We adopt deep learning models to directly optimise the portfolio Sharpe ratio. The framework we present circumvents the requirements for forecasting expected returns and allows us to directly optimise portfolio weights by updating model parameters. Instead of selecting individual assets, we trade Exchange-Traded Funds (ETFs) of market indices to form a portfolio. Indices of different asset classes show robust correlations and trading them substantially reduces the spectrum of available assets to choose from. We compare our method with a wide range of algorithms with results showing that our model obtains the best performance over the testing period, from 2011 to the end of April 2020, including the financial instabilities of the first quarter of 2020. A sensitivity analysis is included to understand the relevance of input features and we further study the performance of our approach under different cost rates and different risk levels via volatility scaling.
I discuss some theoretical results with a view to motivate some practical choices in portfolio optimization. Even though the setting is not completely general (for example, the covariance matrix is assumed to be non-singular), I attempt to highlight the features that have practical relevance. The mathematical setting is Stochastic Portfolio Theory, which is flexible enough to describe most realistic assets, and it has been successfully employed for managing equity portfolios since 1987.
The least squares Monte Carlo algorithm has become popular for solving portfolio optimization problems. A simple approach is to approximate the value functions on a discrete grid of portfolio weights, then use control regression to generalize the discrete estimates. However, the classical global control regression can be expensive and inaccurate. To overcome this difficulty, we introduce a local control regression technique, combined with adaptive grids. We show that choosing a coarse grid for local regression can produce sufficiently accurate results.
Conjecture LLC（Conjecture 有限责任公司）；Booth School of Business, University of Chicago（芝加哥大学布斯商学院）；Department of Mathematics, University College London（伦敦大学学院数学系）
We construct a deep portfolio theory. By building on Markowitz's classic risk-return trade-off, we develop a self-contained four-step routine of encode, calibrate, validate and verify to formulate an automated and general portfolio selection process. At the heart of our algorithm are deep hierarchical compositions of portfolios constructed in the encoding step. The calibration step then provides multivariate payouts in the form of deep hierarchical portfolios that are designed to target a variety of objective functions. The validate step trades-off the amount of regularization used in the enco…
University of the Witwatersrand（金山大学）；University of Johannesburg（约翰内斯堡大学）；Institute for Intelligent Systems（智能系统研究所）
The fusion of public sentiment data in the form of text with stock price prediction is a topic of increasing interest within the financial community. However, the research literature seldom explores the application of investor sentiment in the Portfolio Selection problem. This paper aims to unpack and develop an enhanced understanding of the sentiment aware portfolio selection problem. To this end, the study uses a Semantic Attention Model to predict sentiment towards an asset. We select the optimal portfolio through a sentiment-aware Long Short Term Memory (LSTM) recurrent neural network for …
Considering mean-variance portfolio problems with uncertain model parameters, we contrast the classical absolute robust optimization approach with the relative robust approach based on a maximum regret function. Although the latter problems are NP-hard in general, we show that tractable inner and outer approximations exist in several cases that are of central interest in asset management.
In this short note, we consider mean-variance optimized portfolios with transaction costs. We show that introducing quadratic transaction costs makes the optimization problem more difficult than using linear transaction costs. The reason lies in the specification of the budget constraint, which is no longer linear. We provide numerical algorithms for solving this issue and illustrate how transaction costs may considerably impact the expected returns of optimized portfolios.
We discuss a class of risk-sensitive portfolio optimization problems. We consider the portfolio optimization model investigated by Nagai in 2003. The model by its nature can include fixed income securities as well in the portfolio. Under fairly general conditions, we prove the existence of optimal portfolio in both finite and infinite horizon problems.
A Comparative Analysis of Portfolio Optimization Using Mean-Variance, Hierarchical Risk Parity, and Reinforcement Learning Approaches on the Indian Stock Market
We consider the problem of finding the efficient frontier associated with the risk-return portfolio optimization model. We derive the analytical expression of the efficient frontier for a portfolio of N risky assets, and for the case when a risk-free asset is added to the model. Also, we provide an R implementation, and we discuss in detail a numerical example of a portfolio of several risky common stocks.
