General Public Release The added value of a stochastic mortality framework Fasecolda Philippe Maeder General Public Release

Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 3 Table of Contents / Agenda ABC of Stochastic Mortality Illustrations Modelling Mortality Shocks Conclusion

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health ABC of Stochastic Mortality From Deterministic to Stochastic Models Random Fluctuation of Mortality 4

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health First models developed by actuaries were deterministic, with more or less sophistication; for a policyholder since age x and having attained age x+t : – Periodic mortality tables: they reflect mortality observed or projected in a given year – Select and ultimate tables: a selection factor intervenes to acknowledge the underwriting effect. – Generation tables (or longitudinal tables): they take into account difference of mortality between generations (defined by birth year Y) and mortality variations (in general: improvements) over time. Most recent developments consisted of adding a stochastic error term reflecting "unexplained" and random variations of mortality over time. – There are several models that consider random fluctuations of mortality either in additive or multiplicative way, so for instance: – This random fluctuation can be autocorrelated in time, or reflect mortality shocks From Deterministic to Stochastic Models 5

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Additive model with normal distribution: – similar to a Brownian "noise": normal distribution with expected value 0 – inconvenience: it can generate negative total mortality rates if not bounded – a variant introduces negatively autocorrelated (ARIMA(0,1,0)) random variations Model with lognormal distribution: – in aggregate format, mortality is a random lognormal variable with expected value qx and a standard deviation determined by calibration – no negative values, but a bigger "tail": resulting probabilities should be less than 1 Multiplicative models – the deterministic mortality rate is multiplied by a random variable with expected value 1; – here as well Gamma or logarithmic random variables are popular because they have positive values only. Random Fluctuation of Mortality 6

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Illustrations Survivorship Group P.V. temporary annuity P.V. Claims for a Term Life P.V. Level Premium Term Life Math. Provision Term Life 7

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Deterministic survivorship group with basis periodic table Stochastic mortality component with lognormal distribution, uncorrelated Calculation of a Survivorship Group 8

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean 92'553 (deterministic: 92'557) and standard deviation 188. Quant (5%) = 92'234; Quant (95%) = 92'853; ±0.33% Distribution of the Number of Survivors 9

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean (deterministic: ) and standard deviation Minimum = ; Maximum = ; ±0.14% Distribution of the P.V. of a Temporary Annuity 10

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean 4.875% (deterministic: 4.872%) and standard deviation 0.115%. Minimum = 4.422%; Maximum = 5.350%; ±9.5% Distribution of the Single Premium of a Term Insurance 11

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean % (deterministic: %) and standard deviation %. Minimum = %; Maximum = %; ±9.6% Distribution of the Level Premium of a Term Insurance 12

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Mathematical Provision - Term Life Policy ( x = 35; n = 30 ) 13

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, at duration 18 (max. prov.), mean % (deterministic: %) and standard deviation %. Minimum = %; Maximum = %; ±9.6% Distribution of the Math. Provision of a Term Insurance 14

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Stochastic mortality introduces a certain complexity in modelling. It barely has any impact on longevity (present value of annuities) and on products where savings and risk both intervene. It brings an additional information for pure Term covers: Single premiums, Annual Premiums and Reserves can vary in the same order of magnitude as the variation coefficient of mortality. What about mortality "shocks"? – they can be introduced in the stochastic mortality model or studied as scenarios; – they should not be modelled for policyholders separately, but strike the whole portfolio at the same time, taking correlation between risks into account. What about the comparative impact of other factors, such as mortality improvements or lapses? What Can We Learn? 15

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Modelling Mortality Shocks Survivorship Group P.V. temporary annuity P.V. Claims for a Term Life P.V. Level Premium Term Life Math. Provision Term Life 16

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health A mortality shock of 3.7 ‰ randomly introduced as a 1 in 30 years event. Calculation of Survivorship Group with Mortality Shocks 17

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean 92'454 and standard deviation 387 (vs. 188) Distribution of the Nb. of Survivors – Mortality Shocks 18

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean and standard dev (vs ). Minimum = ; Maximum = ; [-1.28%; 0.28%] P.V. of a Temporary Annuity with Mortality Shocks 19

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean 4.974% and standard dev % (vs %). Minimum = 4.308%; Maximum = 6.662%; [-13.4%; 33.9%] Single Premium of a Term Insurance – Mortality Shocks 20

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 10'000 simulations, mean % and std. dev % (vs %). Minimum = %; Maximum = %; [-13.6%; 35.3%] Level Premium of a Term Insurance – Mortality Shocks 21

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Math. Provision - Term Life Policy – Mortality Shocks 22

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Conclusion When should Stochastic Models be Used? When should use of Stochastic Models be Questioned? Alternatives to Stochastic Models Disadvantages of Stochastic Models Concluding Remarks 23

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health When required by regulation and / or standards of professionalism – calculation of risk-based capital or statutory liabilities (USA) When analysing extreme outcomes or "tail risks" that are not well understood – scenarios of very unlikely events are difficult to calibrate When using certain risk measures, such as Value at Risk (VaR) or Conditional Tail Expectation (CTE) – Such values are at times used to determine risk-based capital (cf. above) When certain percentiles are required – what would be the extent of a 1 in N years event? When one wants to understand where stress tests fall in the broader spectrum of possible outcomes When should Stochastic Mortality Models be Used? 24

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health When it is difficult / impossible to determine the appropriate probability distribution – normal / lognormal distributions not reasonable? When it is difficult or impossible to calibrate the model – credible historical experience missing When it is difficult or impossible to validate the model When should Use of Stochastic Models be Questioned? 25

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Stress testing / scenario testing – What if…? Collective Risk Models giving analytical and / or numerical answers Static factors – "load factors" developed on other cases Ranges – higher and lower bounds Alternatives to Stochastic Models 26

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health "Black Box" phenomenon – the model must be properly set up and understood before being used – it might give the impression of (false) precision while other basic assumptions are not fixed Improper calibration or validation – What is the purpose of the model? – are historical data relevant and consistent? Uses of inappropriate distributions or parameters – use of standard assumptions (normal, etc.) without thinking enough about the application framework Disadvantages of Stochastic Models 27

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Relative Impact of Stochastic Assumptions 28

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Stochastic Mortality Models should not be the "tree that hides the forest"! Modelling mortality improvements is very important as well Lapse rates have a much higher impact on the present value of claims Selective lapsation – mortality deterioration due to the cancellation of policies of good risks – is also a phenomenon deserving analysis and models. Sensitivity analysis and scenario testing are essential to the actuary to develop an educated judgement about assumptions Questions? Concluding Remarks 29

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Philippe Maeder – Swiss Re: – University of Lausanne: From the IAA: References 30

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health 31

General Public Release Fasecolda | Philippe Maeder | Senior Product Expert Life & Health Legal notice 32 ©2015 Swiss Re. All rights reserved. You are not permitted to create any modifications or derivative works of this presentation or to use it for commercial or other public purposes without the prior written permission of Swiss Re. The information and opinions contained in the presentation are provided as at the date of the presentation and are subject to change without notice. Although the information used was taken from reliable sources, Swiss Re does not accept any responsibility for the accuracy or comprehensiveness of the details given. All liability for the accuracy and completeness thereof or for any damage or loss resulting from the use of the information contained in this presentation is expressly excluded. Under no circumstances shall Swiss Re or its Group companies be liable for any financial or consequential loss relating to this presentation.