Appendix A. Technical Approach for Estimating the Accrued-to-Date Pension Entitlements Included in Table 29
It is crucial to acknowledge that the technical development presented in this appendix is predicated on the authors’ convictions regarding the optimal methodology for calculating pension liabilities. This differs from the approach outlined in some technical documents that have been cited as the basis for pension calculations in various countries.
The development presented in this appendix is based on final earnings schemes. In such schemes, pensions are calculated on the basis of final earnings before retirement, as implied by the term. These final earnings may comprise earnings over the entire contribution career or over a shorter period. For notional defined contribution accounts (NDCs) and flat-rate pension schemes, the formulas to calculate the liability to contributors would be different.
In the event that individualised data on contributors and pensioners are not available at the reference date for calculating pension liabilities, which seems unlikely, particularly in more developed countries, the representative contributor approach should be used.
Two main steps are required to estimate accrued-to-date pension liabilities. In the initial step, average pension entitlements for different age and gender groups are calculated. In a second step, these group-specific pension rights are multiplied by the respective age- and gender-specific cohort sizes of the pension scheme members. Step 2 generates the final result, namely the total value of pension entitlements.
In order to accurately assess pension liabilities, it is essential to differentiate between the pension entitlements of current pensioners (liability to pensioners) and those of current contributors (liability to contributors). It is important not to forget that pensioners have already accumulated the full pension benefits to which they are entitled, whereas contributors have only partially accrued their future full pension benefits.
In order to develop the actuarial formulae for the assessment of pension liabilities, it is necessary to ascertain the following transition probabilities, in which no more than one transition in a year is assumed, and we also assume that participants’ lives could last years, where is the maximum lifespan and is the current age of the individual:
is the probability that an active aged of gender in the base year will reach age being active in the year
In general, the values for are to be found in the interval [0, ] and for the year in the interval [, .
is the probability that an active aged of gender in the base year will reach age being disabled.
is the probability that an active aged of gender in the base year will die before reaching age .
is the probability that a disabled person aged of gender in the base year will reach age in the same state in the year . We do not take conversions or recoveries into account, that is conversion and recovery rates are null in our model.
is the probability that a disabled person aged of gender in the base year will die during the year .
Obviously, under the assumption of non-negative probabilities,
The above probabilities are valid for contributors aged
; being
the normal retirement age. For all types of pensioners, the following probabilities also hold:
The yearly probability of dying for the general population for a given group aged
of gender
in the year
can be calculated as a weighted average of the probabilities of dying for both collectives (active and disabled people), the weighting being the disability and active prevalence rates:
Formula (A3) implies that the following probabilities also hold:
where
: Disability prevalence rate for the group aged of gender in the year , which is the ratio between the number of disabled persons and the total population (active + disabled persons) aged .
: Probability that a person (active) aged of gender will reach age in the same state or as a disabled person.
It is well documented in the literature (
Pitacco, 2019;
Ventura-Marco & Vidal-Meliá, 2016) that the mortality of disabled people contains an “extra-mortality” term and can be represented either as a specific mortality (using the appropriate numerical tables or parametric mortality laws) or via adjustments to the standard age pattern of mortality. The “extra-mortality” term is a challenging concept to model, and its overestimation could have significant implications. If the mortality of disabled individuals is underestimated, it would result in an underestimation of the disability liabilities, as disability benefits are considered “living” benefits. Conversely, if the mortality of dependent individuals is overestimated, it would lead to an overestimation of the disability liabilities.
The pensioners at the reference date have already accrued the total of their pension entitlements. Consequently, the present value of the accrued-to-date pension entitlements is equal to the present value of all the pension payments that they would receive until the termination of their pension entitlements, which is generally when the individual dies.
The liability to retirement pensioners can be expressed as follows:
Expected benefits to be paid to the retirement pensioners Being the average direct pension entitlement of a pensioner aged and of gender in the base year , is the average annual pension benefit accrued-to-date for the group of retirement pensioners aged of gender in a future year (), is the probability of a retirement pensioner aged of gender (g) surviving to age in a future year (), is an indexation factor which depends on , the rate on indexation on pensions in payment, and , the annual nominal discount rate and is the maximum lifetime.
Pension entitlements represent the expected amount of pension payments accrued to date. In accordance with this concept, future survival rates are taken into account for the calculation of entitlements.
The element denotes the average annual pension level accrued to date. It is crucial to highlight that only future pension payments are considered in the estimation of pension entitlements. All payments made during or before the base year are excluded. Consequently, the control variable commences with the age , which is one year after the base year, and concludes at age (). In this context, as previously stated, the parameter denotes the highest maximum lifetime considered in the calculations.
