Case Study A.1

The Pension Benefit Guaranty Corporation (PBGC) in the United States charges a flat annual fee of $37 per participant in a multi-employer pension plan in 2024, rising to $52 in 2031 (PBG, 2024, Table M-16). For single-employer plans the flat part of the annual fee is $101 per participant in 2024, with an additional variable fee of up to $686 per participant for unfunded vested liabilities (PBG, 2024, Table S-39). To minimize payments to the PBGC, a pension plan therefore buys annuities to close out the liabilities for the individuals with the smallest pensions. The pensions that are bought out are treated like commutations, with the observation end date being the date of annuity purchase.

A continuous-time model can include the bought-out annuities, as they will be like Case C in Figure 1. Note that the PBGC fee is per participant, so there is a financial benefit from identifying multiple records relating to the same person among retained cases; see the discussion of deduplication in Macdonald et al. (Reference Macdonald, Richards and Currie2018), Section 2.5.

Case Study A.2

A consulting actuary is asked by a reinsurer to set a pricing basis for a longevity swap. However, the Kaplan-Meier curves for males and females exhibit a pattern like that of Figure 12(b), suggesting that the data are corrupted. The consultant informs the reinsurer, which asks the cedant for better-quality data. The cedant refuses, and the consultant recommends that the reinsurer walks away from the transaction, which it does.

Case Study A.3

A consulting actuary is asked by the sponsoring employer of a pension scheme to validate the mortality basis used for funding calculations. The consulting actuary calculates the Kaplan-Meier curves for males and females and, like Case Study A.2, finds a pattern like that of Figure 12(b), suggesting that the data are corrupted. The third-party pension-scheme administrator is ordered to implement a data clean-up programme, which produces better data and thus allows a reliable basis to be derived.

Case Study A.4

The insurer of the annuities in Figure 3 needs to set a mortality basis for the year-end valuation in 2015. A simple multi-year comparison against a standard table is inappropriate, as Figure 3(b) shows that the transferred annuities are not a random sample of the overall portfolio. A $q$ -type analysis is complicated by the exit from observation of 60,000 annuities during 2013. A better solution is a $\mu $ -type analysis, using the date of the last annuity payment on the system as a proxy for the date of right-censoring. This allows both transferred and remaining annuities to receive the same treatment as Cases A and B in Figure 1. Such a solution works best for annuities paid relatively frequently, e.g. monthly.

Case Study A.5

An actuary is tasked with setting a mortality basis for a large annuity portfolio. Eighteen months before the actuary’s appointment, annuities in payment were migrated to a new administration platform and records of prior deaths were lost. The actuary only has one-and-a-half years of experience data. Annuities set up before the migration are therefore like Cases D and E in Figure 1, whereas newer annuities set up post-migration are like Cases A and B.

The portfolio has scale in terms of the number of annuities, but it has no depth in terms of historic data. Using an annual $q$ -type GLM makes this depth problem even worse: the requirement for a complete year of potential exposure means discarding a third of what little historical data is available. The most recent half-year of experience would be discarded in order not to have mortality levels under-stated due to delays in reporting of deaths. Much of the new business written would therefore also be excluded from $q$ -type analysis, robbing the actuary of information on possible selection effects.

In contrast, a continuous-time solution is to model ${\mu _x}$ and use all eighteen months of data, including the new business. Delays in reporting of recent deaths could be handled either by using a parametric OBNR term (Richards, Reference Richards2022b) or using the approach of Richards (Reference Richards2022c), as illustrated in Figure A1. In the latter case, the level of mortality to use for a long-term purpose like valuation would have to be decided by judgement, say by picking the modelled (and thus smoothed) mortality level mid-way between a winter peak and a summer trough.

Figure A1.

Mortality multiplier over time for French annuity portfolio, with multipliers standardized at 1 for October 2009. The data extract was taken in mid-2010, and the impact of reporting delays (OBNR) is very pronounced.

Source: Own calculations using flexible B-spline basis for mortality levels described in Richards (Reference Richards2022c).

Case Study A.6

A pricing analyst is looking for risk factors to use for annuitant mortality. One of the data fields available is whether an annuity has an attaching spouse’s benefit, i.e. a contingent annuity that commences on death of the first life. Statistical modelling suggests that the presence or absence of a spouse’s benefit is highly significant, and has a mortality effect on a par with the sex of the annuitant. The direction of the effect is plausible, with single-life annuitants exhibiting higher mortality (Ben-Shlomo et al., Reference Ben-Shlomo, Smith, Shipley and Marmot1993).

However, the analyst is aware of the potential for record corruption from Case Study A.2. Specifically, if the main annuitant dies and the spouse is then discovered to have pre-deceased them, the record of the contingent benefit might be removed during death processing. This would make the annuity look like a single-life annuity from outset, and thus distort the apparent mortality differential. The analyst emails the IT department with his concerns and is told that they are unfounded. Sceptical, he makes a further telephone enquiry and is again assured that the feared corruption is not present.

Believing that he may have found an important new risk factor for pricing, the analyst writes a note to the chief actuary. The analyst then remembers that these quality assurances come from the same IT department that lost all the mortality data in Case Study A.5. The analyst performs a third investigation, and discovers that his original suspicions were correct. The excess mortality for single-life annuitants has been grossly exaggerated by the deletion of pre-deceased spouse records.

Case Study A.7

An insurer with a large annuity portfolio has an administration system that automatically deletes records of annuitants who died more than two years in the past. Annuities are set up like Cases A and B in Figure 1, but after two years a Case A annuity becomes like Case D, while Case B annuities simply disappear from the administration system entirely. An urgent request is made to stop this rolling deletion of valuable data.

Similar to Case Study A.5, an annual $q$ -type GLM only permits the use of a single year of data – the most recent six months are ignored due to late-reported deaths, leaving 1.5 years of data, but a further half-year of unaffected data also has to be discarded to meet the requirement of only using complete years.

As in Case Study A.5, a continuous-time solution is to model ${\mu _x}$ and use all two years of data. Delays in reporting of recent deaths could be handled either by using a parametric OBNR term (Richards, Reference Richards2022b) or using the approach of Richards (Reference Richards2022c), as illustrated in Figures 6 and A1. As in Case Study A.5, the level of mortality would be chosen by judgement, such as a modelled value from spring or autumn.

Case Study A.8

A UK insurer has historically written annuities based solely on the experience of its own maturing money-purchase pension policies. The insurer decides to enter the open market for non-underwritten annuities. However, there is a parallel open market for underwritten annuities, where better terms are offered to lives with demonstrable health conditions (Ainslie, Reference Ainslie2000). The insurer is concerned that the existence of the underwritten market creates selection effects in the new open-market standard annuities that it is writing. The experience data of annuitants from its own maturing pension policies are not believed to have such selection effects, so the new mortality pricing basis contains some guesswork (Richards and Robinson, Reference Richards and Robinson2005).

At the end of the first year of the new strategy the insurer wants to urgently (i) validate its pricing assumptions with respect to selection and (ii) set a valuation mortality basis for the new open-market annuities. This cannot be done particularly cleanly using ${q_x}$ models, as a full year of exposure has not yet been achieved for the new business. In contrast, a continuous-time analysis based on ${\mu _x}$ can easily handle the fractional exposures of the annuities written during the year, with no special modifications required; see Richards (Reference Richards2020), Section 6 for an example.

See also the related discussion around Figure 14.

Case Study A.9

A medium-sized UK pension scheme provides data for analysis. Inspection of the data suggests that temporary and commuted pensions have been erroneously marked as deaths; see Figure A2, where almost all 20 year-olds are allegedly die. This indicates that the data have not been properly processed. In particular, the mortality of recipients of small pensions may be considerably over-stated due to commuted cases being misclassified as deaths.

