Pub Date : 2020-10-29DOI: 10.13140/RG.2.2.21502.61766
C. Franco, Johann Nicolle, H. Pham
We study a discrete-time portfolio selection problem with partial information and maxi-mum drawdown constraint. Drift uncertainty in the multidimensional framework is modeled by a prior probability distribution. In this Bayesian framework, we derive the dynamic programming equation using an appropriate change of measure, and obtain semi-explicit results in the Gaussian case. The latter case, with a CRRA utility function is completely solved numerically using recent deep learning techniques for stochastic optimal control problems. We emphasize the informative value of the learning strategy versus the non-learning one by providing empirical performance and sensitivity analysis with respect to the uncertainty of the drift. Furthermore, we show numerical evidence of the close relationship between the non-learning strategy and a no short-sale constrained Merton problem, by illustrating the convergence of the former towards the latter as the maximum drawdown constraint vanishes.
{"title":"Discrete-time portfolio optimization under maximum drawdown constraint with partial information and deep learning resolution","authors":"C. Franco, Johann Nicolle, H. Pham","doi":"10.13140/RG.2.2.21502.61766","DOIUrl":"https://doi.org/10.13140/RG.2.2.21502.61766","url":null,"abstract":"We study a discrete-time portfolio selection problem with partial information and maxi-mum drawdown constraint. Drift uncertainty in the multidimensional framework is modeled by a prior probability distribution. In this Bayesian framework, we derive the dynamic programming equation using an appropriate change of measure, and obtain semi-explicit results in the Gaussian case. The latter case, with a CRRA utility function is completely solved numerically using recent deep learning techniques for stochastic optimal control problems. We emphasize the informative value of the learning strategy versus the non-learning one by providing empirical performance and sensitivity analysis with respect to the uncertainty of the drift. Furthermore, we show numerical evidence of the close relationship between the non-learning strategy and a no short-sale constrained Merton problem, by illustrating the convergence of the former towards the latter as the maximum drawdown constraint vanishes.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124480825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-26DOI: 10.26233/heallink.tuc.76532
A. Georgantas
The field of portfolio selection is an active research topic, which combines elements and methodologies from various fields, such as optimization, decision analysis, risk management, data science, forecasting, etc. The modeling and treatment of deep uncertainties for future asset returns is a major issue for the success of analytical portfolio selection models. Recently, robust optimization (RO) models have attracted a lot of interest in this area. RO provides a computationally tractable framework for portfolio optimization based on relatively general assumptions on the probability distributions of the uncertain risk parameters. Thus, RO extends the framework of traditional linear and non-linear models (e.g., the well-known mean-variance model), incorporating uncertainty through a formal and analytical approach into the modeling process. Robust counterparts of existing models can be considered as worst-case re-formulations as far as deviations of the uncertain parameters from their nominal values are concerned. Although several RO models have been proposed in the literature focusing on various risk measures and different types of uncertainty sets about asset returns, analytical empirical assessments of their performance have not been performed in a comprehensive manner. The objective of this study is to fill in this gap in the literature. More specifically, we consider different types of RO models based on popular risk measures and conduct an extensive comparative analysis of their performance using data from the US market during the period 2005-2016.
