Abstract
The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu (2018), presented in Part I of this series. As an alternative to classical portfolio risk measures such as the standard deviation, we, in particular, construct risk measures related to the “current” drawdown of the portfolio equity. In contrast to references Chekhlov, Uryasev, and Zabarankin (2003, 2005), Goldberg and Mahmoud (2017), and Zabarankin, Pavlikov, and Uryasev (2014), who used the absolute drawdown, our risk measure is based on the relative drawdown process. Combined with the results of Part I, Maier-Paape and Zhu (2018), this allows us to calculate efficient portfolios based on a drawdown risk measure constraint.
Highlights
Modern portfolio theory due to Markowitz (1959) has been the state of the art in mathematical asset allocation for over 50 years
In Part I of this series (see Maier-Paape and Zhu (2018)), we generalized portfolio theory such that efficient portfolios can be considered for a wide range of utility functions and risk measures
The so found portfolios provide an efficient trade-off between utility and risk, just as in the Markowitz portfolio theory
Summary
Modern portfolio theory due to Markowitz (1959) has been the state of the art in mathematical asset allocation for over 50 years. According to the theory of Part I Maier-Paape and Zhu (2018), such positively homogeneous risk measures provide—as in the CAPM model—an affine structure of the efficient portfolios when the identity utility function is used Often in this situation, even a market portfolio, i.e., a purely risky efficient portfolio, related to drawdown risks can be provided as well. We provide the basic market setup for application of the generalized portfolio theory of Part I Maier-Paape and Zhu (2018), and it is used as a link between Part I and Part II It shows how the theory of Part I can be used with the risk measures constructed in this paper. A simple argument using the compactness of S1M−1 yields that G is bounded as well
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