Case Study A.10

A Canadian pension plan will not provide full dates of death, as Canadian privacy laws apply to those who have died in the past twenty years (PIP, 2018). Instead, only the year and month of death are provided.

Since we only know the month in which death occurs, this is interval censoring (Macdonald, Reference Macdonald1996, Section 2.3). A continuous-time model can be built assuming that deaths take place on the 15 $^{{\rm{th}}}$ of the month. The distortion will be small, with the actual date of death being on average only a week away from this assumption (and with average deviations being close to zero). Note, however, that the tracking statistic of Richards (Reference Richards2022b) will be affected by this loss of information, and the Kaplan-Meier survival curve will have a more ridged appearance than would be the case with exact dates of death.

Case Study A.11

An insurer is thinly capitalized with a small solvency margin. In the hope of lowering its capital requirements, the insurer asks for reinsurance quotations for its large annuity portfolio. However, even the cheapest reinsurance quotation has a mortality basis that is stronger than the insurer’s existing reserving basis. This would require the insurer to pay an up-front premium, which it cannot afford.

While the insurer can of course decline to reinsure, it cannot now leave its annuitant mortality basis unchanged. It has just discovered the market price for its annuitant mortality, and must now strengthen its mortality basis.

Case Study A.12

An insurer with a large annuity portfolio needs to plan staffing levels. In particular, the work processing death notifications is often upsetting and stressful, and limits are in place for the amount of time an individual member of staff can be assigned to this task. The insurer uses a mortality model with a season term (see Section 4.4) to make short-term forecasts of likely deaths, thus informing staffing requirements and rotas.

Case Study A.13

An insurer has assumed the annuity liabilities of another insurer. In the UK this is a court-sanctioned process known as a Part VII transfer (FSM, 2000, PartVII). The IT department sets up the transferred annuities on the administration system with the original commencement date. This makes it difficult to separately identify the transferred annuities, which is problematic because there are periods of survival between commencement and transfer without corresponding deaths.

A continuous-time solution is to use the date of the first annuity payment on the new system as a proxy for the transfer-in date. This allows the new liabilities to be treated like Cases D and E in Figure 1. As in Case Study A.4, such a solution works best for annuities paid relatively frequently, e.g. monthly.

Case Study A.14

In contrast to Case Study A.13, a system migration takes place where the commencement date on the new system is the date of migration, not the original date of retirement (which is then lost).

The migrated cases look like Cases A and B in Figure 1, but in reality they are like Cases D and E. The exposure period can be calculated, but no modelling of duration or selection effects can take place due to not having the actual retirement date.

Case Study A.15

An insurer runs a programme to identify unreported annuitant deaths. Instead of using the actual date of death, the insurer processes the discovered cases en masse using the date of processing, resulting in a large spike in deaths recorded for that particular date.

Here the survival times are unreliable for the mass-processed cases. Superficially, it might seem preferable to use a model that specifically discards precise survival times, i.e. a $q$ -type analysis. However, this is only likely to be true if such exercises are carried out sufficiently regularly that the inaccuracy of the date of death is modest. There is a strong financial incentive to run regular existence-verification exercises, say quarterly, as each identified case will release reserves. For this reason it is particularly beneficial to run such exercises prior to the year-end statutory valuation.

Case Study A.16

A UK insurer has written a lot of bulk annuity business and thus has significant liabilities for both pensions in payment and deferred pensions. The actuary is tasked with setting a basis for both sub-types of business, which are administered on the same computer system.

The occurred-but-not-reported (OBNR) deaths for pensions in payment is a relatively contained problem. While there are reporting delays, they can be measured and allowed for, as described in Section 4.3. Part of the reason for this is that the insurer has a constant feedback mechanism – most UK pension payments are made via automated bank credit (BACS Direct Credit, 2019), and there is a specific rejection code for an attempt to pay money into the account of a decedent (ARUCS2 – Beneficiary Deceased).

However, the situation for deferred pensions is different – with no premiums to collect or payments to make, the reporting of deaths of deferred pensioners is typically sparse with very long delays. It is not uncommon for the insurer to only find out about a past death when it tries to make the first pension payment at retirement age. It is therefore common for the mortality data for deferred liabilities to be unreliable, and for a basis to be derived without reference to experience data.

The situation for deaths of deferred pensions can sometimes be improved by the regular use of third-party services, which provide matches to known deaths of varying degrees of confidence based on name, date of birth and last known address. However, it is important to use such external data thoughtfully, as clumsy application can lead to bad publicity (Tims, Reference Tims2024).

Case Study A.17

The German multi-employer pension scheme in Richards et al. (Reference Richards, Kaufhold and Rosenbusch2013) stores only the year of death. The data collected by the pension scheme have been designed for $q$ -type analysis only. The exact date of death could be obtained from another source, but the third-party administrator demands payment for bespoke work not covered by the service contract. An additional issue is that pensioners dying during the calendar year of retirement are not recorded at all, being neither in the in-force at the start of the year nor in the deaths.

If this were an insurer concerned about anti-selection, as in Case Study A.8, then it might be worth paying for the exact dates of death for a continuous-time model. However, for a pension scheme with slowly changing membership, the cost was not felt to be justified. A ${q_x}$ model could be built with the data, or else a ${\mu _x}$ model assuming that all deaths occur in the middle of the year. Two further consequences of not knowing the date of death are that (i) the tracking statistic of Richards (Reference Richards2022b) cannot be used, and (ii) the Kaplan-Meier survival curves will take a pronounced step shape, as exact ages of death are not known.

Case Study A.18

An annuity portfolio has been administered continuously since outset on the same computer system without any record deletion, archiving of historic deaths or migration onto or off the system. All annuity records are like Cases A, B and C in Figure 1, and it would be possible to model mortality along these lines if (i) mortality levels were stable, (ii) mortality differentials were stable and (iii) selection effects were stable. However, in practice these things change over time, so most organizations restrict their modelling to the most recent experience, such as the past five years. This means that older annuities are deliberately turned from Cases A and B into Cases D and E, respectively (and very old Cases B and C are excluded entirely). See also Figure 11 for an illustration of the application of such an investigation “window.”

Note that such continuity of experience can also be found with some very large, self-administered pension schemes. The outsourcing of pension-scheme administration is invariably bad news for the mortality experience data, especially changes of administrator.

Case Study A.19

An analyst at a UK insurer is directed to use the annual valuation extracts for analysis. The extracts are policy-orientated and have been validated for year-end statutory balance sheet calculations. However, the valuation extracts contain neither names nor postcodes, so the analyst cannot perform the deduplication necessary for statistical modelling (Macdonald et al., Reference Macdonald, Richards and Currie2018, Section 2.5). The absence of postcodes further means that geodemographic mortality modelling cannot be done either (Macdonald et al., Reference Macdonald, Richards and Currie2018, Section 2.8.1). Finally, the annual “snapshot” nature of the valuation extracts means that policies that commence and terminate during the calendar year are not included, thus hampering the modelling of anti-selection.

The analyst therefore arranges for a direct extract of policy records from the administration system. Also, since mortality models routinely use geodemographic profiles (Madrigal et al., Reference Madrigal, Matthews, Patel, Gaches and Baxter2011), a request is made to add postcodes to future valuation extracts.

