Geographical Diversification and Longevity Risk Mitigation in Annuity Portfolios

This paper provides a method to assess the risk relief deriving from a foreign expansion by a life-insurance company. We build a parsimonious continuous-time model for longevity risk, that captures the dependence across different ages in domestic versus foreign populations. We provide three measures of the diversification effects of expanding an annuity portfolio toward a foreign population. The reduction in the risk margin, computed à la Solvency II, provides a regulation-consistent measure of the benefit in the tail risk. The change in the volatility of the average mortality intensity of a portfolio provides an intuitive measure of the change in the longevity risk of a portfolio. The Diversification Index provides a synthetic assessment of the diversification benefit of combining different populations in one portfolio. We calibrate the model to portray the case of a UK annuity portfolio expanding internationally towards Italian policyholders. Our application shows that the longevity risk diversidication benefits of an international expansion are sizable, in particular when interest rates are low.


Introduction
In the last twenty years, insurance companies have been expanding internationally, via subsidiaries operating in dierent countries or via cross-border mergers and acquisitions. The largest insurers and re-insurers are indeed multinational companies, with subsidiaries and branches located in several countries. Between 1990 and2003, namely before the introduction of Solvency II, the internationalization of banks and insurance companies followed similar patterns, as argued by Focarelli and Pozzolo, 2008. As of 2012, namely after the adoption of Solvency II, instead, Schoenmaker and Sass (2016) argue that the share of cross-border activity in the insurance sector is higher than in the banking one, and that the degree of internationalization of the 25 largest European insurers increased over the period 2000-2012, despite the nancial crisis.
A possible explanation for the higher level of internationalization of insurance companies relative to banks after the adoption of Solvency II in 2009 is the fact that, on top of nancial risks, whose diversication costs and benets may be similar between banks and insurers, through internationalization insurers and re-insurers may reap also longevity risk diversication benets, and the consequent regulatory burden relief.
Longevity risk is the risk of experiencing losses, due to unexpected uctuations in mortality rates. It aects annuity providers in particular when, as occurred in the last decades, policyholders' longevity exceeds the expectations. Why is international diversication of life insurance portfolios benecial in terms of risk? Because, even if in expectation longevity has been steadily increasing on a worldwide scale, idiosyncratic longevity risks of dierent populations are dierent and may be nonperfectly correlated across countries. As a result, pooling portfolios of policies written on the lives of dierent populations allows to diversify longevity risk and reduce the regulatory capital required by Solvency to face it.
The present paper aims at lling a gap in the literature by providing three measures of the diversication eects deriving from longevity risk pooling across populations.
As an example of the extra-benet of international diversication proper of life insurers and re-insurers, we consider an annuity provider, who can decide to expand her portfolio by selling policies to members of a population dierent from the one to which she is currently exposed. Our goal is to quantify the diversication benet deriving from such an expansion, relative to an expansion of the portfolio not involving internationalization.
To this end, we rst introduce a novel parsimonious model for the joint mortality dynamics of policyholders in dierent countries, which extends the model presented in De Rosa et al. (2017) to a multi-population setting. We set ourselves in the continuoustime framework, which has gained increasing popularity, alongside the more traditional discrete-time one, because of its analytical tractability. The model we propose is a stochastic, continuous-time multi-population extension of the deterministic Gompertz mortality law, a benchmark in the classical modelling of mortality arrival rates. It allows to compute survival probabilities and hedge ratios in closed form, dierently from models à la Lee-Carter. At the same time, it allows a rich description of the mortality dynamics of multiple populations, and generations within them. In the continuous-time framework, some models cope with the longevity risk of two cohorts or populations (Dahl et al., 2008), or several cohorts within one population (Blackburn and Sherris, 2013;Jevti¢ et al., 2013). Up to our knowledge, only Sherris et al. (2018) and Jevti¢ and Regis (2019)have attempted to combine the description of the mortality intensities of multiple populations and generations together in a continuoustime setting. Both these papers apply models driven by three independent Brownian motions (risk factors) and entail Gaussian intensities. Our model is richer, because we assume as many dependent risk factors as domestic generations and an idiosyncratic source that drives the mortality intensity of the foreign population. Thus, we are able to capture the correlation structure of dierent generations within and across populations accurately, while preserving a good level of parsimony. Our intensities follow square-root processes, and therefore can not become negative. Building on our model, we provide three measures for the longevity risk of a portfolio, which we use to describe the eects of geographical diversication on a portfolio.
The rst is the percentage risk margin, computed à la Solvency II. To evaluate it, we dene the value of the portfolio as the sum of the actuarial value of the policies (best estimate) and of a risk margin, i.e. an amount that the insurer has to set aside to cover up for the unhedgeable risks. The risk margin is dened as the value at risk (VaR) of the unexpected loss in the portfolio value at a certain condence level. If it decreases, after an international expansion, its change provides a dollar-based measure of the benets of diversication, because its reduction is the capital requirement relief for the insurer. The second measure is the standard deviation of the portfolio mortality intensity, which measures the volatility of the cohort-based intensity, weighted by the relevance of each cohort in the portfolio. It gives an indication of the longevity risk of the portfolio, because it measures the dispersion around the mean of the average mortality intensity. The third measure, that we call Diversication Index is an average of the dissimilarities between the same cohorts in dierent populations, present in both the initial portfolio and in the portfolio after the expansion, weighted by the percentage of policies belonging to the cohort. The three measures, analyzed together, provide three complementary views to assess the risk eects of the international expansion. The risk margin reduction oers a measure of the mitigation of the tail risk, because it represents the loss in a worst-case scenario occurring with a low probability. The standard deviation of the portfolio mortality intensity provides a measure of volatility of the portfolio, due to longevity risk. The diversication index, nally, assesses in a simple way how dissimilar a portfolio after an international expansion is relative to the initial one.
In a numerical application, which portrays the situation of a UK annuity provider that can expand to Italy, we rst assess that the model is able to t well the observed mortality rates of individuals aged 65-75 in the two populations, while capturing, using the Gaussian mapping technique, the imperfect correlations observed across ages and populations. Based on our model estimates, we then compute our international diversication measures for dierent portfolio expansions. We show that the risk margin reduction can be as high as 3% as a proportion of the actuarial value, in the case of a foreign expansion, targeted to those cohorts in the Italian population who have low covariance with the initial annuity portfolio. We also highlight that longevity risk mitigation eects are more sizable when the interest rate a at term structure, for simplicity is lower. The expansion can be performed, at a practical level, by starting foreign branches, acquiring foreign undertakings or, as shown in an appendix, through the use of longevity derivatives.
The paper unfolds as follows. Section 2 describes the set up and the problem of the insurer. Section 3 presents the longevity risk model. Section 4 denes the longevity risk measures we propose and their application to assess the benets of geographical diversication. Section 5 provides a calibrated application, describing our model calibration procedure, computes the diversication measures of various portfolio choices and provides sensitivity analysis to relevant parameters. Appendix A details the Gaussian mapping technique used to estimate the correlation structure, while Appendix B compares the two ways of achieve the international diversication: a physical one, in which a foreign aliate is opened, and a synthetic one, through a longevity swap.