{"title":"Robust Optimization Approaches for Portfolio Selection: A Computational and Comparative Analysis","authors":"A. Georgantas","doi":"10.26233/heallink.tuc.76532","DOIUrl":"https://doi.org/10.26233/heallink.tuc.76532","url":null,"abstract":"The field of portfolio selection is an active research topic, which combines elements and methodologies from various fields, such as optimization, decision analysis, risk management, data science, forecasting, etc. The modeling and treatment of deep uncertainties for future asset returns is a major issue for the success of analytical portfolio selection models. Recently, robust optimization (RO) models have attracted a lot of interest in this area. RO provides a computationally tractable framework for portfolio optimization based on relatively general assumptions on the probability distributions of the uncertain risk parameters. Thus, RO extends the framework of traditional linear and non-linear models (e.g., the well-known mean-variance model), incorporating uncertainty through a formal and analytical approach into the modeling process. Robust counterparts of existing models can be considered as worst-case re-formulations as far as deviations of the uncertain parameters from their nominal values are concerned. Although several RO models have been proposed in the literature focusing on various risk measures and different types of uncertainty sets about asset returns, analytical empirical assessments of their performance have not been performed in a comprehensive manner. The objective of this study is to fill in this gap in the literature. More specifically, we consider different types of RO models based on popular risk measures and conduct an extensive comparative analysis of their performance using data from the US market during the period 2005-2016.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126350441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
During the last few years, there has been an interest in comparing simple or heuristic procedures for portfolio selection, such as the naive, equal weights, portfolio choice, against more "sophisticated" portfolio choices, and in explaining why, in some cases, the heuristic choice seems to outperform the sophisticated choice. We believe that some of these results may be due to the comparison criterion used. It is the purpose of this note to analyze some ways of comparing the performance of portfolios. We begin by analyzing each criterion proposed on the market line, in which there is only one random return. Several possible comparisons between optimal portfolios and the naive portfolio are possible and easy to establish. Afterwards, we study the case in which there is no risk free asset. In this way, we believe some basic theoretical questions regarding why some portfolios may seem to outperform others can be clarified.
{"title":"Which portfolio is better? A discussion of several possible comparison criteria","authors":"H. Gzyl, Alfredo J. Ríos","doi":"10.2139/SSRN.3179577","DOIUrl":"https://doi.org/10.2139/SSRN.3179577","url":null,"abstract":"During the last few years, there has been an interest in comparing simple or heuristic procedures for portfolio selection, such as the naive, equal weights, portfolio choice, against more \"sophisticated\" portfolio choices, and in explaining why, in some cases, the heuristic choice seems to outperform the sophisticated choice. We believe that some of these results may be due to the comparison criterion used. It is the purpose of this note to analyze some ways of comparing the performance of portfolios. We begin by analyzing each criterion proposed on the market line, in which there is only one random return. Several possible comparisons between optimal portfolios and the naive portfolio are possible and easy to establish. Afterwards, we study the case in which there is no risk free asset. In this way, we believe some basic theoretical questions regarding why some portfolios may seem to outperform others can be clarified.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129284275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Ciliberti, Emmanuel S'eri'e, G. Simon, Yves Lemp'eriere, J. Bouchaud
We find that when measured in terms of dollar-turnover, and once $beta$-neutralised and Low-Vol neutralised, the Size Effect is alive and well. With a long term t-stat of $5.1$, the "Cold-Minus-Hot" (CMH) anomaly is certainly not less significant than other well-known factors such as Value or Quality. As compared to market-cap based SMB, CMH portfolios are much less anti-correlated to the Low-Vol anomaly. In contrast with standard risk premia, size-based portfolios are found to be virtually unskewed. In fact, the extreme risk of these portfolios is dominated by the large cap leg; small caps actually have a positive (rather than negative) skewness. The only argument that favours a risk premium interpretation at the individual stock level is that the extreme drawdowns are more frequent for small cap/turnover stocks, even after accounting for volatility. This idiosyncratic risk is however clearly diversifiable.
{"title":"The \"Size Premium\" in Equity Markets: Where is the Risk?","authors":"S. Ciliberti, Emmanuel S'eri'e, G. Simon, Yves Lemp'eriere, J. Bouchaud","doi":"10.2139/ssrn.3018454","DOIUrl":"https://doi.org/10.2139/ssrn.3018454","url":null,"abstract":"We find that when measured in terms of dollar-turnover, and once $beta$-neutralised and Low-Vol neutralised, the Size Effect is alive and well. With a long term t-stat of $5.1$, the \"Cold-Minus-Hot\" (CMH) anomaly is certainly not less significant than other well-known factors such as Value or Quality. As compared to market-cap based SMB, CMH portfolios are much less anti-correlated to the Low-Vol anomaly. In contrast with standard risk premia, size-based portfolios are found to be virtually unskewed. In fact, the extreme risk of these portfolios is dominated by the large cap leg; small caps actually have a positive (rather than negative) skewness. The only argument that favours a risk premium interpretation at the individual stock level is that the extreme drawdowns are more frequent for small cap/turnover stocks, even after accounting for volatility. This idiosyncratic risk is however clearly diversifiable.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115756229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Trading strategies that were profitable in the past often degrade with time. Since unlucky streaks can also hit "healthy" strategies, how can one detect that something truly worrying is happening? It is intuitive that a drawdown that lasts too long or one that is too deep should lead to a downward revision of the assumed Sharpe ratio of the strategy. In this note, we give a quantitative answer to this question based on the exact probability distributions for the length and depth of the last drawdown for upward drifting Brownian motions. We also point out that both managers and investors tend to underestimate the length and depth of drawdowns consistent with the Sharpe ratio of the underlying strategy.
