Josep Lledó,Jose M. Pavía,Jorge Sánchez Salas
An alternative approach to manage mortality catastrophe risks under Solvency II
Solvency Capital Requirements,Best Estimate Liabilities,Loaded probabilities,Pandemics
The Solvency II Directive, in its standard formula, proposes that a permanent 15% increase should be applied to mortality rates for covering normal deviations of mortality and that the rates should be increased uniformly over just the following year by 1.5‰ to cover catastrophic mortality deviations. The latter increase means disproportionally raising the rates for some ages, especially young people, with hardly any modification for others. In this paper we show, using data from Spain, the inadequacy of these loadings and this timing (for 1 year only) in protecting against a pandemic. As a solution, we propose a new methodology that allows the catastrophic risk of life to be modelled/calibrated through two parameters: a parameter to mainly cover catastrophic risks linked to specific extreme disasters (such as earthquakes, hurricanes or nuclear explosions) and another parameter to mainly cover catastrophic risks linked to pandemics. After establishing the relationships between these parameters in three different scenarios where the aggregate levels of risk/mortality associated with the Solvency II catastrophic shock are kept constant, we study its effect in terms of Solvency Capital Requirements (SCR) on a real life insurance portfolio. The results obtained show that the new methodology leads to SCRs, over total BEL, less dependent on the age structure of the portfolio.
An alternative approach to manage mortality catastrophe risks under Solvency II

The Solvency II Directive, in its standard formula, proposes that a permanent 15% increase should be applied to mortality rates for covering normal deviations of mortality and that the rates should be increased uniformly over just the following year by 1.5‰ to cover catastrophic mortality deviations. The latter increase means disproportionally raising the rates for some ages, especially young people, with hardly any modification for others. In this paper we show, using data from Spain, the inadequacy of these loadings and this timing (for 1 year only) in protecting against a pandemic. As a solution, we propose a new methodology that allows the catastrophic risk of life to be modelled/calibrated through two parameters: a parameter to mainly cover catastrophic risks linked to specific extreme disasters (such as earthquakes, hurricanes or nuclear explosions) and another parameter to mainly cover catastrophic risks linked to pandemics. After establishing the relationships between these parameters in three different scenarios where the aggregate levels of risk/mortality associated with the Solvency II catastrophic shock are kept constant, we study its effect in terms of Solvency Capital Requirements (SCR) on a real life insurance portfolio. The results obtained show that the new methodology leads to SCRs, over total BEL, less dependent on the age structure of the portfolio.

Since the Solvency II Directive became fully applicable on 1 January, 2016, the monitoring and explicit quantifying of the risks inherent in the insurance business has become one of the most important activities in the sector. The new framework promotes transparency, comparability and competitiveness in the insurance sector and, by introducing prudential requirements tailored to the specific risks an insurer bears, helps actuaries and risk managers to know how the potential different risks could impact the company. Currently, issues such as an increase in costs, an escalation of interest rates or even the risk of a pandemic need to be considered and eventually quantified. All these risks are handled individually, keeping in mind their relationships, through a modular approach whose main economic indicators are the Solvency Capital Requirements (hereinafter, SCRs). Each SCR is computed as the change that the net value of assets minus liabilities would experience between a baseline scenario and a stressed scenario defined by the related risk. SCRs are set at levels that guarantee that insurers and reinsurers can meet their obligations to policyholders and beneficiaries over the following 12 months with a 0.995 probability. Default levels of risk were defined by the European Insurance and Occupational Pensions Authority (EIOPA) through different consultations, called QIS (e.g. CEIOPS 2010), with the main European insurers. Currently, most European companies and their subsidiaries outside Europe follow the standard guidelines set by the regulations at the end of each accounting period. The possibility of using alternatives to those proposed in the regulations, through the so-called internal models, has been an exception rarely used and, if used, they have been almost always implemented by large corporations. Despite them being open to all firms, the cost and complexity of developing internal models deter smaller companies from using them. Since the beginning of 2016, multiple events have occurred that offer the opportunity to assess whether the different sub-risks were correctly calibrated. Current inflation in costs and falling interest rates are two examples. But without a doubt, in recent times the COVID-19 pandemic has been the most relevant event linked to the so-called catastrophic life-risk sub-module, which represents one of the most important underwriting risks affecting any company that sells life insurance. This paper analyses and looks more in-depth at how catastrophic life risk is managed under the current regulation. The aim of this paper is twofold. On the one hand, the paper sets out to evaluate, for the first time in the literature, the catastrophic life risk defined by Solvency II and the limitations in its theoretical definition. We note that the level established by the regulations is insufficient for some ages and excessive for others and also that the duration of the risk does not correspond to the real time that a catastrophic event of pandemic nature can last. On the other hand, we propose a methodology that allows correction/mitigation of the limitations of the current regulations and study their impact on a real life insurance portfolio. The rest of the paper is structured as follows. “Background” section contextualizes the catastrophic life shock in the framework of Solvency II and provides a description of catastrophic events. “The life catastrophe risk sub-module under Solvency II” section explains the main limitations of the current shock, specifying its implicit assumptions, and shows its inadequacy with real data. “Methodological alternative” section proposes a new approach to deal with these through a combination of a fixed and a variable component and derives the relations between the components in three different scenarios that, for comparisons purposes, maintain the aggregate levels of risks at the levels established by Solvency II. “Empirical comparisons” section shows the impact of applying the new proposal in a real portfolio. Finally, “Discussion and concluding remarks” section summarizes and discusses findings, proposing new lines of research.

