Single and Cross-Generation Natural Hedging of Longevity and Financial Risk

Abstract

The paper provides natural hedging strategies among death benefits and annuities written on a single and on different generations. It obtains closed-form Delta and Gamma hedges, in the presence of both longevity and interest rate risk. We present an application to UK data on survivorship and bond dynamics. We first compare longevity and financial risk exposures: Deltas and Gammas for longevity risk are greater in absolute value than the corresponding sensitivities for interest rate risk. We then calculate the optimal hedges, both within and across generations. Our results apply to both asset and asset-liability management.

Assessing the Solvency of Insurance Portfolios Via a Continuous Time Cohort Model

“…2 Given the results in Luciano et al (2012), the fair-value of a whole-life annuity A(x, R), (Z A x ), is the expected present value of the future payments to the annuitant, discounted appropriately given the term structure of interest rates and survivorship…”

“…In the numerical section, we also explore the case in which, instead of implementing a portfolio allocation strategy of the type (δ M (t i ), δ B (t i )), we Delta-hedge the portfolio according to Luciano et al (2012). The Appendix summarizes the theory behind the strategy.…”

“…Stevens et al (2011) highlighted the importance of considering the interactions between longevity risk and financial risks, as such an omission might lead to overestimation of the natural hedging potential. Luciano et al (2012) proposed a Delta-Gamma hedging strategy, accounting for the effects of natural hedging on the interest-rate risk exposure of the company. This paper is the first to propose an analysis of the effectiveness of natural hedging strategies in the context of an ALM model of the insurance company in which liabilities are evaluated at market values and affected by interest-rate and longevity risks alike.…”

“…We couple a standard description of the financial market by means of the well-known Vasicek (1977) model with a parsimonious description of mortality risk via a continuous-time cohort based stochastic model, following Luciano and Vigna (2008). This choice, in addition to being reasonably accurate in describing the evolution of mortality and interest rates, allows us to obtain the fair-value reserves of liabilities and their sensitivities (Greeks) to relevant risk factors in closed form (Luciano et al (2012)). This permits us to account for multiple risk sources, while considerably reducing the computational effort.…”

“…While interest-rate risk is the most relevant risk source at short horizons, systematic longevity risk largely affects the variability of portfolio value in the medium-long run and needs to be managed for solvency purposes. Second, we analyze the effects of natural Delta-hedging strategies as proposed by Luciano et al (2012). They are effective in reducing longevity risk and in improving the solvency of an annuity portfolio, especially when it is welldiversified (i.e.…”