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

  Citation: De La Peña, J.I.; FernándezRamos, M.C.; Garayeta, A.; Martín, I.D. Transforming Private Pensions: An Actuarial Model to Face LongTerm Costs. Mathematics 2022,10, 1082. https://doi.org/10.3390/ math10071082 Academic Editor: Anatoliy Swishchuk Received: 15 February 2022 Accepted: 22 March 2022 Published: 28 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Transforming Private Pensions: An Actuarial Model to Face Long-Term Costs J. Iñaki De La Peña 1,* , M. Cristina Fernández-Ramos 2, Asier Garayeta 1and Iratxe D. Martín1 1Financial Economics I Department, Faculty of Business and Economics, University of the Basque Country, 48015 Bilbao, Spain; asier[email protected] (A.G.); [email protected] (I.D.M.) 2Department of Education, Junta Castilla-León, 47011 Valladolid, Spain; [email protected] *Correspondence: [email protected]; Tel.: +34-946-013-876 Abstract: A common response in public pension systems to population ageing is to link pensions to observed longevity. This creates an automatic stabiliser that arises from the valuation of a private actuarially funded system. However, no private pension plan mechanism has been articulated to adapt to this ageing in relation to the increased costs it entails. Private pension plans focus on saving for retirement; capital is accumulated to pay for it. However, perceptions of health status change over time and, as retirement age approaches, concerns about long-term care (LTC) increase. Moreover, there is not enough time to plan for it sufficiently in advance. This paper proposes to incorporate a mechanism to add an allowance to the financial pension (retirement, disability, rotation) to cover LTC within a private defined benefit pension plan, in the case of a pensioner becoming dependent. Depending on a pensioner’s health status, both the expected number of payments and their intensity are transformed. For this purpose, a mechanism is defined (through Markov chains) to adapt the amount of LTC support to a beneficiary’s health-related life expectancy. The study’s main contribution is that it establishes a private pension plan model that offers to incorporate dependency aid through this mechanism into the economic pensions without increasing the total cost of the plan. It adapts to life expectancy according to a person’s state (healthy, disabled, dependent). Keywords: ageing; dependency; long-term care; private pensions MSC: 91G05 1. Introduction Long-term care (LTC) is defined as those expenditures devoted to the care of older people over a period of time [ 1 ]. This care is the direct support for activities of daily living (bathing, dressing, eating, etc.) or through support for instrumental activities (preparing meals, cleaning, managing money, etc.). LTC mainly arises from the loss of autonomy caused by old age [ 2 , 3 ]. The growing number of older people who need healthcare thus increases the financial pressure on healthcare systems [ 4 – 6 ]. At the same time, demand for better access to higher quality services is growing [ 7 ]. To realise the importance, according to [ 8 ], total LTC spending in the United States in 2012 was 8.7% of total health care expenditure (USD 220 billion). Between one-third and one-half of United States retirees needed nursing and care services, and between 10% and 20% of these required it for more than five years [ 9 ]. It is estimated that 43% of the European Union’s population will be over 65 by 2025, reaching 129.8 million inhabitants; this will increase health and care expenditure [10]. This is now an area of special relevance in the scientific literature [11]. Health care systems differ widely between countries and taken reforms have changed the approach to LTC. As a result, some authors [ 12 , 13 ] have classified care systems based on integrated dependency on a welfare state and the relationship with other institutions. This allows LTC to be included gradually into both public and private systems. Mathematics 2022,10, 1082. https://doi.org/10.3390/math10071082 https://www.mdpi.com/journal/mathematics Mathematics 2022,10, 1082 2 of 17 Although dependency costs can be considered as a natural extension of health insurance, dependency is a contingency that should be considered as important as retirement [ 14 ] so that the insured can be protected from the risk of outliving their resources after retirement [ 15 ]. In this sense, the literature proposes linking dependency coverage to pensions to extend its effect [16–18]. LTC coverage should be integrated into an individual’s particular pension strategy, such