Mortality is a demographic phenomenon that evolves smoothly and continuously over time, except when extreme catastrophes or a medical breakthrough occur. According to Solvency II, “[c]atastrophe risk stems from extreme or irregular events whose effects are not sufficiently captured in the other life underwriting risk sub-modules. Examples could be a pandemic event or a nuclear explosion” (CEIOPS 2010, p. 161). To cover the sub-risk of mortality, the standard model imposes as default a permanent 15% increase in mortality rates for each age and each policy where the payment of benefits (either lump sum or multiple payments) is contingent on mortality risk (CEIOPS 2010, p. 149), and likewise to use a (permanent) 20% decrease in mortality rates for each age and each policy where the payment of benefits (either lump sum or multiple payments) is contingent on longevity risk (CEIOPS 2010, p. 151), for the risk of longevity. For the life catastrophe risk sub-module (CEIOPS 2010, p. 162), an absolute increase was established of 1.5‰ in the rate of policyholders dying over the following year. The catastrophic loading sub-risk applied is therefore homogeneous, regardless of the level of maturity or the characteristics (composition) of a given insurance portfolio. From our perspective this is questionable, since it is assumed that any catastrophic risk affects all companies/ages in a homogeneous way. This means that exactly the same capital amount in absolute terms would be required for two equal portfolios that only differ in their age structure, despite the fact that their differences in relative terms could be enormous. For example, if we take the portfolio that we analyse in “Empirical comparisons” section and assume that all the policies correspond to either 25-year-old policyholders or to 65-year-old insured, the first portfolio would show the SCR under Solvency II to represent 313% of the total Best Estimate Liabilities (hereinafter, BEL) and the second portfolio a mere 25%. Our main point is that the impacts of catastrophic risk under Solvency II are not fairly distributed among ages, as many catastrophic risks are age dependent, and that a more balanced distribution in terms of age could be reached. The SCR of the catastrophic life risk is calculated as the change in the net value of assets less liabilities, , after an increase in absolute terms of the mortality rate of the insured during the following year of 1.5 per thousand (only applicable to policies which are contingent on mortality): . In terms of Solvency II, the difference is calculated between the BEL of a scenario that considers for each policy an increase in mortality risk of 1.5 individuals per thousand and the BEL of the base scenario (without catastrophic risk). This calculation is done for any life-risk product (insurance covering death) for all ages and for both sexes. The occurrence of a catastrophic risk can cause considerable human losses. These events can affect a certain geographical area (e.g. a volcanic eruption) or an entire population (e.g. a pandemic). Whatever the situation, it causes a significant growth in claims and, therefore, an increase in technical provisions. However, exposure to this risk is not homogeneous among insurance companies, nor is it similar for all countries. In addition to differences inherent in the type of catastrophe (which may depend, for example, on the size of the population living relatively close to a nuclear power plant or in an area with high seismic activity), there are also differences in terms of coverage by public programmes or entities. In some countries there are numerous state programmes whose purpose is to cover extraordinary events (OECD 2021, pp. 26–29), although their coverage is usually quite limited, covering just a small number of risks. In the structure of an insurance company, in the life sub-module, a catastrophic risk occurs when a high number of deaths is observed due to an extraordinary event. This can be caused, for instance, by a terrorist attack (such as the September 11 attacks) or by a natural disaster. For example, in terms of deaths, in Europe 1672 reported disasters caused 159,438 deaths during the 1970–2019 period, with floods (38%) and storms (32%) being the most reported causes of disasters. However, extreme temperatures led to the highest proportion (93%) of deaths, impacting significantly more on older people (WMO 2019). The heat waves observed in Europe during the summer of 2022 have also led to an increase in deaths. In addition to extreme natural events, numerous pandemics have been recorded over time and all over the world since the sixth century (Benedictow 2004; Björk 2018). The 1918 Great Influenza epidemic, commonly known as the Spanish Flu, is among the first significant pandemics by number of deaths in recorded cases. In this pandemic, between 50 and 100 million people died and nearly a third of the global population was infected (Spreeuwenberg et al. 2018). The Spanish Flu was an exceptionally deadly global influenza pandemic that relatively impacted more in young adults and during winter, when the climatologic conditions were more severe (Pavía and Lledó 2022). More recently, the global HIV/AIDS epidemic is estimated to have already caused the death of 36.3 million people worldwide (UNAIDS 2022). Finally, we come to the widely known COVID-19 pandemic which, to date, has caused