Set up
We consider a ltered probability space (Ω, F, P), endowed with the usual properties, where F is the ltration containing the information regarding all the relevant variables and P is the historical probability measure. In this probability space, the mortality intensities of individuals are described as stochastic processes, and longevity risk, i.e. the risk of unexpected uctuations in the likelihood of deaths of individuals, arises. In what follows, we will consider longevity risk as the only source of risk in our setup.
We consider an Annuity Provider, or Life-Insurer, based in a certain country (that we call Domestic), having a portfolio of deferred annuities written on dierent cohorts belonging to the Domestic Population. Let X = {x 1 , . . . , x m } be the set of annuitants' ages at time zero, and let n i , for i = 1, . . . , m, be the number of annuities sold to people aged x i . When an annuity is sold at time zero, the annuitant pays an initial premium. We compute the actuarial value of the liabilities net of that premium. After signing the contract, the annuitant will receive a series of xed annual instalments R, starting from the year-end of his 65-th birthday if x i < 65, or immediately if x i ≥ 65, until his death, that may happen at most when he reaches a nal age ω, at which he will die with probability 1.

Portfolio value
In Europe, the life-insurance business falls under the Solvency II regulation, that requires insurers to value their liabilities at market value and set aside VaR-based risk margins with respect to the sources of risk that aect these valuations. These risk margins are amounts prudentially set aside by the insurer, meant as nancial covers for the unhedgeable risk that the insurer bears. We consider, then, that the overall value Π 0 (t) of the liability portfolio of a life insurer at time t is the sum of two components: the Actuarial Value AV Π 0 (t), which is the sum of the actuarial values of each individual contract N i (t) and represents a best estimate of the liabilities of the insurer, and the Risk Margin RM Π 0 (t) of the portfolio itself. In formulas, we have that: (1) We now detail further the assumptions we make to compute the two components.
The actuarial value of the contract is its fair premium. To compute it, we rst dene the number of years before the individual i aged x i reaches age 65 as τ = max(65 − x i , 0). If τ > 0, then the contract is a deferred annuity, while if τ = 0 the contract is an immediate annuity. Because we consider no risk source other then longevity risk, the actuarial value of an annuity can be expressed as where D(t, s), s ≥ t denotes the deterministic nancial discount factor, D(t, s) = e −r(s−t) , r ∈ R and S i (t, ·) is the time-t survival probability curve of the individual aged x i at time t.
We dene the portfolio risk margin RM Π 0 (t) as the discounted Value-at-Risk, at a given condence level α ∈ (0, 1), of the unexpected portfolio's future actuarial value at a given time horizon T : where P(·) denotes the probability of the event that the future actuarial value exceeds its time-t expected value by more than l.

Portfolio Expansion
In our setup, we consider the case in which the Insurer wants to expand the size of her annuity portfolio and can choose between two alternative strategies. The rst one consists simplt in selling new contracts to her own Domestic population. In this case, we denote with n i the number of new contracts sold to individuals aged x i , with Π D the portfolio composed of just these new annuities, and with Π 1 the portfolio after the expansion, composed of the old and the new contracts. The actuarial value of the new portfolio is simply and The value of the total portfolio Π 1 is the sum of the actuarial value of the old portfolio, the actuarial value of the new portfolio and the risk margin of the total portfolio: The second possible strategy is to acquire a new portfolio of annuities Π F , written on a foreign population. We assume that, for each age x i , the number of annuities written on people aged x i in the foreign population is n f i . The actuarial value of We denote with Π 2 the portfolio obtained after the expansion towards the foreign country. The actuarial value of such portfolio is and its overall value is Notice that the original portfolio and the one obtained after the expansion do not have the same actuarial value, neither when the expansion is domestic nor foreign.
The risk margin of the two portfolios is dierent as well.
Our aim is to measure the eects of the two alternative strategies on the longevity risk prole of the insurer. To this end, in the next sections we introduce a novel longevity risk model and three measures of the diversication eects.

