Optimal design of equity-linked products with a probabilistic constraint

Abstract

There is a rich variety of tailored investment products available to the retail investor. These products combine upside participation in bull markets with downside protection in bear markets. Examples include the equity-linked products sold by insurance companies and the structured products marketed by banks. This paper examines a particular contract design for products of this nature. The paper finds the optimal design from the investor's viewpoint. It is assumed that the investor wishes to maximize expected utility of the terminal wealth subject to certain constraints. These constraints include a guaranteed rate of return as well as the opportunity to outperform a benchmark portfolio with a given probability. We derive the explicit form of the optimal design assuming both constraints apply and we illustrate the nature of the solution using some specific examples.

Keywords:

• Equity-Indexed Annuities
• Structured products
• Optimal design
• Optimal portfolio selection

JFL Classification:

• G11
• G13

Mathematics Subject Classification (2000):

• 60G44
• 93E20

Acknowledgements

The authors thank the referee for constructive suggestions. The paper has benefited from the comments of seminar participants at the University of Texas at Austin and Birkberk College (University of London). The authors acknowledge research support from the Natural Sciences and Engineering Research Council of Canada.

Notes

¹For a general discussion of structured products see Kat (1998).

²See for instance Jorion (2003) and Basak et al. (2006).

³Boyle & Tian (2008).

⁴See Nelsen (1998) and Cont & Tankov (2004) for a general discussion of Lévy process and its application to insurance contract design.

⁵We thank the referee for pointing out this extension.

⁶For simplicity, we ignore investment fees and all transaction costs as well as mortality and lapses.

⁷Cont & Tankov (2008) extend Black & Perold's analysis to Lévy process. Vanduffel, Chernih and Maj (2008) derive some suboptimality results of path-dependent payoffs for Lévy process.

⁸Of course in drawing up the contract in practice it is much simpler to specify a cap than a probability.

⁹The extension to a stochastic guaranteed amount is standard and we omit the details.

¹⁰The approach can be extended to any CRRA utility function with some technical modifications. We use log utility to simplify the notation while conveying the essential insights. For similar computations on the CRRA utility we refer to Basak & Shapiro (2005) and Basak et al. (2006).

¹¹Boyle & Tian (2008)