The emergence of robust optimization has been driven primarily by the necessity to address the demerits of the Markowitz model. There has been a noteworthy debate regarding consideration of robust approaches as superior or at par with the Markowitz model, in terms of portfolio performance. In order to address this skepticism, we perform empirical analysis of three robust optimization models, namely the ones based on box, ellipsoidal and separable uncertainty sets. We conclude that robust approaches can be considered as a viable alternative to the Markowitz model, not only in simulated data but also in a real market setup, involving the Indian indices of S&P BSE 30 and S&P BSE 100. Finally, we offer qualitative and quantitative justification regarding the practical usefulness of robust optimization approaches from the point of view of number of stocks, sample size and types of data.
Our study focuses on determining the presence of abnormal returns for physical momentum portfolios in the context of the Indian market. The physical momentum portfolios, comprising stocks from the NSE 500, are constructed for the daily, weekly, monthly, and yearly timescales. In the aforementioned timescales, we empirically evaluate the historical returns and varied risk profiles of these portfolios for the years 2014-2021. It has been observed that the best-performing physical momentum portfolios from each of the four timescales achieved higher returns and better risk measures when compared t…
Portfolio optimization is a challenging problem that has attracted considerable attention and effort from researchers. The optimization of stock portfolios is a particularly hard problem since the stock prices are volatile and estimation of their future volatilities and values, in most cases, is very difficult, if not impossible. This work uses three ratios, the Sharpe ratio, the Sortino ratio, and the Calmar ratio, for designing the mean-variance optimized portfolios for six important sectors listed in the National Stock Exchange (NSE) of India. Three portfolios are designed for each sector m…
A Comparative Analysis of Portfolio Optimization Using Mean-Variance, Hierarchical Risk Parity, and Reinforcement Learning Approaches on the Indian Stock Market
Designing an optimum portfolio that allocates weights to its constituent stocks in a way that achieves the best trade-off between the return and the risk is a challenging research problem. The classical mean-variance theory of portfolio proposed by Markowitz is found to perform sub-optimally on the real-world stock market data since the error in estimation for the expected returns adversely affects the performance of the portfolio. This paper presents three approaches to portfolio design, viz, the minimum risk portfolio, the optimum risk portfolio, and the Eigen portfolio, for seven important …
The paper predicts an Efficient Market Property for the equity market, where stocks, when denominated in units of the growth optimal portfolio (GP), have zero instantaneous expected returns. Well-diversified equity portfolios are shown to approximate the GP, which explains the well-observed good performance of equally weighted portfolios. The proposed hierarchically weighted index (HWI) is shown to be an even better proxy of the GP. It sets weights equal within industrial and geographical groupings of stocks. When using the HWI as proxy of the GP the Efficient Market Property cannot be easily …
The stock market offers a platform where people buy and sell shares of publicly listed companies. Generally, stock prices are quite volatile; hence predicting them is a daunting task. There is still much research going to develop more accuracy in stock price prediction. Portfolio construction refers to the allocation of different sector stocks optimally to achieve a maximum return by taking a minimum risk. A good portfolio can help investors earn maximum profit by taking a minimum risk. Beginning with Dow Jones Theory a lot of advancement has happened in the area of building efficient portfoli…
In this paper, we introduce a matrix-valued time series model for foreign exchange market. We then formulate trading matrices, foreign exchange options and return options (matrices), as well as on-line portfolio strategies. Moreover, we attempt to predict returns of portfolios by developing a cross rate method. This leads us to construct an on-line portfolio selection algorithm for this model. At the end, we prove the profitability and the universality of our algorithm.
We study a portfolio selection problem in a continuous-time Itô-Markov additive market with prices of financial assets described by Markov additive processes which combine Lévy processes and regime switching models. Thus the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. For this reason the market is incomplete. We complete the market by enlarging it with the use of a set of Markovian jump securities, Markovian power-jump securities and impulse regime switching securities. Moreover, we give conditions under which the market is asymptotic-ar…
There is a great number of factors to take into account when building and managing an investment portfolio. It is widely believed that a proper set-up of the portfolio combined with a good, robust management strategy is the key to successful investment. In this paper, we aim at an analysis of two aspects that may have an impact on investment performance: diversity of assets and inclusion of cash in the portfolio. We also propose two new management strategies based on the MACD and RSI factors known from technical analysis. Monte Carlo simulations within the Heston model of a market are used to …
The problem of data uncertainty has motivated the incorporation of robust optimization in various arenas, beyond the Markowitz portfolio optimization. This work presents the extension of the robust optimization framework for the minimization of downside risk measures, such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We perform an empirical study of VaR and CVaR frameworks, with respect to their robust counterparts, namely, Worst-Case VaR and Worst-Case CVaR, using the market data as well as the simulated data. After discussing the practical usefulness of the robust optimizatio…
We develop a novel five-component decomposition of optimal dynamic portfolio choice, which reveals the simultaneous impacts from market incompleteness and wealth-dependent utilities. Under the HARA utility and a nonrandom interest rate, we can explicitly solve for the optimal policy as a combination of a bond holding scheme and the corresponding simpler CRRA strategy. Under a stochastic volatility model estimated on US equity data, we use closed-form solution to demonstrate the sophisticated impacts from the wealth-dependent utilities, including cycle-dependence and hysteresis effect in optima…
