We propose a data-driven portfolio selection model that integrates side information, conditional estimation and robustness using the framework of distributionally robust optimization. Conditioning on the observed side information, the portfolio manager solves an allocation problem that minimizes the worst-case conditional risk-return trade-off, subject to all possible perturbations of the covariate-return probability distribution in an optimal transport ambiguity set. Despite the non-linearity of the objective function in the probability measure, we show that the distributionally robust portfolio allocation with side information problem can be reformulated as a finite-dimensional optimization problem. If portfolio decisions are made based on either the mean-variance or the mean-Conditional Value-at-Risk criterion, the resulting reformulation can be further simplified to second-order or semi-definite cone programs. Empirical studies in the US equity market demonstrate the advantage of our integrative framework against other benchmarks.
Blockchain Research Center（区块链研究中心）；Humboldt-Universität zu Berlin（柏林洪堡大学）；Xiamen University（厦门大学）；Wang Yanan Institute for Studies in Economics（王亚南经济研究院）；Singapore Management University（新加坡管理大学）；Sim Kee Boon Institute for Financial Economics（沈基文金融经济研究院）；National University of Singapore（新加坡国立大学）；Asia Competitiveness Institute（亚洲竞争力研究所）；Charles University（查理大学）；National Yangming Jiaotong University（阳明交通大学）；ByteDance AI Lab（字节跳动人工智能实验室）；University of Cambridge（剑桥大学）；Janeway Institute（詹韦研究所）；Hochschule für Technik und Wirtschaft Berlin（柏林工程与应用技术大学）；ETH Zürich（苏黎世联邦理工学院）
Markowitz mean-variance portfolios with sample mean and covariance as input parameters feature numerous issues in practice. They perform poorly out of sample due to estimation error, they experience extreme weights together with high sensitivity to change in input parameters. The heavy-tail characteristics of financial time series are in fact the cause for these erratic fluctuations of weights that consequently create substantial transaction costs. In robustifying the weights we present a toolbox for stabilizing costs and weights for global minimum Markowitz portfolios. Utilizing a projected g…
Accurately estimating risk measures for financial portfolios is critical for both financial institutions and regulators. However, many existing models operate at the aggregate portfolio level and thus fail to capture the complex cross-dependencies between portfolio components. To address this, a new approach is presented that uses vine copulas in combination with univariate ARMA-GARCH models for marginal modelling to compute conditional portfolio-level risk measure estimates by simulating portfolio-level forecasts conditioned on a stress factor. A quantile-based approach is then presented to o…
In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfo…
We build the time series of optimal realized portfolio weights from high-frequency data and we suggest a novel Dynamic Conditional Weights (DCW) model for their dynamics. DCW is benchmarked against popular model-based and model-free specifications in terms of weights forecasts and portfolio allocations. Next to portfolio variance, certainty equivalent and turnover, we introduce the break-even transaction costs as an additional measure that identifies the range of transaction costs for which one allocation is preferred to another. By comparing minimum-variance portfolios built on the components…
Financial portfolios are often optimized for maximum profit while subject to a constraint formulated in terms of the Conditional Value-at-Risk (CVaR). This amounts to solving a linear problem. However, in its original formulation this linear problem has a very large number of linear constraints, too many to be enforced in practice. In the literature this is addressed by a reformulation of the problem using so-called dummy variables. This reduces the large number of constraints in the original linear problem at the cost of increasing the number of variables. In the context of reinsurance portfo…
Risk management is very important for individual investors or companies. There are many ways to measure the risk of investment. Prices of risky assets vary rapidly and randomly due to the complexity of finance market. Random interval is a good tool to describe uncertainty with both randomness and imprecision. Considering the uncertainty of financial market, we employ random intervals to describe the returns of a risk asset and consider the tail risk, which is called the interval-valued Conditional Value at Risk (ICVaR, for short). Such an ICVaR is a risk measure and satisfies subadditivity. Un…
Over the decades, the Markowitz framework has been used extensively in portfolio analysis though it puts too much emphasis on the analysis of the market uncertainty rather than on the trend prediction. While generative adversarial network (GAN), conditional GAN (CGAN), and autoencoding CGAN (ACGAN) have been explored to generate financial time series and extract features that can help portfolio analysis. The limitation of the CGAN or ACGAN framework stands in putting too much emphasis on generating series and finding the internal trends of the series rather than predicting the future trends. I…
Over the decades, the Markowitz framework has been used extensively in portfolio analysis though it puts too much emphasis on the analysis of the market uncertainty rather than on the trend prediction. While generative adversarial network (GAN) and conditional GAN (CGAN) have been explored to generate financial time series and extract features that can help portfolio analysis. The limitation of the CGAN framework stands in putting too much emphasis on generating series rather than keeping features that can help this generator. In this paper, we introduce an autoencoding CGAN (ACGAN) based on d…
The asymptotic distribution of the Markowitz portfolio is derived, for the general case (assuming fourth moments of returns exist), and for the case of multivariate normal returns. The derivation allows for inference which is robust to heteroskedasticity and autocorrelation of moments up to order four. As a side effect, one can estimate the proportion of error in the Markowitz portfolio due to mis-estimation of the covariance matrix. A likelihood ratio test is given which generalizes Dempster's Covariance Selection test to allow inference on linear combinations of the precision matrix and the …
