Geographical Diversification of life-insurance companies: evidence and diversification rationale

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1 of life-insurance companies: evidence and diversification rationale 1 joint work with: Luca Regis 2 and Clemente De Rosa 3 1 University of Torino, Collegio Carlo Alberto - Italy 2 University of Siena, Collegio Carlo Alberto - Italy 3 Scuola Normale Superiore, Pisa - Italy Paris Intl Risks Forum, March 25, 2018 Paris Intl Risks Forum, March 25,

2 Introduction Introduction Paris Intl Risks Forum, March 25,

3 Introduction Motivation Motivation so far International expansion is a well established phenomenon in the Insurance Industry. Motivations studied so far: Balancing business cycles Managing costs more efficiently..but often business cycles are not so far apart, and expansion is not cost effective Figure. Geographic distribution of insurance premium income for global top 10 insurers (2008) 1. Paris Intl Risks Forum, March 25, Source: Internationalization: ageographical path to high Diversification performance for insurers in uncertain

4 Introduction Motivation Internationalization of largest Insurers Table. World s largest Insurers ranked by foreign insurance income in million of dollars 2 (2003). Insurance Income Employment N. Host Rank TNC Home Country Foreign Total Foreign Total Countries 1 Allianz Germany 75, , , , AXA France 65, , , , ING Netherlands 47, , , , Zurich Switzerland 45, , 920 n.a. 58, Generali Italy 38, , , , AIG US 32, , 319 n.a. 86, Munich Re Germany 27, , , , Aviva UK 26, , , , Swiss Re Switzerland 25, , 940 n.a. 7, Winterthur Switzerland 19, , , , Source: Outreville, J. F. (2008). Foreign affiliates of the largest insurance groups: Location- specific advantages. Journal of Risk and Insurance 75(2), Paris Intl Risks Forum, March 25,

5 Introduction Motivation Number of Life Insurance Undertakings Figure. Number of life insurance undertakings Source: OECD.Stat. Paris Intl Risks Forum, March 25,

6 Introduction Motivation Gross Premiums Life Insurance Undertakings Figure. Gross premiums life insurance undertakings Source: OECD.Stat. Paris Intl Risks Forum, March 25,

7 Introduction Economic Question Unexplored motivation We focus on an additional driver, longevity diversification. We consider an annuity provider who wishes to increase the size of her annuity portfolio and can choose between selling new contracts: to his/her domestic population, to a foreign population. Economic Question How can we measure the diversification of an annuity portfolio? Does geographical longevity diversification provide true economic benefit, namely lower risk margin)? Paris Intl Risks Forum, March 25,

8 Introduction Longevity Risk Longevity Risk 1. Is the risk of unexpected improvements in survivorship. Figure. Source: Dowd K, Blake D, Cairns AJG. Facing Up to Uncertain Life Expectancy: The Longevity Fan Charts. Demography. 2010;47(1): Paris Intl Risks Forum, March 25,

9 Introduction Longevity Risk Mortality Intensity 2. To model longevity risk we need mortality intensity to be a stochastic process: Mortality Intensity Time in Years Figure. Mortality Intensity simulations UK 65y males. The red line represents its non-stochastic version. Paris Intl Risks Forum, March 25,

10 Introduction Longevity Risk Survival Probability 3.1 Current survival probability at a given horizon is computed as an expectation over the intensity paths in the previous slide. 3.2 Future survival probabilities are random because they depend both on the future initial value of the intensity (say λ(1) at time 1) and the paths of the intensity afterwards S(t,10) Time t Figure. 10 years Survival Probability simulations UK 65y males. The red line represents its non-stochastic version. Paris Intl Risks Forum, March 25,

11 Theoretical Setup Theoretical Setup Paris Intl Risks Forum, March 25,

12 Theoretical Setup Aim Aim We provide a mortality model that: Accounts for different generations and populations parsimoniously, Permits to compute the similarity between the longevity of different populations explicitly, Allows to compute correlations between populations, Is analytically tractable, Can be coupled with one of the best known models for interest rate risk and still gives analytic solutions, Allows the computation of sensitivities and hedging ratios (greeks) explicitly. Paris Intl Risks Forum, March 25,

13 Theoretical Setup Mortality Model Mortality Model Domestic population: The mortality intensity of generation x (see De Rosa et al. (2016), SAJ) is: dλ d x (t) = (a + bλ d x (t))dt + σ λ d x (t)dw x (t), (1) Gompertz Mortality Paris Intl Risks Forum, March 25,

14 Theoretical Setup Mortality Model Mortality Model Foreign population: For generation x λ f x = δλ d x + (1 δ)λ x, (2) where dλ x = (a + b λ x)dt + σ λ xdw x, (3) Delta Specification 2 Paris Intl Risks Forum, March 25,

15 Theoretical Setup Mortality Model Correlation between populations Assuming 0 u t, the conditional correlation between λ d x i (t) and λ f x j (t) is given by: [ Corr u λ d xi (t), λ f x j (t) ] = δ j Cov u (λ d x i (t), λ d x j (t)), (4) Var u (λ d x i (t)) Var u (λ f x j (t)) where Cov u (λ d x i (t), λ d x j (t)) is computed using the Gaussian mapping technique, Gaussian Mapping Variance Paris Intl Risks Forum, March 25,