Case Study A.20

Many valuation systems are heavily restricted in the mortality risk factors they can cope with. It is common to have the flexibility to upload a new mortality table of ${q_x}$ rates, but this typically limits the calculations to using just age and sex as risk factors. Beyond this, such systems sometimes also allow the application of a multiplier to the ${q_x}$ rates for portfolio-specific customization. This is a problem for modern mortality analysis, which typically uses many more risk factors than just age and sex – see the list in Section 1.

One solution is the equivalent-reserve method popularized by Willets (Reference Willets1999). This involves calculating an annuity factor using the multi-factor mortality model, then finding the percentage of the given mortality table that produces the same value. This is done for various risk factor combinations in Richards et al. (Reference Richards, Kaufhold and Rosenbusch2013), Table 13. However, this approach can be extended to the entire portfolio valuation, thus automatically weighting the mortality risk factors by their impact on the liabilities. The resulting table percentages can then be fed into the valuation system. An important point to note is that financial distortions occur if the resulting table percentage falls far outside the 85–115% range. If this is the case, then it is crucially important to change the reference table so that the percentages are closer to 100%.

Case Study A.21

A US insurer is bidding for a multi-billion dollar buy-out of a large pension plan. The proposed transaction is so large that on its own it will meet the insurer’s new business target for this type of risk. The pricing committee is both keen to win the business, but also worried about the financial impact of possibly getting the mortality basis wrong. A request is made for a statistically rigorous confidence interval around the best estimate of current mortality in the pension plan (the future-improvement assumption is set centrally at the corporate level).

The pricing basis was derived using an individual survival model calibrated to the pension plan’s experience data. This allowed the analysts to use the methodology of Richards (Reference Richards2016) to measure the sensitivity of the liability value to the uncertainty surrounding the parameter estimates. A process of simulating alternative parameter estimates consistent with the covariance matrix produced a thousand alternative valuations of the liabilities. The relevant quantiles of these simulated liabilities were then back-solved using the equivalent-reserve method of Case Study A.20 to produce a confidence interval for the best-estimate mortality pricing assumption.

Case Study A.22

An underfunded UK pension scheme is approached by an advisor suggesting that health questionnaires should be sent to pensioners. Those pensioners qualifying for enhanced annuities due to poor health will then have annuities bought from a specialist insurer. Since the enhanced annuities will typically cost less than the notional reserve held in respect of these pensioners, the advisor argues that this will improve the scheme’s funding position.

However, if pensioners in poorer health are selected out in this way, the remaining lives are by definition the healthier, longer-lived ones. Thus, the scheme’s mortality basis therefore needs to be strengthened for the remaining members after the enhanced-annuity exercise, and there should be no net improvement in the funding position. Furthermore, carrying out this exercise limits future buy-out options for the scheme – some insurers in the UK specifically ask about past ill-health buy-outs, and refuse to quote if such an exercise has been carried out in the preceding five years.

Case Study A.23

An insurer decides that it no longer wants to write annuities for its portfolio of vesting pension policies. It decides instead to operate a small panel of annuity providers and take commission as an introducer. This way the insurer can make money from the annuities without having to carry the liabilities on its balance sheet. The insurer provides the mortality experience of its existing annuity portfolio to prospective annuity writers seeking to join the panel. The mortality experience data contain details of the pension product type and whether a guaranteed annuity rate (GAR) is present.

The panel annuity writers need to calibrate a mortality model specific to the profile of this portfolio, which can be done using survival models; see Richards (Reference Richards2008). It is important to create a differentiated mortality model that, where statistically justified, uses risk factors beyond age, sex, pension size and postcode; see Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011) and Richards et al. (Reference Richards, Kaufhold and Rosenbusch2013). Product type, distribution channel and annuity options selected by the policyholder are often signals for higher or lower mortality. The panel annuity writer with the least sophisticated mortality basis will be selected against by winning mainly mispriced business (Ainslie, Reference Ainslie2000).

Case Study A.24

A long-established company has a large pension scheme. The company’s market value has steadily shrunk over the years, while the pension scheme has become rather large. The company wants the pension scheme to buy backing annuities to reduce the volatility in its published financial statements. The nature of the company’s business has changed radically over time, and the workforce and pensioners have changed along with it. The older pensioners are mainly drawn from a formerly blue-collar workforce with occupational exposure to dust and chemicals. In contrast, recent retirees are from a more white-collar workforce running a more services-orientated business.

Pension size is a less-reliable proxy for socio-economic differentials here. Former blue-collar workers tended to have longer periods of service with lower salaries. However, their pension amounts overlap substantially with former higher-salaried white-collar staff, who also had shorter periods of service. A crude comparison against a standard table would not pick up the shift in the nature of the socio-economic profile. This is particularly important here because the older, blue-collar pensioners dominate the death counts, while the younger, white-collar pensioners dominate the liabilities. The preferred solution is to fit a scheme-specific survival model, using the postcode-driven geodemographic profile as a means of identifying the socio-economic differentials; see Richards (Reference Richards2008), Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011) and Macdonald et al. (Reference Macdonald, Richards and Currie2018), pages 32–36 for details.

Case Study A.25

A UK insurer wants to reinsure a large block of annuity business exceeding $${\unicode{x00A3}}$$ 10 billion in reserves. To reduce exposure to a single counterparty, and to open the bidding to smaller and medium-sized reinsurers, the portfolio is split into four different blocks to be bid on separately. The blocks have not been selected randomly, but according to internal criteria of the insurer. The insurer provides the mortality experience of the entire annuity portfolio to prospective bidders.

Even if they are only planning on bidding for one or two blocks, bidders will use the entire data set to calibrate a mortality model for pricing. It is important that all relevant risk factors are built into the model, since bidders using simplistic pricing are at risk of winning only the blocks they misprice (Ainslie, Reference Ainslie2000).

Case Study A.26

A large UK pension scheme wants to buy annuities to back the pensions in payment. There is an executive sub-scheme containing former executives, senior managers and the surviving spouses of both. However, the executive sub-scheme is too small for a stand-alone analysis, especially a crude comparison against a standard table. Instead, a mortality model is calibrated by the bidding insurer using age, sex, pension size, geodemographic profile and a binary flag denoting membership of the executive sub-scheme. Unsurprisingly, members of the executive sub-scheme have lower mortality than non-members, even after allowing for risk factors like pension size and geodemographic type. This turns out to be significant for pricing the buy-out, as the executive sub-scheme has a lower average age and thus accounts for a larger share of reserves than is implied by their share of annual pensions in payment.

Case Study A.27

A UK pension scheme has far heavier levels of mortality than predicted by a standard commercial model allowing for age, sex, pension size and geodemographic profile. Much of the workforce in question had past occupational exposure to asbestos, so there are grounds to believe that this is a real result. However, the office-based members of the workforce did not have this exposure. The question is whether the experience data is credible enough to justify a weaker mortality basis than the standard commercial model? And, if so, what is statistically justifiable?

A bespoke mortality basis is derived for the pension scheme using survival models; see Richards et al. (Reference Richards, Kaufhold and Rosenbusch2013) and Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011) for examples of scheme-specific risk factors that can be used.

Case Study A.28

A UK insurer wishes to improve its mortality model by using postcodes and geodemographic profiles. However, upon extracting the mortality experience data it finds that almost all deaths have the same postcode, namely that of the insurer’s head office. It transpires that servicing staff change the address of deceased annuitants to avoid accidentally sending post to the dead. From a strict data perspective this is a misuse of the address field, but the business requirement of not further upsetting surviving spouses takes precedence.

It is impossible to calibrate a geodemographic mortality model when only survivors have a residential postcode. The insurer needs to extract the history of address changes for the annuitants, and select the residential postcode for geodemographic profiling. However, it is also possible for an address to be changed to that of a solicitor for an annuitant entering care. Fortunately, good geodemographic profilers in the UK typically distinguish between residential and non-residential postcodes. If a history of addresses is available, the most recent residential profile should be used for modelling.