as retirement [ 1 ], rather than being considered an additional health service [ 14 ]. Financial support for LTC is therefore a logical extension of the purpose of pension plans: to provide an adequate complement to meet retirement needs, no matter the individual’s health status. Pension plans should therefore finance LTC needs [ 19 ]: on the one hand they should provide an income to make up for the lack of a salary and, on the other, should supplement the pension income. In this way, as the individual ages, the needs change; when the baby boom generation reaches an age where LTC is needed, there will be resources to meet them [20]. Reference [ 21 ] claims that dependency coverage is already into retirement planning so that the probability of becoming dependent is already included in the planning itself. Other authors [ 22 , 23 ] argue that dependency and mortality are negatively correlated, thus creating a natural demand for each product. Two affecting factors are undoubtedly age and health status, with a degree of uncertainty as to when the insured will become dependent [ 24 ], although there is evidence of adverse risk selection or favourable hedging: individuals who are more risk averse and take better care, typically live longer and are more likely to underwrite LTC products [25,26]. LTC coverage and retirement should be planned well in advance [ 27 , 28 ] to avoid the effects of demographic and social changes. Thus, in most countries, the elderly population will increase and family caregiving will decrease [ 29 – 34 ]. It will therefore be necessary to provide resources, products and services that are adapted to people´s needs [35]. The problem lies in the financing of dependency. As a solution, some authors [ 36 , 37 ] propose to finance LTC coverage by paying the premium from the retirement benefit. In contrast to this approach, other authors [ 38 , 39 ] proposed that LTC should be integrated within the coverage itself, which adapts the pension to the dependent situation through an actuarial factor that relates the pension to a pensioners’ life expectancy [ 40 , 41 ]. Social security in countries such as Sweden, Norway, Italy, Poland and Latvia employ a mechanism in which individuals receive benefits based on both estimated life expectancy and contributions. This reduces pressure on public resources while tailoring the pension to each generation’s characteristics [ 42 , 43 ]. The inclusion of this resource introduces actuarial rationality to the system [44]. This study is novel because, as there is uncertainty as to when a person will become severely or highly dependent, it proposes that beneficiaries prioritise their resources to pay for LTC. Therefore, this paper aims to establish a financial–actuarial model that transforms private economic resources (pensions) by including a supplement to help pay for LTC at a beneficiary’s request. The inclusion of an actuarial model results in a social contribution by adapting the private plan to a beneficiary’s needs, whether they are a retired or disabled pensioner. To meet this objective, the following section presents an actuarial model for valuing pensions. This model, under differential mortality rates according to a beneficiary’s status, determines the actuarial pension correction factor to be applied, which is based on whether a beneficiary is retired or disabled. The choice of correction factor depends on a beneficiary’s life expectancy (healthy, disabled, severely dependent). The third section shows this factor’s results when applied to the Spanish mortality experience for independent persons as well as disabled and dependent persons. The final section discusses both the results and the proposed actuarial model, and proposes future research recommendations. The main results show that expenditure can be adapted to reality. Even while maintaining the original pension, an additional supplement to cover LTC is achieved. As a final result, a dependent person’s quality of life improves when LTC expenses are partly covered. Mathematics 2022,10, 1082 3 of 17 As [ 45 ] highlighted, a dependent person may live longer by improving their functional environment. 