We discuss and extend a powerful, geometric framework to represent the set of portfolios, which identifies the space of asset allocations with the points lying in a convex polytope. Based on this viewpoint, we survey certain state-of-the-art tools from geometric and statistical computing in order to handle important and difficult problems in digital finance. Although our tools are quite general, in this paper we focus on two specific questions.
The first concerns crisis detection, which is of prime interest for the public in general and for policy makers in particular because of the significant impact that crises have on the economy. Certain features in stock markets lead to this type of anomaly detection: Given the assets' returns, we describe the relationship between portfolios' return and volatility by means of a copula, without making any assumption on investor strategies. We examine a recent method relying on copulae to construct an appropriate indicator that allows us to automate crisis detection. On real data, the indicator detects all past crashes in the cryptocurrency market, whereas from the DJ600-Europe index, from 1990 to 2008, the indicator identifies correctly 4 crises and issues one false positive for which we offer an explanation.
Our second contribution is to introduce an original computational framework to model asset allocation strategies, which is of independent interest for digital finance and its applications. Our approach addresses the crucial question of evaluating portfolio management, and is relevant to individual managers as well as financial institutions. To evaluate portfolio performance, we provide a new portfolio score, based on the aforementioned framework and concepts. In particular, our score relies on the statistical properties of portfolios, and we show how they can be computed efficiently.
It is well known that there are asymmetric dependence structures between financial returns. In this paper we use a new nonparametric measure of local dependence, the local Gaussian correlation, to improve portfolio allocation. We extend the classical mean-variance framework, and show that the portfolio optimization is straightforward using our new approach, only relying on a tuning parameter (the bandwidth). The new method is shown to outperform the equally weighted (1/N) portfolio and the classical Markowitz portfolio for monthly asset returns data.
The principal portfolios of the standard Capital Asset Pricing Model (CAPM) are analyzed and found to have remarkable hedging and leveraging properties. Principal portfolios implement a recasting of any correlated asset set of N risky securities into an equivalent but uncorrelated set when short sales are allowed. While a determination of principal portfolios in general requires a detailed knowledge of the covariance matrix for the asset set, the rather simple structure of CAPM permits an accurate solution for any reasonably large asset set that reveals interesting universal properties. Thus f…
We consider the problem of optimizing a portfolio of financial assets, where the number of assets can be much larger than the number of observations. The optimal portfolio weights require estimating the inverse covariance matrix of excess asset returns, classical solutions of which behave badly in high-dimensional scenarios. We propose to use a regression-based joint shrinkage method for estimating the partial correlation among the assets. Extensive simulation studies illustrate the superior performance of the proposed method with respect to variance, weight, and risk estimation errors compare…
Hedge funds have long been viewed as a veritable "black box" of investing since outsiders may never view the exact composition of portfolio holdings. Therefore, the ability to estimate an informative set of asset weights is highly desirable for analysis. We present a compositional state space model for estimation of an investment portfolio's unobserved asset allocation weightings on a set of candidate assets when the only observed information is the time series of portfolio returns and the candidate asset returns. In this paper, we exhibit both sequential Monte Carlo numerical and conditionall…
Portfolio optimization is one of the essential fields of focus in finance. There has been an increasing demand for novel computational methods in this area to compute portfolios with better returns and lower risks in recent years. We present a novel computational method called Representation Portfolio Selection (RPS) by redefining the distance matrix of financial assets using Representation Learning and Clustering algorithms for portfolio selection to increase diversification. RPS proposes a heuristic for getting closer to the optimal subset of assets. Using empirical results in this paper, we…
We employ perturbation analysis technique to study multi-asset portfolio optimisation with transaction cost. We allow for correlations in risky assets and obtain optimal trading methods for general utility functions. Our analytical results are supported by numerical simulations in the context of the Long Term Growth Model.