We propose a Bayesian non-parametric approach for modeling the distribution of multiple returns. In particular, we use an asymmetric dynamic conditional correlation (ADCC) model to estimate the time-varying correlations of financial returns where the individual volatilities are driven by GJR-GARCH models. The ADCC-GJR-GARCH model takes into consideration the asymmetries in individual assets' volatilities, as well as in the correlations. The errors are modeled using a Dirichlet location-scale mixture of multivariate Gaussian distributions allowing for a great flexibility in the return distribut…
The discrete-time mean-variance portfolio selection formulation, a representative of general dynamic mean-risk portfolio selection problems, does not satisfy time consistency in efficiency (TCIE) in general, i.e., a truncated pre-committed efficient policy may become inefficient when considering the corresponding truncated problem, thus stimulating investors' irrational investment behavior. We investigate analytically effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets. More specifically, we derive the semi-analytical expressions for the pre-c…
Signals coming from multivariate higher order conditional moments as well as the information contained in exogenous covariates, can be effectively exploited by rational investors to allocate their wealth among different risky investment opportunities. This paper proposes a new flexible dynamic copula model being able to explain and forecast the time-varying shape of large dimensional asset returns distributions. Moreover, we let the univariate marginal distributions to be driven by an updating mechanism based on the scaled score of the conditional distribution. This framework allows us to intr…
We employ and examine vine copulas in modeling symmetric and asymmetric dependency structures and forecasting financial returns. We analyze the asset allocations performed during the 2008-2009 financial crisis and test different portfolio strategies such as maximum Sharpe ratio, minimum variance, and minimum conditional Value-at-Risk. We then specify the regular, drawable, and canonical vine copulas, such as the Student-t, Clayton, Frank, Joe, Gumbel, and mixed copulas, and analyze both in-sample and out-of-sample portfolio performances. Out-of-sample portfolio back-testing shows that vine cop…
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 propose an iterative gradient-based algorithm to efficiently solve the portfolio selection problem with multiple spectral risk constraints. Since the conditional value at risk (CVaR) is a special case of the spectral risk measure, our algorithm solves portfolio selection problems with multiple CVaR constraints. In each step, the algorithm solves very simple separable convex quadratic programs; hence, we show that the spectral risk constrained portfolio selection problem can be solved using the technology developed for solving mean-variance problems. The algorithm extends to the case where t…
Admissible Portfolio Optimization: Information Constraints, Conditional Efficient Frontiers, and the Price of Causal Identification
Quantitative Analysis and Artificial Intelligence Department, Miralta Finance Bank S.A.（Miralta Finance Bank S.A. 量化分析与人工智能部）；Department of Computer Science, University of Reading（雷丁大学计算机系）
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…
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…
Deep Reinforcement Learning for Investor-Specific Portfolio Optimization: A Volatility-Guided Asset Selection Approach
School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi（印度理工学院曼迪数学与统计科学学院）；Department of Mathematics, Indian Institute of Technology Guwahati（印度理工学院古瓦哈提数学系）
The problem of finding the optimal portfolio for investors is called the portfolio optimization problem. Such problem mainly concerns the expectation and variability of return (i.e., mean and variance). Although the variance would be the most fundamental risk measure to be minimized, it has several drawbacks. Conditional Value-at-Risk (CVaR) is a relatively new risk measure that addresses some of the shortcomings of well-known variance-related risk measures, and because of its computational efficiencies, it has gained popularity. CVaR is defined as the expected value of the loss that occurs be…
The entropic value-at-risk (EVaR) is a new coherent risk measure, which is an upper bound for both the value-at-risk (VaR) and conditional value-at-risk (CVaR). As important properties, the EVaR is strongly monotone over its domain and strictly monotone over a broad sub-domain including all continuous distributions, while well-known monotone risk measures, such as VaR and CVaR lack these properties. A key feature for a risk measure, besides its financial properties, is its applicability in large-scale sample-based portfolio optimization. If the negative return of an investment portfolio is a d…
Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of Neyman-Pearson type binary solution. We add a constraint on expected return to investigate the Mean-CVaR portfolio selection problem in a dynamic setting: the investor is faced with a Markowitz type of risk reward problem at final horizon where variance as a measure of risk is replaced by CVaR. Based on the complete market assumption, we give an analytical solution in general. The novelty of our …
This paper investigates the finite horizon risk-sensitive portfolio optimization in a regime-switching credit market with physical and information-induced default contagion. It is assumed that the underlying regime-switching process has countable states and is unobservable. The stochastic control problem is formulated under partial observations of asset prices and sequential default events. By establishing a martingale representation theorem based on incomplete and phasing out filtration, we connect the control problem to a quadratic BSDE with jumps, in which the driver term is non-standard an…
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…
We analyze conditional optimization problems arising in discrete time Principal-Agent problems of delegated portfolio optimization with linear contracts. Applying tools from Conditional Analysis we show that some results known in the literature for very specific instances of the problem carry over to translation invariant and time-consistent utility functions in very general probabilistic settings. However, we find that optimal contracts must in general make use of derivatives for compensation.
