Survivor Derivatives: A Consistent Pricing Framework

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 LifeMetrics Paul Dawson is at Kent State University. Kevin Dowd is at the Pensions Institute, Cass Business School. Andrew J. G. Cairns is at the Maxwell Institute, Edinburgh and Department of Actuarial Mathematics and Statistics, Heriot-Watt University. David Blake is at the Pensions Institute, Cass Business School. The authors can be contacted via e-mail: [email protected], [email protected], [email protected], [email protected]. The authors are grateful to Hai Lin and the two anonymous referees for helpful comments. 1Dunbar (2006). On February 1, 2010, the Life and Longevity Markets Association (LLMA) was established in London by AXA, Deutsche Bank, J.P. Morgan, Legal & General, RBS, and 579 580 THE JOURNAL OF RISK AND INSURANCE 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 ILS2 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. The “Survivor Swaptions” section extends the pricing framework to price survivor swaptions, caps, and floors, making use of an option pricing formula set out in Dawson et al. (2009). The “Hedging Applications” section gives a number of hedging applications of our pricing framework, and the “Conclusion” section concludes. PRICING VANILLA SURVIVOR SWAPS A Model of Aggregate Longevity Risk It is convenient if we begin by outlining an illustrative model of aggregate longevity risk. Let p(s, t, u, x) be the risk-adjusted probability based on information available at s that an individual aged x at time 0 and alive at time t ≥ s will survive to time u ≥ t (referred to as the forward survival probability by Cairns, Blake, and Dowd, 2006). Our initial estimate of the risk-adjusted forward survival probability to u is therefore Swiss Re. The aim is “to support the development of consistent standards, methodologies and benchmarks to help build a liquid trading market needed to support the future demand for longevity protection by insurers and pension funds.” 2Investors in insurance-linked securities. SURVIVOR DERIVATIVES: A CONSISTENT PRICING FRAMEWORK 581 p(0, 0, u, x), and these probabilities would be used at time 0 to calculate the prices of annuities. We now postulate that, for each s = 1, . . . , t: p(s, t − 1, t, x) = p(s − 1, t − 1, t, x)b(s,t−1,t,x)ε(s), (1) where ε (s) > 0 can be interpreted as a survivorship “shock” at time s for age x, although to keep the notation as simple as possible, we do not make the age dependence explicit (see also Cairns, 2007, Equation (5); Olivier and Jeffery, 2004; Smith, 2005). For its part, b(s, t − 1, t, x) is a normalizing constant, specific to each pair of dates, s and t, and to each cohort, that ensures consistency of prices under our pricing measure.3 It then follows that S(t), the probability of survival to t, is given by: