Stock market returns are typically analyzed using standard regression, yet they reside on irregular domains which is a natural scenario for graph signal processing. To this end, we consider a market graph as an intuitive way to represent the relationships between financial assets. Traditional methods for estimating asset-return covariance operate under the assumption of statistical time-invariance, and are thus unable to appropriately infer the underlying true structure of the market graph. This work introduces a class of graph spectral estimators which cater for the nonstationarity inherent to asset price movements, and serve as a basis to represent the time-varying interactions between assets through a dynamic spectral market graph. Such an account of the time-varying nature of the asset-return covariance allows us to introduce the notion of dynamic spectral portfolio cuts, whereby the graph is partitioned into time-evolving clusters, allowing for online and robust asset allocation. The advantages of the proposed framework over traditional methods are demonstrated through numerical case studies using real-world price data.
Risk diversification is one of the dominant concerns for portfolio managers. Various portfolio constructions have been proposed to minimize the risk of the portfolio under some constrains including expected returns. We propose a portfolio construction method that incorporates the complex valued principal component analysis into the risk diversification portfolio construction. The proposed method is verified to outperform the conventional risk parity and risk diversification portfolio constructions.
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 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…
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…
We consider the problem of minimizing capital at risk in the Black-Scholes setting. The portfolio problem is studied given the possibility that a correlation constraint between the portfolio and a financial index is imposed. The optimal portfolio is obtained in closed form. The effects of the correlation constraint are explored; it turns out that this portfolio constraint leads to a more diversified portfolio.
In this study, we have investigated empirically the effects of market properties on the degree of diversification of investment weights among stocks in a portfolio. The weights of stocks within a portfolio were determined on the basis of Markowitz's portfolio theory. We identified that there was a negative relationship between the influence of market properties and the degree of diversification of the weights among stocks in a portfolio. Furthermore, we noted that the random matrix theory method could control the properties of correlation matrix between stocks; this may be useful in improving …
We relax a number of assumptions in Alexeev and Tapon (2012) in order to account for non-normally distributed, skewed, multi-regime, and leptokurtic asset return distributions. We calibrate a Markov-modulated Levy process model to equity market data to demonstrate the merits of our approach, and show that the calibrated models do a good job of matching the empirical moments. Finally, we argue that much of the related literature on portfolio diversification relies on assumptions that are in tension with certain observable regularities and which, if ignored, may lead to underestimation of risk.
Portfolio diversification and active risk management are essential parts of financial analysis which became even more crucial (and questioned) during and after the years of the Global Financial Crisis. We propose a novel approach to portfolio diversification using the information of searched items on Google Trends. The diversification is based on an idea that popularity of a stock measured by search queries is correlated with the stock riskiness. We penalize the popular stocks by assigning them lower portfolio weights and we bring forward the less popular, or peripheral, stocks to decrease the…
A new framework for portfolio diversification is introduced which goes beyond the classical mean-variance approach and portfolio allocation strategies such as risk parity. It is based on a novel concept called portfolio dimensionality that connects diversification to the non-Gaussianity of portfolio returns and can typically be defined in terms of the ratio of risk measures which are homogenous functions of equal degree. The latter arises naturally due to our requirement that diversification measures should be leverage invariant. We introduce this new framework and argue the benefits relative …
Cryptocurrencies (CCs) have risen rapidly in market capitalization over the last years. Despite striking price volatility, their high average returns have drawn attention to CCs as alternative investment assets for portfolio and risk management. We investigate the utility gains for different types of investors when they consider cryptocurrencies as an addition to their portfolio of traditional assets. We consider risk-averse, return-seeking as well as diversificationpreferring investors who trade along different allocation frequencies, namely daily, weekly or monthly. Out-of-sample performance…
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…
When assets are correlated, benefits of investment diversification are reduced. To measure the influence of correlations on investment performance, a new quantity - the effective portfolio size - is proposed and investigated in both artificial and real situations. We show that in most cases, the effective portfolio size is much smaller than the actual number of assets in the portfolio and that it lowers even further during financial crises.
