Covid-19 mortality: the Proportionality Hypothesis

The pandemic in England fell into three phases: Phase 1 covers the initial period of the pandemic when there was a relatively high degree of infection synchronisation (as shown in Fig. 2) across regions and socio-economic subgroups, and, more importantly, vaccination rollouts had not yet commenced. Phase 2 is the transition period involving the first vaccine rollout, the second and subsequent booster vaccines, together with the sequence of new variants of the virus that brought new waves of infection. Phase 3 (which, at the time of writing, was still ongoing) marks the transition to an endemic state for the virus, with a steady flow of new infections and relatively low numbers of deaths, similar to influenza (flu). The relatively low number of deaths in Phase 3 is also partly explained by the fact that a high percentage of the population now had antibodies, as a result of previous infection or vaccination.

We utilise a range of datasets that are listed in Appendix 1. Much of the data come from the UK’s Office for National Statistics (ONS)² with some additional data from the Human Mortality Database and the UK’s National Health Service (NHS). Wherever possible, we try to use data in a consistent way through time. However, there are inconsistencies between different datasets. For example, infection rates, hospitalisations and death rates are published using different age groupings across the different datasets.

There are also different definitions. We use several distinct measures of Covid-19 deaths that depend on data availability for specific tasks in the paper. For most purposes in this paper, we count a death as a Covid-19 death when Covid-19 is mentioned on the death certificate. We also, in places, make use of death counts where Covid-19 is listed as the underlying or main cause of death. Alongside these differences, death counts can be based on registered deaths during a specified period, or on actual occurrences. While death registrations might not be perfect, they have the advantage of being consistent over time and accurate (for the given definition), apart from the first few weeks of the pandemic when there was some misreporting of cause of death. Mostly, we also use week of registration, but we use week of occurrence as an alternative in one place in the paper.

Frailty is an important concept in this paper, but it can have different meanings to different professional groups. In an actuarial and demographic context, actuarial frailty relates to all the factors (some of which are unobserved) that affect human mortality other than age. It generally involves an adjustment up or down to an individual’s or subgroup’s death rate relative to a standard rate at the same age and sex (see Vaupel et al. [40], and discussions in Carannante et al. [10]). In a medical context, the concept of frailty is quite different: it is an absolute measure of poor health, rather than a measure that is relative to an age-dependent benchmark. As such, it represents an accumulation of visible comorbidities, disabilities, and chronic and acute illnesses, combined with the invisible but gradual deterioration of cells and organs.^3,4

In this paper, we employ two distinct concepts of frailty and, since these differ in definition from actuarial frailty, we give each a precise name. The first concept is ‘biological frailty’. This is measured in absolute terms (and therefore is closer to the medical concept of frailty) and is defined as the 1-year death rate from all causes (in a ‘normal’ non-Covid year), excluding deaths from external causes (e.g., road and other accidents, suicide, and accidental poisoning).⁵ Biological frailty depends on both the impact of age (i.e., the baseline mortality curve) and the variation within an age group (e.g., due to variation in death rates at the individual or subgroup level relative to the average for that age group). We use the term ‘biological frailty’ to match the sister concept of ‘biological age’; where the latter can differ from ‘chronological age’ (see, e.g., Huang et al. [20], and Milevsky [25]).⁶ In our study, biological frailty is based on all-cause mortality rates in a typical pre-Covid year and by socio-economic subgroup.

The key findings are as follows. The main theme running through the paper is the Proportionality Hypothesis, which focuses on the Covid-19 infection fatality rate (IFR). There are two versions depending on the granularity of the data published during the pandemic. The ‘subgroup’ Proportionality Hypothesis states that the IFR is approximately proportional to all-cause death rates (i.e., biological frailty) by both age and subgroup. The ‘aggregate’ Proportionality Hypothesis focuses on age only and states that the IFR is approximately proportional to all-cause death rates by age. As a consequence, and contrary to much media commentary at the time, death rates from Covid-19 did not disproportionately affect the more deprived groups in society. Where we did observe differences in mortality, these were due both to pre-existing health inequalities at the all-cause level and to differences in infection rates between sub-groups.

We find strong evidence showing the benefits of vaccination. First, infection rates fell significantly following vaccination of older groups in advance of younger age groups. Second, for those vaccinated who did become infected with Covid-19, vaccination led, on average, to less severe cases and to fewer deaths (i.e., a lower infection fatality rate). By the end of 2022, the infection fatality rate was 1/20th–1/30th of its value at the end of 2020: the result of the combination of vaccination and/or prior infection alongside improved treatments.

