Valuing variable annuity guarantees on multiple assets

Abstract

Guarantees embedded variable annuity contracts exhibit option-like payoff features and the pricing of such instruments naturally leads to risk neutral valuation techniques. This paper considers the pricing of two types of guarantees; namely, the Guaranteed Minimum Maturity Benefit and the Guaranteed Minimum Death Benefit riders written on several underlying assets whose dynamics are given by affine stochastic processes. Within the standard affine framework for the underlying mortality risk, stochastic volatility and correlation risk, we develop the key ingredients to perform the pricing of such guarantees. The model implies that the corresponding characteristic function for the state variables admits a closed form expression. We illustrate the methodology for two possible payoffs for the guarantees leading to prices that can be obtained through numerical integration. Using typical values for the parameters, an implementation of the model is provided and underlines the significant impact of the assets’ correlation structure on the guarantee prices.

Keywords:

• mortality risk
• variable annuity
• stochastic volatility risk
• correlation risk
• multiple assets
• European option

AMS Subject Classifications:

• C63
• G22
• G13
• G12

Acknowledgements

We would like to thank participants of the CEPAR Long-Term Care and Longevity Insurance Workshop, Sydney, Australia (2014); the 33rd Southern Africa Mathematical Sciences Association Annual Conference, Victoria Falls, Zimbabwe (2014); the 49th Actuarial Research Conference held at University of California Santa Barbara in July 2014; the 22nd Annual Colloquium of Superannuation Researchers held at UNSW Australia in July 2014 for their helpful feedback. The usual caveat applies.

Notes

No potential conflict of interest was reported by the authors.

1 Note that each guarantee is subject to the policyholder paying an associated guarantee fee.

2 This is also called the Guaranteed Minimum Accumulation Benefit (GMAB).

3 See Bacinello et al. (2011) for the pricing of such guarantees.

4 See also Ng & Li (2013) for another kind of multiasset extension of variable annuity guarantees. In a slightly different context, Jin & Ren (2014) and Jin et al. (2014) focus on multivariate claims (or losses), the former work leading them to consider the multivariate Fast Fourier transform as we will do here while the latter involves the moment-generating function of a bivariate variable and an approximation of the density. Regarding affine processes, which will be used in this paper, approximation of their density is presented in Filipović et al. (2013).

5 For notational convenience, we let .

6 Note that for convenience we have suppressed the dependence of the payoff function on and such that . Also, will denote the expectation conditional conditional to the filtration generated by and (i.e. ).

7 Here, .

8 These last numerical results are not reported here, but are available upon request from the authors.