We study how to assess the potential benefit of diversifying an equity portfolio by investing within and across equity sectors. We analyse 20 years of US stock price data, which includes the global financial crisis (GFC) and the COVID-19 market crash, as well as periods of financial stability, to determine the `all weather' nature of equity portfolios. We establish that one may use the leading eigenvalue of the cross-correlation matrix of log returns as well as graph-theoretic diagnostics such as modularity to quantify the collective behaviour of the market or a subset of it. We confirm that f…
Investment returns naturally reside on irregular domains, however, standard multivariate portfolio optimization methods are agnostic to data structure. To this end, we investigate ways for domain knowledge to be conveniently incorporated into the analysis, by means of graphs. Next, to relax the assumption of the completeness of graph topology and to equip the graph model with practically relevant physical intuition, we introduce the portfolio cut paradigm. Such a graph-theoretic portfolio partitioning technique is shown to allow the investor to devise robust and tractable asset allocation sche…
It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies. Indeed, in the U.S. stock market, equal-weighted portfolios, random-weighted portfolios, and other naive, non- optimized portfolios tend to outperform a capitalization-weighted index over the long term. This outperformance is generally attributed to beneficial factor exposures. Here, we provide a deeper, more general explanation of this phenomenon by decomposing portfolio log-returns into an average growth and an excess growth component. Using a rank-based em…
This paper focuses on a dynamic multi-asset mean-variance portfolio selection problem under model uncertainty. We develop a continuous time framework for taking into account ambiguity aversion about both expected return rates and correlation matrix of the assets, and for studying the join effects on portfolio diversification. The dynamic setting allows us to consider time varying ambiguity sets, which include the cases where the drift and correlation are estimated on a rolling window of historical data or when the investor takes into account learning on the ambiguity. In this context, we prove…
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…
Despite the availability of very detailed data on financial market, agent-based modeling is hindered by the lack of information about real trader behavior. This makes it impossible to validate agent-based models, which are thus reverse-engineering attempts. This work is a contribution to the building of a set of stylized facts about the traders themselves. Using the client database of Swissquote Bank SA, the largest on-line Swiss broker, we find empirical relationships between turnover, account values and the number of assets in which a trader is invested. A theory based on simple mean-varianc…
In the past decade many researchers have proposed new optimal portfolio selection strategies to show that sophisticated diversification can outperform the naïve 1/N strategy in out-of-sample benchmarks. Providing an updated review of these models since DeMiguel et al. (2009b), I test sixteen strategies across six empirical datasets to see if indeed progress has been made. However, I find that none of the recently suggested strategies consistently outperforms the 1/N or minimum-variance approach in terms of Sharpe ratio, certainty-equivalent return or turnover. This suggests that simple diversi…
Financial markets, with their vast range of different investment opportunities, can be seen as a system of many different simultaneous games with diverse and often unknown levels of risk and reward. We introduce generalizations to the classic Kelly investment game [Kelly (1956)] that incorporates these features, and use them to investigate the influence of diversification and limited information on Kelly-optimal portfolios. In particular we present approximate formulas for optimizing diversified portfolios and exact results for optimal investment in unknown games where the only available infor…
The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the point of view of statistical learning theory. The occurrence of the instability is intimately related to over-fitting which can be avoided using known regularization methods. We show how regularized portfolio optimization with the expected shortfall …
Geographic diversification is fundamental to risk mitigation among investors and insurers of housing, mortgages, and mortgage-related derivatives. To characterize diversification potential, we provide estimates of integration, spatial correlation, and contagion among US metropolitan housing markets. Results reveal a high and increasing level of integration among US markets over the decade of the 2000s, especially in California. We apply integration results to assess the risk of alternative housing investment portfolios. Portfolio simulation indicates reduced diversification potential and incre…
In this paper, we propose an innovative investment framework incorporating asset allocation and class diversification oriented specifically for the biotechnology industry. With growing interests and capitalization in multiple biotech markets, investors require a more dynamic method of managing their assets within individual portfolios for optimal return efficiency. By selecting a single firm representative of identified industry trends, analyzing financial metrics relevant to the suggested approaches, and assessing financial health, we developed an adaptable investment methodology. We also per…
Cryptocurrency markets have attracted many interest for global investors because of their novelty, wide online availability, increasing capitalization and potential profits. In the econophysics tradition we show that many of the most available cryptocurrencies have return statistics that do not follow Gaussian distributions but heavy--tailed distributions instead. Entropy measures are also applied showing that portfolio diversification is a reasonable practice for decreasing return uncertainty.