Lastly, we discuss some takeaways for insurers. First, insurers should revisit their extreme pandemic mortality scenarios. In particular, models should allow for significant random variation between different geographical areas, age groups and socio-economic subgroups, as well as being consistent with the Proportionality Hypothesis. Second, we conclude that mortality catastrophe bonds might not be as effective as a hedging instrument as had previously been believed, unless they incorporate a linkage to sub-population mortality indices.

The rest of this paper is structured as follows. In Sect. 2, we review the headline data for England, covering the different pandemic waves and regional variations, and the emergence of Covid-19 variants. Section 3 discusses Phase 1, covering the year 2020, in detail, introduces the Proportionality Hypothesis and then assesses it by examining infection and death rates by deprivation decile and region. Section 4 analyses Phase 2, covering the period between the beginning of 2021 and mid-2022, in detail and assesses how well the Proportionality Hypothesis holds up. Section 5 considers Phase 3, the transition to the endemic state of the virus which began in mid-2022 and also reassesses the hypothesis. The impact of Covid-19 on other causes of death is discussed in Sect. 6, while Sect. 7 shows how extrapolative mortality models can be adjusted to allow for future pandemics in a way that is consistent with the Proportionality Hypothesis. Section 8 considers the implications for insurers and Sect. 9 concludes. The data sources are listed in Appendix 1, Appendix 2 summarises the findings of the paper.

In this section, we review the key features of the pandemic between the beginning of 2020 and the Spring of 2023, covering Phases 1–3.

We look, first, at weekly mortality for England and Wales using data from the Short-Term Mortality Fluctuations database available from the Human Mortality Database (see Appendix 1).⁷ Figure 1 shows, for the period since 2014, weekly death rates for males and females in the 75–84 age group, the group that was most badly affected by Covid-19. The pre-pandemic years 2014–19 are represented by grey lines, 2020 in black, 2021 in red, 2022 in green and 2023 in blue. The grey lines give a good indication of the seasonality in mortality in England and Wales in pre-pandemic ‘normal’ years. These typically peak in January and bottom out in the late summer. Winter rates can be seen to be much more variable than summer rates, due, for example, to the severity of seasonal flu. The year 2020 (black line) started off as a relatively benign year, but we can then see the very significant first wave of excess deaths around weeks 15–20 which was much more severe than any of the recent bad flu years. A second wave of excess deaths appeared towards the end of 2020 and peaked in January 2021 (red line). Spring 2021 was characterised by a period of lower-than-normal mortality. This, perhaps, reflected the possibility that many who were already in very poor health and likely to die around this time had died in the previous months with or due to Covid-19.⁸ After that, the weekly mortality curves for males and females fluctuated above and below the pre-pandemic curves as new waves of Covid-19 came and went, partly in response to the emergence of new variants of the virus. But Covid-19-related deaths from mid-2021 onwards can be seen to be relatively modest compared to the first wave in 2021.

In Fig. 2, we use weekly deaths data provided by the ONS which allows analysis across a number of characteristics, including region. In particular, we look at Covid-19-related deaths (where Covid-19 is mentioned on the death certificate and so might not be the underlying cause of death) as a proportion of the 5-year average of all deaths over the 5 years 2015–2019 for the same week of the year (males and females combined, all causes, all ages).

This figure allows us to observe how much variation there was between different parts of England and how this changed during the course of the pandemic. In the upper plot, the first wave of the pandemic was closely synchronised across the nine official regions of England, with the exception that London was a week or two ahead. But, in spite of this synchronisation (and the fact that the first national lockdown was introduced simultaneously across the whole country on 26 March 2020), the magnitude of the first wave varied considerably between regions. London and other regions with dense urban areas experienced significantly higher Covid-19 death rates. Summer 2020 saw very low death rates from Covid-19 (and also very low infection rates).

The second wave consisted of two parts.⁹ The first part (labelled Wave 2A) occurred in the Autumn of 2020 and was largely focused in the northern regions (accompanied by some local lockdowns). But in December 2020, a new variant of the virus, Alpha, emerged, generating much higher peak mortality (with a second national lockdown) in January 2021 (labelled Wave 2B)—with much higher death rates in London and lower death rates in the more-rural South West. This was followed by a period in early summer with very low death rates.

