a e(
); he eis
unanimi y ha  hedependen pe sons’mo ali y a e(

)isdi e en  oandhighe  han
hegene almo ali y a e,asshownin hes anda dmo ali y ablesusedbyinsu e s.I 
is,o cou se,signi ican lyhighe  han hemo ali y a eo insu edpe sons(

󰇛󰇜).







󰇛󰇜(12)
Thispape ini iallys a s omasimpli ied ypeo mul i‐s a e ansi ionmodelbased
ons ochas icMa ko p ocesses[59,60]desc ibing hep obabili iesbe ween a ious
s a es:ac i ewo ke  o e i ed(bo hindependen anddependen ),ac i ewo ke  odisa‐
bled(independen o dependen )and odeceased(Figu e1).Fo anannualpe iod,i isa
mul i‐s a edisc e emodel,whe ei isassumed ha  he ecanbenomo e hanone ansi‐
ionpe yea and he ea eno e u ns op e iouss a es.

Figu e1.T ansi ionp obabili ies.Sou ce:Ownwo k.
Being


󰇛󰇜 :P obabili y ha anac i ewo ke agedx+kli esoneyea mo easac i e
wo ke .


󰇛󰇜 :P obabili y ha anac i ewo ke agedx+kbecomesdisabledinless hanone
yea ,exposed oo he causeso exi (dea hand e i emen ).


󰇛󰇜:P obabili y ha anac i ewo ke agedx+kdiesinless hanoneyea ,exposed
oo he causeso exi (disabili yand e i emen ).


󰇛󰇜 :P obabili y ha anac i ewo ke agedx+k e i esinless hanoneyea ,ex‐
posed oo he causeso exi (dea handdisabili y).
I x+kisp e ious o he e i emen age(x+k<x< ), he ollowingequi alenceis
ob ained
Figu e 1. T ansi ion p obabili ies. Sou ce: Own wo k.
Being
ap(a)
x+k
: P obabili y ha an ac i e wo ke aged x+kli es one yea mo e as ac i e
wo ke .
aq(i)
x+k
: P obabili y ha an ac i e wo ke aged x+kbecomes disabled in less han one
yea , exposed o o he causes o exi (dea h and e i emen ).
aq(m)
x+k
: P obabili y ha an ac i e wo ke aged x+kdies in less han one yea , exposed
o o he causes o exi (disabili y and e i emen ).
aq( )
x+k
: P obabili y ha an ac i e wo ke aged x+k e i es in less han one yea , exposed
o o he causes o exi (dea h and disabili y).
I x+kis p e ious o he e i emen age (x+k<x< ), he ollowing equi alence is
ob ained
ap(a)
x+k+aq(i)
x+k+aq(m)
x+k+aq( )
x+k=1 (13)
This is ue o he whole pe iod o ac i i y.