The parameter
is used to denote the future annual adjustment factor, which reflects the indexation rules in a given country:
is the annuity factor, that is, the present value of a lifetime annuity valued in the year for the retirement pensioner aged () years of 1 monetary unit per year payable at the end of the year and growing at a nominal rate , and with a nominal interest rate equal to . Unless otherwise stated, all actuarial values are computed using these rates and under the assumption that all payments are made regularly at the end of the year.
The annuity factor is calculated in accordance with the methodology employed in the construction of generational tables, which incorporate projected mortality trends. This methodology is not optimal for calculating the liability to pensioners. It would be more advantageous to use periodic tables for a specific cohort of pensioners (
Arnold et al., 2019).
In order to calculate entitlements, it may be advisable to include the indexation practice of the respective pension scheme in the estimation process. This may be achieved by considering those forms of indexation that are referred to as “price indexation” or “wage indexation”.
Besides indexation rules, future pension levels of current retirees might also be altered due to future adjustments in the benefit formula. It would be prudent to consider the potential impact of future alterations to the benefit formula when projecting future pension benefits for current retirees.
Expected benefits to be paid to other beneficiaries (survivor benefits)
In the event of death, survivors’ pensions may be payable to partners or children of old-age and disability pensioners. A widowhood pension is a pension granted to the spouses of disabled and retired workers and pensioners on their death. In general, the widowhood pension is compatible with the retirement pension (or permanent disability) to which one was entitled, provided that the maximum amount is not exceeded.
where
is the probability of a representative individual aged , legal partner of (primary beneficiary) of gender () (initially, the opposite gender of g) surviving to age () in a future year (). Survivor’s pensions may be limited to married people only or may be extended to common law partners.
The age of the retirement pensioner in the base year is denoted by and that of the secondary beneficiary by ; in this specific case, the annuitant (male) and the co-annuitant (female), but obviously other couple combinations are possible.
is the fraction of the benefit paid to the widow(er) or secondary beneficiary.
is the present value of a lifetime annuity valued in year for the beneficiary aged years of 1 monetary unit per year payable at the end of each year.
is the present value of a lifetime annuity valued in year while both partners of the couple remain alive of 1 monetary unit per year payable at the end of each year.
is the probability that the beneficiary, aged () years, will have a spouse at the time of death.
In short, the general formula for calculating the liability to retirement pensioners can be expressed as follows:
is the present value of a life annuity with contingent survivor benefit. With this type of life annuity, a periodic payment at the end of the year is made to the primary beneficiary with an initial age of years, which he/she receives until his/her death. From this moment his/her legal partner, with an initial age of years, assuming she/he has survived until this date, will start to receive an amount calculated as a percentage ( of what the deceased primary beneficiary was receiving.
The methodology used in this instance is analogous to that used in the case of retirement pensioners. However, it is necessary to consider the specific mortality of this particular group of pensioners. Disabled people have a lower life expectancy than active people, but the difference in longevity tends to decrease notably as the individuals increase in age (
Pitacco, 2019;
Ventura-Marco & Vidal-Meliá, 2016,
2014).
In order to calculate the total liability to disability pensioners, it is necessary to apply the general formula, which can be expressed as follows:
The elements of the above formula have the same meaning as those of retirement, with the exception that the superscript indicates disability pensioners, which implies the use of the specific mortality tables for the disabled in the case of direct benefits.
It is crucial to highlight that the disability benefit is contingent upon the degree of disability ascribed to the pensioner (in Spain, there are three degrees: total, absolute and severe disability). It would have been possible to develop a separate formula for each of the recognised degrees of disability.
In the case of widowhood and orphanhood pension schemes and pensions in favour of family members, the entitlement is terminated in certain cases once a certain age has been reached. Therefore, it would be necessary to distinguish between the group of pensions in force at the reference date, which are of a lifelong nature (mainly widowhood and special cases for orphanhood and in favour of family members), and the rest (temporary benefits).
It is evident that the most significant liability is the provision of lifetime annuities to widows and other dependents,
:
where
is the average annual pension benefit for the group of widowhood pensioners aged of gender in the base year (,
is the average annual pension benefit for the group of orphanhood pensioners aged of gender in the base year (,
is the average annual pension benefit for the group of family member pensioners aged of gender in the base year (.
It is important to consider the value of temporary pensions,
, in the event that the age of the beneficiary is below the maximum age for receiving the orphan’s pension (
) or the family benefit (
):
where
is the present value of a temporary annuity valued in year for an orphanhood pensioner aged years of 1 monetary unit per year. As indicated by the subscript, the payments cease at the end of years or, if sooner, at the time of the pensioner’s death.
is the present value of a temporary annuity valued in year for a family member pensioner aged years of 1 monetary unit per year. As indicated by the subscript, the payments cease at the end of years or, if sooner, at the time of the pensioner’s death.
A further distinctive feature is that the death of the beneficiary does not entail the possibility of a total or partial reversal of benefits, except for the case of total orphanhood.