For the remainder of 2021 and through the first half of 2022 (Fig. 2, lower plot), there was a long and irregular third wave which involved the Delta and Omicron variants. Covid-19 death rates were much lower than for Waves 1 and 2.¹⁰ Individual regions had distinct patterns, and January 2022 witnessed the last significant peak, with London death rates clearly higher than other regions.

For the rest of 2022 and into the Spring 2023, the pandemic moved into the endemic phase with relatively regular small waves (but with declining amplitude) approximately every 3 months. Further, the waves reached their troughs at higher levels (indicating more Covid-19 deaths) than those of the Summers of 2020 and 2021.¹¹ In addition, the waves became much more synchronised, with relatively little variation by region. This synchronisation suggests that, as people’s lives returned to normal, new Covid variants could circulate between regions much more quickly, contrasting with Autumn 2020 when travel restrictions were much more severe and this prevented Wave 2A from spreading from the northern regions to the south.

Other cause-of-death data from the ONS (but not publicly available) provides information for the whole of 2020 at the sub-regional level of health boards (Clinical Commissioning Groups; CCGs¹²). Over the data period, there were 106 CCGs across England with an average of around 500,000 people in each. The data allow us to compare deaths from Covid-19 (as the underlying cause of death) for ages 40–89 during 2020 as a percentage of deaths from all causes in 2019. Results are plotted in Fig. 3. As with Fig. 2, the chosen metric implicitly adjusts for socio-economic variation between CCGs. Each CCG is represented by a single dot and shows Covid-19 deaths as a percentage of 2019 deaths for males (x-axis) versus females (y-axis). Dots are coloured according to the urban–rural mix of the CCG: bright green for the most rural CCGs and bright red for the most urban. As might be expected, there is a strong correlation between male and female death rates from Covid-19. But there are other important findings. First, there is much more variation at the sub-regional CCG level than at the regional level (compare with Fig. 2). Second, Fig. 3 reveals the strong association between Covid-19 death rates and the urban–rural mix. The implication of this, again as might be expected, is that Covid-19 spreads more easily in more dense urban areas with consequently higher death rates. This has implications for the development of future (extreme) pandemic scenarios in an insurance context and indicates the importance of knowing a victim’s geographic location, as discussed further in Sect. 8.

Figure 4 shows the dominant variants of Covid-19 over time from the end of 2020. It can be seen that each new variant (e.g., the Delta variant from late Spring 2021) typically takes hold quite rapidly, displacing the previous variant in 1–2 months. There are a number of possible explanations: higher reproductive numbers for new variants (also known as the R number: the average number of new infections caused by each existing infected person); a gradually falling R number for the previous variant, as the proportion of the population still susceptible falls and the population moves towards herd immunity against that variant; and whether or not prior infections give strong or only limited protection against new variants.

The emergence of a new dominant variant is typically associated with a fresh wave of infections and deaths, e.g., Wave 2B being linked to the emerging Alpha variant, and Wave 3 to the Delta variant (Fig. 2).

The period from the end of 2022 was characterised by a sequence of Omicron variants. In some cases, single subvariants took over rapidly (e.g., BA.2), while BA.4, BA.5, BE and BF coexisted for several months. Each time a new dominant subvariant emerges, there was a small wave of new cases and associated deaths (see Fig. 2). However, Nyberg et al. [28] found that the Omicron variant was much less severe and lethal than the Delta variant.

Our starting point in this investigation is to examine Covid-19 death rates by age. These are shown in Fig. 5, where Covid-19 death rates are plotted separately against age for the first and second waves. Wave 1 (solid red line) runs for 26 weeks from March to August 2020 and Wave 2 (dashed red line) a further 26 weeks from September 2020 to February 2021¹³—by which time vaccination was just starting to reduce death rates in the highest age groups. For comparison, Fig. 5 also includes two further curves: all-cause death rates by age in a pre-pandemic year, 2018 (dotted lines), for which we have both accurate exposures and detailed cause-of-death data; and all-cause death rates, with deaths from external causes excluded (black line). All-cause death rates excluding external causes for 2015, 2016 and 2017 are also included for comparison (orange, blue and green lines).

