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The Geometry of Risk Adjustments

Author

Listed:
  • Bermin, Hans-Peter

    (Knut Wicksell Centre for Financial Studies, Lund University)

  • Holm, Magnus

    (Hilbert Group)

Abstract

In this paper we present a geometric approach to portfolio theory, with the aim to explain the geometrical principles behind risk adjusted returns; in particular Jensen’s alpha. We find that while the alpha/beta approach has severe limitations (especially in higher dimensions), only minor conceptual modifications are needed to complete the picture. However, these modifications (e.g. using risk adjusted Sharpe ratios rather than returns) can only be appreciated once a full geometric approach to portfolio theory is developed. We further show that, in a complete market, the so called market price of risk vector is identical to the growth optimal Kelly vector, albeit expressed in coordinates of a different basis. For trading strategies collinear to the growth optimal Kelly vector, we formalise a notion of relative value trading based on the risk adjusted Sharpe ratio. As an application we show that a derivative having a risk adjusted Sharpe ratio of zero has a corresponding price given by the the minimal martingale measure.

Suggested Citation

  • Bermin, Hans-Peter & Holm, Magnus, 2021. "The Geometry of Risk Adjustments," Knut Wicksell Working Paper Series 2021/2, Lund University, Knut Wicksell Centre for Financial Studies.
    • Handle: RePEc:hhs:luwick:2021_002
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    References listed on IDEAS

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    1. L. C. MacLean & W. T. Ziemba & G. Blazenko, 1992. "Growth Versus Security in Dynamic Investment Analysis," Management Science, INFORMS, vol. 38(11), pages 1562-1585, November.
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    More about this item

    Keywords

    Jensen’s alpha; Kelly criterion; market price of risk; option pricing; geometry;
    All these keywords.

    JEL classification:

    • G00 - Financial Economics - - General - - - General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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