Nanyang Business School Forum on Risk Management and Insurance

# Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes

**Tags:** mortality index, longevity risk, K-forward, K-option, longevity hedge, IRFRC

**More from:** Johnny Siu-Hang Li, Jackie Li, Uditha Balasooriya, Kenneth Q. Zhou

**Editor’s Note:**Posted by Johnny Siu-Hang Li, Professor, The University of Melbourne and the University of Waterloo; Jackie Li, Associate Professor, Macquarie University; Uditha Balasooriya, Senior Lecturer, Nanyang Technological University; Kenneth Q. Zhou, PhD Candidate, University of Waterloo

Parametric mortality indexes are mortality indexes that are created using the time-varying parameters in an appropriately chosen stochastic mortality model. They summarize how the mortality curve of a certain population evolves over time, and the random deviations from their expected trajectories reflect the level of longevity risk that the population is subject to.

Compared to non-parametric mortality indexes such as the LLMA’s LifeMetrics and Deutsche Borse’s Xpect Cohort Indexes, parametric mortality indexes are advantageous of being richer in information content, so that non-parallel shifts in the underlying mortality curve over time can be captured with only a few indexes. This property enables the market to better concentrate liquidity, fostering the popularity of less costly index-based longevity hedging solutions. Further, depending on the model from which they are created, parametric mortality indexes are often highly interpretable. For instance, the previously proposed CBD mortality indexes can be understood as the level and gradient of the (logit-transformed) underlying mortality curve, respectively. The ease of interpretation makes securities written on parametric mortality indexes more marketable to capital market investors.

In this paper, we revisit the concept of parametric mortality index, with a focus on three specific objectives. First, we explore security structures other than a zero coupon swap, and examine how the alternative security structures may benefit the hedger. Second, we study risk-neutral valuation of forwards and options written on parametric mortality indexes. This objective distinguishes our paper from previous work that ignored the cost of hedging completely. Third, we consider dynamic hedging with forwards and options

In terms of security structures, we particularly focus on options written on parametric mortality indexes (thereafter referred to as K-options). Longevity securities featuring option-like payoffs enable hedgers to create out-of-the-money hedges against extreme downside longevity outcomes, while retaining the potential upside gain. Our aim is to investigate the feasibility of building out-of-the-money longevity hedges using K-options, and examine how such hedges perform compared to their at-the-money counterparts in different circumstances.

In terms of pricing, we strive to derive analytical (risk-neutral) pricing formulas for K-forwards and options. Thanks to the fact that K-forwards and options are written directly on the time-varying parameters which are assumed follow some tractable time-series processes, analytical pricing formulas for there securities can be derived readily using the structural and statistical properties of the risk-adjusted version of the assumed processes. On the basis of the risk-neutral Cairns-Blake-Dowd model, we derive analytical pricing formulas for K-forwards and options written on the CBD mortality indexes. The resulting formulas are intuitive, and satisfy the put-call parity relationship as the Black-Scholes formula for equity options does.

In terms of hedging, we consider both static and dynamic hedging using the CBD mortality indexes. We consider both cash flow hedges (which focus on the variability of the cash flows arising from the hedged position) and value hedges (which focus on the variability of the values of hedged position at a certain future time point). For both, we measure hedge effectiveness in terms of Value-at-Risk, which takes the cost of hedging into consideration.

We provide real data illustrations of the synthesis of our theoretical contributions. Through a range of sensitivity tests, we arrive at several interesting conclusions. For instance, a K-put hedge is more likely to yield a lower Value-at-Risk than a K-forward hedge when the market prices of risk are high and/or the times-to-maturity of the hedging instruments are long. Another example is that a value hedge is highly robust to changes in the market prices of risk, the risk-free interest rate, and the times-to-maturity of the hedging instruments.

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