more than 6.83 million deaths around the world. In most cases, however, many of those who died from COVID-19 would have died anyway due to other causes of death in the relatively near future (Sasson 2021) as a consequence of their existing frailties or comorbidities (Cairns et al. 2020). Many studies have analysed the COVID-19 impact in different countries (Bauer et al. 2021; Chan et al. 2021; Islam et al. 2021; Undurraga et al. 2021), with all of them pointing to an age-related dependence in the increase of death rates despite observing some differences in age and sex dependency of COVID-19 mortality. Indeed, according to Bauer et al. (2021): “An exponential increase with age is a good model to describe and analyse both COVID-19 and all-cause mortality above 40 years old, where almost all COVID-19 deaths occur”. The impact of the COVID-19 pandemic on the insurance sector is also being addressed and studied. In this vein, Schnürch et al. (2022) analyse the disruption of the decreasing mortality trend present in many mortality modelling applications, especially in life insurance, while Puławska (2021) scrutinizes the impact of the pandemic on asset performance and Nayak and Bhattacharyya (2020) study its effects on key lines of business, such as health insurance. Nevertheless, few studies have analysed the impact of catastrophic risks in insurance portfolios. Fisher et al. (2001) define management of these risks as a continuous learning process and list five steps for identifying, measuring, pricing, controlling and evaluating them. Subsequently, during the development of Solvency II, in the years of definition and evaluation of its different risk modules, some authors pointed out that the mortality catastrophe shock scenario of the standard formula was developed with a focus on pandemic risks (Kraut and Ritcher 2015). Kraut and Richter discuss the standard formula and make proposals on how the capital requirement calculation could be modified to better reflect the characteristics of extreme mortality risks. They advocate for replacing the life catastrophe risk sub-module with a generic internal (partial) model to take into account the characteristics of the different risks that affect the catastrophic life risk. Other studies have analysed catastrophic risks from a financial point of view (Shao et al. 2015) and in non-life insurance (Hürlimann 2013). In this paper, we carry out a deeper analysis focussed on assessing the implicit assumptions of the mortality catastrophe shock scenario under the current Solvency II recommendations and on proposing some methodological alternatives.

The Solvency II catastrophic life shock must include the excess mortality caused by an extreme event such as an earthquake, a hurricane or a pandemic and proposes to cover it by assuming an absolute increase in the rate of dying over the following year of 1.5‰. The nature of the different types of catastrophic events is, however, diverse and its impact on society and on the insurance company will also be diverse. On the one hand, we find that events such as an earthquake or a nuclear explosion are punctual events that affect all population groups in an (almost) uniform way, regardless of the classification variable used (age, gender, socioeconomic level, …). The impact of these events, although it may be high in absolute terms with regard to the number of deaths, is quite limited geographically and has a probability of occurrence that varies significantly between different areas of the planet. In the case of nuclear explosions, moreover, due to radiation exposure there are also long-term impacts that are expected to affect non-uniformly to different ages. On the other hand, other types of events, such as a pandemic, tend to have an uneven impact on different population groups. Indeed, most influenza outbreaks disproportionately kill more of the youngest and oldest in the population, with a higher survival rate for age groups in-between (Gagnon et al. 2013). The 1918 Great Influenza epidemic had an unusually high mortality for young adults and yet the COVID-19 has hardly affected infants directly, however. In the case of COVID-19, despite age not being the unique factor (D'Ascanio et al. 2021), it causes numerous biological changes in the immune system that are mainly linked to age-related illnesses and susceptibility to infectious diseases (Varnica et al. 2021). Mortality from catastrophic risks, therefore, seems to have a constant component, which affects all age groups absolutely and fairly uniformly, and a variable component that depends largely on the age variable. The use of the absolute value of 1.5 per thousand as excess mortality only during the first year and the same value for all ages implies assuming a series of implicit hypotheses (limitations), among which are (i) that the mortality of the second exercise (year two) and successive years is the same as the base mortality, that is, once the catastrophic event has elapsed in year , it is assumed that mortality returns to pre-shock levels in , and (ii) that catastrophic risk affects individuals absolutely regardless of age, assuming that on average 1.5 more individuals will die per thousand for each of the ages regardless of the fact that 1.5‰ has a different weight in the risk of each age. Undoubtedly, the suitability of both hypotheses is debatable and deserves to be studied in more detail.