Case Study A.29

A UK insurer wishes to improve its mortality model by using postcodes and geodemographic profiles. Analysis reveals that UK-resident annuitants without a geodemographic profile have high mortality. Further investigation reveals that postcodes are recorded for these annuitants, but that these postcodes are not listed in the geodemographic profiler database. The geodemographic profiler is used for marketing purposes, and only contains currently valid UK postcodes, of which there are around 1.7 million. There are a further half-million retired postcodes that have been used by the Post Office in years past, but which are no longer in use. However, dead annuitants are more likely to have a retired postcode, and so a bias arises.

The servicing department is unwilling to incur the expense and disruption of updating address details for deceased annuitants. The solution is to get the provider of the geodemographic profiler to back-fill their database with historic postcodes and provide the relevant geodemographic profiles for them. This reduces the bias by assigning geodemographic codes to dead annuitants with out-of-date postcodes.

Case Study A.30

A critical-illness (CI) portfolio has a total of 267 CI claims from almost 130,000 life-years of exposure. A reinsurer wants to assess the extra risk – if any – posed by smokers. However, there are two apparent problems. First, the smokers account for just 56 of the claims and 20,000 life-years of exposure. Second, lapses are vastly more common than CI claims.

The presence of a dominant competing decrement makes a $q$ -type model a poor choice for modelling mortality; see Section 6. The actuary therefore treats lapses as censored observations and fits a $\mu $ -type model for age, sex and smoker status, among other covariates. Despite the small number of smoker CI claims, the model shows conclusively that smokers have a 57% higher CI claim rate than non-smokers, with a standard error of 15%. This gives a p-value of 0.02% for the effect of smoking, i.e. the result is highly significant even for an apparently small number of claims.

Case Study A.31

An investor owns a large portfolio of home-reversion plans. These are residential properties that have been purchased while granting the former owners lifetime tenancy. These are not equity-release mortgages – the investor has purchased the actual properties. Nor are these usufruct arrangements – only the tenants named on the lease can live in the property, and they cannot rent it out to other tenants.

The investor makes money from paying the former owners less than the market value of the property in exchange for the right to live there rent-free. The investor can only take unencumbered possession of the property when the last-named tenant leaves. There are two main decrements of interest to the investor: mortality and inception rates for long-term care (LTC). In the supplied experience data deaths occur nearly four times more frequently than LTC events. However, there is also a number of cases where the tenants bought back their property.

The presence of two competing risks and another decrement makes a $q$ -type model a poor choice; see Section 6. The actuary therefore fits two $\mu $ -type models, one each for mortality and LTC inception. For the mortality model, both LTC and buyback events are treated as censored observations. For the LTC model, both deaths and buybacks are treated as censored observations.

Case Study A.32

An actuary is asked to analyse a second, smaller portfolio of UK home-reversion plans (see Case Study A.31 for background details on this type of business). In addition to the expected decrements of mortality and entry into long-term care, the actuary finds some additional decrements, such as voluntary surrender of the lease and divorce of married tenants. Voluntary surrender of the lease was deemed to be a precursor stage of long-term care, e.g. when an elderly tenant moves in with one of their adult children. Divorces are trickier – they are obviously neither mortality nor care-related, but divorce leads to a named tenant being taken off the lease, thus bringing forward the repossession of the property in the same manner as death or entry into long-term care.

The presence of multiple competing risks makes a $q$ -type model a poor choice; see Section 6. The actuary includes voluntary surrenders as events in the modelling of the care decrement. Divorces are handled as censoring events for the lives taken off tenancy agreements, despite their financial impact being similar to mortality and care events.

A plot of the mortality-only survival curves after allowing for competing risks is shown in Figure A3. This exhibits a gap of eight years in the median survival age of males and females, which is far larger than the equivalent gap in standard mortality tables. The gap in Figure A3 is, however, consistent with the experience data of Case Study A.31 (not shown), suggesting that portfolio-specific analysis is essential for this class of business.

Figure A2.

Lives and “deaths” at young ages for medium-sized UK pension scheme. Temporary pensions to children cease once they leave full-time education, but such cessations have been erroneously labelled as deaths in the data extract.

Source: Own calculations.

Figure A3.

Mortality-only Kaplan-Meier survival curves for lives in a home-reversion portfolio with competing risks including long-term care inception, property buyback, voluntary surrender of lease and divorce.

Case Study A.33

A reinsurer is invited to place a bid for a longevity swap. There are multiple bidders and there will be at least two rounds of bidding; the weakest bids in one round will be dropped from the next round of bidding. A data room is provided for all bidders, including mortality experience data for the portfolio concerned.

During analysis the reinsurer discovers a material issue with the data and asks for a corrected extract. The consultancy managing the bidding process refuses, saying that any updated data will only be provided in a later round. The reinsurer has a dilemma: if it prices too conservatively, it risks not being invited to the second round and never seeing the corrected data; however, if the reinsurer prices using the data as they are, there will be resistance to any unfavourable revision in price based on updated data. The reinsurer is also not entirely sure when (or even if) the cedant will provide updated data.

The reinsurer prices the bid using the data as they are, then decides on an adjustment in respect of the data problems. The reinsurer then communicates the pricing basis with the data-quality adjustment highlighted as a separate item. This makes it explicit to the cedant that there is a cost for poor data quality, and sets a measurable financial incentive to provide corrected data for the mortality experience.

B.1. Standard Bernoulli GLM Likelihood

When a single full year of exposure is present, the standard individual contribution to a Bernoulli log-likelihood, ${\ell ^{{\rm{GLM}}}}$ , is given in Table B1:

Table B1.

Log-likelihoods for ${q_x}$ models, with and without allowance for fractional years of exposure

B.2. Constant Hazard

Under the constant-hazard assumption (Broffitt, Reference Broffitt1984, Table 1), $1{ - _t}{q_x} = {(1 - {q_x})^t}$ . The individual contribution to the Bernoulli log-likelihood, ${\ell ^{{\rm{CH}}}}$ , is given in Table B1. We can see from comparing GLM and CH log-likelihoods that the assumption of a constant hazard to allow for fractional years of exposure cannot be accommodated within a GLM because of the quite different form of the multiplier for $d$ on the right.

B.3. Uniform Distribution of Deaths (UDD)

Under the UDD assumption (Broffitt, Reference Broffitt1984, Table 1), ${{\rm{\;}}_t}{q_x} = t{q_x}$ . The individual contribution to the Bernoulli log-likelihood, ${\ell ^{{\rm{UDD}}}}$ , is given in Table B1. Note that we have dropped an additive term ${t^d}$ as it does not involve the parameter ${q_x}$ , and so does not change inference. We can see from comparing the GLM and UDD log-likelihoods that the UDD allowance for fractional years of exposure cannot be accommodated within a GLM due to the presence of $t$ in ${\rm{log}}\left( {1 - t{q_x}} \right)$ .