Oil is perceived as a good diversification tool for stock markets. To fully understand this potential, we propose a new empirical methodology that combines generalized autoregressive score copula functions with high frequency data and allows us to capture and forecast the conditional time-varying joint distribution of the oil -- stocks pair accurately. Our realized GARCH with time-varying copula yields statistically better forecasts of the dependence and quantiles of the distribution relative to competing models. Employing a recently proposed conditional diversification benefits measure that c…
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 investigate an optimal investment problem with a general performance criterion which, in particular, includes discontinuous functions. Prices are modeled as diffusions and the market is incomplete. We find an explicit solution for the case of limited diversification of the portfolio, i.e. for the portfolio compression problem. By this we mean that an admissible strategies may include no more than m different stocks concurrently, where m may be less than the total number n of available stocks.
Portfolio optimization methods suffer from a catalogue of known problems, mainly due to the facts that pair correlations of asset returns are unstable, and that extremal risk measures such as maximum drawdown are difficult to predict due to the non-Gaussianity of portfolio returns. \\ In order to look at optimal portfolios for arbitrary risk penalty functions, we construct portfolio shapes where the penalty is proportional to a moment of the returns of arbitrary order $p>2$. \\ The resulting component weight in the portfolio scales sub-linearly with its return, with the power-law $w \propto μ^…
We introduce a financial portfolio optimization framework that allows us to automatically select the relevant assets and estimate their weights by relying on a sorted $\ell_1$-Norm penalization, henceforth SLOPE. Our approach is able to group constituents with similar correlation properties, and with the same underlying risk factor exposures. We show that by varying the intensity of the penalty, SLOPE can span the entire set of optimal portfolios on the risk-diversification frontier, from minimum variance to the equally weighted. To solve the optimization problem, we develop a new efficient al…
Deep Reinforcement Learning Framework for Diversified Portfolio Management Across Global Equity Markets
Quantitative Finance Research Group, Faculty of Economic Sciences, University of Warsaw（经济科学学院量化金融研究组，华沙大学）；Quantitative Finance Research Group, Department of Quantitative Finance and Machine Learning, Faculty of Economic Sciences, University of Warsaw（经济科学学院量化金融与机器学习系量化金融研究组，华沙大学）
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 …
Portfolio optimisation typically aims to provide an optimal allocation that minimises risk, at a given return target, by diversifying over different investments. However, the potential scope of such risk diversification can be limited if investments are concentrated in only one country, or more specifically one currency. Multi-currency portfolio is an alternative to achieve higher returns and more diversified portfolios but it requires a careful management of the entailed risks from changes in exchange rates. The deviation between asset and currency exposures in a portfolio is defined as the "…
Diversification of an investment into independently fluctuating assets reduces its risk. In reality, movement of assets are are mutually correlated and therefore knowledge of cross--correlations among asset price movements are of great importance. Our results support the possibility that the problem of finding an investment in stocks which exposes invested funds to a minimum level of risk is analogous to the problem of finding the magnetization of a random magnet. The interactions for this ``random magnet problem'' are given by the cross-correlation matrix {\bf \sf C} of stock returns. We find…
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…
A fund manager invests both the fund's assets and own private wealth in separate but potentially correlated risky assets, aiming to maximize expected utility from private wealth in the long run. If relative risk aversion and investment opportunities are constant, we find that the fund's portfolio depends only on the fund's investment opportunities, and the private portfolio only on private opportunities. This conclusion is valid both for a hedge fund manager, who is paid performance fees with a high-water mark provision, and for a mutual fund manager, who is paid management fees proportional t…
The benefits of portfolio diversification is a central tenet implicit to modern financial theory and practice. Linked to diversification is the notion of breadth. Breadth is correctly thought of as the number of in- dependent bets available to an investor. Conventionally applications us- ing breadth frequently assume only the number of separate bets. There may be a large discrepancy between these two interpretations. We uti- lize a simple singular-value decomposition (SVD) and the Keiser-Gutman stopping criterion to select the integer-valued effective dimensionality of the correlation matrix o…