To study the appropriateness of hypothesis (i), we analyse the evolution of mortality during the years following an epidemic occurring in year , focussing mainly on the immediately ensuing years. The data regarding the lasting effects of radiation after a nuclear explosion (e.g. Chernobyl or Fukushima) are less available. To assess hypothesis (ii), we measure the number of times that the Solvency II catastrophic shock, 0.0015, represents over the total probability of death for each age in a given year. The unsuitability of both hypotheses is shown using data from Spain: the general population statistics for Spain from 1908 to 2021 to assess hypothesis (i) and the baseline Spanish loaded life table to be applied for pricing life-risk products to evaluate hypothesis (ii). On the one hand, to test the first hypothesis, we consider the death probabilities, , recorded for each age during the years (with ) following the outbreak of two extreme events of pandemic type (the Great Influenza epidemic and the COVID-19 pandemic) and compute the excesses (defects) of mortality () by comparing them to the average of death probabilities recorded during years previous to : . On the other hand, to test the second hypothesis, we simply calculate the number of times that the excess shock of mortality proposed by Solvency II represents in terms of for each age : . Total population stocks from 1908 to 2021 as well as death statistics from 1908 to 2020 have been obtained from the Human Mortality Database (2022), whose primary source of information for Spain is the Spanish Official Statistical Agency (INE). The ages, , available in this source span the age range from 0 to 110 years. The statistics for deaths corresponding to 2021 have been obtained directly from INE. Since the beginning of the COVID-19 pandemic, the INE has compiled each week the number of accumulated deaths registered for any cause in five-year age groups for ages ranging between 0 and 89 years of age (INE 2022). The number of deaths of each age during the year 2021 has been estimated assuming within each five-year group the same mortality distributions for 2021 as registered for the year 2020. In both cases, the data are differentiated by sex. From these data, the probabilities of death by age and year have been calculated, and from these the statistics , with , which are presented in Figs. 1 and 2. Given that under normal circumstances mortality rates decrease, taking represents a conservative approach that underestimates mortality increases. Indeed, even stronger evidence to support our point is observed taking or All the computations have been performed using the free statistical software R, version 4.1.0 (R Core Team 2021). Figure 1 shows the distribution of for by sex for the cases of the Great Influenza epidemic (upper panel) and the COVID-19 pandemic (lower panel), for and , respectively. Although it is not simple to date outbreaks and declines of infectious diseases (Veres-Ferrer and Pavía 2021), we follow the consensus and assume as outbreak years 1918 and 2020, respectively. As can be seen in the upper panels of Fig. 1, mortality in 1919, 1920 and 1921 registered levels higher than the average of the 3 years (1915, 1916 and 1917) prior to the outbreak of the epidemic. For example, mortality for the years 1919 and 1920 was between 10 and 20% higher for young ages and between 5 and 10% for adult ages. The differences for the year 1921 are notably reduced. Regarding COVID-19 (lower panels), mortality in 2021, with the exception of the age of 0 years and the age group around 50 years, did not decrease to the levels prior to the pandemic, but rather shows an excess of mortality in the range of 5–10%. The differences between the sexes are not significant in either pandemic, although mortality levels increase at a slower pace in women than in men. At the time of carrying out this study, the data show that Spain has still not yet recovered the mortality levels prior to the outbreak of COVID-19, an issue that should not be particularly surprising since the data available for the Great Influenza epidemic show that a hundred years ago it took about three years to return to earlier mortality levels, as can be seen in Fig. 2. Figure 2 shows, for the 12 years after 1918 (from 1919 to 1931), the mean differences for , by decennial age groups , where , and . The death probabilities eventually continued to progressively reduce, once the disruption caused by this epidemic diminished. However, recovery of mortality in subsequent years is not uniform across age groups and by sex. Mortality levels did recover in 1920–1921 for adult age groups (over 40 years of age), while it was not until 1922–1923 that younger age groups recorded lower mortality than before the epidemic. That said, although mortality recovers later in young people, this improvement is more pronounced for the years analysed. By sex, women recover previous levels before men. In any case, in view of the results, it can be concluded that the assumption of hypothesis (i) does not seem reasonable. After an epidemic, mortality seems to require more than a year to return to its previous levels. This is in line with the observation/conjecture stated by Schnürch et al. (2022). Regarding hypothesis (ii), Solvency II establishes a uniform and constant increase of 1.5 per thousand of qx for all ages. This implies a very heterogeneous relative impact by age and gender. Its impact is much greater in women and for people between 3 and 40 years old and practically nil for older people. Figure 3 shows the results of calculating the number of times that a 0.0015 mortality rate represents with respect to the base probability, θx = 0.0015/qx, taking as reference the mortality table PASEM2019_second_order (BOE, 2020), recommended by the Spanish supervisor to be applied in life-risk insurance. Figure 3 displays both the base and the catastrophic risk (Solvency II) loaded probabilities in log-scale (main axis) as well as the values of the θx statistic (secondary axis). Figure 3 shows that for young ages the shock recommended by Solvency II can represent up to 30 (40) times the mortality of the insurance group in men (women). For ages between 20 and 30 years, the number of times is around 5 (15) times for men (women), being practically null for ages over 50 years. The analysis carried out shows a clear decompensation in the catastrophic event by age that does not agree with the evidence presented (see Figures 1 and 2) during the last two pandemics.