In this work, we consider the optimal portfolio selection problem under hard constraints on trading volume amounts when the dynamics of the risky asset returns are governed by a discrete-time approximation of the Markov-modulated geometric Brownian motion. The states of Markov chain are interpreted as the states of an economy. The problem is stated as a dynamic tracking problem of a reference portfolio with desired return. We propose to use the model predictive control (MPC) methodology in order to obtain feedback trading strategies. Our approach is tested on a set of a real data from the radi…
Stock price prediction is a challenging task and a lot of propositions exist in the literature in this area. Portfolio construction is a process of choosing a group of stocks and investing in them optimally to maximize the return while minimizing the risk. Since the time when Markowitz proposed the Modern Portfolio Theory, several advancements have happened in the area of building efficient portfolios. An investor can get the best benefit out of the stock market if the investor invests in an efficient portfolio and could take the buy or sell decision in advance, by estimating the future asset …
We study the problem of dynamically trading futures in a regime-switching market. Modeling the underlying asset price as a Markov-modulated diffusion process, we present a utility maximization approach to determine the optimal futures trading strategy. This leads to the analysis of the associated system of Hamilton-Jacobi-Bellman (HJB) equations, which are reduced to a system of linear ODEs. We apply our stochastic framework to two models, namely, the Regime-Switching Geometric Brownian Motion (RS-GBM) model and Regime-Switching Exponential Ornstein-Uhlenbeck (RS-XOU) model. Numerical examples…
There is substantial empirical evidence showing the fundamental portfolio outperforming the market portfolio. Here a theoretical foundation is laid that supports this empirical research. Assuming stock prices revert around fundamental prices with sufficient strength and symmetry, the fundamental portfolio outperforms the market portfolio in expectation. If reversion toward the fundamental price is not sufficiently strong, then the fundamental portfolio underperforms the market portfolio in expectation.
Optimal capital allocation between different assets is an important financial problem, which is generally framed as the portfolio optimization problem. General models include the single-period and multi-period cases. The traditional Mean-Variance model introduced by Harry Markowitz has been the basis of many models used to solve the portfolio optimization problem. The overall goal is to achieve the highest return and lowest risk in portfolio optimization problems. In this paper, we will present an optimal portfolio based the Markowitz Mean-Variance-Skewness with weight constraints model for sh…
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…
In this work, we consider the optimal portfolio selection problem under hard constraints on trading amounts, transaction costs and different rates for borrowing and lending when the risky asset returns are serially correlated. No assumptions about the correlation structure between different time points or about the distribution of the asset returns are needed. The problem is stated as a dynamic tracking problem of a reference portfolio with desired return. Our approach is tested on a set of a real data from Russian Stock Exchange MICEX.
We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market portfolio and an additional path of finite variation. The second version is formulated within the functional pathwise Itô calculus of Dupire (2009) and Cont \& Fournié (2010) and allows for portfolio-generating functionals that may depend additionally on the entire path of the market…
The objective of this paper is to measure the degree of home bias (HB) within holdings portfolio and to identify their determining factors. By following literature and an international capital asset pricing model, we have chosen quite a number of susceptible factors that impact HB. This model is, hence, estimated for 20 countries, with cross-section econometrics, between 2008 and 2013. Our results show that all countries have recorded a high level of HB in their holdings portfolio. After that, we test if the HB of the emerging markets and that of the developed markets react differently to the …
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 …
Energy markets are strategic to governments and economic development. Several commodities compete as substitutable energy sources and energy diversifiers. Such competition reduces the energy vulnerability of countries as well as portfolios' risk exposure. Vulnerability results mainly from price trends and fluctuations, following supply and demand shocks. Such energy price uncertainty attracts many market participants in the energy commodity markets. First, energy producers and consumers hedge adverse price changes with energy derivatives. Second, financial market participants use commodities 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 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…
We maximize the expected utility from terminal wealth for an HARA investor when the market price of risk is an unobservable random variable. We compute the optimal portfolio explicitly and explore the effects of learning by comparing it with the corresponding myopic policy. In particular, we show that, for a market price of risk constant in sign, the ratio between the portfolio under partial observation and its myopic counterpart increases with respect to risk tolerance. As a consequence, the absolute value of the partial observation case is larger (smaller) than the myopic one if the investor…
We consider portfolio optimization in futures markets. We model the entire futures price curve at once as a solution of a stochastic partial differential equation. The agents objective is to maximize her utility from the final wealth when investing in futures contracts. We study a class of futures price curve models which admit a finite-dimensional realization. Using this, we recast the portfolio optimization problem as a finite-dimensional control problem and study its solvability.
