EXPERIENCE-BASED LONGEVITY ASSESSMENT

Transcription

1 EXPERIENCE-BASED LONGEVITY ASSESSMENT ERMANNO PITACCO Università di Trieste p. 1/55

2 Agenda Introduction Stochastic modeling: the process risk Uncertainty in future mortality trend The uncertainty risk: a static approach The uncertainty risk: a dynamic approach Concluding remarks Presentation mostly based on research and teaching material, jointly with Annamaria Olivieri (University of Parma) p. 2/55

3 INTRODUCTION Actuarial mathematics: the need for stochastic models Consider for example the following traditional actuarial formulae (according to the usual actuarial notation): Actuarial value of a whole life insurance A x0 = E [ (1 + i) (K x 0 +1) ] = + t=0 Actuarial value of a life annuity in arrears a x0 = E [ ] a Kx0 which can also be expressed as follows: = t 1q x0 (1 + i) (t+1) + t 1q x0 a t t=1 a x0 = + t=1 tp x0 (1 + i) t p. 3/55

4 Introduction (cont d) Expected number of survivors at time t in a given cohort of n 0 insureds or annuitants initially age x 0 E[N x0 +t] = n x0 tp x0 Similar formulae in the time-continuous framework Probabilities are involved, but only used to determine actuarial values, i.e. expected present values classical actuarial formulae provide a deterministic setting of insurance problems Actuarial values are traditionally used to calculate premiums and reserves impact of risks on portfolio results are accounted for only via (implicit) safety loadings p. 4/55

5 Introduction (cont d) Traditional actuarial formulae do not allow us to capture important problems concerning life insurance and life annuities, originated by new (and evolving) scenarios mortality / longevity dynamics volatility in financial markets legislation shift from DB to DC pension plans solvency issues financial rorting standards... p. 5/55

6 Introduction (cont d) Risks in life insurance, annuities and pensions Life insurers and annuity providers take, according to policy conditions (options and guarantees) financial risks biometric risks (mortality / longevity and disability risks) Focus on mortality / longevity risks Risk arising from individual lifetimes is a process risk (originated by random mortality fluctuations), called individual mortality / longevity risk, and can be diversified by increasing the portfolio size or via reinsurance arrangements, i.e. inside the traditional insurance-reinsurance process Risk arising from average lifetime in the portfolio (in particular originated by future unknown mortality trend) is a systematic risk, called aggregate mortality / longevity risk, and cannot be diversified inside the traditional insurance-reinsurance process p. 6/55

7 STOCHASTIC MODELING: THE PROCESS RISK See: Olivieri and Pitacco [2012] The basics Refer to a cohort initially consisting of n x0 individuals age x 0. Define: T (j) x 0 = random lifetime of individual j (j = 1, 2,...,n x0 ) N x0 +t = n x0 j=1 I {T (j) x 0 >t} D x0 +t = N x0 +t N x0 +t+1 Assume random lifetimes T (j) x 0, j = 1, 2,...,n x0, are i.i.d., with probability distribution provided by the life table {l x } x=0,1,...,ω Then: N x0 +t Bin(n x0, t p x0 ) p. 7/55

8 Stochastic modeling: the process risk (cont d) For z > t, conditional on N x0 +t = n x0 +t: [N x0 +z n x0 +t] Bin(n x0 +t, z t p x0 +t) [D x0 +t n x0 +t] Bin(n x0 +t, q x0 +t) Poisson distribution often adopted as an approximation to the binomial distribution: [D x0 +t n x0 +t] Pois(n x0 +t q x0 +t) with E[D x0 +t n x0 +t] = n x0 +t q x0 +t (under both the Binomial and the Poisson assumption) Example x 0 = 65; q x = G Hx 1+G H x with G = , H = See following Figures p. 8/55

9 Stochastic modeling: the process risk (cont d) n 65 = 100 t = 5 n 65 = t = 10 probability t = 15 probability N 65+t n 65 N 65+t n 65 t = 10 t = 15 t = 5 Fig. 1 Probability distribution of N 65+t n 65 ; t = 5,10, 15 p. 9/55

10 Stochastic modeling: the process risk (cont d) Example of insurance application Focus on: Y [P] 0 = random present value at time 0 of the benefits which will be paid by a portfolio of life annuities (individual annual amount b) or, equivalently: We have: ω x 0 Y [P] 0 = b t=1 Y [P] 0 = b N x0 +t (1 + i) t n x0 j=1 a K (j) x 0 [ E Y [P] 0 ] = b ω x 0 t=1 E[N x0 +t] (1+i) t = b ω x 0 t=1 n x0 tp x0 (1+i) t = n x0 b a x0 p. 10/55

