A strategy for hedging risks associated with period and cohort effects using q-forwards

Abstract

The stochastic nature of future mortality arises from both period (time-related) and cohort (year-of-birth-related) effects. Existing index-based longevity hedging strategies mitigate the risk associated with period effects, but often overlook cohort effects. The negligence of cohort effects may lead to sub-optimal hedge effectiveness, if the liability being hedged is a deferred pension or annuity which involves cohorts that are not covered by the data sample. In this paper, we propose a new hedging strategy that incorporates both period and cohort effects. The resulting longevity hedge is a value hedge, reducing the uncertainty surrounding the $\tau$-year ahead value of the liability being hedged. The proposed method is illustrated with data from the male population of England and Wales. It is found that the benefit of incorporating cohort effects into a longevity hedging strategy depends heavily on the persistence of cohort effects and the choice of q-forwards.

Introduction

Solutions for hedging pure longevity risk can be broadly divided into two categories: customized and index-based. A customized longevity hedge is based on the actual mortality experience of the individuals associated with the liability being hedged, whereas an index-based longevity hedge is linked to a broad-based mortality index which reflects the actual mortality experience of a larger pool of individuals, such as a national population. Each of these two types of longevity hedge has its pros and cons.

A customized hedge eliminates all longevity risk, but it is more costly and difficult to unwind. It is also less attractive to capital markets investors, because it lacks liquidity and transparency. Typically, a customized hedge is structured as a ‘cash flow hedge’, in which net payments are made period by period to immunize the longevity risk associated with every single cash flow of the hedger. An example of a customized cash flow hedge is the longevity swap that was executed between J.P. Morgan and Canada Life in 2009. Under this 40-year longevity swap, Canada Life receives from J.P. Morgan the actual payments it must pay to its annuitants and, in return, makes a series of fixed payments to J.P. Morgan (Blake et al., 2013).

In contrast, an index-based hedge is not perfect, leaving behind residual risks such as small sample risk and population basis risk. The problems arising from these residual risks have been analyzed by researchers including Cairns et al. (2014), Coughlan et al. (2011), Haberman et al. (2014), Li and Hardy (2011) and Ngai and Sherris (2011). Nevertheless, an index-based hedge is potentially more liquid and consequently cheaper due to a lower illiquidity premium. If the hedge is adjusted periodically, then the hedging instruments used can be shorter-dated, better meeting investors’ preference and reducing the hedger’s exposure to counterparty default risk at the expense of increased rollover risk (see, e.g., Cairns, 2011; Zhou and Li, 2014). Furthermore, because the value of a pension or annuity liability depends on the level of the relevant mortality index at the time of valuation, index-based hedges can be structured as a ‘value hedge’, which immunizes the risk associated with the liability’s value rather than the liability’s cash flows. An example of an index-based value hedge is the q-forward contract that Lucida executed with J.P. Morgan to lock in the value of its annuity liability at the end of the hedging horizon (Blake et al., 2013).

As Coughlan et al. (2013) pointed out, index-based value hedges are very well suited for de-risking deferred liabilities, which involve no cash flow during the period of deferral. They permit pension plan sponsors with a large number of non-pensioner members (who are still years from receiving their pension benefits) to offload their longevity risk exposures. As an index-based value hedge is quite flexible, it can be adjusted from time to time according to, for example, the changes in the composition of plan sponsor’s workforce and the options (e.g., a lump sum vs. lifetime payments) chosen by the plan members. Index-based value hedges also provide a way for insurers selling advanced-life delayed annuities (i.e., deeply deferred life annuities) to reduce their longevity risk exposures and hence the associated solvency capital.¹ The world’s first index-based value hedge for deferred members of a pension plan was the q-forward deal executed between the Pall (UK) Pension Fund and J.P. Morgan in 2011 (Blake et al., 2014). The hedge was calibrated to lock in the value of the deferred pension liability over a 10-year horizon.