We need to estimate the average pension entitlements for a group of contributors aged
, of gender
, who are likely to retire at the age of
in a future year
):
where
is the expected average annual pension benefits for the group of contributors aged , of gender , who is likely to retire at the age of in a future year ). The determination of this element is based on the projected benefit obligation (PBO) approach, which incorporates salary increases over time.
The traditional definition of the normal retirement age is considered, that is, the first age at which retirement can occur without any reduction in the benefits calculated according to the retirement benefit formula.
In general, there are two approaches to calculating the initial benefit to which a current contributor is entitled (
Eurostat, 2020). One approach assumes that all contributors have a homogeneous contribution career (homCC). This approach differs from previous methods in that future pension levels for current contributors are not estimated using past contribution data. Instead, future retirement benefits are approximated based on current pension levels. This approach has the advantage of significantly limiting the input data required for estimations, allowing for partially disregarding data on past contributions.
In the absence of or in the case of limited contribution data, the homCC approach is recommended. Alternatively, the heterogeneous contribution careers (hetCC) approach may be considered. This approach requires comprehensive data on past contributions. The hetCC approach has the advantage of reflecting cohort-specific employment careers. Consequently, its application may result in more accurate estimations than the homCC approach.
The average liability for the group of contributors aged
, of gender
, who are likely to retire at the age of
in a future year
) valued in the base year
can be calculated as follows:
is the accrual factor (prorate accrual factor), defined as the proportion of the total pension entitlement that has been accrued by a beneficiary of gender
and age
at the end of year
, who is likely to retire at the age of
in a future year
). The quotient is calculated by dividing the number of years contributing to the pension until the reference year by the expected total number of years contributing until the acquisition of the condition of pensioner (or disabled person, in the case of a disability scheme).
being
the age of entry into the system.
is the gender--specific probability that a contributor aged years will reach age being active in a future year ). The contributor could reach age in a state of disability in a future year ). This includes the decreases due to death and disability associated with ages at the interval time , with no possibility of a return to active life being considered.
However, where possible, heterogeneous retirement ages should be taken into account.
The presence of disability among active contributors precludes their eligibility for retirement benefits. Consequently, the associated costs are reduced. However, the extent of the reduction may be greater or less than the cost of disability, depending on the legislative benefits associated with the disability pension.
The average liability by disability for the group of contributors aged
, of gender
, who are likely to retire at the age of
) in a future year
) valued in the base year
can be calculated as follows:
where
is the expected average annual disability benefits for the group of contributors aged of gender in a future year () if disability occurs during age .
is the gender--specific probability that a contributor aged years will reach age being active in a future year ).
is the probability that a contributor of gender ( aged ( will enter into a state of disability this year ().
is the probability that a contributor of gender aged , initially active, reaches age in the disability state in a future year ). It can occur because the individual remains active for the next ( years, becomes disabled at age ( and then lives as an invalid for the number of years remaining in the interval .
is the present value of 1 monetary unit per year payable in advance of a life annuity with contingent survivor benefit. With this type of life annuity, a periodic payment is made to the primary annuitant (disabled, with the initial age of years), which he/she receives until his/her death. From this moment his legal partner (active, with the initial age of years), assuming she/he has survived until this date, will start to receive an amount calculated as a percentage ( of what the deceased annuitant was receiving.
is the present value of a lifetime annuity for the disabled person aged years of 1 monetary unit per year payable in advance and growing at a nominal rate , and with a nominal interest rate equal to . This annuity is based on disabled life mortality.
is the present value of a lifetime annuity while both members of the couple remain alive of 1 monetary unit payable in advance. The primary annuitant is disabled, with an initial age of years, and the legal partner is active, with the initial age of years.
As previously stated in the context of the liability to disability pensioners, it would have been possible to develop a separate formula for each of the recognised degrees of disability in the case of the liability to contributors.
Mortality among contributors eliminates the retirement benefit obligation, but mortality prior to retirement may create another form of pension obligation; for instance, the commencement of periodic payments to a surviving spouse for life.
The average liability by survivorship contingencies (before retirement) for the group of contributors aged
, of gender
, who are likely to retire at the age of
) in a future year
) valued in the base year
can be calculated as follows:
where
is the expected average annual survivor’s benefits for the group of contributors aged of gender in a future year () if the death occurs during age .
is the probability that an active contributor of initial age will die in the course of year or at age () is the so-called deferred probability of death.
Having described the simplified approach to calculating average pension entitlements for the several groups of pensioners and current contributors, it is now necessary to consider the number of pensioners () and contributors () by age and gender at the end of year . This will enable the total pension liabilities of the system to be calculated.
