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Risk management strategies for finding universal portfolios

  • S.I.: APMOD 2014
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We consider Cover’s universal portfolio and the problem of risk management in a distribution-free setting when learning from experts. We aim to find optimal portfolios without modelling the financial market at the outset. Although it exists, the price distribution of the constituent assets is neither known nor given as part of the input. We consider the portfolio selection problem from the perspective of online algorithms that process input piece-by-piece in a serial fashion. Under the minimax regret criterion, we propose two risk-adjusted algorithms that track the expert with the lowest maximum drawdown. We obtain upper bounds on the worst-case performance of our algorithms that equal the bounds obtained by Cover (Math Finance 1(1):1–29, 1991). We also present computational evidence using NYSE data over a 22-year period, which shows superior performance of investment strategies that take risk management into account.

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  1. \(\textit{CRP}\) holds an a priori fixed portfolio, i.e., maintains a constant fraction of the total wealth in each of the underlying m assets. Widely-used is the Uniform \(\textit{CRP}\) (\(\textit{UCRP}\)), where \(b_t=(\frac{1}{m},\dots ,\frac{1}{m})\). \(\textit{UCRP}\) is also known as \(\big (\frac{1}{m}\big )\)-portfolio and/or naive diversification.

  2. For a model allowing short sales see Vovk and Watkins (1998).

  3. Equivalent to the average logarithmic wealth (\(\frac{1}{T}\ln W_T(P)\)) (Borodin et al. 2000).

  4. Not to be mistaken with the game in which the players make wagers on the outcome of a pair of dice (Turner 2005).

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  6. \(R_f\) risk-free return fixed at 0 % in this work; \(\hat{\sigma }_i\) standard deviation of the (daily) returns of the i-th asset; \(\hat{\sigma }^2_{ij}\) sample covariance between two assets i and j.

  7. For further details on the performance of the considered algorithms see Dochow (2016).


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The authors thank the editors, the anonymous referees, and Günter Schmidt for their insightful comments, which have significantly improved this paper. Preparation of this paper was supported, in part, by the ‘RiSC—Research Seed Capital Program’ of the Ministry of Science, Research and the Arts (MWK) Baden-Württemberg and the University of Mannheim, Germany. The APMOD conference presentation of this paper was funded by the Julius-Paul-Stiegler Gedächtnis Stiftung.

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Correspondence to Esther Mohr.

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The material in this paper was presented in part at the International Conference on Operations Research (OR2013), Erasmus University Rotterdam, NL; see also Dochow et al. (2014).

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Mohr, E., Dochow, R. Risk management strategies for finding universal portfolios. Ann Oper Res 256, 129–147 (2017).

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