In order to overcome the identified issues (the potential inappropriateness of the implicit hypotheses assumed in the life catastrophe risk sub-module of Solvency II), we aim to handle this risk using a different methodology: as a combination of a fixed and a variable component. Our proposal combines a fixed part, , that captures the baseline risk which is the same for all ages and makes it possible to collect mortality from specific catastrophic phenomena such as hurricanes or earthquakes, and a variable part, , which depends on age, for catastrophic events that affect (generally) any type of population, such as a pandemic. The variable part, whose parameter we express in relative terms (as a proportion of ), does not necessarily have to be different for each age. As with mortality or longevity risks under Solvency II, it could be specified as a constant proportion of the probability of death at each age : . The combination of both coefficients enables the catastrophic risk to be modelled/calibrated according to each region and the intensity of the type of extreme event. In particular, we consider that each company should use a different value for , depending on the geographical-risk distribution of its portfolio, with the values for the () parameters being more general and less case dependent. In our view, a specific study should be implemented to establish the general levels of risks that would correspond to each region (as a function of the characteristics of the area, for instance, its seismologic activity, relative proximity to a nuclear plant, hurricane susceptibility, etc.) in order for them to be applied in each portfolio, taking into account the relative percentages of insured population affected in the portfolio. In the following three subsections, we establish the relationships for the and values that, in three different scenarios, maintain the aggregate levels of risk/mortality at the levels established by Solvency II. Note that this does not necessarily entail keeping SCRs constant. We consider these constrained scenarios not because we believe the risks should be mechanically maintained in these levels, but to have a benchmark for comparison. The constrained scenarios considered distribute the Solvency II catastrophic risk between a fixed parameter and a variable parameter with both parameters being mutually dependent, that is, they cannot be jointly identified. The first scenario is where the risks (the death probabilities) are equal to the defaults for all ages (Scenario 1: ). The second scenario is where a constant proportional variable loading is assumed by sex for all ages, keeping the aggregate of risks fixed (Scenario 2: ). And a final scenario is one in which the excess deaths which would cause the shock, being kept constant in a given portfolio, are distributed between ages and sexes under the assumption of a constant proportional variable surcharge (Scenario 3: ). The proposal of a more flexible model (with fixed and variable loadings) is a response to the inadequacy of hypothesis (ii) under normal conditions. In response to the inadequacy of hypothesis (i) in the event of a pandemic, and as long as the probabilities of death are not updated, that is, they do not capture the impact of a pandemic that began in year , our proposal would be to additionally recharge the probabilities of death to be used in the immediately subsequent years, ; with the extra loadings, , to be added to at each age and year after the year of the outbreak, dynamically determined by exploiting the information available on the current and previous pandemics. Note that this extra charge would be conditional on the fact that a pandemic started in year . We address this issue in greater detail in the fourth subsection.

Our first scenario considers that for each age the mortality of the catastrophic life shock remains at the levels of the standard model proposed by Solvency II, but assuming that the new risk is reached as a result of applying a fixed increase, , and a variable one, . In this case, the mathematical equation that relates to the current methodology and the new proposed methodology would be given by (1). In this scenario, a greater or lesser risk is not assumed at each age, but rather a new distribution, which is different from Solvency II, through the new coefficients and . In a simple calculation, from fixed (which will depend on the risk propensity of specific events of the portfolio and it is the same for all ages) the coefficient can be obtained, as a function of , for each age through (2). Note that takes its maximum value when , while when the new methodology will coincide with the current Solvency II methodology, given that This scenario can be interpreted as a re-parametrization of the Solvency II scenario. The new risk specification of this scenario does not resolve the inadequacy of hypothesis (ii) detected in the empirical analysis in the event of a pandemic, since it maintains the probabilities of death for each age of the standard model. In other words, it maintains excess mortality at some ages, especially where is low, and it is not capable of capturing the excess mortality that usually occurs in the event of a pandemic for ages 50 and older. These issues are addressed in Scenario 2.