2. Materials and Methods 2.1. The Actuarial Model This paper extends the model initially proposed by [46–48] obtaining complemented model to pay LTC needs for higher degrees of dependency. The initial assumption takes a defined benefit (DB) plan and the individual lacks information about his/her future health status. Therefore, the individual’s contribution history is independent of the future health status: the plan depends solely on the labour career. Moreover, any additional information about their true health status that emerges over time does not really affect the pension at retirement. In contrast, if the plan were a defined contribution (DC) plan, an individual would progressively acquire information about his/her health status and could therefore underwrite a coverage (insurance, annuity, reverse mortgage, etc.) suited to their LTC needs [49–51]. The classification of degrees of dependency depends on institutional factors in each country. In any case, as the degree of dependency increases, more LTC is needed. In fact, milder degrees of dependency may result in disability pensions as they do not allow a worker to perform their usual work. As derived from the study of populations in the United Kingdom [ 52 , 53 ] and Norway [ 54 ], it is also appreciated that mortality for dependent persons is proportional to their age and the level of care needed. It is therefore necessary to focus on the calculation of the probability of a dependent person’s death that limits the duration of LTC payments. Therefore, let Xbe the random variable “age of death of a new born”. Frepresents the death distribution function F(x)=P(X≤x)(1) where x≥0 and F(0)=0. The complementary is the survival function. For each age x, it gives the probability that a new born will reach that age alive. That is ∀x≥0. s(x)=P(X>x)=1−F(x)(2) The derivative function f(x)of the death function F(x)results in f(x)=dF(x) d x =−ds(x) d(x)=−s0(x)(3) being (µx) the instantaneous mortality rate. µx=f(x) 1−F(x)(4) As µx≥0 and f(x)=−s0(x), then µx=−s0(x) s(x)=−d Ln (s(x)) dx (5) The probability that a person of age xwill live tyears or more can therefore be defined as tpx=e−Rx+t xµxdz (6) Likewise, vT is the financial discount factor from the t-th instant to the origin or zero moment, where the financial discount function is defined by the discounting process at the instantaneous rate of interest δ(t). vT=e−RT 0δ(t)dt (7) Mathematics 2022,10, 1082 4 of 17 The present value of the compensation at time t-th will be ZT=bT·vT(8) and both magnitudes bTand vTdepend on the time-to-death. If the survival function ( s(x) ), the payoff function ( bT ) and the financial function ( vT ) are known, it is possible to estimate the present value of future pension benefits (assuming a duration from, e.g., retirement age rto maximum life expectancy age w) or actuarial value at age r(PVFBr) as PVFBr=E(ZT)=E(bT·vT)=Zw rbt·e−Rw rµtdt·e−Rw rδ(t)dt·dt (9) This study assumes that when a beneficiary becomes severely or highly dependent at an intermediate age x, between age rand age w, the pension automatically increases by the factor λd x . As a result, the new amount helps to pay for their LTC. This factor, which increases the pension, is counterbalanced by a dependent person’s inherent mortality difference. At each age there is a different factor that is affected only by the different mortality probabilities: general mortality and dependent´s mortality. When an individual becomes dependent, bx·λd x replaces bx (the pension due at age x). Thus, the pension is automatically increased to provide a higher pension to help to pay for LTC costs. 2.2. The Actuarial Equity Factor This factor means that there is no additional funding. Therefore, at an age x>r, if a beneficiary becomes dependent and decides to transform his/her pension, the present value of future pension benefits has to be equal to the present value of future pension benefits combined with LTC assistance for a dependent person. PVFBx=PVFLTCx(10) PVFBx : Actuarial value of future pensions to be received by an independent pensioner valued at age x, such that x>r. PVFLTCx : Actuarial value of a future pension, including the new LTC aid at age x, such that x>r. The result is the conversion factor at age x, such that x > r. The factor depends on the differential of mortality tables (general versus dependent’s mortality) discounted to the expected return of the pension fund. λd x=Rw xe−Rt+1 tµtdt·e−Rt+1 rδ(t)dt·dt Rw xe−Rt+1 tµd tdt·e−Rt+1 rδ(t)dt·dt =am x dam x (11) e−Rt+1 tµd tdt : Probability that a dependent person of age twill live to age t+ 1 as a dependent person. e−Rt+1 tµtdt : Probability that a person of age twill live to age t+ 1, based on a general mortality table. am x : Actuarial life annuity of a healthy person at age x. It can be variable or constant, depending on whether it is indexed to an external benchmark. dam x : Actuarial life annuity of a dependent person at age x. It can be variable or constant, depending on whether it is indexed to an external benchmark. The resulting LTC complement