11 Stochastic modeling: the process risk (cont d) n 65 = probability n 65 = 100 Y [P] 0 n 65 Fig. 2 Probability distribution of Y [P] 0 (n n 65 = 100; n 65 = 1 000) 65 p. 11/55

12 UNCERTAINTY IN FUTURE MORTALITY TREND See, for example: Pitacco et al. [2009] Main features of mortality trends Decreasing infant mortality Decreasing annual probabilities of death, over wide age ranges, in particular at old ages Increasing life expectancy (both at the birth, and at old ages) Increasing modal duration of life (Lexis point) Changes in the shape of survival functions: rectangularization (but at old ages; see next Table) expansion p. 12/55

13 Uncertainty in future mortality trend (cont d) lx SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM age Fig. 3 Survival curves. Source: ISTAT - Italian population tables (Males) p. 13/55

14 Uncertainty in future mortality trend (cont d) 5000 dx SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM age Fig. 4 Curves of deaths. Source: ISTAT - Italian population tables (Males) p. 14/55

15 Uncertainty in future mortality trend (cont d) marker e0 Lexis e Fig. 5 Life expectancy and Lexis point. Source: ISTAT - Italian pop. tables (Males) p. 15/55

16 Uncertainty in future mortality trend (cont d) Observed mortality trends construction of projected life tables (or projected survival functions), including a forecast of future mortality See, for example: Medved et al. [2011] (construction of Slovenian projected life tables for annuities) Whatever projection adopted, future mortality trend is unknown systematic risk borne by the annuity provider In particular: overall rectangularization lower volatility uncertainty in future expansion higher systematic risk Systematic longevity risk also called uncertainty risk, referring to uncertainty in the rresentation of future age-patterns of mortality Both random fluctuations and systematic deviations affect mortality at old ages p. 16/55

17 Uncertainty in future mortality trend (cont d) 6000 dx SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM age Fig. 6 Curves of deaths (x > 65). Source: ISTAT - Italian population tables (Males) p. 17/55

18 Uncertainty in future mortality trend (cont d) p d f IQR[T 65 ] 65 age x 1st quartile Median Lexis 3rd quartile Fig. 7 Probability density function of 65 + T 65 p. 18/55

19 Uncertainty in future mortality trend (cont d) Looking at the behavior of the interquartile range of the survival function, at age 65 importance of random fluctuations (process risk) SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM 2002 Median st quartile rd quartile IQR Table 1 Trends. Source: ISTAT - Italian Population Tables (Males) Random fluctuations in lifetimes should not be underestimated, in particular when dealing with life annuities and pensions p. 19/55

20 THE UNCERTAINTY RISK: A STATIC APPROACH A very extensive literature concerning stochastic mortality modelling See, for example: Pitacco et al. [2009] for a list of references A simple approach See: Olivieri [2001], Olivieri and Pitacco [2003], Olivieri and Pitacco [2002] (the last one for extension to a continuous set of scenarios, defined by parameters of the force of mortality) The approach basically consists of two sts 1. Choose a set of, say, r scenarios, in order to express alternative hypotheses about future mortality evolution: H = {H 1, H 2,...,H r } p. 20/55

21 The uncertainty risk: a static approach (cont d) Each scenario: a projected life table (or a survival function, or a force of mortality, etc.) Actuarial applications: scenario testing, assessing the range of variation of quantities such as cash flows, profits, portfolio reserves, etc. sensitivity analysis 2. Assign non-negative normalized weights to the mortality scenarios a probability distribution on the space H: Example ρ 1, ρ 2,...,ρ r Actuarial application: a stochastic approach can be adopted unconditional (i.e. non conditional on a particular scenario) variances, percentiles, etc., of the value of future cash flows, profits, etc. See the following Figure p. 21/55

22 The uncertainty risk: a static approach (cont d) probability best-estimate scenario mixture of three scenarios Y [P] 0 n 65 Fig. 8 Conditional and unconditional probability distribution of Y [P] 0 (n n 65 = 100) 65 p. 22/55

