# Quadratic hedging: an actuarial view extended to solvency control

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## Abstract

An investment strategy or portfolio is uniquely determined by an exposure process specifying the number of shares held in risky assets at any time and a cost process representing deposits into and withdrawals from the portfolio account. The strategy is a hedge of a contractual payment stream if the payments are currently deposited on/withdrawn from the portfolio account and the terminal value of the portfolio is 0 (ultimate settlement of the contractual liabilities). The purpose of the hedge is stated as an optimization criterion for the investment strategy. The purpose of the present paper is two-fold. Firstly, it reviews the core of quadratic hedging theory in a scenario where insurance risk can partly be offset by trading in available insurance-linked derivatives (e.g. catastrophe bonds or mortality bonds) and relates it to actuarial principles of premium rating and provision of reserves. Working under a martingale measure and some weak integrability conditions allows simple proofs based on orthogonal projections: quadratic hedging theory without agonizing pain. Secondly, it is pointed out that certain quadratic hedging principles lead to the same optimal exposure process but different optimal cost processes, special cases being mean-variance hedging and risk minimization. It is shown that these results are preserved if the value of the portfolio is required to coincide with a given adapted process, a case in point being the capital requirement introduced through regulatory regimes like the Basel accords and Solvency II.

## Keywords

Mean-variance hedging Risk minimization Constrained portfolio value Solvency control## Notes

### Acknowledgments

The author thanks the BNP Paribas Cardif Chair *“Management de la modélisation”* for financial support. The views expressed in this document are the author’s own and do not necessarily reflect those endorsed by BNP Paribas Cardif. Special thanks are due to an anonymous referee or editor whose diligence saved the author from blundering in public.

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