Basis risk in static versus dynamic longevity-risk hedging

Abstract

This paper provides a tractable, parsimonious model for assessing basis risk in longevity and its effect on the hedging strategies of Pension Funds and annuity providers. Basis risk is captured by a single parameter, that measures the co-movement between the portfolio and the reference population’s longevity. The paper sets out the static, full and customized swap-hedge for an annuity, and compares it with a dynamic, partial, and index-based hedge. We calibrate our model to the UK and Scottish populations. The effectiveness of static versus dynamic strategies depends on the rebalancing frequency of the second, on the relative costs, and on basis risk, which does not affect fully-customized, static hedges. We show that appropriately calibrated dynamic hedging strategies can still be reasonably effective, even at low rebalancing frequencies.

Keywords:

• Longevity risk
• customized vs. indexed hedge
• longevity swaps
• longevity bonds
• rebalancing frequency

AMS Subject Classifications:

• G22
• G32

Acknowledgements

The Authors would like to thank conference participants to the IRMC Conference 2015, the 19th IME Conference, the Longevity 11 Conference. The Authors gratefully acknowledge financial support by the Global Risk Institute, Canada.

Notes

No potential conflict of interest was reported by the authors.

1 For a description of a customized and an index-based transaction involving JP Morgan as a buyer of longevity risk, see Barrieu et al. (2012).

2 Let the filtration be the filtration generated by the two Brownian motions.

3 In details, given two time instants , then the distribution of conditional on is(5)

where denotes the density of a noncentral chi-square random variable with d degrees of freedom, where(6)

and the noncentrality parameter is(7)

Similarly, the distribution of r(t) conditional on r(u) is(8)

where denotes the density of a noncentral chi-square random variable with degrees of freedom, with(9)

and the noncentrality parameter is(10)

4 It is easy to show that(16) (17)

5 Using the fact that , (20(20) ) becomes(19)

Hence, we have an expression for the survival equivalent to (13(13) ).

6 As we know, the forecast error in longevity has been substantial over the last decades. Approximately, expected lifetime improvement has been underestimated by 3 years over the last century (see International Monetary Fund (IMF) (2012)).

7 Here(23) (24)

8 Indeed, for simplicity, we abstract from idiosyncratic risk and consider the annuity as a perfectly-diversified portfolio of annuities issued on homogeneous individuals.

9 An alternative would be to fix a unique swap rate for all periods, . In this case fairness would be guaranteed by setting K equal to the following value:

10 We can use a number of other instruments to cover the annuity, starting from life assurances or death bonds, which pay the benefit in case of death. We restrict the attention to longevity bonds for the sake of simplicity, whose payoff for the maturity T is

11 Reinsurance companies have less constraints in this respect. For instance, they can swap pure endowments or issue longevity bonds: see for instance Cowley & Cummins (2005).

12 We remark that this result is obtained in the absence of transaction costs, which we neglect here and will be higher, the higher the frequency.

Additional information

Funding

This work was supported by the Global Risk Institute, Canada.