A linear programming model for selection of sparse high-dimensional multiperiod portfolios

Any given set of asset parameters yields a specific mean–variance optimal tangency portfolio. Yet, when the number of assets is large, there are some general characteristics of optimal portfolios that hold ‘almost surely’. This paper investigates these characteristics. We analytically show that the proportion of assets held short converges to 50% as the number of assets grows. This is a fundamental and robust property of mean–variance optimal portfolios, and it does not depend on the parameter estimation method, the investment horizon, or on a special covariance structure. While it is known that optimal portfolios may all have positive weights in some special situations (e.g. uncorrelated assets), the analysis shows that these cases occupy a zero measure in the parameter space, and therefore should not be expected to be observed empirically. Thus, our analysis offers a general explanation for the empirical finding of many short positions in optimal portfolios.

Summary This study demonstrates how portfolio insights can be created by combining well-known optimization methods that are generally used individually—genetic algorithm (GA), linear programming (LP), and portfolio filtering (PF). An integrated optimization approach combines the advantages of individual methods while mitigating their shortcomings. Effective portfolio management requires a comprehensive understanding of the tradeoffs between different portfolio choices. Given a set of constraints, portfolio-optimization techniques based on LP and GAs can be applied to identify an optimal portfolio. However, this optimal portfolio might not be the preferred portfolio. Decision makers have to understand the tradeoffs between generally conflicting objectives and constraints before one portfolio can be identified as the preferred option. Such assessment of the overall search space is not made with LP and GAs when used to identify a single best solution, and many portfolio options will have been eliminated before an understanding of these alternatives has been developed. Markowitz's mean-variance (M-V) approach and the traditional “rank and cut” approach are used typically to establish a relationship between a portfolio's value and its variance or associated development cost. Although these methods enable decision makers to compare and contrast different options, the optimization is limited to the portfolio value measure and a single other metric. This latter limitation is overcome by the more recently developed PF approach. This method is practical and transparent and allows for a quick development of strategic portfolio alternatives while considering a large number of portfolio attributes. Its main drawback is that the analyzed set of portfolios generally represents a subset of the total search space. Thus, as the number of feasible portfolio options increases, so does the chance that the optimal portfolio is not present in the population of sampled portfolios.

Mean-variance portfolio optimization is widely used by financial professionals as a fundamental strategy for constructing portfolios that achieve the highest returns for a given degree of risk tolerance. This traditional approach suffers from computational instability issues, which are often solved by integrating regularization-based methods. However, regularized mean-variance models overlook sector considerations, emphasizing a need for market-informed portfolio optimization. In this paper, we propose an extended mean-variance portfolio selection framework that incorporates a new adaptive sparse-group least absolute shrinkage and selection operator (LASSO) regularization as a penalty parameter. In the proposed model, the sparse optimal portfolio is selected considering both sectors and assets simultaneously. Moreover, the inclusion of adaptive weights in the regularization, estimated based on chosen criteria, enables the use of more market data in a Markowitz portfolio model. We also develop an efficient alternating direction method of multipliers (ADMM) algorithm that guarantees convergence, for finding the optimal sparse portfolio. The effectiveness of the suggested portfolio selection model is validated by numerical results derived from the constituents of the Standard and Poor’s 500. These results demonstrate superior performance over our selected benchmark models across various evaluation measurements.

This paper incorporates Bayesian estimation and optimization into a portfolio selection framework, particularly for high-dimensional portfolios in which the number of assets is larger than the number of observations. We leverage a constrained minimization approach, called the linear programming optimal (LPO) portfolio, to directly estimate effective parameters appearing in the optimal portfolio. We propose two refinements for the LPO strategy. First, we explore improved Bayesian estimates, instead of sample estimates, of the covariance matrix of asset returns. Second, we introduce Bayesian optimization (BO) to replace traditional grid-search cross-validation (CV) in tuning hyperparameters of the LPO strategy. We further propose modifications in the BO algorithm by (1) taking into account the time-dependent nature of financial problems and (2) extending the commonly used expected improvement acquisition function to include a tunable trade-off with the improvement’s variance. Allowing a general case of noisy observations, we theoretically derive the sublinear convergence rate of BO under the newly proposed EIVar and thus our algorithm has no regret. Our empirical studies confirm that the adjusted BO results in portfolios with higher out-of-sample Sharpe ratio, certainty equivalent, and lower turnover compared to those tuned with CV. This superior performance is achieved with a significant reduction in time elapsed, thus also addressing time-consuming issues of CV. Furthermore, LPO with Bayesian estimates outperforms the original proposal of LPO, as well as the benchmark equally weighted and plugin strategies.