We develop the idea of using Monte Carlo sampling of random portfolios to solve portfolio investment problems. In this first paper we explore the need for more general optimization tools, and consider the means by which constrained random portfolios may be generated. A practical scheme for the long-only fully-invested problem is developed and tested for the classic QP application. The advantage of Monte Carlo methods is that they may be extended to risk functions that are more complicated functions of the return distribution, and that the underlying return distribution may be computed without …
Community detection methods can be used to explore the structure of complex systems. The well-known modular configurations in complex financial systems indicate the existence of community structures. Here we analyze the community properties of correlation-based networks in worldwide stock markets and use community information to construct portfolios. Portfolios constructed using community detection methods perform well. Our results can be used as new portfolio optimization and risk management tools.
Portfolio optimization has been a broad and intense area of interest for quantitative and statistical finance researchers and financial analysts. It is a challenging task to design a portfolio of stocks to arrive at the optimized values of the return and risk. This paper presents an algorithmic approach for designing optimum risk and eigen portfolios for five thematic sectors of the NSE of India. The prices of the stocks are extracted from the web from Jan 1, 2016, to Dec 31, 2020. Optimum risk and eigen portfolios for each sector are designed based on ten critical stocks from the sector. An L…
Although modern portfolio theory has been in existence for over 60 years, fund managers often struggle to get its models to produce reliable portfolio allocations without strongly constraining the decision vector by tight bands of strategic allocation targets. The two main root causes to this problem are inadequate parameter estimation and numerical artifacts. When both obstacles are overcome, portfolio models yield excellent allocations. In this paper, which is primarily aimed at practitioners, we discuss the most common mistakes in setting up portfolio models and in solving them algorithmica…
In this paper, different approaches to portfolio optimization having higher moments such as skewness and kurtosis are classified so that the reader can observe different paradigms and approaches in this field of research which is essential for practitioners in Hedge Funds in particular. Several methods based on different paradigms such as utility approach and multi-objective optimization are reviewed and the advantage and disadvantageous of these ideas are explained. Keywords: multi-objective optimization, portfolio optimization, scalarization, utility
We prove a general duality result for multi-stage portfolio optimization problems in markets with proportional transaction costs. The financial market is described by Kabanov's model of foreign exchange markets over a finite probability space and finite-horizon discrete time steps. This framework allows us to compare vector-valued portfolios under a partial ordering, so that our model does not require liquidation into some numeraire at terminal time. We embed the vector-valued portfolio problem into the set-optimization framework, and generate a problem dual to portfolio optimization. Using re…
In this paper, we propose a market model with returns assumed to follow a multivariate normal tempered stable distribution defined by a mixture of the multivariate normal distribution and the tempered stable subordinator. This distribution is able to capture two stylized facts: fat-tails and asymmetry, that have been empirically observed for asset return distributions. On the new market model, we discuss a new portfolio optimization method, which is an extension of Markowitz's mean-variance optimization. The new optimization method considers not only reward and dispersion but also asymmetry. T…
Kelly's Criterion is well known among gamblers and investors as a method for maximizing the returns one would expect to observe over long periods of betting or investing. These ideas are conspicuously absent from portfolio optimization problems in the financial and automation literature. This paper will show how Kelly's Criterion can be incorporated into standard portfolio optimization models. The model developed here combines risk and return into a single objective function by incorporating a risk parameter. This model is then solved for a portfolio of 10 stocks from a major stock exchange us…
Minimum-variance portfolio optimizations rely on accurate covariance estimator to obtain optimal portfolios. However, it usually suffers from large error from sample covariance matrix when the sample size $n$ is not significantly larger than the number of assets $p$. We analyze the random matrix aspects of portfolio optimization and identify the order of errors in sample optimal portfolio weight and show portfolio risk are underestimated when using samples. We also provide LoCoV (low dimension covariance voting) algorithm to reduce error inherited from random samples. From various experiments,…
Risk aversion plays a significant and central role in investors' decisions in the process of developing a portfolio. In this framework of portfolio optimization we determine the portfolio that possesses the minimal risk by using a new geometrical method. For this purpose, we elaborate an algorithm that enables us to compute any generalized Euclidean distance to a standard simplex. With this new approach, we are able to treat the case of portfolio optimization without short-selling in its entirety, and we also recover in geometrical terms the well-known results on portfolio optimization with al…
We investigate a hybrid quantum-classical solution method to the mean-variance portfolio optimization problems. Starting from real financial data statistics and following the principles of the Modern Portfolio Theory, we generate parametrized samples of portfolio optimization problems that can be related to quadratic binary optimization forms programmable in the analog D-Wave Quantum Annealer 2000Q. The instances are also solvable by an industry-established Genetic Algorithm approach, which we use as a classical benchmark. We investigate several options to run the quantum computation optimally…
This paper proposes analytic forms of portfolio CoVaR and CoCVaR on the normal tempered stable market model. Since CoCVaR captures the relative risk of the portfolio with respect to a benchmark return, we apply it to the relative portfolio optimization. Moreover, we derive analytic forms for the marginal contribution to CoVaR and the marginal contribution to CoCVaR. We discuss the Monte-Carlo simulation method to calculate CoCVaR and the marginal contributions of CoVaR and CoCVaR. As the empirical illustration, we show relative portfolio optimization with thirty stocks under the distress condi…
We apply numerical dynamic programming techniques to solve discrete-time multi-asset dynamic portfolio optimization problems with proportional transaction costs and shorting/borrowing constraints. Examples include problems with multiple assets, and many trading periods in a finite horizon problem. We also solve dynamic stochastic problems, with a portfolio including one risk-free asset, an option, and its underlying risky asset, under the existence of transaction costs and constraints. These examples show that it is now tractable to solve such problems.
