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A market- and time-consistent extension for the EIOPA risk-margin

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Abstract

In this paper, we investigate market- and time-consistent valuation of life-insurance liabilities, which are long-dated by nature. To obtain a market- and time-consistent value, the “two-step market evaluation” introduced by Pelsser and Stadje (Math Finance 24:25–65, 2014) is used to evaluate a hybrid payoff with underlying hedgeable financial and (partially) unhedgeable actuarial risks. The resulting time-consistent and market-consistent (TCMC) price captures the dynamics of the risk drivers over the lifetime of the contract. We show that the EIOPA standard-formula for the risk-margin is not time-consistent, and we construct a time-consistent version of the risk-margin that captures the extra uncertainties from the process dynamics. EIOPA’s standard-formula for the Risk-Margin is compared to the TCMC price for a simple unit-linked contract and we show that the effects of time-inconsistency are increasing with maturity and are significant for long-dated contracts.

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Notes

  1. See example (3.4) in Pelsser and Stadje [37] for examples of two-step actuarial operators under the previously described representation.

  2. Also called the “risk-neutral” probability.

  3. The EIOPA Technical Specification [21] and in specific articles https://www.asf.com.pt/NR/rdonlyres/359F79DF-586C-42D0-8064-97811541C23F/0/A__Technical_Specification_for_the_Preparatory_Phase__Part_I_.pdf released by the European Insurance and Occupational Pension Authority (EIOPA) provides advice and formulations on the calculation of the risk-margin on top of the best-estimate for long-term liabilities in a multi-period setting.

  4. A one-year discount should be applied to the pay-off \(h(y_{t+k})\) under \(\mathbb {V}\text {aR}\) operator.

  5. Note that all \(y_t\) values may also be represented by the “discounted” quantities relative to the money-market account process B(t) defined in Sect. 3.1 with which the discount factors can be taken off from the formula.

  6. Note that the conditional expectation is time-consistent.

  7. The discount factor is omitted due to use of \(x_t\) as the discounted financial risk driver. Further explanation is in Sect. 3.1

  8. Considering the stochastic evolution of mortality risk through time, a more precise concept is “the remaining lifetime at the beginning of the calendar year t” for which the notation is \(T_{x}(t)\).

  9. In the notation, if we assume the present \(t=0\), the notation t is representative of the past “calendar times” \(\{t_0, t_0 +1, \dots , 0\}\) in the model.

  10. In case the function f is non-increasing, then any probability p under \(\mathbb {V}\text {aR}\) function will be turned into \(1-p\).

  11. These parameters are estimated on the basis of mortality data aggregated for “men and women” of the Netherlands during the calendar years 1960–2006 (47 years).

  12. Accessible via ECB web-page, https://www.ecb.europa.eu/stats/money/yc/html/index.en.html.

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Authors and Affiliations

  1. KU Leuven-Faculty of Economics and Business (FEB), Leuven, Belgium

    Ahmad Salahnejhad Ghalehjooghi

  2. Maastricht University (UM), Maastricht, The Netherlands

    Antoon Pelsser

  3. University of Amsterdam (UvA), Netspar and Risk At Work, Amsterdam, The Netherlands

    Antoon Pelsser

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  1. Ahmad Salahnejhad Ghalehjooghi
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Correspondence to Ahmad Salahnejhad Ghalehjooghi.

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The authors are grateful for the constructive and helpful comments and directions given by Prof. Dr. Jan Dhaene (KU Leuven) on this paper.

The research leading to these results has received funding from the European Union Seventh Framework Programme ([FP7/2007-2013] [FP7/2007-2011]) under Grant agreement \(n^\circ\) 289032.

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Salahnejhad Ghalehjooghi, A., Pelsser, A. A market- and time-consistent extension for the EIOPA risk-margin. Eur. Actuar. J. 13, 517–539 (2023). https://doi.org/10.1007/s13385-023-00343-7

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