The total liability to pensioners at the end of year
is calculated as follows:
for
, males and
females. While the total liability to contributors at the end of year
is
Appendix B. Main Formulas Used to Transform Table 29 into an ABS and Compile Its Associated Income Statement (IS)
This Technical appendix shows the main formulas used to transform Table 29 into an ABS and compile its associated income statement (IS).
The change in net worth can (
be detailed as follows:
1. Change in the fund/financial asset (
):
where
is the income from sponsor contributions for non-contributory rights (NCRs),
is total pension disbursements,
is other outflows,
is the net return on funded capital, and
is the value of the change in financial loans.
2. Change in the total contribution asset (
):
where
is the total contribution asset (contribution asset plus public contribution asset; see Formulas (A28) and (A29)),
is the total income from contributions (income from sponsor contributions for non-contributory rights plus employer’s and employee’s actual social contributions), and
is the turnover duration (see Formula (A28)).
3. Change in pension liability (
):
where
is the employer’s and employee’s actual social contributions plus the property income earned or imputed in the schemes,
is other (actuarial) changes of pension entitlements in social security pension schemes,
is the change in entitlements due to negotiated changes in the scheme’s structure,
is changes in entitlements due to revaluations, and
is changes in entitlements due to other changes in volume.
4. Change in financial liabilities:
where
is the value of the change in financial loans,
is loans for the period, and
is the payment of loans for the period.
5. In the case of Spain, the (new) pension liability is estimated based on the values published by the INE (
), for the base scenario and the alternative scenarios 1 and 2, taking into account the change
between the new discount rate (
) and the original value used for estimating pension entitlements (
), and the change
between the new indexation rate of pensions in payment (
) and the original value (
). It can be expressed as follows:
where
and
.
A two-stage numerical derivation procedure and a sensitivity analysis are applied from a polynomial adjustment of the values published by the INE for each of the different scenarios taken from Table 29 (
Garvey et al., 2023).
Our procedure is superior to
Rauh’s (
2017) method of approximating the change in value of a liability when the discount interest rate changes. We calculate the change in value of the liability simultaneously for two parameters (the discount rate and the indexation rate of pensions in payment), rather than relying solely on the duration and convexity of the pension entitlements.
These are parameters that allow for an approximation of the change in value of a liability when the interest rate used to discount that liability is changed. Measures of the duration and convexity of pension benefits help to understand the impact of changes in discount rates on the present value of liabilities (
Novy-Marx & Rauh, 2011).
Table 29 disclosures oblige countries to reveal pension liabilities (
) based on different discount rate assumptions: 1 percentage point higher (
) and 1 percentage point lower (
). The duration (
), expressed in years, can be calculated as follows:
The convexity, (
) can be calculated as follows:
Duration is not a perfect measure of a liability’s change in value because it assumes a linear relationship when in reality it exhibits a sloped or “convex” shape. A liability has positive convexity if its duration increases as the discount rate declines. A liability with positive convexity will experience larger increases in value due to a decline in discount rates rather than decreases in value due to an increase in discount rates.
To estimate the value of the liability, taking into account the change
between the new discount rate (
) and the original value used for estimating pension entitlements (
), is calculated as follows:
As previously mentioned, we must calculate the change in liability value for two parameters simultaneously. Therefore, we should use Formula (A24) instead of Formula (A27).
6. The contribution asset (
):
where
is the income from ordinary contributions,
is the turnover duration,
is the weighted average age of the pensioners, and
is the weighted average age of the contributors.
To preserve solvency, the asset counterpart that underwrites the liabilities caused by SBs (the commitments deriving from supplementing benefits (SBs) to achieve the minimum pension benefit (MPB) in force) can appear as a special type of public contribution asset (), financed by general government revenues.
It is also straightforward to obtain the value of the system’s public contribution asset. It is the product of the system’s turnover duration and the value of the public contributions made in the period by the sponsor in order to pay for the minimum pension benefits (MPBs) in force:
where
is the income from sponsor contributions for SBs.
7. The system’s solvency indicator is the ratio between its assets and liabilities (
):
where
is the contribution asset,
is the buffer fund,
is the public contribution asset,
is the liability to contributors,
is the liability to pensioners,
is the liability to contributors due to non-contributory rights (NCRs),
is the liability to pensioners for NCRs, and
is the amount of the financial loans.
It is denoted as primary
if the public contribution asset is excluded (
). At the valuation date the balance ratio value will be 1 if the net worth is zero or greater.
A solvency ratio of 1 indicates that a system has enough assets to meet all scheduled accrued benefits payable, while a ratio of 0.5 means that only half of the promised benefits can be covered.
8. Another valuable indicator of the system’s sustainability is the required growth rate (
), which actuarially matches the pension system’s liabilities with its assets at the valuation date:
To put it another way, this is the expected discount rate necessary to achieve the system’s solvency (
). The required growth rate is denoted as primary (
) if the public contribution asset is excluded from the calculation.