The mean-variance method developed by Markowitz (1959) was aimed at obtaining optimizing portfolios. But selection of portfolio in the real world mostly deviates from this optimal criterion. In this paper we have considered this issue from an altogether different aspect and developed means for aiming at nearly optimum portfolio. We considered the risk taking propensity as the main driving force and presented a heuristic method to reach the near to the optimal state. For doing so, we have introduced the coefficient of optimism in the decision making process and simultaneously considered conditional optimum portfolio and corresponding heuristic portfolio. In the extreme situations three different human value systems can be considered as optimistic, pessimistic and risk planner. To examine the closeness between the heuristic and optimum portfolios we have carried out empirical analysis covering ten years data of fifteen companies from Nifty (2000-09). Regarding the choice of companies we have adopted random selection technique. From empirical study we have found that for moderate values of the coefficient of optimism a heuristic investor’s decision nearly coincides with the corresponding optimum portfolio. However, for extreme situations i.e. optimistic and pessimistic situations heuristic portfolio differs from optimum portfolio. Keywords: Expected return, risk, optimum portfolio, heuristic portfolio, coefficient of optimism.

In this paper, we consider the problem of an optimal pension fund portfolio given the heterogeneous risk preferences of pension fund participants. The relative risk aversion of a pension fund tends to be a decreasing function of the level of aggregate wealth. We find that the dynamic optimal portfolio is simply characterized as the weighted sum of the optimal portfolio for each participant. Our model helps successfully establish the microfoundation of asset liability management models. A numerical example using recent Japanese data indicates the significant total welfare losses of adopting a suboptimal portfolio strategy and an inefficient risk‐sharing rule.

Effects of Price of Gold on Bombay Stock Exchange Sectoral Indices: New Evidence for Portfolio Risk Management

Portfolio Selection with Inflation-linked Bonds and Indexation Lags

Portfolio selection with inflation-linked bonds and indexation lags

We introduce a novel framework for the portfolio selection problem in which investors aim to target a return distribution, and the optimal portfolio has a return distribution as close as possible to the targeted one. The proposed framework can be applied to a variety of investment objectives. In this paper, we focus on improving the higher moments of mean-variance-efficient portfolios by designing the target so that its first two moments match those of the chosen efficient portfolio but has more desirable higher moments. We show theoretically that the optimal portfolio is in general different from the mean-variance portfolio, but remains mean-variance efficient when asset returns are Gaussian. Otherwise, it can move away from the efficient frontier to better match the higher moments of the target distribution. An extensive empirical analysis using three characteristic-sorted datasets and a dataset of 100 individual stocks indicates that the proposed framework delivers a satisfying compromise between mean-variance efficiency and improved higher moments.

Optimal portfolios with regime switching and value-at-risk constraint

Portfolio construction is an important practical problem in finance. In the traditional approach, introduced by Markowitz, one assumes normally distributed returns and constructs a portfolio with a minimum risk (measured by the standard deviation of portfolio returns) for a specified (and minimally acceptable) return.In practice, returns are not normally distributed and have heavy tails. As a result, the normality assumption severely underestimates risk. It has been long suggested that a more appropriate way is to model returns by using alpha-stable distributions.In this paper, we use elliptical stable distributions for optimal stable portfolio construction. We illustrate this by considering portfolios from S&P 500 sector exchange-traded funds (ETFs).Our main results indicate that stable portfolios are in general comparable with standard Markowitz portfolios. But we discovered a few properties of stable distributions that are promising for optimal portfolio selection problem. Stable risk estimation for next year is much closer to the actual portfolio risk, stable portfolios are more balanced, so they better handle maximum restriction on component weights, and stable portfolios are more resilient to extreme market conditions.Both stable and normal optimal portfolios outperform S&P 500 index and equally-weighted ETF portfolio in most years.We propose an investment strategy that uses both stable and normal distributions for building optimal portfolios. The return of this strategy exceeds the return of the strategies which use only stable or only normal distributions.

The question how to optimize consumption and portfolio choice over the life cycle has been widely discussed in the literature so far. In this paper we concentrate on a retiree's optimal consumption and portfolio selection over his remaining years of life. We apply the logistic model of mortality thus modeling the empirically observed increase of mortality during the retirement period. The optimal consumption strategy and portfolio choice are established by reducing the Hamilton-Jacobi-Bellmann equation to the explicit solution of an ordinary differential function (ODF) that includes the mortality rate. A general finding is that the Merton-Samuelson result of constant portfolio choice for a constant mortality is confirmed for arbitrary mortality. The portfolio choice is only influenced by risk and return of assets and the retirees’ risk aversion. To get the specific optimal consumption strategy in a realistic situation the logistic model of mortality has been fitted to the data of the Statistical Yearbook for the Federal Republic of Germany 2006/2008. The optimal initial value for the ODF is obtained by numerical methods. The solution provides a large increase in the ratio of optimal consumption to wealth up to about 92 years followed by a sharp decrease. A bequest motive dampens the magnitudes of the ups and downs of the consumption ratio but does not change the basic shape.

In this paper we derive a closed-form solution for a representative investor who optimally allocates her wealth among the following securities: a credit-risky asset, a default-free bank account, and a stock. Although the inclusion of a credit-related financial product in the portfolio selection is more realistic, no closed-form solutions to date are given in the literature when a recovery value is considered in the event of a default. While most authors have assumed some recovery scheme in their initial model set up, they do not address the portfolio problem with a recovery when a default actually occurs. Given the tractability of the recovery of market value, we solved the optimal portfolio problem for the representative investor whose utility function is a Constant Relative Risk Aversion utility function.

A constrained portfolio selection model at considering risk-adjusted measure by using hybrid meta-heuristic algorithms