Once external causes have been excluded, we can see a striking, near-parallel relationship between (log) death rates from all causes in a normal year and Covid-19 death rates in both Waves 1 and 2.¹⁴ It can be seen that the pre-Covid all-cause death rate curves for 2015 to 2018 are almost identical meaning that the near-parallel relationship is not sensitive to the choice of ‘normal’ pre-Covid year. The key implication of this is that a birth cohort’s ability to fight a Covid-19 infection is linked to that cohort’s ‘biological frailty’ (defined earlier as its 1-year all-cause death rate).

We now build on this initial link between Covid-19 death rates and biological frailty in two ways. First, at any given age, there is evidence of significant heterogeneity in death rates. For example, within the same age group, some socio-economic groups experience higher death rates than other groups (see, e.g., Wen et al. [44], [45]¹⁵, and Cairns et al. [7], and references therein). This reflects, for example, differences between subgroups in the prevalence of various health-related risk factors. For instance, more deprived areas will typically have a higher proportion of people who smoke. In turn, this leads to higher death rates from smoking-related diseases. So, we extend what we infer at the aggregate level from Fig. 5 to include subgroups. Thus, we postulate that Covid-19 death rates will be proportional to all-cause death rates by age and by socio-economic or other subgroup.

Second, we know that Covid-19 infection rates vary significantly between socio-economic subgroups and between age groups. Everything else being equal, if the infection rate is doubled, then we would anticipate that the Covid-19 death rate would also double approximately.

Building on this, we can decompose Covid-19 death rates (annualised) as:where, for subgroup \(i\) at age \(x\), We need to be careful about interpreting the various components of Eq. (1) in relation to the measurement period, \(\tau\). In particular, the proportion of the total population who die as a result of an infection during this time period is \(1-{\text{exp}}\left(-{m}_{C}(i,x)\right)\). Annualisation (e.g., as shown in Fig. 5) implies that we assume that new infections continue to occur for a full year at the same rate.

Subgroups might be based on a number of different geodemographic measures, e.g., the Index of Multiple Deprivation (IMD),¹⁶ ethnic group, employment group, region and urban–rural location. It is well known (see, e.g., Wen et al. [45] and references therein) that, at a given fixed age \(x\), average all-cause mortality in these subgroups varies considerably. Cairns et al. [7] find that most of this variation in pre-Covid all-cause mortality was due to socio-economics factors, implying that region or other locational identifiers had relatively little additional impact. This is a very important finding which we utilise below.

The infection rate, \(IR\left(i,x\right)\), also varies in potentially different ways between subgroups. The pattern of variation in infection rates between socio-economic subgroups might be quite different from the pattern of variation in all-cause death rates in the same subgroups: for example, infection rates show significant dependence on regional and other factors in a way that is not evident in all-cause death rates (e.g., Ward et al. [43], and, by inference, regional variation in Fig. 2 and urban–rural variation in Fig. 3).

Next, we define the ‘infection fatality rate’, a quantity that has received considerable attention since the early stages of the pandemic (see, e.g., Verity et al. [41]):which measures the (approximate) proportion of people in group \(i\) at age \(x\) who are newly infected and who subsequently die from Covid-19. This implies that Eq. (1) can be rewritten

Further, note that the IFR, as defined in Eq. (2), is a ‘rate’, from which we can derive the (exact) probability of death from Covid-19, given an individual has just been infected, as \(1-{\text{exp}}\left(-IFR(i,x)\right)\).

We are now in a position to make a formal statement of the Proportionality Hypothesis. There are two versions (subgroup and aggregate) depending on the granularity of the data that are available.

The subgroup Proportionality Hypothesis states that:

PH_SG: The Covid-19 infection fatality rate, \(IFR\left(i,x\right)\), is approximately proportional to the all-cause death rate, \({m}_{A}\left(i,x\right)\), by both age and subgroup.

PH_SG can also be expressed in two alternative, but equivalent, ways: In Sects. 4 and 5, we will consider the aggregate Proportionality Hypothesis.¹⁷ This version of the hypothesis is of interest and relevance when suitable data by subgroup are not available. In this case, we can still model Covid-19 death rates by age, but at the national rather than subgroup level. In this case, Eq. (1) simplifies towhere \(\widetilde{IR}(x)\) is the annualised infection rate by age, \(IR(x)/\tau\).

The aggregate Proportionality Hypothesis is then formally stated as:

PH_A: The Covid-19 infection fatality rate, \(IFR\left(x\right)\), is approximately proportional to the all-cause death rate, \({m}_{A}\left(x\right)\), by age.