Traditional approaches to financial asset allocation start with returns forecasting followed by an optimization stage that decides the optimal asset weights. Any errors made during the forecasting step reduce the accuracy of the asset weightings, and hence the profitability of the overall portfolio. The Portfolio Transformer (PT) network, introduced here, circumvents the need to predict asset returns and instead directly optimizes the Sharpe ratio, a risk-adjusted performance metric widely used in practice. The PT is a novel end-to-end portfolio optimization framework, inspired by the numerous…
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…
Portfolio management problems are often divided into two types: active and passive, where the objective is to outperform and track a preselected benchmark, respectively. Here, we formulate and solve a dynamic asset allocation problem that combines these two objectives in a unified framework. We look to maximize the expected growth rate differential between the wealth of the investor's portfolio and that of a performance benchmark while penalizing risk-weighted deviations from a given tracking portfolio. Using stochastic control techniques, we provide explicit closed-form expressions for the op…
We show how to reduce the problem of computing VaR and CVaR with Student T return distributions to evaluation of analytical functions of the moments. This allows an analysis of the risk properties of systems to be carefully attributed between choices of risk function (e.g. VaR vs CVaR); choice of return distribution (power law tail vs Gaussian) and choice of event frequency, for risk assessment. We exploit this to provide a simple method for portfolio optimization when the asset returns follow a standard multivariate T distribution. This may be used as a semi-analytical verification tool for m…
Diversification return is an incremental return earned by a rebalanced portfolio of assets. The diversification return of a rebalanced portfolio is often incorrectly ascribed to a reduction in variance. We argue that the underlying source of the diversification return is the rebalancing, which forces the investor to sell assets that have appreciated in relative value and buy assets that have declined in relative value, as measured by their weights in the portfolio. In contrast, the incremental return of a buy-and-hold portfolio is driven by the fact that the assets that perform the best become…
The Sharpe ratio is a way to compare the excess returns (over the risk free asset) of portfolios for each unit of volatility that is generated by a portfolio. In this paper we introduce a robust Sharpe ratio portfolio under the assumption that the risk free asset is unknown. We propose a robust portfolio that maximizes the Sharpe ratio when the risk free asset is unknown, but is within a given interval. To compute the best Sharpe ratio portfolio all the Sharpe ratios for any risk free asset are considered and compared by using the so-called cross-efficiency evaluation. An explicit expression o…
Despite the fact that the Euler allocation principle has been adopted by many financial institutions for their internal capital allocation process, a comprehensive description of Euler allocation seems still to be missing. We try to fill this gap by presenting the theoretical background as well as practical aspects. In particular, we discuss how Euler risk contributions can be estimated for some important risk measures. We furthermore investigate the analysis of CDO tranche expected losses by means of Euler's theorem and suggest an approach to measure the impact of risk factors on non-linear p…
Given a set of assets and an investment capital, the classical portfolio selection problem consists in determining the amount of capital to be invested in each asset in order to build the most profitable portfolio. The portfolio optimization problem is naturally modeled as a mean-risk bi-criteria optimization problem where the mean rate of return of the portfolio must be maximized whereas a given risk measure must be minimized. Several mathematical programming models and techniques have been presented in the literature in order to efficiently solve the portfolio problem. A relatively recent pr…
We propose a novel risk matrix to characterize the optimal portfolio choice of an investor with tail concerns. The diagonal of the matrix contains the Value-at-Risk of each asset in the portfolio and the off-diagonal the pairwise Delta-CoVaR measures reflecting tail connections between assets. First, we derive the conditions under which the associated quadratic risk function has a closed-form solution. Second, we examine the relationship between portfolio risk and eigenvector centrality. Third, we show that portfolio risk is not necessarily increasing with respect to stock centrality. Forth, w…
This paper proposes a new method for financial portfolio optimization based on reducing simultaneous asset shocks across a collection of assets. This may be understood as an alternative approach to risk reduction in a portfolio based on a new mathematical quantity. First, we apply recently introduced semi-metrics between finite sets to determine the distance between time series' structural breaks. Then, we build on the classical portfolio optimization theory of Markowitz and use this distance between asset structural breaks for our penalty function, rather than portfolio variance. Our experime…
The portfolio optimization problem in which the variances of the return rates of assets are not identical is analyzed in this paper using the methodology of statistical mechanical informatics, specifically, replica analysis. We define two characteristic quantities of an optimal portfolio, namely, minimal investment risk and concentrated investment level, in order to solve the portfolio optimization problem and analytically determine their asymptotical behaviors using replica analysis. Moreover, numerical experiments were performed, and a comparison between the results of our simulation and tho…