B.4. Balducci Assumption

Also known as the hyperbolic assumption, the Balducci assumption is based on flawed reasoning (Hoem, Reference Hoem1984) and implies a decreasing hazard with age (Broffitt (Reference Broffitt1984), page 85 described the Balducci assumption as leading to “some unreasonable results”). It is therefore deprecated in favour of more realistic assumptions like the constant hazard or UDD. We include it here merely for historical curiosity; although it often appears in exercises for actuarial students, the Balducci assumption should never be used in real work. Under the Balducci assumption (Broffitt, Reference Broffitt1984, Table 1):

(9) $${\;_t}{q_x} = {{t{q_x}} \over {1 - \left( {1 - t} \right){q_x}}}.$$

The contribution to the Bernoulli log-likelihood, ${\ell ^{\rm{B}}}$ , is given in Table B1. There is an additive ${t^d}$ term that does not affect inference for the parameter ${q_x}$ and is therefore omitted. Once again, we can see from comparing the GLM and Balducci log-likelihoods that the Balducci assumption to allow for fractional years of exposure cannot be accommodated within a GLM because of the extra third term on the right of ${\ell ^{\rm{B}}}$ .

B.5. Weighting the Log-Likelihood

Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011), Section 4.3.1 claimed that fractional years of exposure could be included in a $q$ -type model by “weight[ing] the contribution of each of the membership records according to its exposure to risk in a year.” However, this does not allow for fractional years of exposure, as discussed in Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011), page 61. Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011), page 62 claim that their individual contribution to the Bernoulli log-likelihood, ${\ell ^{\rm{M}}}$ , is as given in Table B1 (after some re-arrangement). However, ${\ell ^{\rm{M}}}$ cannot be part of a GLM because of the $\left( {t - d} \right)$ coefficient of ${\rm{log}}\left( {1 - {q_x}} \right)$ . Furthermore, weighting observations by the interval length is not a valid way to allow for fractional exposures.

B.6. Modifying the data

For grouped-counts q-type models it would be possible to create an artificial equivalent number of lives to allow for less-than-complete years of exposure. This would typically involve a somewhat dubious assumption of a non-integer number of lives. However, this option makes little sense when modelling mortality at the level of individual lives.

B.7. Conclusions

As per Figure B1, the only way to have a one-year $q$ -type GLM is to discard fractional years of exposure. If the actuary insists on a $q$ -type model, but relaxes the requirement to have a GLM, then it is possible to directly use a likelihood with either the constant-hazard or UDD assumptions. Apparent alternatives are either deprecated (such as the Balducci assumption (Hoem, Reference Hoem1984)) or invalid (such as weighting the exposures as in Madrigal et al. (Reference Madrigal, Matthews, Patel, Gaches and Baxter2011, Section 4.3.1).

However, a one-year $q$ -type model still involves compromises, such as the inability to simultaneously model period effects and selection effects demonstrated in Section 4.2. If an analyst is willing to directly maximize a likelihood, then a survival model for ${\mu _x}$ is both simpler and less restrictive.

B.8. Open-Source Software Implementations

The R (R Core Team, 2021) stats package contains the glm() function for fitting ordinary GLMs for $q$ -type and $\mu $ -type models. However, $q$ -type GLMs can only be fitted where the exposure intervals are of identical length, i.e. no fractional years of exposure.

C.1. Background

Methods based on the estimation of ${q_x}$ came naturally in early studies of mortality, since the aim was to construct a life table. Notably, the seminal attempt to give an explanatory model of mortality (Gompertz, Reference Gompertz1825) was based on ${\mu _x}$ , but this lead was pursued only sporadically in the following two centuries. Even in 1988, the landmark paper of its time (Forfar et al., Reference Forfar, McCutcheon and Wilkie1988) treated the estimation of ${\mu _x}$ and ${q_x}$ equally. There are, of course, situations where the data make a binomial model most appropriate (see Case Study A.17 in Appendix A) but otherwise the $q$ -type modeller faces a choice between discarding data that do not suit the hypothesis, or adapting the hypothesis to suit the actual data.

When starting out in mortality studies, it is often stated, or simply assumed, that all data must be included in the analysis – that discarding some selected data must bias the results – thus presenting the $q$ -type modeller with a problem. However, this is rarely demonstrated, which is understandable, as doing so requires much of the apparatus which has not yet been seen. In this appendix, we consider the legitimacy of discarding parts of the data. Our approach is informal, drawing on the more formal arguments in Andersen et al. (Reference Andersen, Borgan, Gill and Keiding1993), which are based on counting processes. We refer the reader there for a more rigorous treatment.

For simplicity we confine attention to an interval of age $\left[ {x,x + 1} \right]$ , and suppress the initial age $x$ , so we consider the interval of time or duration $t \in \left[ {0,1} \right]$ . We assume a constant force of mortality $\mu $ on this interval, therefore also a binomial model with parameter $q = 1 - {\rm{exp}}\left( { - \mu } \right)$ . This is enough for our purpose because:

1. (a) if $\hat \mu $ is the maximum likelihood estimate of $\mu $ then $1 - {\rm{exp}}\left( { - \hat \mu } \right)$ is the maximum likelihood estimate of $q$ ; and

2. (b) our analysis can be extended to non-constant parameters ${\mu _{x + t}}$ and ${q_{x + t}}$ but at the cost of much more notation and the use of product integrals (see Macdonald and Richards (Reference Macdonald and Richards2025)).

C.2. Data and Likelihood

We observe $M$ individuals, and for the $i$ th individual we have the following data:

1. (a) durations ${r_i}$ of entry to observation and ${t_i}$ of exit from observation, with $0 \le {r_i} \lt {t_i} \le 1$ ; and

2. (b) an integer-valued indicator ${d_i}$ of the reason for observation ceasing; ${d_i} = 1$ if the reason was death, and ${d_i} = 0$ if the reason was right-censoring.

The $i$ th individual is observed on the interval $\left[ {{r_i},{t_i}} \right]$ , which we denote by ${{\rm{\Delta }}_i}$ , and we denote the interval $\left[ {0,1} \right]$ by ${\rm{\Delta }}$ . From Macdonald and Richards (Reference Macdonald and Richards2025) we borrow the process ${Y^i}\left( t \right)$ , equal to 1 if the $i$ th individual is under observation “just before” time $t$ , and equal to 0 otherwise (the “just before” is a technicality we can ignore).

To extend the model to allow for voluntary lapsation, all that is needed is to (i) assume that a constant force of lapsing $\nu $ operates alongside $\mu $ , and (ii) to define another indicator ${w_i}$ , taking the value 1 if lapsing is the reason for observation ceasing, and 0 otherwise.

The contribution of the $i$ th individual to the likelihood is the factor ${L_i}$ defined as:

(C1) $${L_i} = {\rm{exp}}\left( { - \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}\left( t \right){\rm{\;}}\left( {\mu + \nu } \right){\rm{\;}}dt} \right){\rm{\;}}{\mu ^{{d_i}}}{\rm{\;}}{\nu ^{{w_i}}},$$

the total likelihood, denoted by $L$ , is:

(C2) $$L = \mathop \prod \limits_i {L_i},$$

and the MLE, denoted by $\hat \mu $ , is:

(C3) $$\hat \mu = {{\mathop \sum \nolimits_i {d_i}} \over {\mathop \sum \nolimits_i \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}\left( t \right){\rm{\;}}dt}}.$$

C.3. Obligate versus Random Right-Censoring

Observation of the $i$ th individual is left-truncated if ${r_i} \gt 0$ . Observation is right-censored if ${d_i} = 0$ , but we distinguish two kinds of right-censoring:

1. (a) random right-censoring means the policy lapsed, ${w_i} = 1$ ; and

2. (b) obligate right-censoring means the individual was observed to survive until some predetermined time (in a sense to be made precise shortly) at which observation necessarily ended. Examples of such times are:

   1. (1) the end of the interval $\left[ {0,1} \right]$ (or more generally, the rate interval);

   2. (2) the date of data extraction;

   3. (3) the date of transfer out of the portfolio; or

   4. (4) the end date of the investigation.