depends on: •The age of decision making. •A cohort’s expected mortality. •A pension plan’s performance. Mathematics 2022,10, 1082 5 of 17 •A dependent person’s expected mortality. •The amount of pension that a beneficiary receives. With the exception of a dependent person’s expected mortality, all other parameters are standard in the development of a private pension plan. 2.3. The Mortality of Dependent Persons In the literature [ 55 – 58 ], there is consensus that at a given age, the disabled persons’ mortality rate ( iqm x ) is higher than the general population mortality rate ( qm x ); there is unanimity that the dependent persons’ mortality rate ( dqm x ) is different to and higher than the general mortality rate, as shown in the standard mortality tables used by insurers. It is, of course, significantly higher than the mortality rate of insured persons (aq(m) x). dqm x>iqm x>qm x>aq(m) x(12) This paper initially starts from a simplified type of multi-state transition model based on stochastic Markov processes [ 59 , 60 ] describing the probabilities between various states: active worker to retired (both independent and dependent), active worker to disabled (independent or dependent) and to deceased (Figure 1). For an annual period, it is a multistate discrete model, where it is assumed that there can be no more than one transition per year and there are no returns to previous states. Mathematics2022,10,xFORPEERREVIEW5of17   TheresultingLTCcomplementdependson:  Theageofdecisionmaking.  Acohort’sexpectedmortality.  Apensionplan’sperformance.  Adependentperson’sexpectedmortality.  Theamountofpensionthatabeneficiaryreceives. Withtheexceptionofadependentperson’sexpectedmortality,allotherparameters arestandardinthedevelopmentofaprivatepensionplan. 2.3.TheMortalityofDependentPersons Intheliterature[55–58],thereisconsensusthatatagivenage,thedisabledpersons’ mortalityrate(  )ishigherthanthegeneralpopulationmortalityrate( );thereis unanimitythatthedependentpersons’mortalityrate(  )isdifferenttoandhigherthan thegeneralmortalityrate,asshowninthestandardmortalitytablesusedbyinsurers.It is,ofcourse,significantlyhigherthanthemortalityrateofinsuredpersons(  󰇛󰇜).        󰇛󰇜(12) Thispaperinitiallystartsfromasimplifiedtypeofmulti‐statetransitionmodelbased onstochasticMarkovprocesses[59,60]describingtheprobabilitiesbetweenvarious states:activeworkertoretired(bothindependentanddependent),activeworkertodisa‐ bled(independentordependent)andtodeceased(Figure1).Foranannualperiod,itisa multi‐statediscretemodel,whereitisassumedthattherecanbenomorethanonetransi‐ tionperyearandtherearenoreturnstopreviousstates.  Figure1.Transitionprobabilities.Source:Ownwork. Being   󰇛󰇜 :Probabilitythatanactiveworkeragedx+klivesoneyearmoreasactive worker.   󰇛󰇜 :Probabilitythatanactiveworkeragedx+kbecomesdisabledinlessthanone year,exposedtoothercausesofexit(deathandretirement).   󰇛󰇜:Probabilitythatanactiveworkeragedx+kdiesinlessthanoneyear,exposed toothercausesofexit(disabilityandretirement).   󰇛󰇜 :Probabilitythatanactiveworkeragedx+kretiresinlessthanoneyear,ex‐ posedtoothercausesofexit(deathanddisability). Ifx+kisprevioustotheretirementage(x+k<x<r),thefollowingequivalenceis obtained Figure 1. Transition probabilities. Source: Own work. Being ap(a) x+k : Probability that an active worker aged x+klives one year more as active worker. aq(i) x+k : Probability that an active worker aged x+kbecomes disabled in less than one year, exposed to other causes of exit (death and retirement). aq(m) x+k : Probability that an active worker aged x+kdies in less than one year, exposed to other causes of exit (disability and retirement). aq(r) x+k : Probability that an active worker aged x+kretires in less than one year, exposed to other causes of exit (death and disability). If x+kis previous to the retirement age (x+k<x<r), the following equivalence is obtained ap(a) x+k+aq(i) x+k+aq(m) x+k+aq(r) x+k=1 (13) This is true for the whole period of activity. Mathematics 2022,10, 1082 6 of 17 Once an active worker becomes disabled and is receiving a disability pension, for an age x+k, the following is fulfilled ip(i) x+k+iq(m) x+k+iq(d) x+k=1 (14) where: ip(i) x+k: Probability that a disabled person aged x+klives one year more as disabled. iq(m) x+k : Probability that a disabled person aged x+kdies in less than one year, exposed to other causes of exit (dependency). iq(d) x+k : Probability that a disabled person aged x+kbecomes dependent in less than one year, exposed to other causes of exit (death). This is a binomial value (0; 1). Zero value when a beneficiary decides to continue receiving his/her pension