16 Theoretical Setup Annuity Portfolio Annuity Portfolio Π(t) : Portfolio Value at time t AV Π (t) : Actuarial value, or best estimate RM Π (t) : Risk Margin Risk Margin Π(t) = AV Π (t) + RM Π (t). (5) The portfolio risk margin RM Π (t) is the discounted Value-at-Risk, at a confidence level α (0, 1) of the unexpected portfolio s future actuarial value change at a given time horizon T : RM Π (t) = D(t, t + T ) VaR α ( AVΠ (t + T ) E t[av Π (t + T )] ), (6) Paris Intl Risks Forum, March 25,

17 Theoretical Setup Annuity Portfolio Annuity Portfolio Expansion An Insurer is exposed to the domestic population Π 0 = AV Π 0 + RM Π 0 and can choose between acquiring: a new domestic portfolio Π 0, ending up with Π 1 = Π 0 + Π 0, a new foreign portfolio Π F, ending up with Π 2 = Π 0 + Π F. Paris Intl Risks Forum, March 25,

18 Theoretical Setup Similarity and Diversification Index Similarity and Diversification Index Let ni f be the number of annuities sold to cohort x i in the foreign population, and let n i = ni d + ni f and m f the number of generations in the foreign portfolio. Diversification Index: DI = 1 f m ( m f 1 nd i + ni f δ ) i. (7) n i i=1 Similarity Index: SI = 1 DI. (8) Property 1 If δ i = 1 for every i SI = 1 2 If δ i = 0 for every i and n f i while n d i remains constant DI 1 Paris Intl Risks Forum, March 25,

19 Empirical Application Empirical Application Paris Intl Risks Forum, March 25,

20 Empirical Application UK vs Italy Parameters Estimation The estimation of parameters is performed using a 3-step procedure: 1 Estimate UK parameters using 20 years of UK death rates data for each generation ( , source: HMD), and minimizing RMSE. 2 Estimate ITA parameters using the corresponding data for Italy 3 Estimate instantaneous correlations ρ i,j between UK generations using 54 years of UK central mortality rates (period data) for each generation ( , source: HMD). We employ the Gaussian mapping technique. Paris Intl Risks Forum, March 25,

21 Empirical Application UK vs Italy Parameters Estimation The estimation of parameters is performed using a 3-step procedure: 1 Estimate UK parameters using 20 years of UK death rates data for each generation ( , source: HMD), and minimizing RMSE. 2 Estimate ITA parameters using the corresponding data for Italy 3 Estimate instantaneous correlations ρ i,j between UK generations using 54 years of UK central mortality rates (period data) for each generation ( , source: HMD). We employ the Gaussian mapping technique. Paris Intl Risks Forum, March 25,

22 Empirical Application UK vs Italy Parameters Estimation The estimation of parameters is performed using a 3-step procedure: 1 Estimate UK parameters using 20 years of UK death rates data for each generation ( , source: HMD), and minimizing RMSE. 2 Estimate ITA parameters using the corresponding data for Italy 3 Estimate instantaneous correlations ρ i,j between UK generations using 54 years of UK central mortality rates (period data) for each generation ( , source: HMD). We employ the Gaussian mapping technique. Paris Intl Risks Forum, March 25,

23 Empirical Application UK vs Italy Empirical Estimation: UK vs Italy 1 UK Population 1 Italian Population Survival Probability y 66y 67y 68y 69y 70y 71y 72y 73y 74y 75y Fit Time Survival Probability y 66y 67y 68y 69y 70y 71y 72y 73y 74y 75y Fit Figure. Fit of Survival probabilities Time Paris Intl Risks Forum, March 25,

24 Empirical Application UK vs Italy Correlation between populations Table. Correlation between populations. Rows are UK generations, columns are Italian generations. Colored cells highlight the minimum of each row Paris Intl Risks Forum, March 25,

25 Empirical Application UK vs Italy Covariance matrix between populations Figure. Covariance matrix between Italian and UK generations. Paris Intl Risks Forum, March 25,

26 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π 0 is the initial portfolio: UK ITA Paris Intl Risks Forum, March 25,

27 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π F is the foreign portfolio: UK ITA Paris Intl Risks Forum, March 25,

28 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π 1 = Π 0 + Π 0 is the portfolio after domestic expansion: UK ITA Paris Intl Risks Forum, March 25,

29 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π 2 = Π 0 + Π F is the portfolio after foreign expansion: UK ITA Paris Intl Risks Forum, March 25,

30 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π 3 = Π 0 + 2Π F is the portfolio after a more aggressive foreign expansion: UK ITA Paris Intl Risks Forum, March 25,

31 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π 1 opt is the portfolio after domestic expansion with optimal composition: UK ITA Paris Intl Risks Forum, March 25,

32 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 2%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Π 2 opt is the portfolio after foreign expansion with optimal composition: UK ITA Paris Intl Risks Forum, March 25,