Longevity Risk Modeling
We now turn to the description of the source of uncertainty that aects the value of the Insurer's portfolio: the risk of longevity, i.e. the risk that her policyholders live longer than expected. We set ourselves in the well-established continuous-time stochastic mortality setting initiated by Milevsky and Promislow (2001) that models the death of individuals as a Cox process. The time to death of an individual belonging to cohort x i is the rst jump time of a Poisson process with stochastic intensity. This intensity is indeed the force of mortality of the individual. When we consider dierent populations, and dierent cohorts within each population, it is reasonable to assume that their mortality intensities processes will be dierent, even though they may be (even closely) related one another. In this section, we propose a novel, parsimonious model to describe the evolution of the mortality intensities of several cohorts in two dierent populations. The parsimony of our approach stems from making the intensity of one population (the foreign one) a linear combination of the other, benchmark, population's intensity (the domestic one) and of an idiosyncratic risk factor. This makes the whole correlation structure across populations dependent on the weight of the linear combination.
To preserve tractability, allowing for closed form expressions for the survival probabilities, but at the same time ensuring non-negativity of the intensities, we adopt stochastic processes belonging to the ane family, of the Cox et al. (1985) type. These models have been used in single-country longevity modeling by Dahl et al. (2008) and Luciano et al. (2012).

Mortality intensities and survival probabilities
Let us consider two populations, each containing m dierent cohorts. The rst population is called the Domestic population and the second one is called the Foreign population. A given cohort i, with i = 1, . . . , m, belonging to one of the two populations, is identied by the (common) initial age x i at time zero. The set X of initial ages is common to the two populations.

Domestic Population
The mortality intensity of each cohort x i , for i = 1, . . . , m, belonging to the Domestic population is denoted with λ d i , and follows a non-mean reverting CIR process: where a i , b i , σ i , λ d i (0) ∈ R ++ are strictly positive real constants and the W i 's are instantaneously correlated standard Brownian Motions: dW i (t)dW j (t) = ρ ij dt with i, j ∈ {1, . . . , m}. As a consequence, the mortality intensities of two dierent cohorts belonging to the Domestic Population are instantaneously correlated, as soon as

Foreign Population
The mortality intensity of cohort x i belonging to the Foreign population is denoted with λ f i , and is given by the convex combination of the mortality intensity of the corresponding cohort belonging to the Domestic population λ d i and an idiosyncratic component λ , which aects the Foreign population only and that depends on the initial age x i in a deterministic way 1 , i.e.
where dλ (t; with δ i ∈ [0, 1]. 2 The functions a , b and σ are positive constants, while W is a standard Brownian Motion, that is assumed to be independent of W i for each i = 1, . . . , N . Intuitively, the idiosyncratic risk source W is population-specic, in the sense that it is common to all the cohorts of the Foreign population. Nonetheless, each foreign cohort x i has a specic sensitivity to the idiosyncratic component λ (t; x i ), that is given by the parameter δ i , which is, instead, cohort-specic. The mortality intensities of two dierent cohorts of the Foreign population are correlated, and the correlation between λ f i and λ f j depends both on the correlation between λ d i and λ d j and on the weights δ i and δ j . Moreover, thanks to the presence of the idiosyncratic component λ aecting the Foreign population, our model allows to account for the non-perfect correlation between cohorts across the two populations. The correlation structure among the dierent cohorts of the two populations will be derived in Section 5.2.
From (11) we have that the survival probability of generation x i in the Domestic population is given by: 1 For the empirical application in Section 5 we will consider the easiest case of no dependence on the initial age x i . 2 In principle, linear ane coecients a , b and σ could be chosen. where i . Similarly, for the Foreign population we have: where The time-t survival probability curves of the two populations are thus both available in closed form, and depend on the parameters of the model.

Variance Covariance Structure
The model we proposed allows the computation of the variance of each generation's mortality intensity, as well as the covariance between generations within and across population.
The time-t variance of the intensity of a generation i belonging to the Domestic population, λ d i , conditional on the information at time 0 is available in closed form and is equal to Similarly, the conditional variance of λ where Since the mortality of the Domestic generations follow a square-root process, there is no closed form expression for the covariance between the intensities two generations i and j. However, we can obtain a closed form approximation using the Gaussian Mapping technique described in Section 5.2 and in Appendix B. Indeed, referring the reader to those sections for further details, we have that where σ V i and σ V j are the instantaneous volatilities resulting from the mapping of λ d i and λ d j into Gaussian processes.
From (11) and (12) we have that the covariance between the same generation i belonging to the Domestic and Foreign population can be written as: Considering, instead, two dierent generations i and j belonging to the Foreign population, we have that Finally, the covariance between the mortality intensity of generation i belonging to the Foreign population and generation j belonging to the Domestic is given by: From (26), it is interesting to notice that the covariance between λ f i and λ d j depends both on δ i , which measures the dependence between the same generation i across the two populations, and on Cov 0 λ d i , λ d j , which instead measures the dependence between the generations i and j within the Domestic population.
Diagram (27) visualizes that, when computing the covariance λ f i and λ d j , we are able to disentangle the eect of the two types of dependence: the within-population and the cross-population ones. The importance of (27) can be explained with a simple example. Suppose there are two portfolios belonging to two populations f 1 and f 2 that are competing targets of a foreign expansion, and suppose that each portfolio is composed of annuities sold only to one generation k. The objective of the expansion is to nd a foreign portfolio that minimizes Cov λ f * k , λ d j , to obtain the maximum level of longevity risk diversication. Since we cannot change the covariance structure of the Domestic population, the solution to the problem is to nd the portfolio Π f * such that f * = arg min Then, it is sucient to compare the δ's of the two competing foreign populations. For instance, if δ f 1 k < δ f 2 k , then the optimal foreign expansion target portfolio is Π f 1 .

