Longevity Seminar. Forward Mortality Rates. Presenter(s): Andrew Hunt. Sponsored by

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1 Longevity Seminar Sponsored by Forward Mortality Rates Presenter(s): Andrew Hunt

2 Forward mortality rates SOA Longevity Seminar Chicago, USA 23 February 2015 Andrew Hunt

3 Agenda Why forward mortality rates? Defining forward mortality rates Market-consistent measure Hedging Discussion 2

4 Why forward mortality rates? Valuing technical provisions and pricing longevity-linked securities requires consistent expectations of future mortality rates C.f. forward interest rates embedded in yield curve for bond pricing Other approaches to forward mortality rates Continuous time Bauer et al (2008,2012) Non-parametric Zhu and Bauer (2011a,b,2014) Olivier-Smith model Olivier and Jeffrey (2004), Smith (2005) 3

5 Defining forward mortality rates Hypothetical market in longevity zeros with price Define Forward mortality rates in discrete time 4

6 Defining forward mortality rates We identify Approximation due to Jensen s inequality but tested numerically and reasonable (within 0.1%) across most ages and years Assume that the force of mortality in the real-world measure is modelled by an age/period/cohort mortality model Hunt and Blake (2014d) Then 5

7 Defining forward mortality rates Assume random walk with drifts for the period functions Deterministic functions may be included in drift, identifiability reasons Hunt and Blake (2014b,c), for Therefore 6

8 Defining forward mortality rates Use Bayesian approach to model and project the cohort parameters Fitted parameter estimates based on partial information Assume annual observations of each cohort providing new information Cohort parameter only known with certainty once observed over its entire life 7

9 Defining forward mortality rates Details get quite involved see Hunt and Blake (2014a) 8 However, this approach is necessary for measuring risk, as discussed later

10 Defining forward mortality rates Together, these give the forward mortality surface Difference < 0.05%, due to rounding errors in simulations 9

11 Market-consistent measure In order to value liabilities or value securities, we need to convert the forward mortality surface from the real-world to a market-consistent measure Use Esscher transformation, see Gerber and Shiu (1994), Bühlmann et al (1996) 10

12 Market-consistent measure Values of market prices of longevity risk found from: prices of traded longevity securities (if they exist) or deterministic projection of mortality (e.g., CMI Projection Model) Once market-consistent measure calibrated, surface of forward mortality rates can be used to value many different longevity-linked securities consistently Currently unable to value longevity-linked options (work in progress) 11

13 Market-consistent measure 1 25 LC 0.9 CBDX APC 0.8 RP GP LC CBDX APC RP GP s-forwards Annuities Year Age 12 q-forwards LC CBDX APC RP GP CMI 1.75% Year e-forwards LC CBDX APC RP GP Observed Year

14 Hedging For many purposes, we need to know how the forward mortality surface updates E.g., Value at Risk, hedging This depends upon how the period and cohort functions update with one year s extra observations NB by tower property of conditional expectations, have Period functions are straightforward 13

15 Hedging Cohort functions, need to use Bayesian approach and assumed data generating process 14

16 Hedging Using this framework, we can update the forward mortality surface by one year and recalculate liability values Value at Risk Applications to Solvency II SCR and Risk Margin % VaR 99% VaR 99.5% VaR Economic Capital Ratio Age Age

17 Hedging Can also recalculate longevity-linked asset prices in consistent fashion Forward values (per 100 nominal) q-forward s-forward e-forward

18 Hedging Because liabilities and securities prices are valued from the same forward mortality surface, they will be updated consistently with one another Useful for investigating hedge effectiveness for different value hedging strategies Single hedging instrument to hedge annuity portfolio Hedge ratio chosen to minimise variance 17

19 Hedging Empirical distribution of liability value for different hedging instruments Hedged - q-forward Hedged - s-forward Hedged - e-forward Unhedged

20 Hedging Risk measure (as % of liabilities) Unhedged q-forward s-forward e-forward VaR(95%) 2.44% 0.84% 1.08% 0.21% % reduction - 65% 55% 91% TVaR(95%) 3.06% 1.07% 1.37% 0.27% % reduction - 66% 55% 91% 19 Relatively high reductions in risk for very simple (single instrument) hedging strategies Model dependent (though valuation will be mark-to-model for foreseeable future) No allowance for basis risk

21 Discussion Forward mortality rates provide a useful framework for many of the issues with the valuation / risk management of longevity risk We have introduced a discrete time forward mortality rate framework which: Is consistent with models of the short mortality rate Can be calibrated easily to available data Can be used with a variety of underlying mortality models Can be extended for basis risk and longevity-linked options 20

22 Selected References 21 Bauer, D., Benth, F. E., Kiesel, R., Modeling the forward surface of mortality. SIAM Journal on Financial Mathematics 3 (1), Bauer, D., Börger, M., Ruß, J., Zwiesler, H., The volatility of mortality. Asia-Pacific Journal of Risk and Insurance 3 (1), 10. Bühlmann, H., Delbaen, F., Embrechts, P., Shiryaev, A. N., No-arbitrage, change of measure and conditional Esscher transforms. CWI Quarterly 9, Coughlan, G. D., Epstein, D., Sinha, A., Honig, P., q-forwards: Derivatives for transferring longevity and mortality risks. JPMorgan Pension Advisory Group. Dawson, P., Dowd, K., Cairns, A. J. G., Blake, D., Options on normal underlyings with an application to the pricing of survivor swaptions. Journal of Futures Markets 29 (8), Gerber, H., Shiu, E., Option pricing by Esscher transforms. Transactions of the Society of Actuaries 46, Hunt, A., Blake, D., 2015a. Consistent mortality projections allowing for trend changes and cohort effects. Work in Progress. Hunt, A., Blake, D., 2015d. On the structure and classification of mortality models. Work in Progress. Li, J. S.-H., Luo, A., Key Q-duration: A framework for hedging longveity risk. ASTIN Bulletin 42 (2), Olivier, P., Jeffrey, T., Stochastic mortality models. URL Norberg, R., Forward mortality and other vital rates - Are they the way forward? Insurance: Mathematics and Economics 47, Smith, A., Stochastic mortality modelling. URL 01_PensionerMortality/ _Stochastic%20MortalityModelling/ Zhu, N., Bauer, D., 2011a. Applications of forward mortality factor models in life insurance practice. Geneva Papers on Risk and Insurance 36, Zhu, N., Bauer, D., 2011b. Coherent modeling of the risk in mortality projections: A semi parametric approach. Tech. Rep. 678, Georgia State University. Zhu, N., Bauer, D., A cautionary note on natural hedging of longevity risk. North American Actuarial Journal 18 (1),

23 Questions? Thank you very much for your attention and your feedback 22

24 Addendum Solvency II SCR is the 99.5% VaR of the technical provisions Therefore, forward rate model can calculate SCR Avoids nested sims for SCR Compare with Solvency II standard model - 20% shock to 0.16 mortality to proxy for VaR Forward mortality rates 0.14 SII standard model C.f., Börger (2010) 0.12 Economic Capital Ratio Age

25 Addendum Calculation of risk margin suffers from calculation problems (nested simulations) Various different approaches proposed Approach SCR (% liabilities) Risk Margin (% liabilities) Total Risk Capital Nested simulations 4.0% 3.3% 7.3% Duration 4.0% 2.5% 6.5% Standard model 4.0% 5.3% 9.3% Proportional 4.0% 4.0% 8.0% Median 4.0% 3.5% 7.5% 24 10model points 4.0% 3.5% 7.5%