Jan Dhaene and Karim Barigou acknowledge the financial support of the Onderzoeksfonds KU Leuven (GOA/13/002). Karim Barigou is a PhD fellow of the Research Foundation -Flanders (FWO) [grant number 1146118N].
A general class of fair valuations which are both market-consistent (mark-tomarket for any hedgeable part of a claim) and actuarial (mark-to-model for any claim that is independent of financial market evolutions) was introduced in Dhaene et al. [Insurance: Mathematics & Economics, 76, 14-27 (2017)] in a single period framework. In particular, the authors considered mean-variance hedge-based (MVHB) valuations where fair valuations of insurance liabilities are expressed in terms of mean-variance hedges and actuarial valuations. In this paper, we generalize this MVHB approach to a multi-period dynamic investment setting. We show that the classes of fair valuations and MVHB valuations are equivalent in this generalized setting. We derive tractable formulas for the fair valuation of equity-linked contracts and show how the actuarial part of their MVHB valuation decomposes into a diversifiable and a non-diversifiable component.
Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.
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