NORMALIZING RISK MEASURES IN RISK-BASED PORTFOLIOS THROUGH COVARIANCE MISSPECIFICATION ERROR ANALYSIS

Portfolio weights solely based on risk avoid estimation errors from the sample mean, but they are still affected from the misspecification in the sample covariance matrix. To solve this problem, we shrink the covariance matrix towards the Identity, the Variance Identity, the Single-index model, the Common Covariance, the Constant Correlation, and the Exponential Weighted Moving Average target matrices. Using an extensive Monte Carlo simulation, we offer a comparative study of these target estimators, testing their ability in reproducing the true portfolio weights. We control for the dataset dimensionality and the shrinkage intensity in the Minimum Variance (MV), Inverse Volatility (IV), Equal-Risk-Contribution (ERC), and Maximum Diversification (MD) portfolios. We find out that the Identity and Variance Identity have very good statistical properties, also being well conditioned in high-dimensional datasets. In addition, these two models are the best target towards which to shrink: they minimise the misspecification in risk-based portfolio weights, generating estimates very close to the population values. Overall, shrinking the sample covariance matrix helps to reduce weight misspecification, especially in the Minimum Variance and the Maximum Diversification portfolios. The Inverse Volatility and the Equal-Risk-Contribution portfolios are less sensitive to covariance misspecification and so benefit less from shrinkage.

In this article we extend the research on risk-based asset allocation strategies by exploring how using an SRI universe impacts the properties of risk-based portfolios. We focus on four risk-based asset allocation strategies: the Equally Weighted (EW), the Most Diversified Portfolio (MDP), the Minimum Variance (MV) and the Equal Risk Contribution (ERC). Using different estimators of the matrix of covariances, we apply these strategies to the EuroStoxx universe of stocks, the Advanced Sustainability Performance Index (ASPI) and the complement of the ASPI in the EuroStoxx universe from March 15, 2002 to May 1, 2012. We observe several impacts but one is particularly important in our mind. We observe that risk-based asset allocation strategies built on the entire universe, concentrate their solution on non-SRI stocks. Such risk-based portfolio is therefore under-weighted in socially responsible firms.

We introduce a new specification of the dynamic conditional correlation (DCC) model, where its parameters are estimated with the use of closing and additionally low and high prices. Such prices are often commonly available for many financial series and contain more information about the variation of returns. We construct the model with the range-based estimator of variance but more importantly also with the range-based estimator of covariance. The latter estimator and as a consequence the proposed DCC model require, however, that the range of a portfolio return is given. We compare the model with three other specifications of the DCC models and evaluate them based on Monte Carlo experiments and currencies rates from the Forex market. We show that the use of low and high prices can improve estimation of covariance and correlation matrices of returns and increase the accuracy of forecasts of covariance and correlation matrices based on this model, compared with using closing prices only. The proposed model is superior not only to the standard DCC model, but also to the competing range-based DCC model.

In the field of portfolio management, practitioners are focusing increasingly on risk-based portfolios rather than on mean-variance portfolios. Risk-based portfolios are constructed based solely on covariance matrices, and include methods such as minimum variance (MV), risk parity (RP), and maximum diversification (MD). It is well known that the performance of a mean-variance portfolio depends on the accuracy of the estimations of the inputs. However, no studies have examined the relationship between the performance of risk-based portfolios and the estimated accuracy of covariance matrices. In this research, we compare the performance of risk-based portfolios for several estimation methods of covariance matrices in the Japanese stock market. In addition, we propose a highly accurate estimation method called cDCC-NLS, which incorporates nonlinear shrinkage into the cDCC-GARCH model. The results confirm that (1) the cDCC-NLS method shows the best estimation accuracy, (2) the RP and MD do not depend on the estimation accuracy of the covariance matrix, and (3) the MV does depend on the estimation accuracy of the covariance matrix.

Summary The primary purpose of this paper is to propose empirical measures of the structural differences between two communities of plants or animals composed of the same species. Structure is defined to consist of; 1) the species in the community, 2) the pattern of interactions as represented by the covariance or correlation matrix of successive observations on each species, and 3) the mean abundances of each species in each of the two communities. Statistical tests are proposed for testing whether the covariance matrices and the vectors of mean densities for each community are equal and empirical measures of the differences between the covariance matrices and mean vectors are proposed. Given unequal covariance or correlation matrices the factor analysis model is used to derive empirical measures of the degree to which each variable of the ecosystem is responsible for the observed defferences in the pattern of interactions in each community. These tests and measures were applied to data gathered by Hunter (1966) on the abundances of six species of Drosophila censused monthly over a period of approximately two and a half years in two adjacent, but different habitats near Bogota, colombia. The two covariance matrices were significantly different indicating different patterns of interactions in the two Drosophila communities.

