QUADRATIC PROGRAMMING: AN OPTIMIZATION TOOL FOR BUILDING GLOBAL MINIMUM VARIANCE PORTFOLIO WITH NO SHORT SALE

Quantitative method in portfolio selection is a fascinating issue to make a decision in investment. Portfolio optimization is a very important to manage investment risk. There are many papers dealing with the Markowitz portfolio model, but not all of the papers studied about positive weight portfolio or no short sale constrained portfolio. Positive weight portfolio describes that short sale is allowed for the investor. While, short sale is banned in a certain economic condition due to its ability in decreasing stock market index. Besides, Islamic capital market does not allow speculative transaction such as short selling. Hence, portfolio with no short sale constraint is needed. This study aims to build Global Minimum Variance Portfolio (GMVP) with no short sale constraint. The GMVP with positive asset allocation based on Markowitz model can be built by using quadratic programming with interior point method. The main theory applied in this research is Markowitz portfolio optimization model. Mean and variance of stocks closing price are two things that should be considered in this model. The result shows that the positive weight of GMVP includes 0% of ADRO shares; 2, 65% of ANTM shares; 0% of CTRA shares; 30,27% of EXCL shares; 37,21% of ICBP shares; 3,37% of INCO shares; 13,89% of KLBF shares; 0% of PGAS shares; and 12,61% of PTBA shares.

Regularized Maximum Diversification Investment Strategy

The maximum diversification has been shown in the literature to depend on the vector of asset volatilities and the inverse of the covariance matrix of the asset return covariance matrix. In practice, these two quantities need to be replaced by their sample statistics. The estimation error associated with the use of these sample statistics may be amplified due to (near) singularity of the covariance matrix, in financial markets with many assets. This, in turn, may lead to the selection of portfolios that are far from the optimal regarding standard portfolio performance measures of the financial market. To address this problem, we investigate three regularization techniques, including the ridge, the spectral cut-off, and the Landweber–Fridman approaches in order to stabilize the inverse of the covariance matrix. These regularization schemes involve a tuning parameter that needs to be chosen. In light of this fact, we propose a data-driven method for selecting the tuning parameter. We show that the selected portfolio by regularization is asymptotically efficient with respect to the diversification ratio. In empirical and Monte Carlo experiments, the resulting regularized rules are compared to several strategies, such as the most diversified portfolio, the target portfolio, the global minimum variance portfolio, and the naive 1/N strategy in terms of in-sample and out-of-sample Sharpe ratio performance, and it is shown that our method yields significant Sharpe ratio improvements.

Portfolio Construction by Using Different Risk Models: A Comparison among Diverse Economic Scenarios

We aim to construct portfolios by employing different risk models and compare their performance in order to understand their appropriateness for effective portfolio management for investors. Mean variance (MV), semi variance (SV), mean absolute deviation (MaD) and conditional value at risk (CVaR) are considered as risk measures. The price data were extracted from the Pakistan stock exchange, Bombay stock exchange and Dhaka stock exchange under diverse economic conditions such as crisis, recovery and growth. We take the average of GDP of the selected period of each country as a cut-off point to make three economic scenarios. We use 40 stocks from the Pakistan stock exchange, 92 stocks from the Bombay stock exchange and 30 stocks from the Dhaka stock exchange. We compute optimal weights using global minimum variance portfolio (GMVP) for all stocks to construct optimal portfolios and analyze the data by using MV, SV, MaD and CVaR models for each subperiod. We find that CVaR (95%) gives better results in each scenario for all three countries and performance of portfolios is inconsistent in different scenarios.

Portfolio Strategies to Track and Outperform a Benchmark

I investigate the question of how to construct a benchmark replicating portfolio consisting of a subset of the benchmark’s components. I consider two approaches: a sequential stepwise regression and another method based on factor models of security returns’ first and second moments. The first approach produces the standard hedge portfolio that has the maximum feasible correlation with the benchmark. The second approach produces weights that are proportional to a “signal-to-noise” ratio of factor beta to idiosyncratic volatility. Using a factor model of securities returns allows the use of a larger number of securities than the number of time periods used to estimate the parameters of the factor model. I also consider a second objective that maximizes expected returns subject to a target tracking error variance. The security selection criterion naturally extends to the product of the information ratio and the signal-to-noise ratio. The optimal tracking portfolio is either a one-fund or a two-fund portfolio rule consisting of the optimal hedging portfolio, the tangent portfolio or the global minimum variance portfolio, depending on what constraints are imposed on the objective function. I construct buy-and-hold replicating portfolios using the algorithms presented in the paper to track a widely followed stock index with very good results both in-sample and out-of-sample.

Optimization of special cryptocurrency portfolios

Purpose This paper aims to elaborate on the optimization of two particular cryptocurrency portfolios in a mean-variance framework. In general, cryptocurrencies can be classified to as coins and tokens where the first can be thought of as a medium of exchange and the latter accounts for security or utility tokens depending upon its design. Design/methodology/approach Against this backdrop, this empirical study distinguishes, in particular, between pure coin and token portfolios. Both portfolios are optimized by maximizing the Sharpe ratio and, subsequently, compared with alternative portfolio strategies. Findings The empirical findings demonstrate that the maximum utility portfolio of coins, with a risk aversion of λ = 10, outweighs alternative frameworks. The portfolios optimized by maximizing the Sharpe ratio for both coins and tokens indicate a rather poor performance. Testing the maximized utility for different levels of risk aversion confirms the findings of this empirical study and confers them more robustness. Research limitations/implications Further investigation is strongly recommended as tokens represent a new phenomenon in the cryptocurrency universe, for which only a limited amount of data are available, which restricts the sampling. Furthermore, future study is to include more sophisticated optimization models using different constraints in portfolio creation. Practical implications In light of the persistently substantial volatility in cryptocurrency markets, the empirical findings assert that portfolio managers are advised to construct a global minimum variance portfolio. In the absence of sophisticated optimization models, private investors can invest according to the market values of cryptocurrencies. Despite minor differences in the risk and reward ratios of the portfolios tested, tokens tend to be more speculative, especially, if the Tether token is excluded, which may require enhanced supervision and investor protection by regulating authorities. Originality/value As the current literature investigates on diversification effects of blended cryptocurrency portfolios rather than making an explicit distinction, this paper reflects one of the first to explore the investability and role of diversifying coins and tokens using a classic Markowitz approach.

Properties of the beta coefficient of the global minimum variance portfolio

The paper is devoted to the investigation of statistical properties of the sample estimator of the beta coefficient in the case when the weights of benchmark portfolio are constant and for the target portfolio, the global minimum variance portfolio is taken. We provide the asymptotic distribution of the sample estimator of the beta coefficient assuming that the asset returns are multivariate normally distributed. Based on the asymptotic distribution we construct the confidence interval for the beta coefficient. We use the daily returns on the assets included in the DAX index for the period from 01.01.2018 to 30.09.2019 to compare empirical and asymptotic means, variances and densities of the standardized estimator for the beta coefficient. We obtain that the bias of the sample estimator converges to zero very slowly for a large number of assets in the portfolio. We present the adjusted estimator of the beta coefficient for which convergence of the empirical variances to the asymptotic ones is not significantly slower than for a sample estimator but the bias of the adjusted estimator is significantly smaller.