We analyze the benefit to the insured of newly traded, innovative life insurance contracts. On a sequence of yearly reference days, the insured can choose between a guaranteed return (linked to the insurer’s asset result) and a capped index participation. The cap is adjusted at the beginning of each year such that both alternatives have the same value and the option to select is costless (product structuring condition). We point out that this condition cannot always be met. If the guaranteed return exceeds the upper bound of the capped index participation, the insurer can make a side profit. We show that a rather low insurance result also implies a rather low stock exposure, even if the insured opts for the index participation. Concerning the impact of the index dynamics, we emphasize that it is important to distinguish between jump and diffusion risk because the pricing of jump risk has an impact on cap rates that can be offered to an insured. Finally, we show that the optimal decision strategy of a CRRA investor implies an index selection even if it is unfairly priced such that the insurer indeed makes a side profit.
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In practice, products also allow to mix both alternatives. Insured may choose to increase the account value partially according to guaranteed return and capped index participation.
The asset price dynamics under the pricing measure \(\mathcal Q\) [cf. Equation (7)] and the real world measure \(\mathcal P\) are linked by means of a state-price density. Except for our assumption of a constant interest rate r (dividend rate \(\delta \), respectively) and a constant asset volatility \(\sigma _D\), the asset dynamics under consideration resemble the ones given in Pan (2002). For the state price density we thus refer to Pan (2002), Appendix A. With respect to measure transformations for Lévy processes, the interested reader is referred to Cont and Tankov (2004), Chapter 9.
This is in line with the related literature. For example, Bauer et al. (2006) find in their setting that the contract values of standard life insurances exceed initial investments of the insured.
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Mahayni, A., Muck, M. The benefit of life insurance contracts with capped index participation when stock prices are subject to jump risk. Rev Deriv Res 20, 281–308 (2017). https://doi.org/10.1007/s11147-017-9131-9
- Life insurance
- Interest rate guarantees
- Capped index participation
- Monthly sum cap
- Select products