Abstract
In this paper, we provide an alternative trend (time)-dependent risk measure to Ruttiens’ accrued returns variability (Ruttiens in Comput Econ 41:407–424, 2013). We propose to adjust the calculation procedure to achieve an alternative risk measure. Our modification eliminates static mean component and it is based on the deviation of squared dispersions, which reflects the trend (time factor) precisely. Moreover, we also present a new perspective on dependency measures and we apply a PCA to a new correlation matrix in order to determine a parametric and nonparametric return approximation. In addition, the two-phase portfolio selection strategy is considered, where the mean–variance portfolio selection strategies represent the first optimization. The second one is the minimization of deviations from their trend leading to identical mean and final wealth. Finally, an empirical analysis verify the property and benefit of portfolio selection strategies based on these trend-dependent measures. In particular, the ex-post results show that applying the modified measure allows us to reduce the risk with respect to the trend of several portfolio strategies.
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Notes
We also refer to the accrued returns variability as trend-dependent risk measure, time-dependent risk measure, or dynamic risk measure.
All observations in 2002 are used for the initial optimization, therefore the total investment period is reduced by one year.
While the first strategy (Strategy 1) is a global minimum variance portfolio (GMV), the last one (Strategy 40) is a maximum expected return portfolio (MER). Therefore, for the rest of the strategies, we compute the lower bound of the expected return \(M \in (x_{GMV}^{\prime }r, x_{MER}^{\prime }r)\) with the equidistant difference d calculated as \(d=\frac{(x_{MER}^{\prime }r-x_{GMV}^{\prime }r)}{N-1}\), where N is the number of strategies. Recall that in this empirical analysis we use \(N=40\).
According to our preliminary analysis, even when we include more factors explaining the given variability, the composition of the portfolios is not substantially different. In general, the portfolio statistics do not change significantly.
In order to measure the performance of the portfolio, we selected the usually used Sharpe ratio (SR) ( Sharpe 1994; Biglova et al. 2004). This indicator was chosen due to its explanatory power based on the entire distribution of returns. The Sharpe ratio expresses the expected excess return for the unity of risk measured as standard deviation calculated as \(SR=\frac{E(x^{\prime }r-r_b)}{\sigma _{x^{\prime }r}}\), where \(\sigma _{x^{\prime }r}\) denotes the standard deviation of the portfolio and \(r_b\) is the benchmark return, see Rachev et al. (2008). Following Ortobelli et al. (2017), we define new types of trend-dependent ratios TDR1 and TDR2. These two measures indicate the value of excess wealth per unit of various kinds of risk, static or trend-dependent. Specifically, the formulations of TDR1 and TDR2 are given by \(TDR1=\frac{W_T-1}{ARV_{\text {mod}}(x^{\prime }r)}\) and \(TDR2=\frac{W_T-1}{\sigma _{x^{\prime }r}+ARV_{\text {mod}}(x^{\prime }r)}\).
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Acknowledgements
This paper was supported through the Czech Science Foundation (GACR) under Project 20-16764S and through SP2023/019, a SGS research project of VSB–TU Ostrava.
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Neděla, D., Ortobelli, S. & Tichý, T. Mean–variance vs trend–risk portfolio selection. Rev Manag Sci (2023). https://doi.org/10.1007/s11846-023-00660-x
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DOI: https://doi.org/10.1007/s11846-023-00660-x