33 Empirical Application Risk Margin Reduction Effects of Table. Effects of geographical diversification (r = 0%) Portfolio AV RM Π %RM DI Π % - Π F % - Π % 0 Π % Π % Π 1 opt % 0 Π 2 opt % Paris Intl Risks Forum, March 25,

34 Conclusions Conclusions Paris Intl Risks Forum, March 25,

35 Conclusions Conclusions The empirical application shows that: Our proposed model: Fits well the empirical data, Has endogenous correlations within and across populations but a parsimonious number of parameters as a whole, Allows to compute similarity and diversification indices of insurance companies liability portfolios, Geographical diversification reduces risk margins, The magnitude of the reduction depends on the similarities of the two populations, Low interest rates amplify the effect of geographical diversification. Paris Intl Risks Forum, March 25,

36 Conclusions Paris Intl Risks Forum, March 25,

37 Conclusions References I Accenture. Internationalization: a path to high performance for insurers in uncertain times. Report Biener, Christian, Martin Eling, and Ruo Jia (2015). Globalization of the Life Insurance Industry: Blessing or Curse? In: Brigo, Damiano and Fabio Mercurio (2001). Interest rate models : theory and practice. Springer finance. Berlin, Heidelberg, Paris: Springer. isbn: url: De Rosa, Clemente,, and Luca Regis (2016). Basis risk in static versus dynamic longevity-risk hedging. In: Scandinavian Actuarial Journal 0.0, pp doi: / Dowd, Kevin, David Blake, and Andrew J.G. Cairns (2010). Facing Up to Uncertain Life Expectancy: The Longevity Fan Charts. In: Demography 47, pp Paris Intl Risks Forum, March 25,

38 Conclusions References II Fung, Man Chung, Katja Ignatieva, and Michael Sherris (2014). Systematic mortality risk: An analysis of guaranteed lifetime withdrawal benefits in variable annuities. In: Insurance: Mathematics and Economics 58, pp Luciano, Elisa and Elena Vigna (2005). Non mean reverting affine processes for stochastic mortality. In: ICER Applied Mathematics Working Paper. Outreville, J Francois (2008). Foreign Affiliates of the Largest Insurance Groups: Location-Specific Advantages. In: Journal of Risk and Insurance 75.2, pp (2012). A note on geographical diversification and performance of the world s largest reinsurance groups. In: Multinational Business Review 20.4, pp Paris Intl Risks Forum, March 25,

39 Gompertz Mortality dλ d x (t) = (a + bλ d x (t))dt + σ λ d x (t)dw (t) (9) If a = σ = 0, then the mortality intensity is deterministic and we have: dλ d x (t) = bλ d x (t)dt, (10) that after simple integration becomes: λ d x (t) = λ d x (0)e bt (11) Back which is the usual Gompertz model. Paris Intl Risks Forum, March 25, / 5

40 Delta λ f x = δ λ d x + (1 δ) λ x }{{} Common Factor }{{} Idiosyncratic Factor, (12) The parameter δ measures the dependence between the two populations: 1 δ = 1 The two population are the same perfect dependence 2 0 < δ < 1 The two population are different partial dependence 3 δ = 0 The two population are different perfect independence Back Paris Intl Risks Forum, March 25, / 5

41 Idiosyncratic component specification Specification 2: A different specification for λ x i is: with a, b, σ, γ > 0. Back a(x i ) = a x i, b(x i ) = b, σ(x i ) = σ e γ x i, For each x i, λ x i is different but has the same functional form and the same set of parameters. This allows the model to be parsimonious. Since a > 0, the drift of the mortality intensity is increasing with age. γ > 0 captures the empirical evidence that the volatility of mortality tends to increase with age (see also Fung et al. (2014)). Paris Intl Risks Forum, March 25, / 5

42 Variance Var u (λ d x i (t)) = a iσ 2 i 2b 2 i ( e b i (t u) 1 ) 2 + σ 2 i b i e bi (t u)( e bi (t u) 1 ) λ d x i (u) (13) Var u (λ (t; x i )) = a(x i; a )σ(x i ; σ, γ ) 2 ( e b(x i ;b )(t u) 2b(x i ; b ) 2 1 ) σ(x i; σ, γ ) 2 b(x i ; b e b(xi ;b )(t u) ( e b(xi ;b )(t u) 1 ) λ (u; x i ) (14) ) Back Paris Intl Risks Forum, March 25, / 5

43 Gaussian Mapping 3 For each generation x i, we map the CIR dynamic dλ d x i = (a i + b i λ d i )dt + σ i λ d i dw i into a Vasicek dynamics which is as "close" as possible, i.e dλ V i = (a i + b i λ V i )dt + σ V i dw i, λ V i (0) = λ d i (0), where σ V i is such that S d i (t, T ) = S V i (t, T ; σ V i ). Corr 0 (λ d i, λ d j ) Corr 0 (λ V i, λ V j ) Back 3 For more details see Brigo and Mercurio (2001) Paris Intl Risks Forum, March 25, / 5