23 The uncertainty risk: a static approach (cont d) Why static? Uncertainty is expressed just in terms of a set H of assumptions, and the relevant probability distribution ( which one of the assumptions is the best one for describing the aggregate mortality in the cohort) No future shift from such a trend is allowed for in the stochastic model Critical aspect: assumptions about the temporal correlation of changes in the probabilities of death are implicitly involved Possible mortality shocks are not embedded into the static rresentation (not a problem when dealing with life annuities) Updates of the weights ρ s based on experience could be introduced, while keing the setting as a static one See: Olivieri and Pitacco [2002] for an example in a continuous scenario space p. 23/55

24 THE UNCERTAINTY RISK: A DYNAMIC APPROACH See: Olivieri and Pitacco [2009], Olivieri [2011], Olivieri and Pitacco [2012] Preliminary ideas Refer to a portfolio of life annuities (one cohort or multi-cohort) Assume that: a life table, providing a best-estimate of annuitants mortality, is available to the insurer the insurer has no access to data sets and the methodology underlying the construction of the life table awareness of uncertainty in future mortality trends the life table is used as the basic input of a stochastic mortality model p. 24/55

25 The uncertainty risk: a dynamic approach (cont d) The model Notation t 0 = starting time of the portfolio x 0 = age at entry t = portfolio duration since time t 0, t = 0, 1, 2,... D x,t = random number of deaths in year (t 1, t) for those aged x at time t 1 N x,t = random number of individuals age x at time t possible (and observed) outcomes of the random variables D x,t, N x,t denoted by d x,t, n x,t p. 25/55

26 The uncertainty risk: a dynamic approach (cont d) Probability distribution: [D x,t q x,t ; n x,t 1 ] Bin(n x,t 1, q x,t ) with q x,t = assumed probability of death (possibly the best-estimate) Approximation: [D x,t q x,t ; n x,t 1 ] Pois(n x,t 1 q x,t ) Uncertainty about the mortality trend Q x,t = random mortality rate Modelling approaches: 1. assign a probability distribution to Q x,t 2. let Q x,t = q x,t Z x,t with Z x,t = (positive) random adjustment to the best-estimate mortality rate q x,t; assign a probability distribution to Z x,t p. 26/55

27 The uncertainty risk: a dynamic approach (cont d) Modelling approach 1 Assume: Q x,t Beta(a x,t, b x,t ) Unconditional distribution of the number of deaths then follows the Beta-Binomial law: ( ) n x,t 1 Γ(a x,t + b x,t ) Γ(a x,t + d) Γ(b x,t + n x,t 1 d) P[D x,t = d n x,t 1 ] = d Γ(a x,t ) Γ(b x,t ) Γ(a x,t + b x,t + n x,t 1 ) where Γ( ) is the incomplete Gamma function We have: E[Q x,t ] = a x,t a x,t + b x,t E[D x,t n x,t 1 ] = n x,t 1 a x,t a x,t + b x,t p. 27/55

28 The uncertainty risk: a dynamic approach (cont d) Note that: E[D x,t q x,t; n x,t 1 ] = n x,t 1 q x,t Then E[D x,t n x,t 1 ] E[D x,t q x,t; n x,t 1 ] dending on the comparison between q x,t and E[Q x,t ] Modelling approach 2 Assume: Z x,t Gamma(α x,t, β x,t ) It turns out: Q x,t Gamma ( α x,t, β ) x,t qx,t p. 28/55

29 The uncertainty risk: a dynamic approach (cont d) and [D x,t n x,t 1 ] NegBin ( α x,t, ) θ x,t θ x,t + 1 ( ) with θ x,t = β x,t n x,t 1 q x,t Result ( ) generalizes the well known Poisson-Gamma structure We have: E[Q x,t ] = α x,t β x,t q x,t E[D x,t n x,t 1 ] = α x,t θ x,t = α x,t β x,t n x,t 1 q x,t p. 29/55

30 The uncertainty risk: a dynamic approach (cont d) Note that: E[D x,t n x,t 1 ] E[D x,t q x,t; n x,t 1 ] dending on the value of mortality) α x,t β x,t ( systematic deviations in Advantages / disadvantages of the approach the Gamma distribution does not guarantee that the mortality rate is bounded in (0, 1) (in particular at the oldest ages, when q x,t is high, while n x,t 1 is presumably low) the model leads quite naturally to a dynamic setting, through a Bayesian inferential procedure, allowing to account for correlations among the Z x,t s to update parameters to experience approximation errors at the older ages may become negligible when a portfolio consisting of multiple cohorts is addressed p. 30/55