{"title":"You are in a drawdown. When should you start worrying","authors":"Adam Rej, P. Seager, J. Bouchaud","doi":"10.1002/wilm.10646","DOIUrl":"https://doi.org/10.1002/wilm.10646","url":null,"abstract":"Trading strategies that were profitable in the past often degrade with time. Since unlucky streaks can also hit \"healthy\" strategies, how can one detect that something truly worrying is happening? It is intuitive that a drawdown that lasts too long or one that is too deep should lead to a downward revision of the assumed Sharpe ratio of the strategy. In this note, we give a quantitative answer to this question based on the exact probability distributions for the length and depth of the last drawdown for upward drifting Brownian motions. We also point out that both managers and investors tend to underestimate the length and depth of drawdowns consistent with the Sharpe ratio of the underlying strategy.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116785216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rongju Zhang, Nicolas Langren 'e, Yu Tian, Zili Zhu, F. Klebaner, K. Hamza
A family of sharp target range strategies is presented for portfolio selection problems. Our proposed strategy maximizes the expected portfolio value within a target range, composed of a conservative lower target representing capital guarantee and a desired upper target representing investment goal. This strategy favorably shapes the entire probability distribution of return, as it simultaneously seeks a high expected return, cuts off downside risk, and implicitly caps volatility, skewness and other higher moments of the return distribution. To illustrate the effectiveness of our new investment strategies, we study a multi-period portfolio selection problem with transaction cost, where the results are generated by the Least-Squares Monte-Carlo algorithm. Our numerical tests show that the presented strategy produces a better efficient frontier, a better trade-off between return and downside risk, and a wider range of possible risk profiles than classical constant relative risk aversion utility. Finally, straightforward extensions of the sharp target range are presented, such as purely maximizing the probability of achieving the target range, adding an explicit target range for realized volatility, and defining the range bounds as excess return over a stochastic benchmark, for example, stock index or inflation rate. These practical extensions make the approach applicable to a wide array of investment funds, including pension funds, controlled-volatility funds, and index-tracking funds.
{"title":"Sharp Target Range Strategies with Application to Dynamic Portfolio Selection","authors":"Rongju Zhang, Nicolas Langren 'e, Yu Tian, Zili Zhu, F. Klebaner, K. Hamza","doi":"10.2139/ssrn.2826520","DOIUrl":"https://doi.org/10.2139/ssrn.2826520","url":null,"abstract":"A family of sharp target range strategies is presented for portfolio selection problems. Our proposed strategy maximizes the expected portfolio value within a target range, composed of a conservative lower target representing capital guarantee and a desired upper target representing investment goal. This strategy favorably shapes the entire probability distribution of return, as it simultaneously seeks a high expected return, cuts off downside risk, and implicitly caps volatility, skewness and other higher moments of the return distribution. To illustrate the effectiveness of our new investment strategies, we study a multi-period portfolio selection problem with transaction cost, where the results are generated by the Least-Squares Monte-Carlo algorithm. Our numerical tests show that the presented strategy produces a better efficient frontier, a better trade-off between return and downside risk, and a wider range of possible risk profiles than classical constant relative risk aversion utility. Finally, straightforward extensions of the sharp target range are presented, such as purely maximizing the probability of achieving the target range, adding an explicit target range for realized volatility, and defining the range bounds as excess return over a stochastic benchmark, for example, stock index or inflation rate. These practical extensions make the approach applicable to a wide array of investment funds, including pension funds, controlled-volatility funds, and index-tracking funds.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122222123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-10-04DOI: 10.1007/978-3-030-43384-0_15
J. F. Monge, M. Landete, José L. Ruiz
{"title":"Sharpe Portfolio Using a Cross-Efficiency Evaluation","authors":"J. F. Monge, M. Landete, José L. Ruiz","doi":"10.1007/978-3-030-43384-0_15","DOIUrl":"https://doi.org/10.1007/978-3-030-43384-0_15","url":null,"abstract":"","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122005049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The capitalization-weighted total relative variation $sum_{i=1}^d int_0^cdot mu_i (t) mathrm{d} langle log mu_i rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $mu_i (cdot)$ is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.