The authors advocate the use of agnostic allocation for the construction of long-only portfolios of stocks. 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 classic 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. TOPICS:Portfolio theory, portfolio construction, risk management Key Findings • Long-only risk-based portfolio construction methods that rely on the inverse covariance matrix, such as minimum variance and maximum diversification, are plagued by overconcentration, excess turnover, and exposure to the low-risk factor. • Agnostic allocation portfolios (AAPs) deliver a compromise between risk-based portfolios that are structurally blind to the correlation structure and those that invert the covariance matrix of returns. They significantly reduce turnover and concentration and achieve performance similar to or better than that of traditional methods. • AAPs are supplemented with a pruning technique that discards unnecessary exposures to low-significance statistical factors. This technique, combined with the use of adequately cleaned covariance matrices, improves the outcome of AAPs by further reducing excess trading and concentration.

A key concept in risk and portfolio management is diversification, which has recently come under intense scrutiny following dramatic market movements where historical correlation patterns have collapsed. To counter the devastating effects from these coordinated sell-off events, academics have developed several models to forecast correlation. This article examines two recent dynamic conditional correlation models: an asymmetric dynamic conditional correlation (ADCC) model and a dynamic equicorrelation (DECO) model. Previous research has demonstrated the merits of these advanced models in an artificial setting, but a study in a high-dimensional, though still artificial, setting was inconclusive on the merits. This article casts doubt on the merits of these models in a real-world setting. The results show that neither the ADCC model nor the DECO model provides any additional information over a sample-based approach when evaluated using a simple parametric covariance-variance value at risk (VaR) measurement. Used in risk monitoring, the models provide sound risk estimates but no additional insights into forecasting correlation spikes.

When observed stock returns are obtained from trades subject to friction, it is known that an individual stock's beta and covariance are measured with error. Univariate models of additive error adjustment are available and are often applied simultaneously to more than one stock. Unfortunately, these multivariate adjustments produce nonpositive definite covariance and correlation matrices, unless the return sample sizes are very large. To prevent this, restrictions on the adjustment matrix are developed and a correction is proposed, which dominates the uncorrected estimator. The estimators are illustrated with asset opportunity set estimates where daily returns have trading frictions. A NUMBER OF UNIVARIATE nonsynchronous trading adjustments are available for beta, but no multivariate adjustment has been specifically designed for a general covariance matrix. For example, the univariate trading friction adjustment of Cohen, Hawawini, Maier, Schwartz, and Whitcomb (CHMSW) (1983) is a generalization of Scholes and Williams (1977). The CHMSW procedure adjusts beta to account for price-adjustment delays exceeding one period, where a portion of the true return is reflected in lagged returns. The procedure is extended and applied in Shanken (1987) to estimation of the daily multivariate covariance matrix of stocks' returns. Observed covariances significantly understated true covariances by a factor of two, suggesting that a daily covariance adjustment is necessary. Because of a large number of studies which employ multivariate daily stock returns, it is therefore important that the adjustment results in a good estimator of the covariance matrix. While no evidence is available regarding other properties, it is shown in CHMSW that their univariate adjusted estimator has the desirable statistical properties of consistency and unbiasedness. Unfortunately, when the adjusted covariance estimator is applied in finite samples, the resulting adjusted covariance matrix may not be a covariance matrix and the implied adjusted correlation matrix, independently, may not be a correlation matrix because the adjustment does not restrict the covariance and correlation matrices to be positive definite.1 When these matrices are not positive definite, functions of these matrices can produce very unreliable inference results. This paper provides empirical examples of the violations of the positive definite

This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one. Depending on the case of applications, sparsity priori may occur on the covariance matrix, its inverse or its Cholesky decomposition. We study these three sparsity exploration problems under a unified framework with a general penalty function. We show that the rates of convergence for these problems under the Frobenius norm are of order (s(n) log p(n)/n)(1/2), where s(n) is the number of nonzero elements, p(n) is the size of the covariance matrix and n is the sample size. This explicitly spells out the contribution of high-dimensionality is merely of a logarithmic factor. The conditions on the rate with which the tuning parameter λ(n) goes to 0 have been made explicit and compared under different penalties. As a result, for the L(1)-penalty, to guarantee the sparsistency and optimal rate of convergence, the number of nonzero elements should be small: sn'=O(pn) at most, among O(pn2) parameters, for estimating sparse covariance or correlation matrix, sparse precision or inverse correlation matrix or sparse Cholesky factor, where sn' is the number of the nonzero elements on the off-diagonal entries. On the other hand, using the SCAD or hard-thresholding penalty functions, there is no such a restriction.

This chapter introduces several recent developments for estimating large covariance and precision matrices without assuming the covariance matrix to be sparse. It explains two methods for covariance estimation: namely covariance estimation via factor analysis, and precision Matrix Estimation and Graphical Models. The low rank plus sparse representation holds on the population covariance matrix. The chapter presents several applications of these methods, including graph estimation for gene expression data, and several financial applications. It then shows how estimating covariance matrices of high-dimensional asset excess returns play a central role in applications of portfolio allocations and in risk management. The chapter explains the factor pricing model, which is one of the most fundamental results in finance. It elucidates estimating risks of large portfolios and large panel test of factor pricing models. The chapter illustrates the recent developments of efficient estimations in panel data models.