31 The uncertainty risk: a dynamic approach (cont d) < 1 in year (t 1, t) (where Experience-based rationale: a ratio q x,t qx,t q x,t = realized value of Q x,t ) is quite always followed by a ratio q x+1,t+1 q x+1,t+1 < 1 (and also q x,t+1 q x,t+1 < 1) in the following year One cohort. The inferential procedure Refer to one cohort the Poisson-Gamma model (approach 2) Assume, for all x and all t: Z x,t Gamma ( ᾱ, β ) For example, ᾱ, β such that E[Q x,t ] = q x,t ( one degree of freedom) p. 31/55

32 The uncertainty risk: a dynamic approach (cont d) It follows: [D x0,1 n x0,0] NegBin ( ᾱ, ) θ x0,1 θ x0,1 + 1 where θ x0,1 = β n x0,0 q x 0,1 Let d x0,1 denote the number of deaths observed in year (0, 1) Then, n x0 +1,1 = n x0,0 d x0,1 Posterior probability distribution of Q x0,1 conditional on D x0,1 = d x0,1: ( [Q x0,1 d x0,1] Gamma ᾱ + d x0,1, β q x 0,1 ) + n x0,0 p. 32/55

33 The uncertainty risk: a dynamic approach (cont d) Posterior probability distribution of Z x,t conditional on D x0,1 = d x0,1: [Z x,t d x0,1] Gamma ( ᾱ + d x0,1, β + n x0,0 qx ) 0,1 ( ) Expected values of Z x,t prior E[Z x,t ] = ᾱ β posterior at time 1 E[Z x,t d x0,1] = ᾱ + d x0,1 β + n x0,0 q x 0,1 E[Z x,t d x0,1] E[Z x,t ] dending on the comparison between d x0,1 and the relevant expected value n x0,0 q x 0,1 p. 33/55

34 The uncertainty risk: a dynamic approach (cont d) Valuations performed at time 1 involving the next year: and hence with [Q x0 +1,2 d x0,1] Gamma ( [D x0 +1,2 n x0,0, d x0,1] NegBin ᾱ + d x0,1, ( ᾱ + d x0,1, β + n x0,0 q x 0,1 q x 0 +1,2 ) ) θ x0 +1,2 θ x0 +1,2 + 1 θ x0 +1,2 = β + n x0,0 q x 0,1 n x0 +1,1 q x 0 +1,2 Similar sts at times t = 2, 3,... p. 34/55

35 The uncertainty risk: a dynamic approach (cont d) Focus on the expected number of deaths in each year t at time t 1: E[D x0 +t 1,t n x0,0, d x0,1, d x0 +1,2,...,d x0 +t 2,t 1] = ᾱ + t 1 h=1 d x 0 +h 1,h β + t 1 h=1 n x 0 +h 1,h 1 qx 0 +h 1,h }{{} A n x0 +t 1,t 1 q x 0 +t 1,t }{{} B B = expected value of D x0 +t 1,t conditional on best-estimate q x 0 +t 1,t A = adjustment coefficient, updated to the observed number of deaths in respect of those expected at the beginning of each year experience consistent with what expected coefficient will remain stable in time number of deaths lower than expected coefficient will decrease in time p. 35/55

36 The uncertainty risk: a dynamic approach (cont d) More than one cohort Notation D t = min{x 0 +t 1,ω} x=x 0 D x,t = random numb. of deaths in (t 1, t) N t = min{x 0 +t,ω} N x,t = random numb. of individuals alive at time t Assumptions x=x 0 Conditional on any given q x,t assume the lifetimes of the individuals belonging to one cohort are i.i.d. Assume the Poisson approx [D x,t q x,t ; n x,t 1 ] Pois(n x,t 1 q x,t ) p. 36/55

37 The uncertainty risk: a dynamic approach (cont d) Assume that, conditional on the life table {q x,t }, at any time t the numbers of deaths D x,t are indendent in respect of age, i.e. that individual lifetimes are indendent also among cohorts; hence min{x 0 +t 1,ω} [D t {q x,t }; {n x,t 1 }] Pois q x,t n x,t 1 ( ) x=x 0 Uncertainty about aggregate mortality for any age x and time t, let Q x,t = q x,t Z t Note that Z t expresses systematic deviation in mortality time-specific age-indendent source of uncertainty assumed to be common to all the cohorts p. 37/55

38 The uncertainty risk: a dynamic approach (cont d) From ( ) we have, conditional on Z t = z: [D t {z qx,t}; {n x,t 1 }] Pois min{x 0 +t 1,ω} x=x 0 n x,t 1 z qx,t Assuming then: Z t Gamma(α t, β t ) [D t {n x,t 1 }] NegBin ( α t, ) θ t θ t + 1 where θ t = β t min{x 0 +t 1,ω} x=x 0 n x,t 1 q x,t p. 38/55