Measuring the longevity risk eects of geographical diversication
In the following paragraphs we introduce some measures of longevity risk in a portfolio, which allow us to appreciate the degree of geographical diversication achieved through a foreign expansion of the annuity portfolio. The rst measure is the Percentage Risk Margin of the portfolio, computed à la Solvency II. Comparing this measure before and after a portfolio expansion allows to appreciate the economic benet of a foreign expansion. A reduction in the percentage risk margin is connected with a reduction of tail risk, evaluated as the portfolio losses in a worst-case scenario. This measure follows the principle with which capital requirements are computed in the current regulation. Reducing the Percentage Risk Margin of a portfolio can thus be connected to a reduction in the regulatory capital requirement for longevity risk.
The second measure we propose is the Standard Deviation of the Portfolio Mortality Intensity. We dene the portfolio mortality intensity as a weighted average of the cohort-based mortality intensities entering a portfolio, with weights equal to the percentage of policies written on each generation. A reduction of this quantity indicates a stronger concentration of the distribution of the portfolio mortality intensity around its mean, denoting a reduction of longevity risk. Finally, the Diversication Index is an average of the degree of dissimilarity of the mortality intensities of the cohorts in dierent populations. This measure is a synthetic way of quantifying the level of diversication achieved by a foreign expansion.

Percentage Risk Margin
To be able to compare the eects of an expansion, we consider rst a normalized quantity, i.e. the ratio of the risk margin and the actuarial value of a portfolio Π, which we call percentage risk margin: A lower percentage risk margin denotes a lower percentage loss in the worst-case scenario, relative to portfolio value. Hence, reducing this measure is benecial for the company in two respects. First, it indicates a mitigation in the risk connected to adverse scenarios. In this sense, the risk margin can be considered as a measure of the systemic risk that the company may generate, by triggering losses that will hit its creditors. Second, it represents a capital requirement reduction, which frees up resources. Because the risk margin can be interpreted as both a capital requirement and a measure of the loss the company can generate at a given level of condence among its creditors, it is then conceivable that minimizing the percentage risk margin aligns the interests of both the insurance company and its regulators. In what follows we take the point of view of the insurer, taking for granted the alignment of her interest with the ones of the regulator.

Standard Deviation of the Portfolio Mortality Intensity
Another measure of the diversication eects deriving from longevity risk pooling across populations can be derived by looking at the change in the standard deviation of the portfolio mortality intensity pre-and post-foreign expansion. Given an annuity portfolio Π, we dene its portfolio mortality intensity λ Π as the weighted average of the mortality intensities of each generation in the portfolio, where the weights are the percentages of contracts written on each generation. Considering the initial Domestic portfolio Π 0 , let n d i be the number of contracts sold to generation i belonging to the Domestic population, and let n d = m i=1 n d i be the total number of contracts in the portfolio. Then, we dene λ Π 0 as: where ω d i = n d i n d is the weight for each generation i of the domestic population. Similarly, let n = n d + n f be the total number of contracts in the portfolio, Π 2 , after a foreign expansion in which n f contracts are written on the target foreign population, n f i on each generation i. The mortality intensity of the portfolio Π 2 is given by: Starting with the initial Domestic portfolio Π 0 and its mortality intensity λ Π 0 dened in (30), we have that: Thus we dene the standard deviation of the portfolio Π 0 mortality intensity as: Similarly, considering the post expansion portfolio Π 2 , we have that: and A foreign expansion provides a diversication benet if This can happen because, after the expansion, λ Π 2 (t) depends on λ d i , but also on the dierent risk source λ f i that may be non perfectly correlated with λ d i for i = 1, . . . , m.
Moreover, if there are multiple target portfolios for a foreign expansion, a possible way to decide about the optimal expansion target would be for instance to choose the portfolio that provides the lowest σ λ (Π * ).

