ari, N., A. De Waegenaere, B. Melenberg, and T. E. Nijman (2008). Longevity risk in portfolios of pension annuities. Insurance: Mathematics and Economics 42, 505–519.
Bauer, D., M. B orger, and J. Russ (2010). On the pricing of longevity-linked securities. Insurance: Mathematics and Economics 46, 139–149.
Blake, D., A. J. G. Cairns, and K. Dowd (2006). Living with mortality: longevity bonds and other mortality-linked securities. British Actuarial Journal 12, 153–228.
- Boonen, T. (2010). Bargaining for over-the-counter risk redistributions. Master’s Thesis, Department of Econometrics and Operations Research, Tilburg University, Tilburg, available at http://arno.uvt.nl/show.cgi?fid=107606.
Paper not yet in RePEc: Add citation now
- Borch, K. (1962). Equilibrium in a reinsurance market. Econometrica 30, 424–444.
Paper not yet in RePEc: Add citation now
- Brouhns, N., M. Denuit, and J. Vermunt (2002). Measuring the longevity risk in mortality projections. Bulletin of the Swiss Association of Actuaries, 105–130.
Paper not yet in RePEc: Add citation now
- C Lee-Carter model In this appendix, we describe the Lee-Carter model (1992). The probability that an individual of age x at time t survive the next year is modeled as px,t = exp(−mx,t), (48) where mx,t represents the central death rate of a men with age x at time t (see, e.g., Pitacco et al., 2009). The central death rate is given by mx,t = Dx,t/Ex,t, where Dx,t is the observed number of deaths in year t in the cohort aged x at the beginning of year t, and Ex,t is the 10The rationale for young participants of a death benefit insurer is that death benefit insurance is often obliged if individuals buy a mortgage. 20 30 40 50 60 70 80 90 100 0.01 0.02 0.03 0.04 0.05 Density Age 0.2 0.4 0.6 0.8 Accrued pension rights
Paper not yet in RePEc: Add citation now
Cairns, A. J. G., D. Blake, and K. Dowd (2006). A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance 73, 687–718.
- Cairns, A. J. G., D. Blake, K. Dowd, G. Coughlan, and D. Epstein (2007). A quantitative comparison of stochastic mortality models using data from England & Wales and the United States. Pensions Institute Discussion Paper PI-0701.
Paper not yet in RePEc: Add citation now
- Cairns, A. J. G., D. Blake, K. Dowd, G. Coughlan, D. Epstein, and M. Khalaf-Allah (2008). Mortality density forecasts: An analysis of six stochastic mortality models. Pensions Institute Discussion Paper PI-0801.
Paper not yet in RePEc: Add citation now
Compte, O. and P. Jehiel (2010). The coalitional nash bargaining solution. Econometrica 78, 1593–1623.
- Cossette, H., A. Delwarde, M. Denuit, F. Guillot, and E. Marceau (2007). Pension plan valuation and dynamic mortality tables. North American Actuarial Journal 11, 279–298.
Paper not yet in RePEc: Add citation now
- Coughlan, G., D. Epstein, A. Ong, A. Sinha, J. Hevia-Portocarrero, E. Gingrich, M. KhalafAllah, and P. Joseph (2007). Lifemetrics: A toolkit for measuring and managing longevity and mortality risks. Technical report, JPMorgan.
Paper not yet in RePEc: Add citation now
Dowd, K., A. J. G. Cairns, D. Blake, G. D. Coughlan, D. Epstein, and M. Khalaf-Allah (2010). Evaluating the goodness of fit of stochastic mortality models. Insurance: Mathematics and Economics 47, 255–265.
Dowd, K., D. Blake, A. J. G. Cairns, and P. Dawson (2006). Survivor swaps. Journal of Risk and Insurance 73, 1–17.
- Dowd, K., D. Blake, and A. J. G. Cairns (2008). Facing up to uncertain life expectancy: The longevity fan charts. Discussion Paper PI-0703, The Pensions Institute, Cass Business School.
Paper not yet in RePEc: Add citation now
- Gerber, H. and G. Pafumi (1998). Utility functions: from risk theory to finance. North American Actuarial Journal 2, 74–91. H
Paper not yet in RePEc: Add citation now
- Kalai, E. (1977). Nonsymmetric Nash solutions and replications of 2-person bargaining. International Journal of Game Theory 6, 129–133.
Paper not yet in RePEc: Add citation now
- Lee, R. D. and L. Carter (1992). Modelling and forecasting the time series of U.S. mortality. Journal of the American Statistical Association 87, 659–671.
Paper not yet in RePEc: Add citation now
Nash, J. F. (1950). The bargaining problem. Econometrica 18, 155–162.
- OECD (2005). Ageing and pension system reform: Implications for financial markets and economic policies. Financial Market Trends 1.
Paper not yet in RePEc: Add citation now
Olivieri, A. (2001). Uncertainty in mortality projections: an actuarial perspective. Insurance: Mathematics and Economics 29, 231–245.
Pitacco, E., M. Denuit, S. Haberman, and A. Oliveiri (2009). Modelling longevity dynamics for pensions and annuity business. Oxford University Press.
Plat, R. (2009). On stochastic mortality modeling. Insurance: Mathematics and Economics 45, 393–404.
- Proof of Theorem 6. Scarf (1967) considers the correspondence b V defined as b V (S) = n a ∈ IRS ∃(X post i )i∈S ∈ b F(S) such that a ≤ (b ui(Xpost i ))i∈S o , (43) where b F(S)= (Xpost i )i∈S ∈ IRS : P i∈S Xpost i = P
Paper not yet in RePEc: Add citation now
Riddell, W. C. (1981). Bargaining under uncertainty. American Economic Review 71, 579– 590.
- Scarf, H. E. (1967). The core of an n person game. Econometrica 35, 50–69.
Paper not yet in RePEc: Add citation now
Stevens, R., A. De Waegenaere, and B. Melenberg (2011). Calculating capital requirements for longevity risk in life insurance products. using an internal model in line with Solvency II. Working paper, Tilburg University.
Tsai, J., J. Wang, and L. Tzeng (2010). On the optimal product mix in life insurance companies using conditional value at risk. Insurance: Mathematics and Economics 46, 235–241.
Wang, J., H. Huang, S. Yang, and J. Tsai (2010). An optimal prodict mix for hedging longevity risk in life insurance companies: The immunization theory approach. Journal of Risk and Insurance 77, 473–497. A Proofs Proof of Proposition 1. (i) The fact that Stability implies No Pareto Improvement follows immediately from (5) and (6). To see that Stability implies Individual Rationality, note that for all i ∈ N, it holds that Xi ∈ F({i}) and ∆Ui(Xi) = 0. Moreover, it follows from (6) that if (Xpost i )i∈N satisfies Stability, then Xpost i ∈ NI({i}) for all i ∈ N. Combined with (4), this implies that if (Xpost i )i∈N satisfies Stability, then ∆Ui(Xpost i ) ≥ ∆Ui(Xi) = 0 for all i ∈ N. (ii) it suffices to show that No Pareto Improvement and Individual Rationality implies Stability. This follows immediately from (5) and (6), and the fact that Individual Rationality implies that Xpost i ∈ NI({i}) for all i ∈ N.