Ma hema ics 2022,10, 1082 6 o 17
Once an ac i e wo ke becomes disabled and is ecei ing a disabili y pension, o an
age x+k, he ollowing is ul illed
ip(i)
x+k+iq(m)
x+k+iq(d)
x+k=1 (14)
whe e:
ip(i)
x+k: P obabili y ha a disabled pe son aged x+kli es one yea mo e as disabled.
iq(m)
x+k
: P obabili y ha a disabled pe son aged x+kdies in less han one yea , exposed
o o he causes o exi (dependency).
iq(d)
x+k
: P obabili y ha a disabled pe son aged x+kbecomes dependen in less han
one yea , exposed o o he causes o exi (dea h). This is a binomial alue (0; 1). Ze o alue
when a bene icia y decides o con inue ecei ing his/he pension and one alue when he
bene icia y decides o ans o m he pension.
As in he case o disabili y, om he age o e i emen (x> ), a e i emen pension is
paid i he bene icia y is ali e. The bene icia y can apply o a supplemen in case o se e e
dependency. The e o e
p( )
x+k+ q(m)
x+k+ q(d)
x+k=1 (15)
p( )
x+k
: P obabili y ha a e i emen pension bene icia y aged x+kli es one yea mo e
as e i ed.
q(m)
x+k
: P obabili y ha a e i emen pension bene icia y aged x+kdies in less han one
yea , exposed o o he causes o exi (dependency).
q(d)
x+k
: P obabili y ha a e i emen pension bene icia y aged x+kbecomes dependen
in less han one yea , exposed o o he causes o exi (dea h). This is a binomial alue (0; 1).
Ze o alue when a bene icia y decides o con inue ecei ing his/he e i emen pension
and a alue o one when he bene icia y decides o ans o m he pension.
Finally, o de e mine dependen pe sons:
dpd
x+k
: P obabili y ha a e i emen pension bene icia y who is dependen aged x+k
li es one yea mo e as a dependen pe son.
dqm
x+k
: P obabili y ha a e i emen pension bene icia y who is dependen aged x+k
dies in less han one yea .
E iden ly, he sum is uni y a age x+k.
dpd
x+k+dqm
x+k=1 (16)
I he model applies a ac o (
λd
x
) o a pension bene i when becoming a dependen
pe son, only he p obabili y o dea h as a dependen should be de e mined.
3. The Spanish Expe ience Resul s
3.1. Dependency Deg ees and Mo ali y Tables
Ins i u ional LTC sys ems o ganise dependency acco ding o he deg ee o se e i y:
om he mildes o he mos se e e dependence and depending on he numbe and ype
o ac i i ies o daily li ing ha an indi idual can pe o m. The classi ica ion has a di ec
impac on he public aid ecei ed; bo h he classi ica ion and amoun o aid di e s o each
coun y.
In Spain, he co e age o he highes le els o dependency is o e ed [
61
] ei he
h ough insu ance p oduc s o h ough pension plans. In ac , dependency co e age [
62
]
was added as a con ingency in a p i a e pension plan and was exclusi ely o deg ees o
se e e o high dependency [48,63].
The PERM/F 2000 mo ali y ables o he gene al popula ion we e chosen—speci ically
o he yea 2008 [64]. The e is one li e able o men and one o women.
Fo he disabled pe sons g oup, he Spanish social secu i y ac ua ial ables o pension-
e s ecei ing a li e annui y o disabili y [
65
] p o ide in o ma ion on he en i e popula ion
wi h pe manen disabili y, whe he o no hey a e dependen . All ages om 16 o 108 yea s
Ma hema ics 2022,10, 1082 7 o 17
a e included. The da a ega ding gende we e ob ained om he 2008 Spanish popula ion
census [66].
3.2. Mo ali y o Se e ely and Highly Dependen Pe sons
In Spain, only he na ional su ey on disabili y, pe sonal au onomy and depen-
dency [
67
] p o ides da a o his g oup o people. I is impossible o de e mine deg ees o
dependency om his su ey because i ocuses solely on he habi s and ca e o dependen
pe sons. Howe e , he e a e s udies [
46
] ha ha e es ablished he li e expec ancy o an
indi idual su e ing om he mos se e e s ages o dependency. In hese wo ks, based
on an o e all mo ali y, a dependen indi idual has an excess mo ali y exp essed by a
mul iplica i e co ec ion (θ)
dqm
x=θ·qm
x(17)
This co ec ion can be a iable a each age, al hough [
68
] indica ed ha a ixed co ec-
ion adjus s he mo ali y o olde dependen s be e han o he ypes o app oxima ions.
Howe e , he mul iplica i e co ec ion ends o o e es ima e he mo ali y o a depen-
den indi idual a lowe ages and unde es ima e i a highe ages. I is he e o e mo e
co ec o make an addi i e adjus men (
ε
) o he o e all mo ali y by conside ing age as an
independen a iable in a unc ional o m [69]
dqm
x=qm
x+ε(18)
whe e ε= (x).
Thus, mo ali y a es a e lowe a younge ages and a e inc easing wi h he le el o
dependency. Fo lowe dependency le els, no excess mo ali y applies [70].
Re e ence [
71
] de e mined he p obabili y o dea h o se e ely and highly dependen
pe sons; hey used gene al mo ali y ables and adjus ed hem o HID 98-01 s a is ics
o F ance.
They ound ha excess mo ali y di e en ials wi h espec o o e all mo ali y de-
c eased om age 96 onwa ds. To cap u e his e ec , hey included a a ia ion o he
Rickayzen and Walsh o mula, which is based on a mixed co ec ion o o e all mo ali y,
o model dependen mo ali y. In his mixed co ec ion, his s udy’s au ho s conside ed
an addi i e modi ica ion unde he Rickayzen and Walsh exp ession and a mul iplica i e
co ec ion on he o e all mo ali y numbe s ha e lec s he decline in absolu e mo ali y
di e en ials in olde ages. The unc ion is
dqm
x=(qm
x+δ
1+γxi−x∀xi<95
qm
x·(1+β)+δ
1+γxi−x∀xi≥95 (19)
δ
: Maximum alue o be inco po a ed acco ding o he age a which i asymp o ically
con e ges.
γ: Slope ac o
xi
: Age a he in lec ion poin whe e he cu e changes shape om con ex o conca e.
β: Mul iplica i e ac o on o e all mo ali y.
The alues ob ained wi h an o dina y leas squa e p ocedu e wi h espec o he c ude
alues o se e e dependency es ima ed o Spain a e gi en in Table 1.
Table 1.
Dependen excess mo ali y ac o s o he le el o se e e and high dependency in Spain.
Sou ce: [71].
Fac o s Men Women
δ0.245 0.165
γ1.135 1.09
xi62.50 58.61
β0.1142 0.0962
Ma hema ics 2022,10, 1082 8 o 17
Mo ali y a es o se e ely and highly dependen pe sons a e highe han he o e all
mo ali y o all ages (Figu e 2). The e o e, exp ession (12) is ul illed in he Spanish
mo ali y expe ience. Figu e 2shows he di e en mo ali y a es o di e en pe iods and
gende s o he 10 yea s p io o e i emen o bo h men (a) and women (b). In all cases,
male mo ali y alues a e highe han emale mo ali y alues.
Ma hema ics2022,10,xFORPEERREVIEW8o 17