Similarity/Diversication index
Building up on the characteristics of the longevity model described in the previous section, nally, we propose a synthetic measure to describe the similarity/dissimilarity between the annuity portfolios written on two populations, that we dene as Similarity and Diversication index. Let n d i be the number of annuities written on cohort x i belonging to the domestic population, n f i the number of annuities written on cohort x i belonging to the foreign population, n i = n d i + n f i and m the number of generations in the initial, domestic portfolio. Then the Diversication Index (DI) is equal to: and the Similarity Index (SI) 3 is: The Diversication Index represents a weighted average of the dissimilarities between the same cohorts in dierent populations 4 , present in both the initial portfolio and in the portfolio after the expansion. Dissimilarities are captured by the complement to 1 of δ i , the generation-specic parameter that captures the degree of correlation between the same generation of the dierent populations. The weights, n f i /n i , are given, for each cohort in the initial portfolio, by the number of annuities in the foreign population (after the expansion) relative to the total number of annuities written on that cohort in both populations. We average the weighted dissimilarities across all the m cohorts of the domestic population initially present in the annuity portfolio.
Our proposed indicator has the following properties. First, 0 ≤ DI ≤ 1. If δ i = 1 for every i, i.e. the two portfolios are written on perfectly correlated populations, then, obviously, SI = 1 and DI = 0. On the other hand, if δ i = 0, for every i, which means that the intensities of the foreign population are independent of the risk factor of the domestic, the DI does not go to 1 independently of the portfolio composition.
If n f i → ∞ and n d i remains constant, then SI → 0, DI → 1. This happens because the longevity risk of the foreign population is completely idiosyncratic and therefore diversication is reaped only enlarging the foreign portfolio as much as possible. This shows that the Diversication Index appropriately reects both the properties of the intensity correlation structure and the portfolio mix chosen by the underwriter.
Let us conclude this section with some intuition behind the derivation of the DI and a comparison with σ λ (Π 2 ). From the denition of λ Π 2 (t) and from (12) we observe that: The last term in (46) can be interpreted as the source of the diversication benet, and each coecient of the summation w f,Π 2 i (1 − δ i ) can be interpreted as the diversication contribution of each foreign generation i. Hence the DI can be seen as the average diversication contribution of each generation in the foreign portfolio.
Recalling diagram (27) for the dependence structure between the foreign and domestic generations, we could say that σ λ (Π 2 ) captures both the horizontal and the vertical dependence, while DI only focuses on the rst one. Further insights on the properties and indications deriving from the three measures presented will emerge from the application in Section 5, but let us comment briey on them before going on.
The percentage risk margin has the advantage of being expressed in economic terms, allowing a comparison between the economic benet of a foreign expansion and its implementation cost, and between the benets of competing target portfolios. Among the three measures, the percentage risk margin is the only one that can capture the impact of the term structure of interest rates on the economic benet of geographical longevity risk diversication. However, computing it for our proposed model, at least requires Monte Carlo simulations, making it the most computationally expensive measure among the ones presented.
The standard deviation of the portfolio mortality intensity does not require Monte Carlo simulations and can provide similar information to the risk margin when comparing dierent expansion strategies. It is able to capture the entire dependence structure between the domestic and foreign generations, but it is simply a distributional property and not a monetary measure.
The Diversication Index is the easiest measure to compute, because it does not require Monte Carlo simulations or the estimation of a correlation matrix. However, it does not capture the entire dependence structure between the domestic and foreign generations and, therefore, can provide useless indications when the target foreign portfolio population shows low dependence across generations and when the diversication benet of grouping dierent cohorts belonging to dierent populations is large.

Application
In this section, we calibrate our proposed model and try to quantify the diversication gains deriving from an international expansion towards Italy of an initially UK-based annuity portfolio. The situation we consider is that of a UK annuity provider who has the option of expanding her business either in her home country or abroad, selling additional policies to Italian policyholders. In practice, this expansion can be performed by creating an Italian branch or acquiring an Italian undertaking. As an alternative, the geographical diversication can be obtained through the use of longevity derivatives (see  for instance), which allow the insurer to gain some exposure to the mortality development of a dierent population. We explore this case in Appendix A.