We investigate the use of Kelly's strategy in the construction of an optimal portfolio of assets. For lognormally distributed asset returns, we derive approximate analytical results for the optimal investment fractions in various settings. We show that when mean returns and volatilities of the assets are small and there is no risk-free asset, the Kelly-optimal portfolio lies on Markowitz Efficient Frontier. Since in the investigated case the Kelly approach forbids short positions and borrowing, often only a small fraction of the available assets is included in the Kelly-optimal portfolio. This…
In the market place, diversification reduces risk and provides protection against extreme events by ensuring that one is not overly exposed to individual occurrences. We argue that diversification is best measured by characteristics of the combined portfolio of assets and introduce a measure based on the information entropy of the probability distribution for the final portfolio asset value. For Gaussian assets the measure is a logarithmic function of the variance and combining independent Gaussian assets of equal variance adds an amount to the diversification. The advantages of this measure i…
The investment economy is a main characteristic of prosperous society. The investment portfolio management is a main financial problem, which has to be solved by the investment, commercial and central banks with the application of modern portfolio theory in the investment economy. We use the learning analytics together with the integrative creative imperative intelligent conceptual co-lateral adaptive thinking with the purpose to advance our scientific knowledge on the diversified investment portfolio management in the nonlinear dynamic financial system. We apply the econophysics principles an…
Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, the seller, on the other hand, can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses in this case we need to develop new methods. One of the steps moving the theory towards practical needs is to take into account the time lag of the liquidation of an illiquid asset. This task became especially significant for the practitioners in the time of the global financial crises. …
The debate between active and passive investment strategies has been ongoing for many years and is far from being over. In this paper, we show that the choice of an optimal portfolio management strategy depends on an investment climate, which we measure via the parameters of the Heston model calibrated to the real stock market data. Depending on the values of those parameters, the passive strategy may namely outperform the active ones or vice versa. The method is tested on three stock market indices: S\&P500, DAX and WIG20.
Markowitz (1952, 1959) laid down the ground-breaking work on the mean-variance analysis. Under his framework, the theoretical optimal allocation vector can be very different from the estimated one for large portfolios due to the intrinsic difficulty of estimating a vast covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to the accumulation of estimation errors. We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the theoreti…
The investment risk minimization problem with budget and return constraints has been the subject of research using replica analysis but there are shortcomings in the extant literature. With respect to Tobin's separation theorem and the capital asset pricing model, it is necessary to investigate the implications of a risk-free asset and examine its influence on the optimal portfolio. Accordingly, in this work, we explore the investment risk minimization problem in the presence of a risk-free asset with budget and return constraints. Moreover, we discuss opportunity loss, the Pythagorean theorem…
This paper studies the investment decision of the Spanish households using a unique data set, the Spanish Survey of Household Finance (EFF). We propose a theoretical model in which households, given a fixed investment in housing, allocate their net wealth across bank time deposits, stocks, and mortgage. Besides considering housing as an indivisible and illiquid asset that restricts the portfolio choice decision, we take into account the financial constraints that households face when they apply for external funding. For every representative household in the EFF we solve this theoretical proble…
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 …
In this paper, we document a novel machine learning based bottom-up approach for static and dynamic portfolio optimization on, potentially, a large number of assets. The methodology applies to general constrained optimization problems and overcomes many major difficulties arising in current optimization schemes. Taking mean-variance optimization as an example, we no longer need to compute the covariance matrix and its inverse, therefore the method is immune from the estimation error on this quantity. Moreover, no explicit calls of optimization routines are needed. Applications to equity portfo…
Managing investment portfolios is an old and well know problem in multiple fields including financial mathematics and financial engineering as well as econometrics and econophysics. Multiple different concepts and theories were used so far to describe methods of handling with financial assets, including differential equations, stochastic calculus and advanced statistics. In this paper, using a set of tools from the probability theory, various strategies of building financial portfolios are analysed in different market conditions. A special attention is given to several realisations of a so cal…
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.