Fac o sMenWomen
0.2450.165
1.1351.09
62.5058.61
0.11420.0962
Mo ali y a es o se e elyandhighlydependen pe sonsa ehighe  han heo e all
mo ali y o allages(Figu e2).The e o e,exp ession(12)is ul illedin heSpanishmo ‐
ali yexpe ience.Figu e2shows hedi e en mo ali y a es o di e en pe iodsand
gende s o  he10yea sp io  o e i emen  o bo hmen(a)andwomen(b).Inallcases,
malemo ali y aluesa ehighe  han emalemo ali y alues.

(a)(b)
Figu e2.Mo ali ydi e en ialbys a usandgende in helas 10yea so wo kac i i y.(a)Men;
(b)women.Sou ce:Ownwo k.Da abase:PERM/F2000mo ali y ables o  hegene alpopula ion
[64];Spanishsocialsecu i yac ua ial ables o disabili y[65];se e eandhighdependencymo al‐
i y ablesinSpain[71].
Thisinequali ycanalsobeseenin hepe ioda e  e i emen (Figu e3) o bo hmen
(a)andwomen(b).In hiscase,disabledpe sons end o eachamo ali y a ein hei 
la e yea s,whichisclose o ha o se e elyandhighlydependen pe sons.Logically, he
degene a iono  hehumanbodymakes hemo ali ys a uso bo hcoincide(Figu e3).

