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The geometry of risk adjustments

Author

Listed:
  • Hans-Peter Bermin

    (Lund University)

  • Magnus Holm

    (Hilbert Group)

Abstract

We present a geometric approach to portfolio theory with a focus on 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 (e.g., using orthogonal Sharpe ratios rather than risk-adjusted returns) are needed to identify the efficient trading strategies. 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. This implies that a derivative, having an orthogonal Sharpe ratio of zero, has a price given by the minimal martingale measure.

Suggested Citation

  • Hans-Peter Bermin & Magnus Holm, 2024. "The geometry of risk adjustments," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 83-120, June.
    • Handle: RePEc:spr:decfin:v:47:y:2024:i:1:d:10.1007_s10203-023-00421-1
    DOI: 10.1007/s10203-023-00421-1
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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.
    2. Nielsen, Lars Tyge & Vassalou, Maria, 2004. "Sharpe Ratios and Alphas in Continuous Time," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 39(1), pages 103-114, March.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. Hans-Peter Bermin & Magnus Holm, 2023. "Kelly Trading And Market Equilibrium," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 26(01), pages 1-33, February.
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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:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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