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Handbook of Insurance pp 627–645Cite as

Financial Pricing of Insurance

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Abstract

The financial pricing of insurance refers to the application of asset pricing theory, empirical asset pricing, actuarial science, and mathematical finance to insurance pricing. In this chapter we unify different approaches that assign a value to insurance assets or liabilities in the setting of a securities market. By doing so we present the various approaches in a common framework that allows us to discuss differences and commonalities. The presentation is done as simply as possible while still communicating the important ideas with references pointing the reader to more details.

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Notes

  1. 1.

    Different authors within different literatures present this result in terms of different objects such as state price systems, equivalent martingale measures, deflators, or pricing kernels, but the underlying idea is always the same. The most general version is provided by Delbaen and Schachermayer (1994, 1998), who use the lingua of equivalent martingale measures for the statement of their results.

  2. 2.

    The most famous variants are the Capital Asset Pricing Model [CAPM, Sharpe (1964)], which only requires the “market return” as the single factor. Extensions include Arbitrage Pricing Theory [APT, Ross (1976)] and the Intertemporal Capital Asset Pricing Model [ICAPM, Merton (1973)].

  3. 3.

    Note that we do not necessarily require complete markets here. In particular, in an incomplete market, the choice of a parametric form that is identifiable from security data may entail the restriction to a certain subset of all possible state price systems.

  4. 4.

    Potential exceptions are insurance contracts with exercise-dependent features. Here, optimal exercise rules may be influenced by the policyholders’ preferences since they may not face a quasi-complete market.

  5. 5.

    Insurance companies are levered corporations that raise debt capital by issuing a specific type of financial instrument—the insurance policy (cf. Cummins 1988; Phillips et al. 1998).

  6. 6.

    We frame the situation from the policyholder’s point of view. Analogously, we could consider the perspective of the insurance company endowed with a utility function.

  7. 7.

    The assumption of a complete financial market primarily serves to steer clear of the problem of having to “choose” the right pricing kernel in an incomplete financial market that is beyond the scope of this chapter. However, of course generalizations would be possible.

  8. 8.

    The derivation also uses the CAPM pricing relationship for the insurer’s expected asset returns, \(E_{t}\left [R_{t+1}^{(a)}\right ] = {e}^{r_{t}} {+\beta }^{(a)}\left (E_{t}\left [R_{t+1}^{(m)}\right ] - {e}^{r_{t}}\right )\) as well as the relationship \({\beta }^{(e)} ={\beta }^{(a)}(k_{t}s_{t} + 1) {+\beta }^{(u)}s_{t}\).

  9. 9.

    See e.g. Biffis et al. (2010) for a more detailed definition of systematic and unsystematic mortality risks.

  10. 10.

    It is important to note that these price differences may be attributable to factors other than risk charges due to systematic risk. For example, Finkelstein and Poterba (2004) suggest adverse selection plays a prominent role explaining why annuity prices exceed their actuarially fair value.

  11. 11.

    As we will detail in the next section, insurance risk sold from within an insurance company may be subject to adjustments resulting from frictional costs, so that the reliance on insurance contract price data may be problematic. However, resulting estimates should still offer an upper bound for the risk premium (cf. Bauer et al. 2010b).

  12. 12.

    An exception to this characterization is Ibragimov et al. (2010), who remove the assumption of market incompleteness in deriving multi-line capital allocations and insurance prices. The assumption of market completeness is also found in the papers of Phillips et al. (1998) and Sherris (2006), but the important point of departure lies in the assumption of frictional capital costs—which are present in Ibragimov et al. (2010) but absent in the latter articles. Such a maneuver offers an attractive benefit in terms of a uniquely and precisely indicated risk measure for pricing purposes (based on the value of the insurance company’s default option), though it comes with at least two embedded contradictions that must be finessed. Specifically, with a complete market for risk, there is no reason for policyholders to use costly intermediaries (absent a theoretician’s fiat preventing them from accessing the market directly), nor is it efficient for the intermediaries themselves to incur frictional costs by holding assets.

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Authors and Affiliations

  1. Georgia State University, Atlanta, GA, 30303, USA

    Daniel Bauer, Richard D. Phillips & George H. Zanjani

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  1. Daniel Bauer
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  2. Richard D. Phillips
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  3. George H. Zanjani
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Correspondence to Richard D. Phillips .

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  1. Finance, HEC Montreal, Montreal, Québec, Canada

    Georges Dionne

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Bauer, D., Phillips, R.D., Zanjani, G.H. (2013). Financial Pricing of Insurance. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_22

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