Mortality intensities estimation
To calibrate our model, we proceed in two steps. First, we calibrate the parameters of the two intensity processes, of the domestic and of the foreign population respectively.
Then, in a second step, we calibrate the correlation parameters ρ ij . We calibrate the parameters of the mortality model to the generations of UK and Italian males whose age, at 31/12/2012, is between 65 and 75, that is, the cohorts born between 1937 and 1947. We consider thus 11 dierent cohorts present in the initial portfolio: x i = 65, . . . , 75. We use the 1-year×1-year cohort death rates data provided by the Human Mortality Database and recover, using the 20 observations from 1993 and uals alive in 1993. The estimation of the parameters is performed minimizing the Rooted Mean Squared Error (RMSE) between the observed and the model-implied survival probabilities. Tables 1 and 2 report the calibrated parameters for the two populations, while Figures 1 and 2 report the actual and tted survival probabilities and the calibration errors, respectively. The model, although parsimonious, is able to capture well the survival probability curves of the two populations, for all the cohorts considered.  for their correlations in closed form. However, to estimate correlations, we can apply the Gaussian Mapping technique, which has been used extensively in the pricing of Credit Default Swaps (see Brigo and Mercurio, 2001). Such technique allows to obtain a closed-form approximation of the correlations between the intensities of the dierent cohorts, in turn permitting the direct estimate of the correlation parameters ρ ij . Technically, it consists in mapping a CIR process into a Vasicek process that is as close as possible to the original one, i.e. returning the same survival probability.
Since we are able to compute analytically the correlations between each λ d i and λ d j , with i, j = 1, . . . , N in the mapped Vasicek process, we can then retrieve our desired parameters in closed-form.
Starting from the CIR process (11) describing the mortality intensity of cohort x i belonging to the domestic population, we consider a Vasicek process driven by the same Brownian Motion W i (t), having the same drift and the same initial point: The instantaneous volatility coecient σ V i of (47) is then determined by making the two processes as close as possible. Here, having xed a maturity T, by close we mean that the two processes return the same survival probability: Then, we approximate the correlation between λ d i (t) and λ d j (t) by the correlation between λ V i (t) and λ V j (t): since this last correlation can be computed analytically. Each pair-wise correlation is a function of the parameters b i and b j of the mapped Vasicek process and of ρ ij : To estimate the correlation parameters, rst the parameters of the process described by (47) are recovered. Then, using the central mortality rates data available in the UK life tables 6 , we estimate the instantaneous correlations ρ ij between dλ i and dλ j by inverting the approximated correlation expression (50). To compute the correlations between the 11 cohorts involved, we start from the central mortality rates in 1968 of the people aged between 1 and 11, and we follow the diagonal of the life table until we reach the central mortality rates of the people aged between 65 and 75 in 2012. The central mortality rates table constructed this way has dimension 65 × 11 and allows to estimate the correlation coecients which we report in Table   3. The upper and lower condence bounds are computed with bootstrapping from 6 Source: Human Mortality Database. 10, 000 resampled samples with replacement. Each sample has dimension 65 × 11, and is obtained by randomly choosing 65 times with replacement a row of our original central mortality table. As expected, because of the similarity between the UK and the Italian populations, correlations are close to 1 with tight 95% condence bounds, but they tend to decrease with the distance between the initial ages of the two considered cohorts. This behaviour aligns with the intuition that the changes leading to longevity improvements (such as healthy habits or medical advancements) have dierent impact on dierent generations and that cohort eects are at play. Table 4

Evaluating the diversication gains in terms of risk margin
Because the oldest cohort considered in our application is 75, and we assume a maximum life span of ω = 105 years, we x the time horizon of our simulations to 30 years. Consistent with this choice, we consider a constant interest rate of 2%, matching the 30-year risk-free-rate indicated by EIOPA for the calculation of technical provisions.
The choice of a constant interest rate term structure allows us to isolate and capture any possible added benet specically due to the geographical diversication of an annuity portfolio. The time horizon at which the Risk Margin is computed is 15 years.
This choice is justied because we want to focus on the medium-long term benets of geographical diversication. Consistently with the Solvency II regulation, we select a condence level α = 99.5% when calculating the Risk Margin associated to the portfolio.

Initial Portfolio
We consider a UK Insurer with an initial portfolio Π 0 , made of 1000 contracts sold to males whose age, at 31/12/2012, is between 65 and 75. The distribution of contracts among ages reects the proportions of individuals aged between 65 and 75 in the UK national population. For instance, since in the general UK population 69 years old constitute 11.00% of all the people aged between 65 and 75, the domestic portfolio contains 110 contracts sold to 69 years old (see Table 5). The initial Actuarial Value AV Π 0 (0) of the portfolio is: while the Risk Margin computed at time 0 is Hence, the initial portfolio value is The Risk Margin accounts for 8.39% of the initial portfolio Actuarial Value and σ λ (Π 0 ) = 0.00124. Portfolio Π F is exposed to the foreign population only, distributed among ages according to Table 6, useful for comparison. As we did for the initial portfolio, we assume that the policyholders' distribution reects the proportion of individuals belonging to each generation between 65 and 75 in the Italian population (see Table 6). Figure 4 shows the dierent percentage of individuals per cohort in the UK and Italian population. For the foreign portfolio Π F , the risk margin is 7.39% and σ λ (Π F ) = 0.00107.
One could guess that by expanding towards Italy, the UK underwriter could, at most, reduce his risk margin to this level. However, we will show later on that, thanks to the diversication eect, the risk margin of the underwriter can be even lower.

Domestic Expansion
With a Domestic Expansion, we assume that the Insurer doubles the size of her annuity portfolio, selling additional policies to her domestic population, i.e. the UK population. The new portfolio Π 1 is therefore composed of 2000 contracts and is obtained by simply doubling the number of contracts for each generation. Hence, RM Π 1 (0) = 2.3676 · 10 3 , Π 1 (0) = 3.0576 · 10 4 .

Foreign Expansion
In case of a Foreign Expansion, we assume that the Insurer doubles the number of policies in its annuity portfolio by selling contracts written on policyholders belonging to the Foreign population. The composition of the Foreign portfolio per cohort is assumed to follow the same proportions of the Italian population 7 (see Figure 4). The new portfolio Π 2 is, therefore, composed of 1000 contracts sold to the UK population composing the initial portfolio and of 1000 contracts written on the Italian population, Π 2 = Π 0 +Π F (both distributed as described in Table 5). It has the following actuarial value and risk margin: RM Π 2 (0) = 2.2749 · 10 3 , As a consequence, For this portfolio, the Risk Margin accounts for 7.87% of the Actuarial Value, reduced, as expected, by 0.52 percentage points relative to the one of the initial portfolio. The portfolio mortality standard deviation σ λ (Π 2 ) consistently decreases to 0.00115, and the diversication index increases to 0.0746.
The diversication gain provided by the Foreign portfolio just described can be further exploited. We then explore alternative portfolios and summarize the results in terms of actuarial values, risk margins and total values in Table 7.
Portfolio Π 3 represents a more aggressive foreign expansion, where the number of policies sold to each generation of foreign policyholders is twice the number of policies 7 Inserting the exact composition of the UK population is a trivial extension. in Π F . Tilting the portfolio towards the foreign population has the eect of decreasing the percentage risk margin (7.71%) and the portfolio mortality standard deviation (0.00112), while increasing the diversication index (0.0992). However, it is evident that, at most, by increasing the exposure to the Italian population, the risk margin can not be lower than 7.39%, which is the risk margin of the Foreign portfolio. This suggests to optimize the portfolio mix using not only the diversication across populations, but also across generations.
The portfolio Π 1 opt is obtained diversifying within the UK population. Its composition is optimized to obtain the minimum risk margin achievable, under the constraint that the number of new contracts is 1000. It can then be considered as the maximally diversied portfolio, in the absence of geographical diversication. The maximum diversication is thus obtained by selling 1000 annuities to the UK 66 years old, whose mortality intensity process shows the minimum covariance with the other UK cohorts (see the left panel of Figure 3). Notice that the percentage risk margin of this portfolio is 6.61%, which is lower than 7.39%. Being entirely composed of UK annuitants, this portfolio has a null Diversication Index, but due to its composition, is able to reduce σ λ to 0.00096. Similarly, Π 2 opt is obtained allowing for geographical diversication and optimizing the composition of the foreign portfolio. The optimization is performed by looking at the covariance matrix between the two populations(see right panel of Figure 3) and choosing to concentrate the foreign expansion on the Italian 66 years old males, who have the lowest covariance with all the cohorts of the UK population. The risk margin of Π 2 opt is 6.17% and σ λ is 0.00091. The percentage risk margin of portfolio Π 2 opt and its σ λ are the lowest among the portfolios we have considered. The DI of this last portfolio is small compared to the DIs of the other portfolios involving an international expansion, being 0.0121. It is small because the expansion is performed by concentrating the sales of policies in the foreign population in one generation only. Notice that the DI and %RM reduction dier more when the portfolio added to the initial one is optimized across generations than when it is not. This happens because the DI -by denition, to be kept simple -does not capture the eects of putting dierent weights on generations with low within population covariance, while the percentage risk margin and the portfolio mortality standard deviation capture the entire dependence structure between populations and generations. Indeed, the Diversication Index provides a non-dollar measure of diversication which "averages" the contributions of dierent generations and penalizes any concentration in a particular one, even though the latter is justied by a strategy which aims at minimizing the risk margin reduction. This is why we presented all the three measures. Table 8 reports the results for the dierent portfolios considered in Section 5.3, under the assumption of a zero interest rate, i.e. r = 0%. Under this lower interest rate level, the magnitude of longevity risk is more severe, as expected: the percentage Risk

Sensitivity Analysis
Margins are higher for all portfolios, increasing in the best-case scenario to 8.07%, up from 6.17%. However, diversication as measured by the %RM is even more valuable, because the reduction from the initial portfolio to Π 2 opt portfolio is almost 4 percentage points. The Diversication Index and portfolio mortality standard deviation, instead,  (31) and (44) are expressed in nominal terms (number of annuities written on a generation) rather than in value terms (value of the annuity portfolios on the dierent generations, for instance).
We nally assess the impact of the parameter δ i on portfolio diversication following an expansion. In Section 5.3, we considered two countries, the UK and Italy, that belong to the same continent and share many similar features. As a consequence, also their past mortality dynamics were not so dissimilar. We expect, however, that more dierent countries show way lower similarity, and thus lower δ's between cohort intensities. We perform, then, a simulation study where the parameters of the foreign population are set as in Table 2, with the only exception of the parameters δ i , which we assume to be a constant δ for every generation i. The interest rate is set to r = 2%, as in our base case. We exogenously set δ to a value that ranges from 0.1 to 0.9. When δ is close to 0, the dynamics of the mortality intensities of the domestic and the foreign populations are orthogonal. Thus, the international expansion targets a foreign population whose mortality dynamics is very dierent from the domestic one.
In this case, we expect the maximum level of diversication gains from an international expansion. As δ increases, the correlation between the mortality intensities of the two populations increases as well. When δ is close to 1, the mortality dynamics of the domestic and foreign population are perfectly correlated. In this last case, we can expect the lowest level of longevity risk diversication gains from an international expansion strategy. We compute, for each level of δ i , the DI, the portfolio mortality standard deviation and the percentage risk margin reduction, for the portfolios Π 2 and Π 2 opt described in Section 5.3.
As expected, the highest values of both the percentage risk margin reduction and the Diversication Index, for both portfolios, are achieved when δ i is close to 0. The percentage risk margin reduction is 4% in this case, showing that sizable benets from geographical diversication are possible. Such benets, measured in terms of either the risk margin reduction or the Diversication Index, decrease as δ approaches 1. The optimal portfolio expansion Π 2 opt provides consistently higher risk margin reduction than Π 2 , and the gap between the two strategies widens as δ increases (see the left panel of Figure 5). On the contrary, strategy Π 2 shows a higher DI with respect to Π 2 opt , for every δ. The Diversication Index tends to 0 for both portfolios as δ goes to 1. Instead, while the percentage risk margin reduction for Π 2 goes to zero when δ is 1, portfolio expansion Π 2 opt oers a diversication benet relative to the initial portfolio even in that case. This happens because the expansion is targeted in this case to a specic generation. The eects of the international expansion are analogous to those that can be obtained by targeting the domestic expansion to the generation which shows the lowest covariance with the others: Π 2 opt reduces the percentage risk margin as much as Π 1 opt .
Given their properties, and the evidence from this sensitivity analysis, the Diversication Index and the standard deviation of the portfolio mortality intensity can be an extremely easy-to-handle and useful tools when choosing among competing target foreign populations in an international expansion. The percentage risk margin reduction, being a monetary measure of the diversication gains, is better suited, instead, to select the best strategy when dierent alternative foreign portfolio compositions can be targeted, once the candidate foreign population has been selected.

