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The ineffectiveness of coherent risk measures

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  • John Armstrong
  • Damiano Brigo

Abstract

We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term $\rho$-arbitrage for a risk measure $\rho$. We show how to determine analytically whether such $\rho$-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether expected shortfall constraints are ineffective in these markets. We find that the answer depends heavily upon the probability model selected by the risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our results can be applied to value at risk. By contrast, we show that reasonable expected utility constraints are effective in any arbitrage-free market.

Suggested Citation

  • John Armstrong & Damiano Brigo, 2019. "The ineffectiveness of coherent risk measures," Papers 1902.10015, arXiv.org, revised Oct 2020.
    • Handle: RePEc:arx:papers:1902.10015
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    References listed on IDEAS

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    1. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    2. Armstrong, John & Brigo, Damiano, 2019. "Risk managing tail-risk seekers: VaR and expected shortfall vs S-shaped utility," Journal of Banking & Finance, Elsevier, vol. 101(C), pages 122-135.
    3. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    4. John Armstrong, 2016. "The Markowitz Category," Papers 1611.07741, arXiv.org, revised Jun 2018.
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    Cited by:

    1. Martin Herdegen & Nazem Khan, 2022. "Mean‐ρ$\rho$ portfolio selection and ρ$\rho$‐arbitrage for coherent risk measures," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 226-272, January.
    2. Martin Herdegen & Nazem Khan, 2020. "Mean-$\rho$ portfolio selection and $\rho$-arbitrage for coherent risk measures," Papers 2009.05498, arXiv.org, revised Jul 2021.
    3. John Armstrong & Damiano Brigo & Alex S. L. Tse, 2024. "The importance of dynamic risk constraints for limited liability operators," Annals of Operations Research, Springer, vol. 336(1), pages 861-898, May.

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