This paper studies the properties of discrete time stochastic optimal control problems associated with portfolio selection. We investigate if optimal continuous time strategies can be used effectively for a discrete time market after a straightforward discretization. We found that Merton's strategy approximates the performance of the optimal strategy in a discrete time model with the sufficiently small time steps
We study the expected utility portfolio optimization problem in an incomplete financial market where the risky asset dynamics depend on stochastic factors and the portfolio allocation is constrained to lie within a given convex set. We employ fundamental duality results from real constrained optimization to formally derive a dual representation of the associated HJB PDE. Using this representation, we provide a condition on the market dynamics and the allocation constraints, which ensures that the solution to the HJB PDE is exponentially affine and separable. This condition is used to derive an…
Stochastic portfolio theory aims at finding relative arbitrages, i.e. trading strategies which outperform the market with probability one. Functionally generated portfolios, which are deterministic functions of the market weights, are an invaluable tool in doing so. Driven by a practitioner point of view, where investment decisions are based upon consideration of various financial variables, we generalize functionally generated portfolios and allow them to depend on continuous-path semimartingales, in addition to the market weights. By means of examples we demonstrate how the inclusion of addi…
Consider an equity market with $n$ stocks. The vector of proportions of the total market capitalizations that belong to each stock is called the market weight. The market weight defines the market portfolio which is a buy-and-hold portfolio representing the performance of the entire stock market. Consider a function that assigns a portfolio vector to each possible value of the market weight, and we perform self-financing trading using this portfolio function. We study the problem of characterizing functions such that the resulting portfolio will outperform the market portfolio in the long run …
We consider the problem of portfolio optimization in the presence of market impact, and derive optimal liquidation strategies. We discuss in detail the problem of finding the optimal portfolio under Expected Shortfall (ES) in the case of linear market impact. We show that, once market impact is taken into account, a regularized version of the usual optimization problem naturally emerges. We characterize the typical behavior of the optimal liquidation strategies, in the limit of large portfolio sizes, and show how the market impact removes the instability of ES in this context.
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…
This paper describes multi-portfolio `internal' rebalancing processes used in the finance industry. Instead of trading with the market to `externally' rebalance, these internal processes detail how portfolio managers buy and sell between their portfolios to rebalance. We give an overview of currently used internal rebalancing processes, including one known as the `banker' process and another known as the `linear' process. We prove the banker process disadvantages the nominated banker portfolio in volatile markets, while the linear process may advantage or disadvantage portfolios. We describe a…
In this paper, we consider the portfolio optimization problem in a financial market under a general utility function. Empirical results suggest that if a significant market fluctuation occurs, invested wealth tends to have a notable change from its current value. We consider an incomplete stochastic volatility market model, that is driven by both a Brownian motion and a jump process. At first, we obtain a closed-form formula for an approximation to the optimal portfolio in a small-time horizon. This is obtained by finding the associated Hamilton-Jacobi-Bellman integro-differential equation and…
A fractal approach to the long-short portfolio optimization is proposed. The algorithmic system based on the composition of market-neutral spreads into a single entity was considered. The core of the optimization scheme is a fractal walk model of returns, optimizing a risk aversion according to the investment horizon. The covariance matrix of spread returns has been used for the optimization and modified according to the Hurst stability analysis. Out-of-sample performance data has been represented for the space of exchange traded funds in five period time period of observation. The considered …
A geometric analysis of the time series of returns has been performed in the past and it implied that the most of the systematic information of the market is contained in a space of small dimension. Here we have explored subspaces of this space to find out the relative performance of portfolios formed from the companies that have the largest projections in each one of the subspaces. It was found that the best performance portfolios are associated to some of the small eigenvalue subspaces and not to the dominant directions in the distances matrix. This occurs in such a systematic fashion over a…
Almost twenty years ago, E.R. Fernholz introduced portfolio generating functions which can be used to construct a variety of portfolios, solely in the terms of the individual companies' market weights. I. Karatzas and J. Ruf recently developed another methodology for the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. In this paper, both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions are substituted by path-dependent…
We analyze a negative-parameter variant of the diversity-weighted portfolio studied by Fernholz, Karatzas, and Kardaras (Finance Stoch 9(1):1-27, 2005), which invests in each company a fraction of wealth inversely proportional to the company's market weight (the ratio of its capitalization to that of the entire market). We show that this strategy outperforms the market with probability one, under a non-degeneracy assumption on the volatility structure and the assumption that the market weights admit a positive lower bound. Several modifications of this portfolio, which outperform the market un…
Institute for Information Transmission Problems, Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
We present a method for constructing the log-optimal portfolio using the well-calibrated forecasts of market values. Dawid's notion of calibration and the Blackwell approachability theorem are used for computing well-calibrated forecasts. We select a portfolio using this "artificial" probability distribution of market values. Our portfolio performs asymptotically at least as well as any stationary portfolio that redistributes the investment at each round using a continuous function of side information. Unlike in classical mathematical finance theory, no stochastic assumptions are made about ma…
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.
We consider a multi-stock continuous time incomplete market model with random coefficients. We study the investment problem in the class of strategies which do not use direct observations of the appreciation rates of the stocks, but rather use historical stock prices and an a priory given distribution of the appreciation rates. An explicit solution is found for case of power utilities and for a case when the problem can be embedded to a Markovian setting. Some new estimates and filters for the appreciation rates are given.