and one value when the beneficiary decides to transform the pension. As in the case of disability, from the age of retirement (x>r), a retirement pension is paid if the beneficiary is alive. The beneficiary can apply for a supplement in case of severe dependency. Therefore rp(r) x+k+rq(m) x+k+rq(d) x+k=1 (15) rp(r) x+k : Probability that a retirement pension beneficiary aged x+klives one year more as retired. rq(m) x+k : Probability that a retirement pension beneficiary aged x+kdies in less than one year, exposed to other causes of exit (dependency). rq(d) x+k : Probability that a retirement pension beneficiary aged x+kbecomes dependent in less than one year, exposed to other causes of exit (death). This is a binomial value (0; 1). Zero value when a beneficiary decides to continue receiving his/her retirement pension and a value of one when the beneficiary decides to transform the pension. Finally, to determine dependent persons: dpd x+k : Probability that a retirement pension beneficiary who is dependent aged x+k lives one year more as a dependent person. dqm x+k : Probability that a retirement pension beneficiary who is dependent aged x+k dies in less than one year. Evidently, the sum is unity at age x+k. dpd x+k+dqm x+k=1 (16) If the model applies a factor ( λd x ) to a pension benefit when becoming a dependent person, only the probability of death as a dependent should be determined. 3. The Spanish Experience Results 3.1. Dependency Degrees and Mortality Tables Institutional LTC systems organise dependency according to the degree of severity: from the mildest to the most severe dependence and depending on the number and type of activities of daily living that an individual can perform. The classification has a direct impact on the public aid received; both the classification and amount of aid differs for each country. In Spain, the coverage for the highest levels of dependency is offered [ 61 ] either through insurance products or through pension plans. In fact, dependency coverage [ 62 ] was added as a contingency in a private pension plan and was exclusively for degrees of severe or high dependency [48,63]. The PERM/F 2000 mortality tables for the general population were chosen—specifically for the year 2008 [64]. There is one life table for men and one for women. For the disabled persons group, the Spanish social security actuarial tables for pensioners receiving a life annuity for disability [ 65 ] provide information on the entire population with permanent disability, whether or not they are dependent. All ages from 16 to 108 years Mathematics 2022,10, 1082 7 of 17 are included. The data regarding gender were obtained from the 2008 Spanish population census [66]. 3.2. Mortality of Severely and Highly Dependent Persons In Spain, only the national survey on disability, personal autonomy and dependency [ 67 ] provides data for this group of people. It is impossible to determine degrees of dependency from this survey because it focuses solely on the habits and care of dependent persons. However, there are studies [ 46 ] that have established the life expectancy of an individual suffering from the most severe stages of dependency. In these works, based on an overall mortality, a dependent individual has an excess mortality expressed by a multiplicative correction (θ) dqm x=θ·qm x(17) This correction can be variable at each age, although [ 68 ] indicated that a fixed correction adjusts the mortality of older dependents better than other types of approximations. However, the multiplicative correction tends to overestimate the mortality of a dependent individual at lower ages and underestimate it at higher ages. It is therefore more correct to make an additive adjustment ( ε ) to the overall mortality by considering age as an independent variable in a functional form [69] dqm x=qm x+ε(18) where ε=f(x). Thus, mortality rates are lower at younger ages and are increasing with the level of dependency. For lower dependency levels, no excess mortality applies [70]. Reference [ 71 ] determined the probability of death of severely and highly dependent persons; they used general mortality tables and adjusted them to HID 98-01 statistics for France. They found that excess mortality differentials with respect to overall mortality decreased from age 96 onwards. To capture this effect, they included a variation of the Rickayzen and Walsh formula, which is based on a mixed correction for overall mortality, to model dependent mortality. In this mixed correction, this study’s authors considered an additive modification under the Rickayzen and Walsh expression