PH_A can also be expressed in two alternative, but equivalent, ways: Further, if the annualised infection rate, \(\widetilde{IR}(x)\), is approximately constant across ages, then the Covid-19 death rate, \({m}_{C}\left(x\right)\), will be proportional to \({m}_{A}(x)\)—which is what Fig. 5 shows for ages above 40. However, there is no guarantee that the infection rate will always be constant across ages. For this reason, we decided to express the primary version of the Proportionality Hypothesis in terms of the proportionality between the infection fatality rate and the all-cause death rate, since this does not require the constancy of the infection rate across ages to hold.

Assessing the Proportionality Hypothesis is cleanest in Phase 1, since vaccinations had not yet commenced and there were few, if any, reinfections. We now look at two sources of data that, in combination, demonstrate consistency with the subgroup Proportionality Hypothesis, PH_SG in Phase 1.

Phase 2 covers the statistically much more complex period during which a variety of new vaccines were developed and rolled out across the population, new variants emerged (see Fig. 4), improved treatments for severe cases were developed and many in the population caught Covid-19 for a second or third time.

We can now assess the Proportionality Hypothesis in Phase 2. As discussed earlier (Sect. 3.3), in Phase 1 of the pandemic, it was possible to establish that death rates by deprivation level were consistent with the subgroup Proportionality Hypothesis, due to the relative simplicity of the data (i.e., no vaccinations and one dominant variant). In Phases 2 and 3, the data are much more complex.

Let us start with the infection fatality rate (IFR) at the level of the individual. It will depend on several factors: This would then allow us to generalise our definition of relative frailty (by generalising the formulation in Eqs. (1) and (2)) to

In this equation, we have introduced a time element, \(t\), to the IFR that is evident at the aggregate (national) level in Fig. 11. Additionally, the socio-economic subgroups, \(i\), are further subdivided to reflect the additional factors or characteristics listed in the second bullet point above and these are represented in Eq. (5) by the vector \(\theta\).

For this generalised version of the subgroup Proportionality Hypothesis to be true, we would have to establish that, for each set of characteristics \(\theta\), relative frailty is approximately constant across ages and socio-economic subgroups. Unfortunately, data from the ONS are not available at this level of granularity (e.g., infection rates and death counts by socio-economic group and vaccination status are not available). This lack of detail is particularly relevant during Phase 2, when the proportions in the different vaccination groups were changing significantly from month to month (as shown in Fig. 7). This means that we are not able to assess the subgroup Proportionality Hypothesis in Phase 2 (or indeed Phase 3).

We do, however, have data at the national level throughout all three phases and so this does allow us to calculate the aggregate version of relative frailty:where there is no subdivision by socio-economic subgroup, by vaccination status, or other characteristics. RF is shown in Fig. 13. It declined substantially over time (since each line mimics the shape of the corresponding IFR for each age group), but it stabilised from the middle of 2022.

We therefore have the data to assess the aggregate Proportionality Hypothesis. Specifically, we can see from Fig. 13 that relative frailty had some dependence on age in the last quarter of 2020 before vaccination began, with a steady age gradient (e.g., in December 2020, the brown line for ages 75–84 is higher than the orange line for ages 35–44).

The strict age gradient disappeared in the middle in 2021 during the vaccine rollout (e.g., the brown line is no longer consistently above the orange line), but then re-emerged during the early part of 2022, following the full rollout of the booster vaccines. It is clear that vaccination had a significant impact on relative frailty, mimicking the IFR discussion above. But, without more granular data, it is difficult to disentangle precisely the impact of vaccinations from the impact of new variants and the development of new treatments.²⁶ So, although there are some interesting comments to be made on relative frailty, the lack of data granularity means that we cannot formally test for the continuing validity of the aggregate Proportionality Hypothesis during Phase 2 in the way that we were able to do during Phase 1.

Nevertheless, an alternative and powerful graphical illustration of the aggregate Proportionality Hypothesis can be seen in Fig. 14. In the left-hand panel, we plot both the IFR in December 2020 (solid red line) and the all-cause death rate (solid black line) in a pre-Covid year (2018) for ages above 40. The plot shows an approximately parallel relationship between the IFR by age at the end of Phase 1 and the all-cause death rate by age (on a log scale). This mimics what we saw in Fig. 5, but now with the effect of the infection rate filtered out. It is this approximate parallel relationship that justifies one of the ways we express the aggregate Proportionality Hypothesis, namely that infection fatality rates are approximately proportional to all-cause death rates by age.

