Academic journal article Journal of Risk and Insurance

Survivor Derivatives: A Consistent Pricing Framework

Academic journal article Journal of Risk and Insurance

Survivor Derivatives: A Consistent Pricing Framework

Article excerpt

ABSTRACT

Survivorship risk is a significant factor in the provision of retirement income. Survivor derivatives are in their early stages and offer potentially significant welfare benefits to society. This article applies the approach developed by Dowd et al. (2006), Olivier and Jeffery (2004), Smith (2005), and Cairns (2007) to derive a consistent framework for pricing a wide range of linear survivor derivatives, such as forwards, basis swaps, forward swaps, and futures. It then shows how a recent option pricing model set out by Dawson et al. (2009) can be used to price nonlinear survivor derivatives, such as survivor swaptions, caps, floors, and combined option products. It concludes by considering applications of these products to a pension fund that wishes to hedge its survivorship risks.

INTRODUCTION

A new global capital market, the Life Market, is developing (see, e.g., Blake, Cairns, and Dowd, 2008) and "survivor pools" (or "longevity pools" or "mortality pools" depending on how one views them) are on their way to becoming the first major new asset class of the twenty-first century. This process began with the securitization of insurance company life and annuity books (see, e.g., Millette et al., 2002; Cowley and Cummins, 2005; Lin and Cox, 2005). But with investment banks entering the growing market in pension plan buyouts, in the United Kingdom in particular, it is only a matter of time before full trading of "survivor pools" in the capital markets begins. (1) Recent developments in this market include: the launch of the Life-Metrics Index in March 2007; the first derivative transaction, a q-forward contract, based on this index in January 2008 between Lucida, a UK-based pension buyout insurer, and J.P. Morgan (see Coughlan et al., 2007; Grene, 2008); the first survivor swap executed in the capital markets between Canada Life and a group of ILS (2) and other investors in July 2008, with J.P. Morgan as the intermediary; and the first survivor swap involving a nonfinancial company, arranged by Credit Suisse in May 2009 to hedge the longevity risk in UK-based Babcock International's pension plan.

However, the future growth and success of this market depends on participants having the right tools to price and hedge the risks involved, and there is a rapidly growing literature that addresses these issues. The present article seeks to contribute to that literature by setting out a framework for pricing survivor derivatives that gives consistent prices--that is, prices that are not vulnerable to arbitrage attack--across all survivor derivatives. This framework has two principal components: one applicable to linear derivatives, such as swaps, forwards, and futures, and the other applicable to survivor options. The former is a generalization of the swap-pricing model first set out by Dowd et al. (2006), which was applied to simple vanilla survivor swaps. We show that this approach can be used to price a range of other linear survivor derivatives. The second component is the application of the option-pricing model set out by Dawson et al. (2009) to the pricing of survivor options such as survivor swaptions. This is a very simple model based on a normally distributed underlying, and it can be applied to survivor options in which the underlying is the swap premium or price, since the latter is approximately normal. Having set out this framework and shown how it can be used to price survivor derivatives, we then illustrate their possible applications to the various survivorship hedging alternatives available to a pension fund.

This article is organized as follows. The "Pricing Vanilla Survivor Swaps" section sets out a framework to price survivor derivatives in an incomplete market setting and uses it to price vanilla survivor swaps. The "Pricing Other Linear Survivor Derivatives" section then uses this framework to price a range of other linear survivor derivatives: these include survivor forwards, forward survivor swaps, survivor basis swaps, and survivor futures contracts. …

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