39 The uncertainty risk: a dynamic approach (cont d) The inferential procedure for more than one cohort Assume at time 0 Z t Gamma(ᾱ, β) for all future times t. Then: where θ 1 = [D 1 {n x,0 }] NegBin ( ᾱ, β min{x0 +t 1,ω} x=x 0 n x,0 q x,1 ) θ 1 θ = β n x0,0 q x 0,1 Let D 1 = d 1 (i.e. D 1 = d x0,1) = observed number of deaths in (0, 1). Then: [Z t d 1 ] Gamma ᾱ + d 1, β + min{x 0 +t 1,ω} n x,0 q x,1 x=x 0 p. 39/55

40 The uncertainty risk: a dynamic approach (cont d) and [D 2 {n x,0, n x,1 }; d 1 ] NegBin ( ᾱ + d 1, ) θ 2 θ with θ 2 = β + min{x 0 +t 1,ω} x=x 0 n x,0 q x,1 min{x0 +t 1,ω} x=x 0 n x,1 q x,2 = β + n x0,0 q x 0,1 n x0,1 q x 0,2 + n x 0 +1,1 q x 0 +1,2 In general, at time t 1 observe: the annual number of new entrants n x0,0, n x0,1,...,n x0,t 1 the annual number of deaths D 1 = d 1, D 2 = d 2,...,D t 1 = d t 1 the number of survivors n x,h in each cohort at time h, h = 0, 1,...,t 1 p. 40/55

41 The uncertainty risk: a dynamic approach (cont d) We then find: with [D t {n x,0, n x,1,...,n x,t 1 }; d 1, d 2,...,d t 1 ] NegBin ( α t, ) θ t θ t + 1 α t = ᾱ + t 1 h=1 Expected values of Z t prior d h ; θ t = β + t 1 min{x0 +h 1,ω} h=1 x=x 0 min{x0 +t 1,ω} x=x 0 E[Z t ] = ᾱ β n x,t 1 q x,t n x,h 1 q x,h p. 41/55

42 The uncertainty risk: a dynamic approach (cont d) posterior at time s, s = 1, 2,...,t E[Z t d 1, d 2,...,d s 1 ] = β + s 1 h=1 ᾱ + s 1 h=1 d h min{x0 +h 1,ω} x=x 0 n x,h 1 q x,h Hence E[Z t ] E[Z t d 1, d 2,...,d s 1 ] dending on the comparison between the experienced number of deaths and the relevant expected value p. 42/55

43 The uncertainty risk: a dynamic approach (cont d) Numerical findings Refer to the expected systematic deviation E[Z t {d s } s=1,2,...,t ] Choose the best-estimate life table {q } At time 0 set: ᾱ = β ( meaning of the best-estimate life table) β = 100 ( expert s judgment on volatility; see below for alternative choices) Then: E[Q x,t ] = qx,t Var[Q x,t ] = ᾱ ( β) 2 (q x,t) 2 Var[Qx,t ] CV[Q x,t ] = E[Q x,t ] = 1 ᾱ = 10% p. 43/55

44 The uncertainty risk: a dynamic approach (cont d) Some results Figure 9: it is assumed d x,s = n x,s 1 q x,s for s = 1, 2,...,t Figure 10: it is assumed d x,s = 0.75 n x,s 1 q x,s adjustments Figure 11: it is assumed d x,s = 1.25 n x,s 1 q x,s adjustments Figures 12-14: alternative values for β are considered (joint with ᾱ = β) β = 100 CV[Q x,t ] = 0.10 β = 25 CV[Q x,t ] = 0.20 β = 400 CV[Q x,t ] = 0.05 effects of the assumed volatility (in terms of CV) of the mortality rate, on the expected systematic deviation p. 44/55

45 The uncertainty risk: a dynamic approach (cont d) one cohort multiple cohorts expected systematic deviation 1 n 65,0 = 100, 1 000, expected systematic deviation 1 n 65,0 = 100, 1 000, time time Fig. 9 Expected systematic deviation E[Z t {d s } s=1,2,...,t ] ᾱ = β, β = 100; d x,s = n x,s 1 q x,s p. 45/55

