A Mixed Bond and Equity Fund Model for the Valuation of Variable Annuities

Abstract

Variable annuity (VA) policies are typically issued on mutual funds invested in both fixed income and equity asset classes. However, due to the lack of specialized models to represent the dynamics of fixed income fund returns, the literature has primarily focused on studying long-term investment guarantees on single-asset equity funds. This article develops a mixed bond and equity fund model in which the fund return is linked to movements of the yield curve. Theoretical motivation for our proposed specification is provided through an analogy with a portfolio of rolling horizon bonds. Moreover, basis risk between the portfolio return and its risk drivers is naturally incorporated into our framework. Numerical results show that the fit of our model to Canadian VA data is adequate. Finally, the valuation of VAs is illustrated and it is found that the prevailing interest rate environment can have a substantial impact on guarantee costs.

Risk allocation through shapley decompositions, with applications to variable annuities

“…The guaranteed amount can either be fixed, that is, for all t and some constant , or it can evolve stochastically based on a ratchet provision. The ratchet provision considered in this work, when applied, is similar to that considered in Augustyniak et al (2021). In each period at least 10 years prior to the maturity of the contract, it entails increasing the guaranteed amount, at most once a year, to the account value when the latter surpasses the guaranteed amount by 15%: where is the number of times in the current year that the guaranteed amount was increased prior to time .…”

“…This section describes the financial market setup and cash flows associated with variable annuities contracts considered, which are inspired by Trottier et al (2018) and Augustyniak et al (2021).…”

“…Lapse penalties are characterized by a function which is typically decreasing with respect to time in practice. As in Augustyniak et al (2021), lapse penalties are assumed to start at 7% of the account value the first year, and then to decrease by 1% every year until it reaches 0% after the seventh year: where is the integer part function. Assuming a constant force of mortality and lapsation in any period (see Appendix for details), the inflow to the insurer related to lapse penalties is where , the proportion of policyholders active at who lapse their policy in the next period, is For a GMMB policy, the insurer promises to pay to all policyholders active at maturity T the difference between the guaranteed amount and the account value if it is positive: where denotes the guaranteed amount at maturity.…”

“…This section outlines the stochastic models driving market dynamics used in the present work. For the evolution of the term structure, the discrete-time Gaussian three-factor model presented in Augustyniak et al (2021) is used, where -dynamics of the annualized risk-free rate are where , , are constant model parameters, and is a multivariate standard normal process with correlation for some correlation matrix . The advantages of such multi-factor model are twofold.…”

“…The underlying fund of the variable annuity follows the mixed fund model dynamics from Augustyniak et al (2021), which consist in an EGARCH process where the conditional mean is mapped to equity fund returns and term structure factor variations to reflect the presence of both equity and fixed income assets within the fund: …”