In the right-hand panel, we plot RF in December 2020 (solid blue line) and the all-cause death rate (solid black line) in a pre-Covid year (2018) for ages above 40. The plot shows the relative flatness of the RF curve across ages at the end of Phase 1 compared with the all-cause death rate curve.²⁷ It is this approximate constancy of the RF curve that justifies another way in which we express the aggregate Proportionality Hypothesis, namely that relative frailty is approximately constant across ages (relative to the dependence of all-cause death rates on age).

Data from the middle of 2022 onwards can be seen towards the right-hand sides of Figs. 2, 7, 8, 9, 10, 11, 12 and 13. A key feature of these plots is the cyclical nature of infection rates, hospitalisations and death rates. The wave amplitudes are much lower than the initial waves of the pandemic and the troughs are higher. Also, trends flatten out for some variables (e.g., relative frailty in Fig. 13) as the virus transitioned from a pandemic to an endemic state.

Figure 7 also suggests that most of the adult population at all ages had either been vaccinated (with at least three doses) or infected with Covid resulting in a very high percentage carrying antibodies. During Phase 2, there was a rebalancing between age groups, discussed in Sect. 4, in terms of antibody prevalence with a resulting significant impact on the age distribution of infections, hospitalisations and deaths. In Phase 3, the high level of antibody prevalence makes it reasonable to assume that all age groups were approximately the same in terms of their vulnerability to infection without having to account for (as, for example, in early 2021) differing proportions of vaccinated and unvaccinated (or antibody positive and negative).

In younger age groups, there was a higher proportion of people who had not been vaccinated, but who had been infected compared to older age groups. By contrast, in the older age groups, there was a higher proportion of people who had had at least three vaccines—and it seems that it was the third booster vaccine that kept antibody levels high for much longer. Either way, from mid-2022, antibodies in the population stayed high for the remainder of the period covered by the data. A consequence is that relative frailty was much lower in Phase 3 than in Phase 1 (see Figure 13, right-hand panel), reflecting the high levels of prior infections and/or vaccinations as well as improved treatments for serious cases of Covid-19. This therefore provides a good basis for assessing the Proportionality Hypothesis in Phase 3.

Strong support for the aggregate Proportionality Hypothesis in Phase 3 comes from both Figs. 13 and 14. Figure 13 shows that, after July 2022, relative frailty for each age group remains reasonably constant (i.e., has no discernible trend), although there is some volatility across different months. In Fig. 14, the dashed red and blue lines (for July 2022) are approximately parallel to the solid red and blue lines (for December 2020)—although at a lower level—so the same conclusions as in Sect. 4.2 hold, namely that infection fatality rates are approximately proportional to all-cause death rates by age (left-hand panel) and relative frailty is approximately constant across ages (right-hand panel).

As with Phase 2, the data in Phase 3 lack the granularity to allow us to check the validity of the subgroup Proportionality Hypothesis across different socio-economic groups or by vaccination and infection status.

The standard LC model for all-cause death rates is \(\text{log }m\left(t,x\right)={\alpha }_{x}+{\beta }_{x}{\kappa }_{t}+\epsilon (t,x)\), the sum of age and age-period terms plus \(\epsilon (t,x)\), a set of independent, zero-mean error terms. If we first assume a constant Covid-19 infection rate and constant relative frailty across all ages, then the Proportionality Hypothesis would result in an all-cause death rate (including Covid-19) of \(\text{log }m\left(t,x\right)={\alpha }_{x}+{\beta }_{x}{\kappa }_{t}+\delta\). The additional \(\delta\) term is consistent with what we see in Fig. 5, but, even with constant infection rates, this does not capture the time dependency. More generally, in the pandemic setting and being mindful of the Proportionality Hypothesis, the LC model needs to be modified by adding at least one further age-period term³³:

The interpretation of \({I}_{x}\) and \({\delta }_{t}\) and justification for this form is as follows. We start with the aggregate Proportionality Hypothesis (Eq. (4)) modified to incorporate a time dimensionwhere \({\widetilde{m}}_{A}\left(t,x\right)\) represents the underlying all-cause death rate in the absence of Covid. The all-cause death rate including Covid-19 is then \(m(t,x)=\widetilde{m}_{A}(t,x)\big\{1+\widetilde{IR}(t,x)RF(t,x)\big\}\). It follows that