In this paper we develop a concrete and fully implementable approach to the optimization of functionally generated portfolios in stochastic portfolio theory. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. This choice can be motivated by the long term stability of the capital distribution observed in large equity markets, and allows us to circumvent the curse of dimensionality. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the g…
We examine machine learning and factor-based portfolio optimization. We find that factors based on autoencoder neural networks exhibit a weaker relationship with commonly used characteristic-sorted portfolios than popular dimensionality reduction techniques. Machine learning methods also lead to covariance and portfolio weight structures that diverge from simpler estimators. Minimum-variance portfolios using latent factors derived from autoencoders and sparse methods outperform simpler benchmarks in terms of risk minimization. These effects are amplified for investors with an increased sensiti…
The fundamental principle in Modern Portfolio Theory (MPT) is based on the quantification of the portfolio's risk related to performance. Although MPT has made huge impacts on the investment world and prompted the success and prevalence of passive investing, it still has shortcomings in real-world applications. One of the main challenges is that the level of risk an investor can endure, known as \emph{risk-preference}, is a subjective choice that is tightly related to psychology and behavioral science in decision making. This paper presents a novel approach of measuring risk preference from ex…
One of the reasons that higher order moment portfolio optimization methods are not fully used by practitioners in investment decisions is the complexity that these higher moments create by making the optimization problem nonconvex. Many few methods and theoretical results exists in the literature, but the present paper uses the method of successive convex approximation for the mean-variance-skewness problem.
Market conditions change continuously. However, in portfolio's investment strategies, it is hard to account for this intrinsic non-stationarity. In this paper, we propose to address this issue by using the Inverse Covariance Clustering (ICC) method to identify inherent market states and then integrate such states into a dynamic portfolio optimization process. Extensive experiments across three different markets, NASDAQ, FTSE and HS300, over a period of ten years, demonstrate the advantages of our proposed algorithm, termed Inverse Covariance Clustering-Portfolio Optimization (ICC-PO). The core…
We study portfolio optimization of four major cryptocurrencies. Our time series model is a generalized autoregressive conditional heteroscedasticity (GARCH) model with multivariate normal tempered stable (MNTS) distributed residuals used to capture the non-Gaussian cryptocurrency return dynamics. Based on the time series model, we optimize the portfolio in terms of Foster-Hart risk. Those sophisticated techniques are not yet documented in the context of cryptocurrency. Statistical tests suggest that the MNTS distributed GARCH model fits better with cryptocurrency returns than the competing GAR…
Budapest University of Technology and Economics（布达佩斯技术与经济大学）
In this paper we implement a Local Linear Regression Ensemble Committee (LOLREC) to predict 1-day-ahead returns of 453 assets form the S&P500. The estimates and the historical returns of the committees are used to compute the weights of the portfolio from the 453 stock. The proposed method outperforms benchmark portfolio selection strategies that optimize the growth rate of the capital. We investigate the effect of algorithm parameter m: the number of selected stocks on achieved average annual yields. Results suggest the algorithm's practical usefulness in everyday trading.
In this paper Portfolio Optimization techniques were used to determine the most favorable investment portfolio. In particular, stock indices of three companies, namely Microsoft Corporation, Christian Dior Fashion House and Shevron Corporation were evaluated. Using this data the amounts invested in each asset when a portfolio is chosen on the efficient frontier were calculated. In addition, the Portfolio with minimum variance, tangency portfolio and optimal Markowitz portfolio are presented.