and a multiplicative correction on the overall mortality numbers that reflects the decline in absolute mortality differentials in older ages. The function is dqm x=(qm x+δ 1+γxi−x∀xi<95 qm x·(1+β)+δ 1+γxi−x∀xi≥95 (19) δ : Maximum value to be incorporated according to the age at which it asymptotically converges. γ: Slope factor xi : Age at the inflection point where the curve changes shape from convex to concave. β: Multiplicative factor on overall mortality. The values obtained with an ordinary least square procedure with respect to the crude values of severe dependency estimated for Spain are given in Table 1. Table 1. Dependent excess mortality factors for the level of severe and high dependency in Spain. Source: [71]. Factors Men Women δ0.245 0.165 γ1.135 1.09 xi62.50 58.61 β0.1142 0.0962 Mathematics 2022,10, 1082 8 of 17 Mortality rates for severely and highly dependent persons are higher than the overall mortality for all ages (Figure 2). Therefore, expression (12) is fulfilled in the Spanish mortality experience. Figure 2shows the different mortality rates for different periods and genders for the 10 years prior to retirement for both men (a) and women (b). In all cases, male mortality values are higher than female mortality values. Mathematics2022,10,xFORPEERREVIEW8of17   FactorsMenWomen 0.2450.165 1.1351.09 62.5058.61 0.11420.0962 Mortalityratesforseverelyandhighlydependentpersonsarehigherthantheoverall mortalityforallages(Figure2).Therefore,expression(12)isfulfilledintheSpanishmor‐ talityexperience.Figure2showsthedifferentmortalityratesfordifferentperiodsand gendersforthe10yearspriortoretirementforbothmen(a)andwomen(b).Inallcases, malemortalityvaluesarehigherthanfemalemortalityvalues.  (a)(b) Figure2.Mortalitydifferentialbystatusandgenderinthelast10yearsofworkactivity.(a)Men; (b)women.Source:Ownwork.Database:PERM/F2000mortalitytablesforthegeneralpopulation [64];Spanishsocialsecurityactuarialtablesfordisability[65];severeandhighdependencymortal‐ itytablesinSpain[71]. Thisinequalitycanalsobeseenintheperiodafterretirement(Figure3)forbothmen (a)andwomen(b).Inthiscase,disabledpersonstendtoreachamortalityrateintheir lateryears,whichisclosetothatofseverelyandhighlydependentpersons.Logically,the degenerationofthehumanbodymakesthemortalitystatusofbothcoincide(Figure3).  (a)(b) Figure3.Mortalitydifferentialbystatusandgenderafterretirementageat65.(a)Men;(b)women. Source:Ownwork.Database:PERM/F2000mortalitytablesforthegeneralpopulation[64];Spanish socialsecurityactuarialtablesfordisability[65];severeandhighdependencymortalitytablesin Spain[71]. Figure 2. Mortality differential by status and gender in the last 10 years of work activity. ( a ) Men; ( b ) women. Source: Own work. Database: PERM/F 2000 mortality tables for the general population [ 64 ]; Spanish social security actuarial tables for disability [ 65 ]; severe and high dependency mortality tables in Spain [71]. This inequality can also be seen in the period after retirement (Figure 3) for both men (a) and women (b). In this case, disabled persons tend to reach a mortality rate in their later years, which is close to that of severely and highly dependent persons. Logically, the degeneration of the human body makes the mortality status of both coincide (Figure 3). Mathematics2022,10,xFORPEERREVIEW8of17   FactorsMenWomen 0.2450.165 1.1351.09 62.5058.61 0.11420.0962 Mortalityratesforseverelyandhighlydependentpersonsarehigherthantheoverall mortalityforallages(Figure2).Therefore,expression(12)isfulfilledintheSpanishmor‐ talityexperience.Figure2showsthedifferentmortalityratesfordifferentperiodsand gendersforthe10yearspriortoretirementforbothmen(a)andwomen(b).Inallcases, malemortalityvaluesarehigherthanfemalemortalityvalues.  (a)(b) Figure2.Mortalitydifferentialbystatusandgenderinthelast10yearsofworkactivity.(a)Men; (b)women.Source:Ownwork.Database:PERM/F2000mortalitytablesforthegeneralpopulation [64];Spanishsocialsecurityactuarialtablesfordisability[65];severeandhighdependencymortal‐ itytablesinSpain[71]. Thisinequalitycanalsobeseenintheperiodafterretirement(Figure3)forbothmen (a)andwomen(b).Inthiscase,disabledpersonstendtoreachamortalityrateintheir lateryears,whichisclosetothatofseverelyandhighlydependentpersons.Logically,the degenerationofthehumanbodymakesthemortalitystatusofbothcoincide(Figure3).  (a)(b) Figure3.Mortalitydifferentialbystatusandgenderafterretirementageat65.