Risk analysis of annuity conversion options in a stochastic mortality environment Joint work with Alexander Kling and Jochen Russ Research Training Group 1100 Katja Schilling August 3, 2012
Page 2 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Introduction Model framework Numerical results
Page 3 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Introduction Introduction Model framework Numerical results
Page 4 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Introduction Unit-linked deferred annuities Premiums are accumulated in a fund At retirement, the policyholder has the choice between account value as a lump sum converting the account value into an annuity at then prevailing rates Resulting annuity payment highly depends on fund value interest rate expectations mortality expectations
Page 4 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Introduction Unit-linked deferred annuities Premiums are accumulated in a fund At retirement, the policyholder has the choice between account value as a lump sum converting the account value into an annuity at then prevailing rates Resulting annuity payment highly depends on fund value interest rate expectations mortality expectations Annuity conversion options Insurance companies add guarantees to pure unit-linked deferred annuities, e.g. Guaranteed annuity options (GAOs) Guaranteed minimum income benefits (GMIBs) Such options can become unexpectedly valuable (cf. UK)
Page 5 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Introduction Literature Pricing GAOs under deterministic mortality: e.g. Boyle and Hardy (2003), Ballotta and Haberman (2003), Van Haastrecht et al. (2010) Pricing GAOs under stochastic mortality: e.g. Milevsky and Promislow (2004), Biffis and Millossovich (2006), Ballotta and Haberman (2006) Pricing GMIBs: e.g. Bauer et al. (2008), Marshall et al. (2010), Bacinello et al. (2011)
Page 5 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Introduction Literature Pricing GAOs under deterministic mortality: e.g. Boyle and Hardy (2003), Ballotta and Haberman (2003), Van Haastrecht et al. (2010) Pricing GAOs under stochastic mortality: e.g. Milevsky and Promislow (2004), Biffis and Millossovich (2006), Ballotta and Haberman (2006) Pricing GMIBs: e.g. Bauer et al. (2008), Marshall et al. (2010), Bacinello et al. (2011) Research objectives (1) What risk do annuity conversion options imply for the insurer? (2) How does the risk change with different option types? (3) Is it possible to reduce the risk by applying risk management strategies? (4) What risk (fund, interest rate or mortality risk) dominates the total risk?
Page 6 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework Introduction Model framework Numerical results
Page 7 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework Notation 0 T x τ x P 0 A t a T T : deferment period/retirement date x: policyholder s age at inception of the contract (t = 0) τ x : remaining lifetime P 0 : single premium A t : account value a T : value of an immediate annuity with unit amount per year
Page 8 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework Option types Guaranteed annuity option (GAO) certain minimum conversion rate g for converting the account value into a lifelong annuity at time T g: annual annuity per unit account value at time T V GAO T = 1 {τx >T } max {ga T a T A T, 0} GAO with limit Guaranteed minimum income benefit (GMIB)
Page 9 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework Option types Guaranteed annuity option (GAO) certain minimum conversion rate g for converting the account value into a lifelong annuity at time T g: annual annuity per unit account value at time T = 1 {τx >T }ga T max {a T 1g }, 0 V GAO T GAO with limit Guaranteed minimum income benefit (GMIB)
Page 10 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework Option types Guaranteed annuity option (GAO) certain minimum conversion rate g for converting the account value into a lifelong annuity at time T g: annual annuity per unit account value at time T = 1 {τx >T }ga T max {a T 1g }, 0 V GAO T GAO with limit conversion rate g up to a maximum account value L (limit) = 1 {τx >T }g min {A T, L} max {a T 1g }, 0 V Limit T Guaranteed minimum income benefit (GMIB) fixed minimum annuity amount M (= gg) V GMIB T = 1 {τx >T } max {gga T A T, 0}
Page 11 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework Risk management strategies No hedging Hedging No option fee A Option fee B C Strategy B Strategy C Option fee is invested in money market instruments Static hedge against the financial risk during the deferment period Assumption: Option fee = Hedging costs under strategy C
Page 12 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Model framework We analyze...... the insurer s loss distribution at time T for each combination of option type and risk management strategy by performing a Monte Carlo simulation Modeled risk processes Fund value: Geometric Brownian motion ds(t) = (λ S + r(t))s(t)dt + σ S S(t)dW S (t), S(0) > 0. Short rate: one-factor Cox-Ingersoll-Ross model dr(t) = κ(θ r(t))dt + σ r r(t)dw r (t), r(0) > 0. Mortality: 6-factor forward model (cf. Bauer et al., 2008a) dµ(t, T, x) = α(t, T, x)dt + σ(t, T, x)dw µ (t), µ(0, T, x) > 0.
