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The Geometry of Risk Adjustments

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.