(a)Men;(b)women. Source:Ownwork.Database:PERM/F2000mortalitytablesforthegeneralpopulation[64];Spanish socialsecurityactuarialtablesfordisability[65];severeandhighdependencymortalitytablesin Spain[71]. Figure 3. Mortality differential by status and gender after retirement age at 65. ( a ) Men; ( b ) women. Source: Own work. Database: PERM/F 2000 mortality tables for the general population [ 64 ]; Spanish social security actuarial tables for disability [ 65 ]; severe and high dependency mortality tables in Spain [71]. There is a difference in mortality by gender, in all age groups and for the different states. Thus, it can be seen in Figure 4that the percentage of excess mortality in men is much higher. In the 10-year age brackets, a gradual decrease in the excess mortality differential can be seen in all states. In the last years of estimated life, the mortality rate values are almost equal so that excess mortality tends to decrease (d). Mathematics 2022,10, 1082 9 of 17 Mathematics2022,10,xFORPEERREVIEW9of17 Thereisadifferenceinmortalitybygender,inall age groupsandforthedifferent states.Thus,itcanbeseeninFigure4that thepercentage ofexcessmortalityinmenis muchhigher. Inthe10‐year agebrackets, agradualdecreaseinthe excessmortalitydif‐ ferentialcanbeseeninall states.Inthelastyearsofestimated life, themortalityratevalues (a)(b) (c)(d) Figure4.Existing excess mortalityineachstate bygenderandage(a)55to65years;(b)65to75 years;(c)75to85years;(d)85to100years.Source:Ownwork.Database:PERM/F2000mortality tablesforthegeneralpopulation[64];Spanishsocial securityactuarialtablesfordisability [65];se‐ vereandhighdependencymortalitytables inSpain[71]. 3.3. LTCinRetirement Abeneficiaryreceives their retirement pensionaccordingtothe temporarypayment expectationsbasedonthemortalityrate. However, achangeinstatus willnotaffectany accumulatedcapital,butitwillaffect the type ofpensionreceived;theprioritywillnot be toreplacewagesbuttohelpwithLTCexpenses.Therefore, theexpectedpaymentunder thenewcircumstances isaffected,whichleadstoareduction inthe numberofpayments inlinewiththeincreasedprobabilityofdeathasadependent. Tohelptoclarity,Table2referstoretirementoutcomesforpeoplewhowerebornin the1960s.Itnotonly showsthedifferent mortalityratevaluesaccordingtostatus(general versusdependent),butitalsoshowsthevariousfactorstobeappliedwhenanindividual becomesseverely dependent,whichprovidesthevalueofLTCaidandananalysis by gender.Althoughacommondenominatoristheincreasewithageoftheoverallmortality rateforbothmen( )and women( ),inadependentsituationthe sameistrue forthe numberofmen(  )compared tothoseofwomen(  ).However,themortalitygap decreaseswithincreasingageinbothsexes(Δ /  ). Table2.Retirementoutcomesforthe1960sgeneration.Source: Ownwork.Softwareused:Microsoft Excel. MaleMortalityRatesFemaleMortality RatesLTCFactors Figure 4. Existing excess mortality in each state by gender and age ( a ) 55 to 65 years; ( b ) 65 to 75 years; ( c ) 75 to 85 years; ( d ) 85 to 100 years. Source: Own work. Database: PERM/F 2000 mortality tables for the general population [ 64 ]; Spanish social security actuarial tables for disability [ 65 ]; severe and high dependency mortality tables in Spain [71]. 3.3. LTC in Retirement A beneficiary receives their retirement pension according to the temporary payment expectations based on the mortality rate. However, a change in status will not affect any accumulated capital, but it will affect the type of pension received; the priority will not be to replace wages but to help with LTC expenses. Therefore, the expected payment under the new circumstances is affected, which leads to a reduction in the number of payments in line with the increased probability of death as a dependent. To help to clarity, Table 2refers to retirement outcomes for people who were born in the 1960s. It not only shows the different mortality rate values according to status (general versus dependent), but it also shows the various factors to be applied when an individual becomes severely dependent, which provides the value of LTC aid and an analysis by gender. Although a common denominator is the increase with age of the overall mortality rate for both men ( qm x ) and women ( qm y ), in a dependent situation the same is true for the number of men ( dqm x ) compared to those of women ( dqm y ). However, the mortality gap decreases with increasing age in both sexes (∆qm x/dqm x). Analysing the LTC factor, as the individual becomes dependent in younger ages, we obtain higher amounts per euro of retirement pension, for both men ( λd x ) and for women ( λd y ). For men, the values are up to three times higher than the pension they receive in the first years of retirement. Mathematics 2022,10, 1082 16 of 17 22. Murtaugh, C.M.; Spillman, B.C.; Warshawsky, M.J. 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