Scenario 2 considers a situation in which the excess (or defect) of mortality implied by the standard model at each age is distributed among all the ages eligible to take out the insurance, , after assuming that the variable surcharge part is independent of age, i.e. . The equation that, for each sex, equals the respective aggregate risks is given by (3). This equation allows the fixed Solvency II loading to be distributed into a constant and a variable loading to be applied by age keeping constant the sum of death probabilities implied by Solvency II. Working in an independent fashion by sex, Eq. (3) is (probably) the only way of relating the Solvency II fixed loading with a constant and a variable loading by age under the constraint. In aggregate terms, this scenario (in the same fashion as Scenario 1) does not imply a higher risk (a higher sum of the probabilities of death), but rather distributes it via the parameter among all ages. Given fixed, the parameter is calculated, conditional on , using (4). When , takes its maximum value, while when , and again we have the standard model. This scenario allows a parameter to be derived, which keeping the sum of the probabilities of death constant, means the loadings can be distributed more proportionally to the base risks. This distribution makes it possible to reduce the overestimation (underestimation) of the risks at young ages (adults) that the standard model implies in the event of a pandemic, implicit hypothesis (ii). This, from our point of view, should lead to a better measurement of risks.

In the above scenarios, relationships have been calculated between the constant loading, and the variable loading, (, under the assumption of keeping the Solvency II sum of the risks (of the probabilities of death) constant. In this scenario, we consider a situation, where conditional on the portfolio, we study with fixed the relationships between the parameters under the assumption that the figure that will remain constant is the expected number of deaths in the portfolio. In other words, in this scenario, the excesses (deficiencies) of deaths implied by Solvency II for a certain age are transferred to (compensated by) other ages and/or between sexes. Denoting as the number of men and women, with , respectively, with actuarial age in the portfolio and as their probabilities of death, the relations between and can be obtained from Eq. (5). This equation is (probably) the simplest way of relating a constant and a variable loading under the assumption of keeping constant the sum of deaths implied by Solvency II for the portfolio.where, again, a constant parameter has been assumed for all ages. Parameter is related to by Note that Scenario 2 can be viewed as a particular case of this scenario with , treating men and women independently. When , takes its maximum value, while when , and, therefore, this scenario also collapses in Solvency II. In any case, regardless of the values of and verifying (6), in this scenario the expected number of deaths remains the same as that obtained by applying the Solvency II catastrophic shock. The difference is that the deceased are distributed differently by age, as well as there being a redistribution between sexes. How these changes are translated into SCR terms will depend on the specific structure of the portfolio.

Natural disasters tend to have a direct and immediate impact on catastrophic risk. However, other catastrophes can last for a longer time. The empirical analysis carried out in “The life catastrophe risk sub-module under Solvency II” section shows that, in the event of a pandemic, the probabilities of death suffer a shock that extends beyond the year of the outbreak. Solvency II does not address this possibility. Our proposal to negotiate this type of situation is to introduce an additional surcharge, , which would be added to (where without loss of generality we have recovered the possibility of an age-dependent proportional parameter) in the year . This additional surcharge would be calculated based on the information available on the current pandemic and the most similar previous pandemic. Specifically, denoting by (usually with and being case dependent, with resulting in a 5 age-year window) the mean increases recorded by the probabilities of death for ages and during the year of the outbreak and after the most similar past pandemic and by the most current information, , available on the increases registered in the current pandemic, we propose to use , for (usually ), as an additional loading to apply for age . For example, considering the Great Influenza epidemic as the most similar antecedent to the current COVID-19 pandemic, to estimate the excess mortality due to the pandemic to be applied in 2022 at age , we would multiply the average increase registered by the mortality in a given period for ages around in 2021 by the ratio of the mean increases in mortality recorded in the same period between 1920 and 1919. In some ways, this approximation resembles the link ratio method used for reserving in non-life insurance. Obviously, other solutions based on modelling would also be possible. This opens another research line that deserves to be studied in more depth.