(a)(b)
Figu e3.Mo ali ydi e en ialbys a usandgende a e  e i emen agea 65.(a)Men;(b)women.
Sou ce:Ownwo k.Da abase:PERM/F2000mo ali y ables o  hegene alpopula ion[64];Spanish
socialsecu i yac ua ial ables o disabili y[65];se e eandhighdependencymo ali y ablesin
Spain[71].
Figu e 2.
Mo ali y di e en ial by s a us and gende in he las 10 yea s o wo k ac i i y. (
a
) Men;
(
b
) women. Sou ce: Own wo k. Da abase: PERM/F 2000 mo ali y ables o he gene al popu-
la ion [
64
]; Spanish social secu i y ac ua ial ables o disabili y [
65
]; se e e and high dependency
mo ali y ables in Spain [71].
This inequali y can also be seen in he pe iod a e e i emen (Figu e 3) o bo h men
(a) and women (b). In his case, disabled pe sons end o each a mo ali y a e in hei
la e yea s, which is close o ha o se e ely and highly dependen pe sons. Logically, he
degene a ion o he human body makes he mo ali y s a us o bo h coincide (Figu e 3).
Ma hema ics2022,10,xFORPEERREVIEW8o 17


Fac o sMenWomen
0.2450.165
1.1351.09
62.5058.61
0.11420.0962
Mo ali y a es o se e elyandhighlydependen pe sonsa ehighe  han heo e all
mo ali y o allages(Figu e2).The e o e,exp ession(12)is ul illedin heSpanishmo ‐
ali yexpe ience.Figu e2shows hedi e en mo ali y a es o di e en pe iodsand
gende s o  he10yea sp io  o e i emen  o bo hmen(a)andwomen(b).Inallcases,
malemo ali y aluesa ehighe  han emalemo ali y alues.

(a)(b)
Figu e2.Mo ali ydi e en ialbys a usandgende in helas 10yea so wo kac i i y.(a)Men;
(b)women.Sou ce:Ownwo k.Da abase:PERM/F2000mo ali y ables o  hegene alpopula ion
[64];Spanishsocialsecu i yac ua ial ables o disabili y[65];se e eandhighdependencymo al‐
i y ablesinSpain[71].
Thisinequali ycanalsobeseenin hepe ioda e  e i emen (Figu e3) o bo hmen
(a)andwomen(b).In hiscase,disabledpe sons end o eachamo ali y a ein hei 
la e yea s,whichisclose o ha o se e elyandhighlydependen pe sons.Logically, he
degene a iono  hehumanbodymakes hemo ali ys a uso bo hcoincide(Figu e3).

(a)(b)
Figu e3.Mo ali ydi e en ialbys a usandgende a e  e i emen agea 65.(a)Men;(b)women.
Sou ce:Ownwo k.Da abase:PERM/F2000mo ali y ables o  hegene alpopula ion[64];Spanish
socialsecu i yac ua ial ables o disabili y[65];se e eandhighdependencymo ali y ablesin
Spain[71].
Figu e 3.
Mo ali y di e en ial by s a us and gende a e e i emen age a 65. (
a
) Men; (
b
) women.
Sou ce: Own wo k. Da abase: PERM/F 2000 mo ali y ables o he gene al popula ion [
64
]; Spanish
social secu i y ac ua ial ables o disabili y [
65
]; se e e and high dependency mo ali y ables in
Spain [71].
The e is a di e ence in mo ali y by gende , in all age g oups and o he di e en
s a es. Thus, i can be seen in Figu e 4 ha he pe cen age o excess mo ali y in men
is much highe . In he 10-yea age b acke s, a g adual dec ease in he excess mo ali y
di e en ial can be seen in all s a es. In he las yea s o es ima ed li e, he mo ali y a e
alues a e almos equal so ha excess mo ali y ends o dec ease (d).
Ma hema ics 2022,10, 1082 9 o 17
Ma hema ics2022,10,xFORPEERREVIEW9o 17
The eisadi e enceinmo ali ybygende ,inall age g oupsand o  hedi e en
s a es.Thus,i canbeseeninFigu e4 ha hepe cen age o excessmo ali yinmenis
muchhighe . In he10‐yea ageb acke s, ag adualdec easein he excessmo ali ydi ‐
e en ialcanbeseeninall s a es.In helas yea so es ima ed li e, hemo ali y a e alues
(a)(b)
(c)(d)
Figu e4.Exis ing excess mo ali yineachs a e bygende andage(a)55 o65yea s;(b)65 o75
yea s;(c)75 o85yea s;(d)85 o100yea s.Sou ce:Ownwo k.Da abase:PERM/F2000mo ali y
ables o  hegene alpopula ion[64];Spanishsocial secu i yac ua ial ables o disabili y [65];se‐
e eandhighdependencymo ali y ables inSpain[71].
3.3. LTCinRe i emen 
Abene icia y ecei es hei e i emen pensionacco ding o he empo a ypaymen
expec a ionsbasedon hemo ali y a e. Howe e , achangeins a us willno a ec any
accumula edcapi al,bu i willa ec he ype o pension ecei ed; hep io i ywillno be
o eplacewagesbu  ohelpwi hLTCexpenses.The e o e, heexpec edpaymen unde 
henewci cums ances isa ec ed,whichleads oa educ ion in he numbe o paymen s
inlinewi h heinc easedp obabili yo dea hasadependen .
Tohelp ocla i y,Table2 e e s o e i emen ou comes o peoplewhowe ebo nin
he1960s.I no only shows hedi e en mo ali y a e aluesacco ding os a us(gene al
e susdependen ),bu i alsoshows he a ious ac o s obeappliedwhenanindi idual
becomesse e ely dependen ,whichp o ides he alueo LTCaidandananalysis by
gende .Al houghacommondenomina o is heinc easewi hageo  heo e allmo ali y
a e o bo hmen(
)and women(
),inadependen si ua ion he sameis ue o  he
numbe o men(