46 The uncertainty risk: a dynamic approach (cont d) one cohort multiple cohorts expected systematic deviation n 65,0 = 100 n 65,0 = n 65,0 = expected systematic deviation n 65,0 = n 65,0 = 100 n 65,0 = time time Fig. 10 Expected systematic deviation E[Z t {d s } s=1,2,...,t ] ᾱ = β, β = 100; d x,s = 0.75 n x,s 1 q x,s p. 46/55

47 The uncertainty risk: a dynamic approach (cont d) expected systematic deviation n ,0 = n 65,0 = n 65,0 = one cohort expected systematic deviation 1 multiple cohorts 1.25 n 65,0 = n 65,0 = n 65,0 = 100 time time Fig. 11 Expected systematic deviation E[Z t {d s } s=1,2,...,t ] ᾱ = β, β = 100; d x,s = 1.25 n x,s 1 q x,s p. 47/55

48 The uncertainty risk: a dynamic approach (cont d) one cohort multiple cohorts expected systematic deviation 1 β = 25, 100, 400 expected systematic deviation 1 β = 25, 100, 400 time time Fig. 12 Expected systematic deviation E[Z t {d s } s=1,2,...,t ] ᾱ = β; n 65,s = 1 000; d x,s = n x,s 1 q x,s p. 48/55

49 The uncertainty risk: a dynamic approach (cont d) one cohort multiple cohorts expected systematic deviation 1 β = 400 β = β = 25 expected systematic deviation 1 β = 100 β = β = 25 time time Fig. 13 Expected systematic deviation E[Z t {d s } s=1,2,...,t ] ᾱ = β; n 65,s = 1 000; d x,s = 0.75 n x,s 1 q x,s p. 49/55

50 The uncertainty risk: a dynamic approach (cont d) expected systematic deviation one cohort β = 25 β = 100 β = 400 expected systematic deviation multiple cohorts β = 100 β = 400 β = 25 time time Fig. 14 Expected systematic deviation E[Z t {d s } s=1,2,...,t ] ᾱ = β; n 65,s = 1 000; d x,s = 1.25 n x,s 1 q x,s p. 50/55

51 CONCLUDING REMARKS Traditional life insurance mathematics and technique mainly rely on the calculation of expected values (viz in pricing and reserving) An appropriate stochastic approach is however required because of the complexity of some products including options and guarantees See, for example: Bacinello et al. [2011] as regards Variable Annuities products the evolving scenarios a sound assessment of the insurer s risk profile... Starting point of a rigorous stochastic approach to mortality / longevity in life insurance and life annuities: choice of an appropriate probabilistic structure: 1. a best-estimate life table 2. an inferential procedure for adjustments based on monitoring p. 51/55

52 Concluding remarks (cont d) Aim of this talk illustration of a (practicable) approach to point 2 Various generalizations could be conceived; for example: introduction of cohort specificity different probability distributions to express uncertainty MCMC methods to approximate posterior distributions... p. 52/55

53 Our contributions A. R. Bacinello, P. Millossovich, A. Olivieri, and E. Pitacco. Variable annuities: a unifying valuation approach. Insurance: Mathematics & Economics, 49(3): , doi: /j.insmatheco D. Medved, A. Ahcan, J. Sambt, and E. Pitacco. Adoption of projected mortality table for the slovenian market using the poisson log-bilinear model to test the minimum standard for valuing life annuities. EBR (Economic and Business Review), 13(4): , Available online: A. Olivieri. Uncertainty in mortality projections: an actuarial perspective. Insurance: Mathematics & Economics, 29(2): , doi: /S (01) A. Olivieri. Stochastic mortality: experience-based modeling and application issues consistent with Solvency 2. European Actuarial Journal, 1(Suppl 1):S101 S125, doi: /s A. Olivieri and E. Pitacco. Inference about mortality improvements in life annuity portfolios. In Transactions of the 27th International Congress of Actuaries, Cancun (Mexico), 2002 p. 53/55

54 Our contributions (cont d) A. Olivieri and E. Pitacco. Solvency requirements for pension annuities. Journal of Pension Economics & Finance, 2(2): , doi: /S A. Olivieri and E. Pitacco. Stochastic mortality: the impact on target capital. ASTIN Bulletin, 39(2): , doi: /AST A. Olivieri and E. Pitacco. Life tables in actuarial models: from the deterministic setting to a Bayesian framework. AStA Advances in Statistical Analysis, 96(2): , doi: /s y E. Pitacco, M. Denuit, S. Haberman, and A. Olivieri. Modelling Longevity Dynamics for Pensions and Annuity Business. Oxford University Press, 2009 p. 54/55

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