In the previous section, a new methodology was proposed to specify the alternative catastrophic risk shock to Solvency II, and its relationship with the standard solution was studied in three different constrained scenarios. Each scenario allows the catastrophic life risk to be modelled/modulated as a combination of a fixed component and a component proportional to the probability of death, assuming the same aggregate levels of risk/mortality as the standard Solvency II formula. The objective of this section is to exemplify each of these alternatives and compare them with the current Solvency II methodology through their application to a real dataset of a Spanish insurance company that markets a classic life-risk product. The dataset consists of 76,102 policies (46,652 men and 28,450 women) and has insurance for all ages that cover the range accepted by the company's underwriting policy, . The policies available in the portfolio have subscription dates between 2000 and 2009 and their main coverage is the risk of death. In exchange for the payment of a premium, the insured cover this risk for a period of one year, a year-term life insurance (), a product that is automatically renewed every year. The insurance ceases to be in force when either the insured decides to cancel the policy or s/he reaches a certain advanced age. The description of the variables and the data as well as more details on the portfolio are available in Lledó and Pavía (2022). The parameter chosen to assess the impact of the new methodology in the three scenarios and to compare with the Solvency II solution is the SCR. The SCR for each scenario and that of Solvency II is calculated as the difference between the BEL resulting from the shock and the base BEL: . The valuation date is set as the end of the accounting period of 2009: 31/12/2009 at 24:00. To calculate the BEL we set constant expenses, , of €25 per policy and we use the so-called PASEM2019 second-order life table (BOE 2020) as the reference mortality table. This table is the one currently recommended by the Spanish insurance supervisor to be applied for reserving processes. The BEL calculation methodology is identical for all regulated insurance companies affected by Solvency II, but the time horizon of the projection of their inflows and outflows depends on the guidelines that the company adopts in the renewals of its policies. (A) The company may project inflows and outflows (taking into account lapse and interest rates) up to the maximum renewal age defined in the underwriting policy if it agrees not to reprice future premiums for its policies. (B1) The company can only project the inflows and outflows up to the renewal date of the policy if it does not waive the repricing of future premiums. In this second case, (B2), the company can also include the so-called tacit renewals, which are those policies with less than 2 months remaining for their renewal. In other words, the company can project this maximum period of two months as well as the following year. At the time of contracting the policy, the company must inform the insured of the evolution of future premiums (which depend, mainly, on age), and this will determine the valuation procedure that can be applied. The choice made by the company determines the result of the BEL. In (A) or (B2) there is the possibility of projecting more than one year, while in (B1) the calculation only takes into account the expected inflows and outflows until the next renewal, less than one year. For the purposes of valuation and results of the SCR, either of the two options (B1) and (B2) would yield similar results. For this study, we choose to use (B1). The technicalities and greater complexities related to option (A), which are similar to those that arise when products with a term of more than 1 year must be valued (multi-term life insurance, , more common in some countries), are addressed in detail in the discussion. In the Appendix, the details of how the BEL and the SCR are calculated under Solvency II and in the three scenarios considered in a regular year (i.e. after assuming are presented. In the case of , the associated SCR coincides with the Solvency II SCR (see equation A.3), while in the case of and it can be expressed as a linear function of and (see equation A.7). The linearity of the SCR in the case of Scenarios 2 and 3 is clearly shown in Table 1, which presents the SCR associated with the analysed portfolio for a selection of values of . Scenario 1 yields the same level of SCR as the standard Solvency II model, given the equivalence of their associated shock risks. Scenario 2, which reveals the inadequacy of hypothesis (ii) in the event of a pandemic, presents in this portfolio SCR levels lower than those of the current regulations. The lower the fixed effect of the parameter , the lower the economic value of the SCR in this portfolio. The result of Scenario 3 is in line with Scenario 2, although with a lower effect in reducing the SCR due to the composition of the analysed portfolio, where the bulk of the insured is concentrated between 40 and 55 years of age. In this portfolio, the use of the new methodology and its specification in the different scenarios, especially Scenario 2, would allow significant savings in terms of SCR even assuming the same aggregate level of catastrophic risk as that used by Solvency II. But, this does not necessarily have to always be the case: SCRs higher than those implied by Solvency II are possible with the new methodology. In order to show this and also disclose more clearly the enormous imbalance that the use of the standard formula can provoke, in Table 2 we present and ratios in our original portfolio and in three newly simulated portfolios (P1, P2 and P3) obtained after artificially modifying the age distribution of the policyholders. In particular, it is assumed that all the insured are either 25 years old or 65 years old in portfolios P1 and P2, respectively, with age distribution having been reversed in portfolio P3. As can be seen in Table 2, in terms of compared to Solvency II (), the new methodology would lead to increases in P1 and P2 and to decreases in P3, even if the aggregate levels of risks (the sum of probabilities of death) are kept constant. Likewise, it would also reduce the differences between the relative effort (measured as ratios) required of the companies when the only difference between portfolios lies in their age distributions. For instance, for , the differences between relative effort to be made in the portfolios P1 and P2 would be reduced from 288 percentage points (312.85–24.66) under Solvency II to 143 (174.71–31.89) points with the proposed methodology. In terms of specific catastrophes (earthquakes, nuclear explosion or hurricanes) and age distributions all the portfolios and/or all the companies are not exposed to the same levels of risk, so using the same parameter, , for all of them does not seem, from our point of view, justified. The is conditioned by the age structure of the company. For example, in a portfolio of adult insured, the ratio is around 30% (depending on the used), whereas in a young portfolio, the ratio exceeds 100% whatever . Note that in these examples the ratios are significantly higher than the ones typically observed in other life portfolios, the main reason being that BEL tends to be small for one-year life insurance risk products since the death benefits for each policy only account for the part of the year until the next renewal (parameter in the Appendix).