Conclusions
In this paper, we discussed the benets of geographically diversied portfolios, due to the non-perfect correlation between the dynamics of the mortality rates of dierent populations. We have considered the problem of an insurer who has to decide whether to expand his portfolio in the country where it is based or in a foreign country. Some diversication gains can be realized when expanding internationally, due to the mitigation of the exposure to domestic longevity risk. To discuss whether these gains may be sizable in an annuity portfolio, we built a longevity risk model that, while being parsimonious, can capture the non-perfect correlations among the dierent cohorts of two dierent populations. We then provided three indicators of the diversication of an international expansion. The percentage risk margin reduction is computed coherently with the Solvency II modeling approach. The standard deviation of portfolio mortality intensity is the volatility of the distribution of a weighted average of the cohort-based mortality intensities, where the weights are the relative contributions of each cohort to the portfolio. The Diversication Index is a weighted average which depends on both the portfolio mix and the weight of the idiosyncratic foreign risk factor. This last measure is a very easy-to-handle indicator which, however may be unreliable when dierent cohorts compose the domestic portfolio and the foreign one.
It is instead very useful when comparing expansions towards the same target portfolios in dierent populations.
Our application, based on an annuity portfolio written on the UK and the Italian populations, shows that the eects of an international diversication are sizable. Expanding internationally decreases the volatility of the portfolio mortality intensity up to 26%. Under a 0% interest rate assumption, we showed that an optimally designed expansion can lower the percentage risk margin, relative to the actuarial value of the portfolio, by almost 4 percentage points.
The example in the paper can be considered as conservative, since the two pop-ulations of UK and Italy present rather similar historical mortality dynamics. The diversication eect is shown to be more relevant the lower the correlation between intensities.
The diversication benets of an international expansion may happen to be counterbalanced by the costs connected to the foreign portfolio acquisition process. These costs, that are -say -the xed costs of opening a foreign aliate, or the fees required by the agents involved in the M & A operation, etc., may be substantial. As an alternative to a physical expansion, the insurer may obtain the same diversication benet operating on the longevity derivatives market. Longevity derivatives, and longevity swaps in particular, are bespoke transactions between (re)insurers and funds or companies, that agree to exchange xed cash ows and cash ows linked to the survivorship of a particular population (see  for instance). The buyer of the protection provided by a longevity swap transfers the longevity risk linked to a given reference population to the seller, who in turn becomes exposed to such risk. In our case, the insurer can expand internationally by receiving a xed periodical fee and paying the realized survivorship of the foreign cohorts. Thus, the risk margin reduction benets of a foreign expansion can be replicated by selling protection through a swap. Even in this case, however, the costs of structuring the agreement and coping with informational asymmetries (Bis et al., 2016), can substantially reduce the diversication gains. Appendix A shows how to compare the physical and swap-based expansions, when the swap fee is fair, based on their costs.
We interpret our results as a possible explanation of the higher degree of internationalization of insurance companies with respect to banks after the adoption of Solvency II. Because of the synthetic possibility to diversify through longevity transfer agreements and longevity swaps, our results also explain the high number of such contracts recently signed in the marketplace and the attention dedicated to the growth of the market capacity (Blake et al., 2018).
to the Italian population. Being the seller of the swap, the UK insurer will receive every year, until the maturity of the contract, a xed amount equal to K and will pay a stochastic amount given by the realized survival rate of the Italian 66 years old males. Let the maturity of the swap be T = ω and assume independence between mortality and interest rate risk. From the point of view of the seller, the value at time t of the longevity swap is: = 1000 where K is the swap rate, S 66 (t, T ) is the (t, T )-Survival probability for a 66 years old Italian male and D(t, T ) is the discount factor. If the swap is fairly priced, the swap rate is chosen in such a way that the value of the contract is zero at inception, that is: In our calibration, assuming a constant interest rate of 2%, we have that K = 0.7295.
The actuarial value ofΠ 2 opt is then The risk margin ofΠ 2 opt is: So, if fairly priced at inception, the longevity swap allows the insurer to achieve the same actuarial value and the same risk margin of a physical sale of annuity contracts to the Italian males. The sales of the longevity swap may entail some initial cost C 0 given, for instance, by the required due diligence actions. Hence, the value of the liability portfolioΠ 2 opt is given by:Π 2 opt = Π 2 opt + C 0 .
As long as C 0 < C 0 , the UK insurer will nd in the synthetic expansion trough the longevity swap a more attractive solution.