We consider the problem of portfolio optimization with a correlation constraint. The framework is the multiperiod stochastic financial market setting with one tradable stock, stochastic income and a non-tradable index. The correlation constraint is imposed on the portfolio and the non-tradable index at some benchmark time horizon. The goal is to maximize portofolio's expected exponential utility subject to the correlation constraint. Two types of optimal portfolio strategies are considered: the subgame perfect and the precommitment ones. We find analytical expressions for the constrained subga…
Reinforcement Learning for Portfolio Optimization with a Financial Goal and Defined Time Horizons
AIMS South Africa /Stellenbosch university, South Africa（AIMS南非 /斯托滕博斯大学，南非）；University of South Africa, South Africa（南非大学，南非）；Michigan State University, USA（密歇根州立大学，美国）
In this paper we present an evolutionary optimization approach to solve the risk parity portfolio selection problem. While there exist convex optimization approaches to solve this problem when long-only portfolios are considered, the optimization problem becomes non-trivial in the long-short case. To solve this problem, we propose a genetic algorithm as well as a local search heuristic. This algorithmic framework is able to compute solutions successfully. Numerical results using real-world data substantiate the practicability of the approach presented in this paper.
This paper investigates performance attribution measures as a basis for constraining portfolio optimization. We employ optimizations that minimize expected tail loss and investigate both asset allocation (AA) and the selection effect (SE) as hard constraints on asset weights. The test portfolio consists of stocks from the Dow Jones Industrial Average index; the benchmark is an equi-weighted portfolio of the same stocks. Performance of the optimized portfolios is judged using comparisons of cumulative price and the risk-measures maximum drawdown, Sharpe ratio, and Rachev ratio. The results sugg…
We consider the optimization of active extension portfolios. For this purpose, the optimization problem is rewritten as a stochastic programming model and solved using a clever multi-start local search heuristic, which turns out to provide stable solutions. The heuristic solutions are compared to optimization results of convex optimization solvers where applicable. Furthermore, the approach is applied to solve problems with non-convex risk measures, most notably to minimize Value-at-Risk. Numerical results using data from both the Dow Jones Industrial Average as well as the DAX 30 are shown.
Using daily returns of the S&P 500 stocks from 2001 to 2011, we perform a backtesting study of the portfolio optimization strategy based on the extreme risk index (ERI). This method uses multivariate extreme value theory to minimize the probability of large portfolio losses. With more than 400 stocks to choose from, our study seems to be the first application of extreme value techniques in portfolio management on a large scale. The primary aim of our investigation is the potential of ERI in practice. The performance of this strategy is benchmarked against the minimum variance portfolio and the…
This paper uses simulation-based portfolio optimization to mitigate the left tail risk of the portfolio. The contribution is twofold. (i) We propose the Markov regime-switching GARCH model with multivariate normal tempered stable innovation (MRS-MNTS-GARCH) to accommodate fat tails, volatility clustering and regime switch. The volatility of each asset independently follows the regime-switch GARCH model, while the correlation of joint innovation of the GARCH models follows the Hidden Markov Model. (ii) We use tail risk measures, namely conditional value-at-risk (CVaR) and conditional drawdown-a…
We study the design of portfolios under a minimum risk criterion. The performance of the optimized portfolio relies on the accuracy of the estimated covariance matrix of the portfolio asset returns. For large portfolios, the number of available market returns is often of similar order to the number of assets, so that the sample covariance matrix performs poorly as a covariance estimator. Additionally, financial market data often contain outliers which, if not correctly handled, may further corrupt the covariance estimation. We address these shortcomings by studying the performance of a hybrid …
We investigate the portfolio selection problem against the systemic risk which is measured by CoVaR. We first demonstrate that the systemic risk of pure stock portfolios is essentially uncontrollable due to the contagion effect and the seesaw effect. Next, we prove that it is necessary and sufficient to introduce options to make the systemic risk controllable by the correlation hedging and the extreme loss hedging. In addition to systemic risk control, we show that using options can also enhance return-risk performance. Then, with a reasonable approximation of the conditional distribution of o…
We introduce a robust variant of the Kelly portfolio optimization model, called the Wasserstein-Kelly portfolio optimization. Our model, taking a Wasserstein distributionally robust optimization (DRO) formulation, addresses the fundamental issue of estimation error in Kelly portfolio optimization by defining a ``ball" of distributions close to the empirical return distribution using the Wasserstein metric and seeking a robust log-optimal portfolio against the worst-case distribution from the Wasserstein ball. Enhancing the Kelly portfolio using Wasserstein DRO is a natural step to take, given …