)compa ed o hoseo women(

).Howe e , hemo ali ygap
dec easeswi hinc easingageinbo hsexes(Δ
/

).
Table2.Re i emen ou comes o  he1960sgene a ion.Sou ce: Ownwo k.So wa eused:Mic oso 
Excel.
MaleMo ali yRa esFemaleMo ali y Ra esLTCFac o s
Figu e 4.
Exis ing excess mo ali y in each s a e by gende and age (
a
) 55 o 65 yea s; (
b
) 65 o 75 yea s;
(
c
) 75 o 85 yea s; (
d
) 85 o 100 yea s. Sou ce: Own wo k. Da abase: PERM/F 2000 mo ali y ables
o he gene al popula ion [
64
]; Spanish social secu i y ac ua ial ables o disabili y [
65
]; se e e and
high dependency mo ali y ables in Spain [71].
3.3. LTC in Re i emen
A bene icia y ecei es hei e i emen pension acco ding o he empo a y paymen
expec a ions based on he mo ali y a e. Howe e , a change in s a us will no a ec any
accumula ed capi al, bu i will a ec he ype o pension ecei ed; he p io i y will no be
o eplace wages bu o help wi h LTC expenses. The e o e, he expec ed paymen unde
he new ci cums ances is a ec ed, which leads o a educ ion in he numbe o paymen s in
line wi h he inc eased p obabili y o dea h as a dependen .
To help o cla i y, Table 2 e e s o e i emen ou comes o people who we e bo n in
he 1960s. I no only shows he di e en mo ali y a e alues acco ding o s a us (gene al
e sus dependen ), bu i also shows he a ious ac o s o be applied when an indi idual
becomes se e ely dependen , which p o ides he alue o LTC aid and an analysis by
gende . Al hough a common denomina o is he inc ease wi h age o he o e all mo ali y
a e o bo h men (
qm
x
) and women (
qm
y
), in a dependen si ua ion he same is ue o he
numbe o men (
dqm
x
) compa ed o hose o women (
dqm
y
). Howe e , he mo ali y gap
dec eases wi h inc easing age in bo h sexes (∆qm
x/dqm
x).
Analysing he LTC ac o , as he indi idual becomes dependen in younge ages, we
ob ain highe amoun s pe eu o o e i emen pension, o bo h men (
λd
x
) and o women
(
λd
y
). Fo men, he alues a e up o h ee imes highe han he pension hey ecei e in he
i s yea s o e i emen .
Ma hema ics 2022,10, 1082 16 o 17
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