We consider the mean--variance portfolio optimization problem under the game theoretic framework and without risk-free assets. The problem is solved semi-explicitly by applying the extended Hamilton--Jacobi--Bellman equation. Although the coefficient of risk aversion in our model is a constant, the optimal amounts of money invested in each stock still depend on the current wealth in general. The optimal solution is obtained by solving a system of ordinary differential equations whose existence and uniqueness are proved and a numerical algorithm as well as its convergence speed are provided. Di…
In behavioral finance, aversion affects investors' judgment of future uncertainty when profit and loss occur. Considering investors' aversion to loss and risk, and the ambiguous uncertainty characterizing asset returns, we construct a distributional robust portfolio model (DRP) under the condition that the distribution of risky asset returns is unknown. Specifically, our objective is to find an optimal portfolio of assets that maximizes the worst-case utility level on the Wasserstein ball, which is centered on the empirical distribution of sample returns and the radius of the ball quantifies t…
Managing a large-scale portfolio with many assets is one of the most challenging tasks in the field of finance. It is partly because estimation of either covariance or precision matrix of asset returns tends to be unstable or even infeasible when the number of assets $p$ exceeds the number of observations $n$. For this reason, most of the previous studies on portfolio management have focused on the case of$ p < n$. To deal with the case of $p > n$, we propose to use a new Bayesian framework based on adaptive graphical LASSO for estimating the precision matrix of asset returns in a large-scale …
We introduce a new set of consistent measures of risks, in terms of the semi-invariants of pdf's, such that the centered moments and the cumulants of the portfolio distribution of returns that put more emphasis on the tail the distributions. We derive generalized efficient frontiers, based on these novel measures of risks and present the generalized CAPM, both in the cases of homogeneous and heterogeneous markets. Then, using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and the…
We advocate the use of Agnostic Allocation for the construction of long-only portfolios of stocks. We show that Agnostic Allocation Portfolios (AAPs) are a special member of a family of risk-based portfolios that are able to mitigate certain extreme features (excess concentration, high turnover, strong exposure to low-risk factors) of classical portfolio construction methods, while achieving similar performance. AAPs thus represent a very attractive alternative risk-based portfolio construction framework that can be implemented in different situations, with or without an active trading signal.
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.