Although the suitability for deferred pension and annuity liabilities is an important feature of index-based value hedges, this feature has not been extensively studied in the literature. From a technical viewpoint, deferred liabilities often involve cohorts that are not covered by the data sample to which the underlying mortality model is calibrated. To illustrate, let us suppose that the underlying mortality model is calibrated to data over ages 50–89(a typical age range to which mortality models for pension valuation are fitted) and a sample period that includes the current year. The data sample does not cover the cohorts who are now age 49 or younger, which means when valuing the deferred liabilities for these cohorts, the cohort effects in the model must be projected. In other words, the deferred liabilities are subject to the stochastic uncertainty surrounding not only period (time-related) effects but also cohort (year-of-birth-related) effects. However, cohort effects are not taken into account in most of the existing methods for calibrating index-based hedges, including those proposed by Cairns (2011), Cairns et al. (2006b), Dahl (2004), Dahl and Møller (2006), Dahl et al. (2008), Liu and Li (2014), Zhou and Li (2014) and Luciano et al. (2012).

So far s we aware, Li and Luo (2012), Cairns et al. (2014) and Cairns (2013) have considered cohort effects in the context of hedging. Li and Luo (2012) incorporated the potential dependence between different cohorts into a static longevity hedge, but their set-up is a cash flow hedge that is not particularly suitable for a deferred liability. Cairns et al. (2014) and Cairns (2013) studied a value hedge for a deferred annuity that is payable to a single cohort of individuals; however, in their set-up, the cohort in question is covered by the data sample used in calibrating the simulation model, which means that the annuity liability is not subject to any cohort effect uncertainty. Therefore, their results do not indicate how cohort effect uncertainty may be mitigated if it is involved in liability being hedged. Also, the way they derive their hedging strategies depends heavily on simulations, and is therefore computationally intensive especially when multiple hedging instruments are included in the hedge portfolio.

This paper complements the literature by contributing a method for hedging the uncertainty surrounding both period and cohort effects. Using the proposed method, one can create a value hedge for a deferred annuity liability which involves cohort effects that are not yet realized as of the time when the hedge is established. The hedging instruments used are q-forwards, which may be linked to cohorts that are different from those that are associated with the annuity liability. At the user’s discretion, the hedge can be executed as a static hedge (which remains unchanged over the hedging horizon) or a dynamic hedge (which is adjusted periodically over the hedging horizon). The proposed method is developed from the stochastic properties of the innovations in the assumed processes for the underlying period and cohort effects. It yields hedge ratios that can be expressed analytically in terms of (i) the variances and covariances of the innovations and (ii) the partial derivatives of the values of the hedge portfolio and the annuity liability with respect to the relevant innovations. As no simulation is required in the calculating the hedge ratios, the execution of our proposed hedging method requires minimal computational effort.

Unlike the hedging strategies proposed by Cairns (2011), Cairns et al. (2006b), Li and Luo (2012) and Zhou and Li (2014), the hedge ratios under our method are not obtained simply by matching the sensitivities of the liability and the hedge portfolio to changes in the underlying mortality. Instead, our method has a closer resemblance to the generalized state-space hedging method proposed by Liu and Li (2014), under which the optimal hedge ratios are calculated on the basis of a specific risk measure.² Compared to the sensitivity matching approaches, this approach to deriving hedge ratios is more suitable for hedgers who have a definite hedging objective. The risk measure we consider is variance, which is commonly used for the purpose of evaluating the effectiveness of longevity hedges (see Cairns (2011), Cairns (2013); Cairns et al., 2014; Coughlan et al., 2011; Li and Hardy, 2011). We illustrate the proposed hedging method using real mortality data from a national population. The empirical work demonstrates the benefit of factoring cohort effects into a longevity hedge under different circumstances.

The remainder of this paper is organized as follows. Section 2 presents the stochastic mortality model that is assumed throughout the paper. Section 3 describes the liability being hedged and the hedging instruments. Section 4 introduces the proposed hedging method and details how the hedge ratios under the proposed method can be calculated. Section 5 defines the metrics for evaluating hedge effectiveness and explains how they can be calculated. Section 6 presents the baseline empirical results and performs several sensitivity tests. Finally, Section 7 concludes the paper with some suggestions for future research.