European Actuarial Journal

, Volume 3, Issue 1, pp 45–68 | Cite as

Quadratic hedging: an actuarial view extended to solvency control

  • Ragnar NorbergEmail author
Original Research Paper


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 The strategy is a hedge of a contractual payment stream if the payments are currently deposited on/withdrawn from the portfolio account and the terminal value of the portfolio is 0 (ultimate settlement of the contractual liabilities).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.


Mean-variance hedging Risk minimization Constrained portfolio value Solvency control 



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.


  1. 1.
    Björk T (2004) Arbitrage theory in continuous time, 2nd edn. Oxford University Press, OxfordCrossRefGoogle Scholar
  2. 2.
    Föllmer H, Sondermann D (1986) Hedging of non-redundant claims. In: Hildebrand W, Mas-Collel A (eds) Contributions to mathematical economics in honor of Gerard Debreu. North-Holland, pp. 205–223Google Scholar
  3. 3.
    Jeanblanc M, Mania M, Santacroce M, Schweizer M (2012) Mean-variance hedging via stochastic control and BSDEs for general semimartingales. Ann Appl Prob 22:2388–2428MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Møller T (2001) Risk-minimizing hedging strategies for insurance payment processes. Financ Stoch 5:419–446CrossRefGoogle Scholar
  5. 5.
    Norberg R (2013) Optimal hedging of demographic risk in life insurance. Financ Stoch 17:197–222. doi: 10.1007/s00780-012-0182-3 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Norberg R, Savina O (2012) A quadratic hedging approach to comparison of catastrophe indices. Int J Theor Appl Financ 15(4):20 doi: 10.1142/S0219024912500306 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Protter P (2004) Stochastic integration and differential equations 2nd edn. Springer, BerlinGoogle Scholar
  8. 8.
    Schweizer M (1991) Option hedging for semimartingales. Stoch Process Appl 37:339-363MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Schweizer M (2001) From actuarial to financial valuation principles. Insur Math Econ 28:31-47MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Schweizer M (2001) A guided tour through quadratic hedging approaches. In: Jouini E, Cvitanic J, Musiela M (eds) Option pricing, interest rates and risk management. Cambridge University Press, Cambridge, pp 538–574Google Scholar
  11. 11.
    Schweizer M (2008) Local risk-minimization for multidimensional assets and payment streams. Banach Center Publications 83, 213-229. Electronic version at

Copyright information

© DAV / DGVFM 2013

Authors and Affiliations

  1. 1.Laboratoire SAFUniversité de Lyon, Université Lyon 1, Fondation Lyon 1LyonFrance

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