In this short report, we discuss how coordinate-wise descent algorithms can be used to solve minimum variance portfolio (MVP) problems in which the portfolio weights are constrained by $l_{q}$ norms, where $1\leq q \leq 2$. A portfolio which weights are regularised by such norms is called a sparse portfolio (Brodie et al.), since these constraints facilitate sparsity (zero components) of the weight vector. We first consider a case when the portfolio weights are regularised by a weighted $l_{1}$ and squared $l_{2}$ norm. Then two benchmark data sets (Fama and French 48 industries and 100 size a…
We consider a portfolio allocation problem for trend following (TF) strategies on multiple correlated assets. Under simplifying assumptions of a Gaussian market and linear TF strategies, we derive analytical formulas for the mean and variance of the portfolio return. We construct then the optimal portfolio that maximizes risk-adjusted return by accounting for inter-asset correlations. The dynamic allocation problem for $n$ assets is shown to be equivalent to the classical static allocation problem for $n^2$ virtual assets that include lead-lag corrections in positions of TF strategies. The res…
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…
The main contribution of the paper is to employ the financial market network as a useful tool to improve the portfolio selection process, where nodes indicate securities and edges capture the dependence structure of the system. Three different methods are proposed in order to extract the dependence structure between assets in a network context. Starting from this modified structure, we formulate and then we solve the asset allocation problem. We find that the portfolios obtained through a network-based approach are composed mainly of peripheral assets, which are poorly connected with the other…
ESG ratings provide a quantitative measure for socially responsible investment. We present a unified framework for incorporating numeric ESG ratings into dynamic pricing theory. Specifically, we introduce an ESG-valued return that is a linearly constrained transformation of financial return and ESG score. This leads to a more complex portfolio optimization problem in a space governed by reward, risk and ESG score. The framework preserves the traditional risk aversion parameter and introduces an ESG affinity parameter. We apply this framework to develop ESG-valued: portfolio optimization; capit…
Classical mean-variance portfolio theory tells us how to construct a portfolio of assets which has the greatest expected return for a given level of return volatility. Utility theory then allows an investor to choose the point along this efficient frontier which optimally balances her desire for excess expected return against her reluctance to bear risk. The means and covariances of the distributions of future asset returns are assumed to be known, so the only source of uncertainty is the stochastic piece of the price evolution. In the real world, we have another source of uncertainty - we est…
This article develops a model that takes into account skewness risk in risk parity portfolios. In this framework, asset returns are viewed as stochastic processes with jumps or random variables generated by a Gaussian mixture distribution. This dual representation allows us to show that skewness and jump risks are equivalent. As the mixture representation is simple, we obtain analytical formulas for computing asset risk contributions of a given portfolio. Therefore, we define risk budgeting portfolios and derive existence and uniqueness conditions. We then apply our model to the equity/bond/vo…
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）
Schumpeter School of Business and Economics（熊彼特商业与经济学院）；University of Wuppertal（伍珀塔尔大学）
This paper proposes a Deep Reinforcement Learning algorithm for financial portfolio trading based on Deep Q-learning. The algorithm is capable of trading high-dimensional portfolios from cross-sectional datasets of any size which may include data gaps and non-unique history lengths in the assets. We sequentially set up environments by sampling one asset for each environment while rewarding investments with the resulting asset's return and cash reservation with the average return of the set of assets. This enforces the agent to strategically assign capital to assets that it predicts to perform …
We explore a decomposition in which returns on a large class of portfolios relative to the market depend on a smooth non-negative drift and changes in the asset price distribution. This decomposition is obtained using general continuous semimartingale price representations, and is thus consistent with virtually any asset pricing model. Fluctuations in portfolio relative returns depend on stochastic time-varying dispersion in asset prices. Thus, our framework uncovers an asset pricing factor whose existence emerges from an accounting identity universal across different economic and financial en…
In this paper, motivated by the celebrated work of Kelly, we consider the problem of portfolio weight selection to maximize expected logarithmic growth. Going beyond existing literature, our focal point here is the rebalancing frequency which we include as an additional parameter in our analysis. The problem is first set in a control-theoretic framework, and then, the main question we address is as follows: In the absence of transaction costs, does high-frequency trading always lead to the best performance? Related to this is our prior work on betting, also in the Kelly context, which examines…
The mean-variance portfolio that considers the trade-off between expected return and risk has been widely used in the problem of asset allocation for multi-asset portfolios. However, since it is difficult to estimate the expected return and the out-of-sample performance of the mean-variance portfolio is poor, risk-based portfolio construction methods focusing only on risk have been proposed, and are attracting attention mainly in practice. In terms of risk, asset fluctuations that make up the portfolio are thought to have common factors behind them, and principal component analysis, which is a…
