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Deferred life annuities: on the combined effects of stochastic mortality and interest rates

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Abstract

The paper analyzes the systematic risk which is inherent in a portfolio of deferred life annuities. We take into account stochastic mortality as well as stochastic interest rates. For the specification of the mortality rate dynamics, we consider a pure diffusion model as well as a compound Poisson jump model. The interest rate dynamics are given by a one-factor Hull–White model. All models, interest rate and mortality rate, are calibrated to financial market as well as demographic data. We use Monte Carlo simulations to approximate the variance of the discounted cash flow and its decomposition into a pooling and a non-pooling risk part. We also consider pricing effects using the principle of zero expected utility and the quantile principle. The estimated risk premiums are benchmarked to the equivalence premium. Finally, we focus on solvency requirements which are based on the investment decisions and the associated shortfall probability of the annuity provider.

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Notes

  1. The redeemed new annuity and pension insurance business contributes regular premiums of about 1.3 billion Euro with an overall share of 31%. The business in force at the end of 2009 consists of 18.3 million contracts (equals a share of 20%), see German Insurance Association (GDV) (2009).

  2. To be precise, mortality risk denotes the risk stemming from mortality rates which are systematically higher than expected and longevity risk refers to the opposite risk scenario.

  3. See Brouhns et al. (2002) and Pitacco (2004a) for an overview over traditional mortality modeling.

  4. Notice that the mortality derivative market is almost illiquid at present and tailor-made reinsurance solutions are cost-intensive. Thus, we do not consider the non-pooling risk to be a traded risk.

  5. Biffis (2005) and Schrager (2006) choose an affine mortality process, Korn et al. (2006) use a stochastic version of the Perks-/Gompertz-Mortality law and Dahl and Møller (2006) a square root diffusion process, whereas Bayraktar et al. (2009) analyze a general diffusion process.

  6. The accumulation phase [0,T[ represents a kind of savings plan, whereas the decumulation phase \([T,\bar{T}]\) forms an immediate life annuity.

  7. Kou (2002) introduces a compound Poisson jump-diffusion process with double asymmetric jump size distribution in the context of option pricing. Bravo (2008) takes this approach to model mortality rates under bidirectional shocks.

  8. The first result follows from Luciano and Vigna (2006) and Bravo (2008). A proof of the second result is given in Luciano and Vigna (2006).

  9. For example, Bravo (2008) estimates the values \(\eta_1=2.8\times10^{-2}\) and \(\eta_2=9\times10^{-5}\) for the survival function of an 65-year old insured and the year 2004. Luciano and Vigna (2006) state that negative jumps are adequate to describe mortality variations. They estimate an average size \(\eta= 3\times10^{-5}\) for the year 1945 and a retiree aged 65.

  10. A swaption is called at-the-money if its strike equals the forward par swap rates.

  11. Recall that, according to Table 5, only σ spot  = 0.0094 is consistent with the swaption straddle volatilities. However, for varying σ spot , we nevertheless calibrate to the same initial term structure of interest, i.e. we use the function θ(t) as in Table 2. Thus, we still have the same expected values, independent of the choice of σ spot .

  12. For dynamic risk minimizing hedging strategies, we refer to Dahl (2004) and Dahl and Møller (2006). The authors consider dynamic risk minimizing hedging strategies under the so called minimal martingale measure introduced by Schweizer (2001).

  13. Among others, Bauer and Weber (2008) stress that it might be unrealistic to finance intermediate shortfalls against a savings account rather than money at the market interest rate.

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Authors and Affiliations

  1. Mercator School of Management, Chair of Insurance and Risk Management, University of Duisburg-Essen, Lotharstr. 65, 47057, Duisburg, Germany

    Antje Mahayni & Daniel Steuten

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  1. Antje Mahayni
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  2. Daniel Steuten
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Correspondence to Daniel Steuten.

Appendix: A market data and calibration results

Appendix: A market data and calibration results

See Table 10 and Fig 7

Table 10 Euro at-the-money swaption straddle implied volatility quotes taken from Thomson Reuters (2010) on 14/06/10
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Fig. 7
figure 7

Long run behavior of the spot rate process (7) using the (estimated) parameters given in Table 2

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Mahayni, A., Steuten, D. Deferred life annuities: on the combined effects of stochastic mortality and interest rates. Rev Manag Sci 7, 1–28 (2013). https://doi.org/10.1007/s11846-011-0066-5

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