The key point about obligate right-censoring is that the time of obligate right-censoring is known as soon as the individual enters observationFootnote ³ . Technically, the time $t'_i$ when observation on the $i$ th individual must end is a random variable with ${r_i} \lt t'_i \le 1$ , but its value becomes known when the $i$ th individual enters observation at time ${r_i}$ (that is, $t'_i$ is ${\mathcal{F}_{{r_i}}}$ -measurable). So we have three possible cases:

1. (a) Case 1: obligate right-censoring, ${d_i} = {w_i} = 0$ , ${t_i} = t'_i$ ;

2. (b) Case 2: random right-censoring, ${d_i} = 0$ , ${w_i} = 1$ , ${t_i} \lt t'_i$ ; and

3. (c) Case 3: death, ${d_i} = 1$ , ${w_i} = 0$ , ${t_i} \lt t'_i$ .

The main result of this Appendix is that the data on selected lives can be discarded, based on the time ${r_i}$ of entry and the time $t'_i$ of obligate right-censoring, without biasing the estimation of $\mu $ . However, randomly right-censored data cannot be so discarded.

Define ${\rm{\Delta'_i }} = \left[ {{r_i},t'_i} \right]$ , the greatest interval on which the $i$ th individual might be observed, and define ${\rm{\Delta }} = \left[ {0,1} \right]$ . Then:

1. (a) observation of the $i$ th individual is randomly right-censored if ${w_i} = 1$ ; and

2. (b) observation of the $i$ th individual is left-truncated or obligate right-censored if ${\rm{\Delta'_i }} \ne {\rm{\Delta }}$ .

C.4. Discarding Left-truncated and Obligate Right-censored Data

If ${\rm{\Delta'_i }} \ne {\rm{\Delta }}$ then the individual cannot complete a full year’s exposure. If the $q$ -type analyst discards such data, the modified likelihood, denoted by ${L^{{\rm{}}Obl{\rm{\;}}}}$ , is:

(C4) $${L^{{\rm{}}Obl{\rm{\;}}}} = \mathop \prod \limits_{{\rm{\Delta'_i }} = {\rm{\Delta }}} {L_i}.$$

Assuming that there is no selective reason for entry to be left-truncated or exit to be obligate right-censored, all such data can be discarded, deaths and lapses along with exposure data, without biasing any likelihood-based estimation. Note that this is an assumption that often seems reasonable, but may be hard to verify. Therefore:

(C5) $${\hat \mu ^{{\rm{}}Obl{\rm{\;}}}} = {{\mathop \sum \nolimits_{{\rm{\Delta'_i }} \ne {\rm{\Delta }}} {d_i}} \over {\mathop \sum \nolimits_{{\rm{\Delta'_i }} \ne {\rm{\Delta }}} \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}\left( t \right){\rm{\;}}dt}} \to \mu .$$

See Section III.2.2, also Example III.2.3, in Andersen et al. (Reference Andersen, Borgan, Gill and Keiding1993) for a formal justification.

C.5. Discarding Randomly Right-censored Data

If randomly right-censored data are discarded the modified likelihood, denoted by ${L^{{\rm{Ran\;}}}}$ , is:

(C6) $${L^{{\rm{Ran\;}}}} = \mathop \prod \limits_{{w_i} \ne 1} {L_i} = \mathop \prod \limits_{{w_i} \ne 1} {\rm{exp}}\left( { - \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}\left( t \right){\rm{\;}}\left( {\mu + \nu } \right){\rm{\;}}dt} \right){\rm{\;}}{\mu ^{{d_i}}}.$$

If ${w_i} = 1$ and the $i$ th individual’s data is excluded, the time $\left| {{{\rm{\Delta }}_i}} \right|$ for which the individual was exposed to the risk of death and lapse, is removed from the exposed-to-risk, but no deaths are removed from the data. The MLE ${\hat \mu ^{\rm Ran{\rm{\;}}}}$ is:

(C7) $${\hat \mu ^{\rm Ran{\rm{\;}}}} = {{\mathop \sum \nolimits_{{w_i} \ne 1} {d_i}} \over {\mathop \sum \nolimits_{{w_i} \ne 1} \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}\left( t \right){\rm{\;}}dt}} = {{\mathop \sum \nolimits_i {d_i}} \over {\mathop \sum \nolimits_{{w_i} \ne 1} \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}(t){\rm{\;}}dt}} \ge {{\mathop \sum \nolimits_i {d_i}} \over {\mathop \sum \nolimits_i \mathop \int \nolimits_{{{\rm{\Delta }}_i}} {Y^i}\left( t \right){\rm{\;}}dt}} = \hat \mu$$

where the last inequality is strict if and only if at least one individual lapses. Therefore, discarding randomly right-censored data will bias the estimate of $\mu $ upwards, and also the estimate of $q$ . Of course, the argument displayed in (C7) is exactly the same as the normal argument for not excluding censored data in a survival analysis.

D.1. Kaplan-Meier Estimator

The methodology of Kaplan and Meier (Reference Kaplan and Meier1958) is non-parametric, and is a re-arrangement of the experience data into a survival curve. This survival curve can be by duration, such as is often the case with medical trials (Collett, Reference Collett2003, Section 2.1.2). However, the most useful definition of the Kaplan-Meier estimator for actuaries is the one by age in Macdonald et al. (Reference Macdonald, Richards and Currie2018, equation 8.3):

(D1) $${\;_t}{\hat p_x} = \prod\limits_{{t_i} \le t} {\left( {1 - {{{d_{x + {t_i}}}} \over {{l_{x + t_i^ - }}}}} \right)}, $$

where $x$ is the outset age for the survival function, $\left\{ {x + {t_i}} \right\}$ is the set of distinct ages at death, ${l_{x + t_i^ - }}$ is the number of lives alive immediately before age $x + {t_i}$ and ${d_{x + {t_i}}}$ is the number of deaths occurring at age $x + {t_i}$ . Informally, and following Section 2, we can think of ${l_{x + t_i^ - }}$ as the number of lives alive just after midnight on a given day, with deaths ${d_{x + {t_i}}}$ happening around mid-day. Figures 12(a) and A3 show examples of Kaplan-Meier curves.

The Kaplan-Meier estimator in Equation (D1) is the product-limit estimator of the survival curve discussed in Macdonald and Richards (Reference Macdonald and Richards2025), Section 4.2. It is also the product integral of the Nelson-Aalen estimator in Equation (D3).

D.2. Nelson-Aalen Estimator

The methodology of Nelson (Reference Nelson1958) is non-parametric, and is a re-arrangement of the experience data into a cumulative risk curve. This cumulative risk can be by duration, such as is often the case in medical studies:

(D2) $$\hat \Lambda \left( t \right) = \sum\limits_{{t_i} \le t} {{{{d_{{t_i}}}} \over {{l_{t_i^ - }}}}}, $$

where $\left\{ {{t_i}} \right\}$ is the set of distinct durations at death, ${l_{t_i^ - }}$ is the number of lives alive immediately before duration ${t_i}$ and ${d_{{t_i}}}$ is the number of deaths occurring at duration ${t_i}$ . Figure 14 shows an example of Equation (D2) being used to monitor differences in mortality for new annuity business.

However, another useful definition for actuaries is one by age:

(D3) $${\hat \Lambda _x}\left( t \right) = \sum\limits_{{t_i} \le t} {{{{d_{x + {t_i}}}} \over {{l_{x + t_i^ - }}}}} .$$

The Nelson-Aalen estimator is the non-parametric estimator of the integrated hazard in Macdonald et al. (Reference Macdonald, Richards and Currie2018, equation 3.23).