The objective of the standard formula is to provide an SCR sufficiently conservative to guarantee the solvency of insurers and the protection of policyholders. In the case of catastrophic risks, however, the current approach is, in our view, extremely conservative if the company has a young portfolio and significantly more relaxed in the opposite case (see Table 2). In this paper, we propose a more balanced and flexible approach by considering both a fixed, constant loading and a relative loading. Our methodological proposal, based on modulating the Solvency II catastrophic risk through the and parameters, is founded on analysis of the behaviour of mortality in the general population in the event of a pandemic, where the examples analysed (Great Influenza epidemic and COVID-19) show that the increase in mortality varies between ages and is not concentrated only in young people. This learning suggests that an age-dependent loading should be considered. Likewise, the analyses performed also show that the mortality rates of different ages are affected differently in the Great Influenza and COVID-19 epidemics, an issue that, added to the sociological changes and medical advances that are experienced over time, advises against using a direct transferring of past data to the future and impels to also include a systematic constant loading. We therefore advocate for using a mixed-approach, where both and are stated as different from zero. Although from the analysed data, one could consider that choosing could be sufficiently conservative, we should comment that our analyses are based on the behaviour of mortality during a pandemic in the general population, which can differ significantly from the mortality in the insured population. However, little is known about the similarity/dissimilarity of the mortality in the general and insured populations under this kind of event. Despite some studies suggesting that COVID-19 has hit hard in large urban areas and in more deprived populations (OECD 2021), we still do not know if this has translated into different relative increases of death for the general and insured populations. More specific analyses are therefore needed in this regard. On other hand, in the previous two sections, we have assumed, explicitly in “Empirical comparisons” section and implicitly in “Methodological alternative” section, that the BEL calculations are carried out by valuing the cash flows during, at most, one year. However, when we have a multi-term life insurance (), the inflows and outflows can be projected for more than a year; up to the maximum subscription age under option (A). In this case, the associated mathematical expressions become significantly more complicated, making it necessary to readapt the formulas. Given that the focus of this study has been on (i) empirically showing the inadequacy of the implicit hypotheses associated with the Solvency II catastrophic life shock and (ii) proposing an alternative methodology to solve the problems associated with them, we do not delve into how the problem should be solved with the new methodology in these more complex situations, but rather we show by way of an example the reasoning to be followed. As an example we develop the financial-actuarial equation equivalent to (3), Scenario 2, for the particular case of a multi-term life insurance with . The complete development of the solutions and their evaluation for the general case, , merits attention in a separate, independent study. As the different reasoning and components involved when are clearer if we identify the life table to be applied in each period univocally, even at the risk of overloading the notation, we denote by the probability of dying at age during year . Hence, assuming that for and , the relationship between risk aggregates under Solvency II and with the new methodology in Scenario 2 with is given by (7).where and denote, respectively, the discount interest rate and the proportion of drop in the portfolio to be applied at time . Although in most products the annual drop in the portfolio is usually assumed to be constant, it could also depend on . In summary, when , for the calculation of the BEL, in addition to the complications associated with the loading in the first year, we would also have to consider other issues, such as drops in the portfolio, the interest rate curve, the revaluation of assets or future premiums (for example, in level premiums). In any case, we consider that similar qualitative conclusions to those reached in this research would have also been attained if we had considered multi-term life insurance policies, or the scheme evaluation (A), since the problem of a constant increase in risk, independent of age, as assumed by Solvency II would also be operating. A new, significantly more complex study would be necessary, nevertheless, to properly quantify this. Likewise, the impact of including extra loadings to incorporate excess mortality in subsequent years after a pandemic outbreak and its impact on also merits further study.