{"title":"Volatility and Arbitrage","authors":"R. Fernholz, I. Karatzas, J. Ruf","doi":"10.1214/17-AAP1308","DOIUrl":"https://doi.org/10.1214/17-AAP1308","url":null,"abstract":"The capitalization-weighted total relative variation $sum_{i=1}^d int_0^cdot mu_i (t) mathrm{d} langle log mu_i rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $mu_i (cdot)$ is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128886490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The conventional wisdom of mean-variance (MV) portfolio theory asserts that the nature of the relationship between risk and diversification is a decreasing asymptotic function, with the asymptote approximating the level of portfolio systematic risk or undiversifiable risk. This literature assumes that investors hold an equally-weighted or a MV portfolio and quantify portfolio diversification using portfolio size. However, the equally-weighted portfolio and portfolio size are MV optimal if and only if asset returns distribution is exchangeable or investors have no useful information about asset expected return and risk. Moreover, the whole of literature, absolutely all of it, focuses only on risky assets, ignoring the role of the risk free asset in the efficient diversification. Therefore, it becomes interesting and important to answer this question: how valid is this conventional wisdom when investors have full information about asset expected return and risk and asset returns distribution is not exchangeable in both the case where the risk free rate is available or not? Unfortunately, this question have never been addressed in the current literature. This paper fills the gap.
{"title":"Risk reduction and Diversification within Markowitz's Mean-Variance Model: Theoretical Revisit","authors":"G. Koumou","doi":"10.2139/SSRN.2595933","DOIUrl":"https://doi.org/10.2139/SSRN.2595933","url":null,"abstract":"The conventional wisdom of mean-variance (MV) portfolio theory asserts that the nature of the relationship between risk and diversification is a decreasing asymptotic function, with the asymptote approximating the level of portfolio systematic risk or undiversifiable risk. This literature assumes that investors hold an equally-weighted or a MV portfolio and quantify portfolio diversification using portfolio size. However, the equally-weighted portfolio and portfolio size are MV optimal if and only if asset returns distribution is exchangeable or investors have no useful information about asset expected return and risk. Moreover, the whole of literature, absolutely all of it, focuses only on risky assets, ignoring the role of the risk free asset in the efficient diversification. Therefore, it becomes interesting and important to answer this question: how valid is this conventional wisdom when investors have full information about asset expected return and risk and asset returns distribution is not exchangeable in both the case where the risk free rate is available or not? Unfortunately, this question have never been addressed in the current literature. This paper fills the gap.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"207 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132139492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to compare the performances of the optimal strategy under parameters mis-specification and of a technical analysis trading strategy. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein-Uhlenbeck process. For both strategies, we provide the asymptotic expectation of the logarithmic return as a function of the model parameters. Finally, numerical examples find that an investment strategy using the cross moving averages rule is more robust than the optimal strategy under parameters mis-specification.
{"title":"Robustness of Mathematical Models and Technical Analysis Strategies","authors":"Ahmed Bel Hadj Ayed, G. Loeper, F. Abergel","doi":"10.2139/SSRN.2774061","DOIUrl":"https://doi.org/10.2139/SSRN.2774061","url":null,"abstract":"The aim of this paper is to compare the performances of the optimal strategy under parameters mis-specification and of a technical analysis trading strategy. The setting we consider is that of a stochastic asset price model where the trend follows an unobservable Ornstein-Uhlenbeck process. For both strategies, we provide the asymptotic expectation of the logarithmic return as a function of the model parameters. Finally, numerical examples find that an investment strategy using the cross moving averages rule is more robust than the optimal strategy under parameters mis-specification.","PeriodicalId":286833,"journal":{"name":"arXiv: Portfolio Management","volume":"148 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122835907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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