D.3. Fleming-Harrington estimator

The methodology of Fleming and Harrington (Reference Fleming and Harrington1991) is non-parametric, and uses the Nelson-Aalen estimator of the cumulative hazard to estimate the survival curve from age $x$ :

(D4) $${{\rm{\;}}_t}{\hat p_x} (t)= {\rm{exp}}\left( { - {{\hat \Lambda }_x}\left( t \right)} \right),$$

where ${\hat \Lambda _x}\left( t \right)$ is the Nelson-Aalen estimator from Equation (D3). The Fleming-Harrington estimator is the non-parametric equivalent of the identity:

(D5) $${{\rm{\;}}_t}{p_x} = {\rm{exp}}\left( { - \mathop \int \nolimits_0^t {\mu _{x + s}}ds} \right) = \mathop \prod \limits_{s \in \left( {0,t} \right]} \left( {1 - {\mu _{x + s}}ds} \right).$$

The Fleming-Harrington estimator is slightly preferable to the Kaplan-Meier estimator for very small data sets, in part because approximate confidence intervals for the latter are not restricted to $\left[ {0,1} \right]$ (Collett, Reference Collett2003, Sections 2.2.1 and 2.2.3). However, for the data sets of the size that actuaries use there is usually no meaningful difference between the two estimators.

D.4. Relationships

There are important relationships between the Kaplan-Meier, Nelson-Aalen and Fleming-Harrington estimators. The product integral of a piecewise-constant function such as Equation (D3) is a discrete product such as Equation (D1). Therefore, the Kaplan-Meier estimator is the product integral of the Nelson-Aalen estimator; this is exact, not approximate. The Fleming-Harrington estimator in Equation (D4) is the exponential of (minus) the Nelson-Aalen estimator in Equation (D3), and is an approximation to the Kaplan-Meier estimator in Equation (D1) (and a good one for the datasets typically used by actuaries).

In some literature (SAS Institute Inc., 2017, page 5336) an alternative to Equation (D3) for small samples is given as:

(D6) $${\hat \Lambda _x}\left( t \right) = \sum\limits_{{t_i} \le t} {\sum\limits_{j = 0}^{{d_{x + {t_i}}} - 1} {{1 \over {{l_{x + t_i^ - }} - j}}} } .$$

Equations (D3) and (D6) are identical if ${d_{x + {t_i}}} = 1$ . Even where some ${d_{x + {t_i}}}$ are greater than 1, Equation (D6) only produces materially different answers with small sample sizes. For actuarial data sets there is little meaningful difference, so we prefer Equation (D3) for simplicity.

D.5. Semi-parametric methods

Richards (Reference Richards2022b) introduced a semi-parametric estimator in calendar time for investigating short-term fluctuations in mortality. The methodology has two stages, the first of which is to calculate the non-parametric Nelson-Aalen estimator of the cumulative hazard with respect to calendar time, rather than age or duration:

(D7) $${\hat \Lambda _y}\left( t \right) = \sum\limits_{{t_i} \le t} {{{{d_{y + {t_i}}}} \over {{l_{y + t_i^ - }}}}}, $$

where $y$ is the outset calendar time for the cumulative risk, $\left\{ {y + {t_i}} \right\}$ is the set of distinct dates of death, ${l_{y + t_i^ - }}$ is the number of lives alive immediately before date $x + {t_i}$ and ${d_{y + {t_i}}}$ is the number of deaths occurring on date $y + {t_i}$ . Informally, and following Section 2, we can think of ${l_{y + t_i^ - }}$ as the number of lives alive just after midnight on a given day, with deaths ${d_{y + {t_i}}}$ happening around mid-day. ${\hat \Lambda _y}\left( t \right)$ is the equivalent of Equation (D3), but with respect to calendar time instead of age. A practical point is that the real-valued calendar time is calculated using the actual number of days in the year, rather than the average of 365.24 given in Section 2. For example, 14th March 2023 would be represented as 2023.1973 (being 72 days into a 365-day year); in contrast, 14th March 2024 would be represented as 2024.1995 (being 73 days into a 366-day year).

The second step is to estimate the portfolio-level hazard in time, ${\mu _y}\left( t \right)$ . This ignores the age of individual lives, and so is only useful for investigating fluctuations over relatively short periods of time, where the age profile of the portfolio does not change radically. To estimate ${\mu _y}\left( t \right)$ from Equation (D7) we need to apply some degree of smoothing over a suitable short interval, $c$ . We thus obtain a semi-parametric estimate of the mortality hazard at a point in time, ${\hat \mu _y}\left( {t,c} \right)$ , using $c$ as a bandwidth parameter. This can be done in a number of ways, but a simple one is to use a central difference in the estimator in Equation (D7) around the time of interest:

(D8) $${\hat \mu _y}\left( {t,c} \right) = {1 \over c}\left( {{{\hat \Lambda }_y}\left( {t + c/2} \right) - {{\hat \Lambda }_y}\left( {t - c/2} \right)} \right),$$

where $c$ is selected by the analyst to smooth random variation. When $c \le 0.5$ , ${\hat \mu _y}\left( {t,c} \right)$ will reveal detail such as seasonal variation (Figure 13(a)), mortality shocks (Richards, Reference Richards2022b, Figure 3) and data-quality issues (Figure 13(b) and (Richards, Reference Richards2022b, Appendix A)). When $c \gg 0.5$ short-term variations are smoothed out. Over longer periods of time ${\hat \mu _y}\left( {t,c} \right)$ will reflect changes in the age distribution of the portfolio (Richards, Reference Richards2022b, Figure 2).

Equation (D8) is calculated using an extract at a point in time, $u$ , so we could write ${\mu _y}\left( {t,c,u} \right)$ to make it explicit that the calculation is with respect to a particular data extract at time $u$ . This can be useful when estimating delays in reporting deaths; see Section 4.3 and Richards (Reference Richards2022b), Section 4, where the ratio ${\mu _y}\left( {t,c,{u_1}} \right)/{\mu _y}\left( {t,c,{u_2}} \right)$ provides an estimate of the proportion of deaths reported up to time ${u_1}$ , under the assumption that the estimate at time ${u_2}$ contains most of the unreported deaths up to time ${u_1}$ . This presupposes that the gap ${u_2} - {u_1}$ is large enough to allow most OBNR cases up to ${u_1}$ to have been subsequently reported. In reality, the estimate of ${\mu _y}\left( {t,c,{u_2}} \right)$ at times before ${u_1}$ will also be affected by OBNR, but this effect can be minimized by a suitably large gap between $u_1$ and ${u_2}$ .

As an alternative to Equation (D8), ${\hat \mu _y}\left( {t,c} \right)$ could be calculated directly in a single step as:

(D9) $${\hat \mu _y}\left( {t,c} \right) = {2 \over c}\mathop \sum \limits_{t - c/2 \lt {t_i} \le t + c/2} K\left( {{{2\left( {{t_i} - t} \right)} \over c}} \right){{{d_{y + {t_i}}}} \over {{l_{y + t_i^ - }}}},$$

where $K\left( u \right)$ is a kernel function for smoothing; see Andersen et al. (Reference Andersen, Borgan, Gill and Keiding1993), Section IV.2.1. Equation (D8) results from using the uniform kernel:

(D10) $$K\left( u \right) = \left\{ {\matrix{ {{1 \over 2}} & {\left| u \right| \le 1} \cr 0 & {{\rm{otherwise}},} \cr } } \right.$$

but other kernel smoothers with greater efficiency are available, such as the Epanechnikov kernel (Epanechnikov, Reference Epanechnikov1969):

(D11) $$K\left( u \right) = \left\{ {\matrix{ {{3 \over 4}\left( {1 - {u^2}} \right)} & {\left| u \right| \le 1} \cr 0 & {{\rm{otherwise}}.} \cr } } \right.$$

The kernel functions in Equations (D10) and (D11) are illustrated in Figure D1. The Epanechnikov kernel is more suitable for applications where measuring short-term fluctuations is important.