Now suppose that we can write \(\widetilde{IR}\left(t,x\right)={I}_{x}^{1}{\delta }_{t}^{1}\). This assumes that the relative age distribution of infections remains stable over time (as a first approximation), while \({\delta }_{t}^{1}\) models the waves of the pandemic over time (e.g., Figs. 2 and 8). Similarly, suppose that we can write \(RF\left(t,x\right)={I}_{x}^{2}{\delta }_{t}^{2}\), where and \({I}_{x}^{2}\) models the relative variation by age in the relative frailty (Fig. 14), and \({\delta }_{t}^{2}\) captures the decline in the relative frailty over time (Fig. 13). The net result of this is that \(\text{log }m\left(t,x\right)\approx {\alpha }_{x}+{\beta }_{x}{\kappa }_{t}+{I}_{x}{\delta }_{t}+\epsilon (t,x)\) where \({I}_{x}={I}_{x}^{1}{I}_{x}^{2}\), and \({\delta }_{t}={\delta }_{t}^{1}{\delta }_{t}^{2}\),

It might be that more than one additional age-period component is required to capture properly the impact of Covid-19 on mortality rates. The assumptions in the preceding paragraph require stability in the relative age-profile of infections and relative frailty. But we saw (Figs. 8 and 13) that these both tilted one way and then the other during 2021, implying that more than one additional age-period component might be necessary. This would be particularly relevant if (as in van Berkum et al. [39]) mortality rates are being modelled on a weekly basis rather than annual and over the full course of the pandemic.

We will take a simple version of the CBD-X models from Dowd et al. [14] without a cohort effect:

This model can capture some of the Covid-19 dynamics in a way that is consistent with the Proportionality Hypothesis. With constant infection rates and relative frailty across ages, the resulting parallel shift (as in the LC model) would be captured by the \({\kappa }_{1t}\) period effect, while the \({\kappa }_{2t}\) period effect in combination with the age effect \((x-\overline{x })\) picks up some of the variation by age in infection rates and relative frailty (as the latter varies through Phase 2). But an additional age-period component similar to the modified LC model (\({I}_{x}{\delta }_{t}\)) might be required if the linear age effect turns out to be insufficient.

Our discussion in the preceding sections suggests the following: when modelling an extreme pandemic-related mortality event, it would be inappropriate to simply model this by adding a fixed percentage to regular death rates.³⁵ Instead, stochastic models need to be developed which allow for significant variation within a country, for example: at regional and sub-regional level; between urban and rural areas; between different socio-economic groups; and between different age groups. This variation will introduce a form of risk diversification, and insurers whose liabilities are well diversified across the groups above should benefit from that diversification through a lower provision for extremes. Conversely, an insurer that holds a more concentrated group of lives (e.g., a high percentage located in London and from a particular socio-economic group) might be expected to post higher reserves against extreme mortality events.

A model that assumes all groups experience the same magnitude of mortality shock effectively assumes that all groups are perfectly correlated, but the Covid-19 pandemic has proved otherwise. The challenge here concerns how much credit should be given to a well-diversified portfolio of lives, and, of course, we really have very little data with which to calibrate a model that allows for variation between groups. This remains an open question, but it should not be ignored.

The last two decades have a seen a small but steady issuance of mortality catastrophe bonds, mainly by insurers and reinsurers; they have been used principally as hedging instruments by the same issuers. Many of these use national mortality indices to determine the payments to the issuer and bondholders, for example, parametric mortality catastrophe bonds (as discussed in Blake et al. [5], or Blake and Cairns, [4]). The linkage to national mortality indices assumes a high correlation (in an extreme scenario) between the national mortality index and the (re)insurer’s own portfolio of lives and associated sums-at-risk. However, the Covid-19 pandemic has revealed significant variation between regions and other groups, and so it is not clear that the assumption of a high correlation between portfolio and national mortality is valid. For example, based on Fig. 3, national mortality might have been 15% higher than normal, but an insurer that is concentrated in highly urban areas might have experienced as much as 25% higher mortality.

This might cause issuers of mortality catastrophe bonds to prefer linkage to a customised mortality index (i.e., based on their own mortality experience). However, an alternative would be to subdivide the national mortality index into regional and socio-economic mortality indices,³⁶ in a way that mimics the current linkage in a parametric bond to several national mortality indices.³⁷