The signal-noise ratio of a portfolio of p assets, its expected return divided by its risk, is couched as an estimation problem on the sphere. When the portfolio is built using noisy data, the expected value of the signal-noise ratio is bounded from above via a Cramer-Rao bound, for the case of Gaussian returns. The bound holds for `biased' estimators, thus there appears to be no bias-variance tradeoff for the problem of maximizing the signal-noise ratio. An approximate distribution of the signal-noise ratio for the Markowitz portfolio is given, and shown to be fairly accurate via Monte Carlo …
The paper Zhao et al. (2015) shows that mean-CVaR-skewness portfolio optimization problems based on asymetric Laplace (AL) distributions can be transformed into quadratic optimization problems under which closed form solutions can be found. In this note, we show that such result also holds for mean-risk-skewness portfolio optimization problems when the underlying distribution is a larger class of normal mean-variance mixture (NMVM) models than the class of AL distributions. We then study the value at risk (VaR) and conditional value at risk (CVaR) risk measures on portfolios of returns with NM…
We study the optimal portfolio allocation problem from a Bayesian perspective using value at risk (VaR) and conditional value at risk (CVaR) as risk measures. By applying the posterior predictive distribution for the future portfolio return, we derive relevant quantiles needed in the computations of VaR and CVaR, and express the optimal portfolio weights in terms of observed data only. This is in contrast to the conventional method where the optimal solution is based on unobserved quantities which are estimated, leading to suboptimality. We also obtain the expressions for the weights of the gl…
Instead of controlling "symmetric" risks measured by central moments of investment return or terminal wealth, more and more portfolio models have shifted their focus to manage "asymmetric" downside risks that the investment return is below certain threshold. Among the existing downside risk measures, the lower-partial moments (LPM) and conditional value-at-risk (CVaR) are probably most promising. In this paper we investigate the dynamic mean-LPM and mean-CVaR portfolio optimization problems in continuous-time, while the current literature has only witnessed their static versions. Our contribut…
In this paper we continue the study of Bian-Miao-Zheng (2011) and extend the results there to a more general class of utility functions which may be bounded and non-strictly-concave and show that there is a classical solution to the HJB equation with the dual control method. We then apply the results to study the efficient frontier of wealth and conditional VaR (CVaR) problem and the turnpike property problem. For the former we construct explicitly the optimal control and discuss the choice of the optimal threadshold level and illustrate that the wealth and the CVaR are positively correlated. …
This paper investigates optimal portfolio strategies in a financial market where the drift of the stock returns is driven by an unobserved Gaussian mean reverting process. Information on this process is obtained from observing stock returns and expert opinions. The latter provide at discrete time points an unbiased estimate of the current state of the drift. Nevertheless, the drift can only be observed partially and the best estimate is given by the conditional expectation given the available information, i.e., by the filter. We provide the filter equations in the model with expert opinion and…
The Markowitz model is still the cornerstone of modern portfolio theory. In particular, when focusing on the minimum-variance portfolio, the covariance matrix or better its inverse, the so-called precision matrix, is the only input required. So far, most scholars worked on improving the estimation of the input, however little attention has been given to the limitations of the inverse covariance matrix when capturing the dependence structure in a non-Gaussian setting. While the precision matrix allows to correctly understand the conditional dependence structure of random vectors in a Gaussian s…
Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approac…