Figure D1.

Sample kernel-smoothing functions in Equations (D10) and (D11).

D.6. Open-source software implementations

The R (R Core Team, 2021) survival package contains the survfit() function for computing the Kaplan-Meier and Fleming-Harrington estimators in Equations (D3) and (D6). The Python module scikit-survival contains an implementation of the Kaplan-Meier estimator. However, many software implementations of Kaplan-Meier cannot handle left truncation by age, which is essential for actuarial work. The R script in Figure D2 can be used as an alternative, as it handles left truncation.

Figure D2.

R script for non-parametric estimates with left-truncated data by age.

Richards (Reference Richards2022b), Appendix C contains R source code for the semi-parametric estimator in Equation (D8).

E.1. Individual lifetimes

Suppose we have the following for life $i$ : the entry age into observation, ${x_i}$ ; the time observed, ${t_i}$ ; and the indicator, ${d_i}$ , which takes the value 1 if life $i$ died at age ${x_i} + {t_i}$ and zero otherwise. The contribution, ${L_i}$ , of life $i$ to the likelihood is then:

(E1) $${L_i}{ = _{{t_i}}}{p_{{x_i}}}\mu _{{x_i} + {t_i}}^{{d_i}}$$

(E2) $$ = {\rm{exp}}\left( { - \mathop \int \nolimits_0^{{t_i}} {\mu _{{x_i} + s}}ds} \right)\mu _{{x_i} + {t_i}}^{{d_i}}$$

(E3) $$ = {e^{ - {{\rm{\Lambda }}_{{x_i}}}\left( {{t_i}} \right)}}\mu _{{x_i} + {t_i}}^{{d_i}},$$

where ${{\rm{\Lambda }}_x}\left( t \right) = \mathop \int \nolimits_0^t {\mu _{x + s}}ds$ is the integrated hazard; see Macdonald et al. (Reference Macdonald, Richards and Currie2018, equation 3.23). The corresponding contribution of life $i$ to the log-likelihood, ${\ell _i}$ , is:

(E4) $${\ell _i} = {\rm{log}}{\,}{L_i}$$

(E5) $$= - {{\rm{\Lambda }}_{{x_i}}}\left( {{t_i}} \right) + {d_i}{\,}{\rm{log}}{\mu _{{x_i} + {t_i}}}.$$

For a data set of $n$ lives, the full likelihood is then $L = \mathop \prod \nolimits_{i = 1}^n {L_i}$ and the corresponding log-likelihood is $\ell = \mathop \sum \nolimits_{i = 1}^n {\ell _i}$ . Modelling individual lifetimes is the most elemental mortality model; Macdonald and Richards (Reference Macdonald and Richards2025) call this a “micro” model. One benefit of modelling individual lifetimes is that more information can be used on individual attributes and covariates. For example, Richards (Reference Richards2022a) showed how to include the exact pension amount as a continuous covariate in mortality modelling.

E.2. Grouped counts

An alternative approach is to create homogeneous sub-groups of shared characteristics. Suppose for sub-group $j$ we have ${n_j}$ lives with total time observed $E_j^c = \mathop \sum \nolimits_{i = 1}^{{n_j}} {t_i}$ and total observed deaths ${D_j} = \mathop \sum \nolimits_{i = 1}^{{n_j}} {d_i}$ . Assuming a constant mortality hazard rate of ${\mu _j}$ , the contribution to the likelihood, ${L_j}$ , is:

(E6) $${L_j} = {e^{ - E_j^c{\mu _j}}}\mu _j^{{D_j}}.$$

Equation (E6) looks like a Poisson likelihood, and this leads to the common misconception that ${D_j}$ has a Poisson distribution with mean parameter $E_j^c{\mu _j}$ . This is not true, as demonstrated in detail by Macdonald and Richards (Reference Macdonald and Richards2025), Section 3. In fact, the random variable is the pair $\left( {{D_{}},E_j^c} \right)$ , which cannot have a Poisson distribution. However, the likelihoods for a Poisson model for ${D_j}$ and the pair $\left( {{D_j},E_j^c} \right)$ are the same up to a multiplicative constant, so the inference will be the same; Macdonald and Richards (Reference Macdonald and Richards2025), Section 3 called the likelihood in Equation (E6) pseudo-Poisson. This means that a pseudo-Poisson grouped-count model like Equation (E6) can be fitted using the machinery of a Poisson GLM, even though the observed death count ${D_j}$ is not Poisson.

Grouped-count models lose information due to grouping. Care is required with the assumption of a constant hazard rate, ${\mu _j}$ , which requires that lifetimes are split into small enough age ranges for this approximation to be justified; see Macdonald and Richards (Reference Macdonald and Richards2025), Appendix A. Even greater care is required with continuous variables like pension size, where the assumption of a constant effect within a discrete band can lead to systematic over-estimation of mortality of pensioners. Figure E1 shows the deviance residuals by pension size-band for a model with age and sex as covariates. The non-random variation suggests that the wealthiest pensioners have significantly lower mortality. If we split the lives into twenty equal-sized groups, Figure E1(a) suggests that the top 10% of amounts are different from the rest. However, Figure E1(c) shows that the assumption of a constant mortality effect within this group is false – there is continuous variation by pension amount. This is important for actuarial purposes, because the assumption of a constant effect risks financial bias – all other factors held equal, the financial impact of those on the right always outweighs the financial impact of those on the left.

Figure E1.

Deviance residuals by pension size-band, with 1 representing the smallest pensions.

Source: Mortality model for age and sex for English pension scheme from Richards (Reference Richards2022a).

One counter-argument sometimes advanced in favour of discretization is that a model could be fitted using the 100 size-bands in Figure E1(b), say with some penalty function to dampen the inevitable volatility in parameter estimates. However, this would be an additional complexity to deal with the problem caused by discretization. It is far simpler to instead treat variables like pension size continuously and not have the problem arise in the first place. Richards (Reference Richards2022a), Tables 7 and 8 showed that treating pension amount as a continuous risk factor for mortality produced a model with the fewest parameters and the lowest-equal information criterion. This is an extension of graduation by mathematical formula, where a smooth progression in mortality rates by age is guaranteed during the model fit (Forfar et al., Reference Forfar, McCutcheon and Wilkie1988). We go a step further and treat all continuous variables this way, not just age.

E.3. Stratification

Irrespective of whether mortality modelling is of individual lifetimes or groups, there is the question of how to handle sub-groups with shared characteristics. One approach is to sub-divide the experience data and fit separate models, i.e. stratification of the data set. In general stratification should be avoided wherever possible, as it increases the overall number of parameters required and weakens the statistical power of the data set (Macdonald et al., Reference Macdonald, Richards and Currie2018, Section 7.3). However, stratification can on rare occasions be justified where the mortality hazard rate has fundamentally different properties, such as where sub-groups have a different shape of mortality curve by age.

E.4. Open-source software for individual lifetimes

The R (R Core Team, 2021) survival package contains the survreg() function for fitting parametric survival models. However, most of the models available do not handle left truncation. Macdonald et al. (Reference Macdonald, Richards and Currie2018), Chapter 5 provide R scripts for fitting models to left-truncated data, with discussion on use of the R’s nlm() and solve() functions. Commercially supported services are also available.