Page 13 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Introduction Model framework Numerical results
Page 14 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Insurer s loss (base case) 1 0.9 0.8 1 0.7 Empirical CDF 0.6 0.5 0.4 GAO A Empirical CDF GAO A 0.3 0.2 0.9 0.1 0 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Insurer s loss at time T 0 0.1 0.2 Insurer s loss at time T Risk of GAOs seems to be quite low for the insurer
Page 15 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Insurer s loss (base case) (2) 1 0.9 0.8 1 0.7 Empirical CDF 0.6 0.5 0.4 GAO A Limit A Empirical CDF GAO A Limit A 0.3 0.2 0.1 0 0.2 0.1 0 0.1 0.2 0.3 0.4 Insurer s loss at time T 0 0.1 0.2 Insurer s loss at time T GAO in-the-money Limit in-the-money
Page 16 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Insurer s loss (base case) (3) 1 0.9 0.8 0.7 Empirical CDF 0.6 0.5 0.4 0.3 0.2 0.1 GAO A Limit A GMIB A 0-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Insurer's loss at time T Risk of GMIBs seems to be much higher than risk of GAOs/Limits
Page 17 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Insurer s loss (very low interest rates) 1 0.9 0.8 0.7 Empirical CDF 0.6 0.5 0.4 0.3 0.2 0.1 GAO A Limit A GMIB A 0-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Insurer's loss at time T Risk of GAOs is now the highest Limit becomes relevant Option values: 0.0157 (GAO A ), 0.0093 (Limit A ), 0.1386 (GMIB A )
Page 18 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Insurer s loss (base case) - Hedging strategies 1 0.9 0.8 0.7 Empirical CDF 0.6 0.5 0.4 0.3 0.2 0.1 GMIB A GMIB B GMIB C 0-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Insurer's loss at time T Strategy B: In many cases low profit, but risk is not significantly reduced Strategy C: Risk is significantly reduced Mortality is not negligible
Page 19 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Main risk driver Sensitivity with respect to... Interest rate level θ Volatility of mortality σ Fund volatility σ S Fund risk premium λ S Option most affected GAO/Limit GAO/Limit GMIB GMIB Interest rate risk and mortality risk seem to dominate GAO and Limit Fund risk seems to dominate GMIB
Page 19 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Main risk driver Sensitivity with respect to... Interest rate level θ Volatility of mortality σ Fund volatility σ S Fund risk premium λ S Option most affected GAO/Limit GAO/Limit GMIB GMIB Interest rate risk and mortality risk seem to dominate GAO and Limit Fund risk seems to dominate GMIB Question of decomposing the risk between the different risk drivers requires further research!
Page 20 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Contact Research Training Group 1100 University of Ulm Katja Schilling katja.schilling@uni-ulm.de Thank you very much for your attention!
Page 21 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results References (1) A. Bacinello, P. Millossovich, A. Olivieri, E. Pitacco (2011): "Variable annuities: A unifying valuation approach", Insurance: Mathematics and Economics, 49(3), 285-297. L. Ballotta, S. Haberman (2003): "Valuation of guaranteed annuity conversion options", Insurance: Mathematics and Economics, 33(1), 87-108. L. Ballotta, S. Haberman (2006): "The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case", Insurance: Mathematics and Economics, 38(1), 195-214. D. Bauer, M. Börger, J. Russ and H.-J. Zwiesler (2008a): "The Volatility of Mortality", Asia-Pacific Journal of Risk and Insurance, 3(1), 172-199. D. Bauer, A. Kling, J. Russ (2008b): "A universal pricing framework for guaranteed minimum benefits in variable annuities", Astin Bulletin, 38(2), 621-651. E. Biffis, P. Millossovich (2006): "The fair value of guaranteed annuity options", Scandinavian Actuarial Journal, 2006(1), 23-41.
Page 22 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results References (2) M. Börger (2010): "Deterministic shock vs. stochastic value-at-risk an analysis of the Solvency II standard model approach to longevity risk", Blätter der DGVFM, 31(2), 1-35. P. Boyle and M. Hardy (2003): "Guaranteed Annuity Options", ASTIN BULLETIN, 33(2), 125-152. S. Graf, A. Kling, J. Russ (2010): "Financial Planning and Risk-return profiles", Working Paper, University of Ulm. C. Marshall, M. Hardy, D. Saunders (2010): "Valuation of a guaranteed minimum income benefit", North American Actuarial Journal, 14(1), 38-58. M. Milevsky, S. Promislow (2001): "Mortality derivatives and the option to annuitise", Insurance: Mathematics and Economics, 29(3), 299-218. A. Van Haastrecht, R. Plat, A. Pelsser (2010): "Valuation of guaranteed annuity options using a stochastic volatility model for equity prices", Insurance: Mathematics and Economics, 47(3), 266-277.
Page 23 Risk analysis of annuity conversion options in a stochastic mortality environment August 3, 2012 Katja Schilling Numerical results Model parameters Description Parameter Value Age x 50 Term to maturity T 15 Single premium P 0 1 Conversion rate g 0.05 Limit L 1 Guaranteed account value G 1 Number of realizations N 10,000 Number of discretization steps n 1,500 GBM initial value S(0) 100 GBM risk premium λ S 0.03 GBM volatility σ S 0.22 CIR initial value r(0) 0.0029 CIR speed of reversion κ ( κ) 0.2 (0.2) CIR mean level θ ( θ) 0.045 (0.045) CIR volatility σ r ( σ r ) 0.075 (0.075) Explicit GBM-CIR correlation ρ 0 Limiting age ω 121