Boundary estimates of random effects covariance matrices commonly arise when using maximum likelihood (ML) estimation in generalized linear mixed effects models, leading to numerical challenges and affecting statistical inference. To mitigate this, we introduce penalties to the likelihood function derived from conditionally conjugate priors for the covariance or precision matrices of the random effects. Our choice of penalties (priors) allows representation through pseudo-observations, enabling implementation of the proposed penalized estimator within the existing ML software by augmenting the original data. We derive a procedure for constructing these pseudo-observations, a non-trivial task because their likelihood contribution must match the functional form of the penalty and depend only on the covariance or precision matrix of the random effects. Our method includes penalty parameters that can be set using existing prior knowledge or, when no reliable prior information is available, via a novel fully data-driven procedure that eliminates the need for prior specification. Through simulation studies under realistic scenarios, we illustrate that the proposed approach can provide improved estimates of random-effects covariance matrices compared with competing methods in the settings considered. The approach is further illustrated on real-world data.

Accurate error covariance is crucial for postprocessing gravity recovery and climate experiment (GRACE) gravity field solutions in terms of spherical harmonic coefficients (SHCs). Unfortunately, most GRACE SHC products only provide formal errors of SHCs due to the large storage requirements of covariance matrices. A covariance matrix can be decomposed into a diagonal matrix and an orthogonal matrix. The orthogonal matrix of a monthly GRACE covariance matrix relates to the monthly repeated ground coverage of the GRACE orbit and remains similar across all monthly covariance matrices. Therefore, we propose a semiparametric approach for reconstructing monthly covariance matrices using a common orthogonal matrix and monthly individual formal errors. Covariance matrices of Tongji-Grace2018 and GFZ RL06 SHC products from April 2002 to December 2016 serve as training data to derive common orthogonal matrices and the model for mapping monthly individual formal errors to diagonal matrices, which are utilized to reconstruct covariance matrices of ITSG-Grace2018, CSR RL06, and JPL RL06 SHC products in this study. The results filtered with the reconstructed covariance matrices closely match with those filtered with actual covariance matrices of ITSG-Grace2018 SHC product and those of CSR and JPL mascon products in global spectral filtering and regional point-mass modeling estimates, significantly better than those estimated using corresponding formal errors. Besides, data storage space for reconstructing covariance matrices is reduced by 98.6% compared to the original matrices. Simulation experiments with ITSG-Grace2018 SHC products demonstrate that the root-mean-square-errors of filtered SHCs and global terrestrial water storage anomalies using reconstructed covariance matrices have minimal differences of 3.8% and 2.7% relative to those using actual covariance matrices, and the errors are significantly reduced by 18.2% and 10.8% compared to those only using formal errors.

Regularized semiparametric estimation of high dimensional dynamic conditional covariance matrices

ABSTRACTThis article examines the economic benefit of using the realized covariance matrix forecasts, for constructing the risk-based portfolios. We use the two-scale realized covariance estimator (TSC), the jump robust two-scale realized covariance estimator (RTSC) and the realized bipower covariance estimator (BPC), to forecast the daily realized covariance matrix. Using these covariance matrix forecasts, we implement three risk-based portfolios: the global minimum variance portfolio, the equal risk contribution portfolio and the most diversified portfolio. There is evidence that the portfolio performance improves by using TSC or RTSC estimators as compared to the daily-returns-based estimator. The performance gains are robust to the choice of risk-based portfolio strategy, the degree of investor’s relative risk-aversion, the market conditions and the choice of time intervals.

This study adapts a variety of techniques derived from multi-spectral image classification to find objects amid cluttered backgrounds in hyperspectral imagery. This study quantitatively compares the algorithms against a standard object search, the matched filter (MF) and recently developed object detector, Adaptive Cosine Estimator (ACE). These object searches require calculating the Mahalanobis distance between the average object spectral signature and the test pixel spectrum and needs the computation of a covariance matrix. The covariance matrix is generated using the entire image (Whitened Euclidean Distance, WED) or using pixels associated with the object (Maximum Likelihood Classifier, MLC). The latter computation requires a relatively large number of pixels to generate a non-singular, accurate covariance matrix. Regularizing object pixels via optimally mixing (likelihood maximization) diagonal, object, and entire image covariance matrices to generate the object covariance matrix estimate. This approximation is called the Regularized Maximum Likelihood Classifier (RMLC). The object searches MF, ACE, WED, MLC, and RMLC were applied to visible/near IR data collected from forest and desert environments. This study searched for objects using object signatures and covariance matrices taken directly from the scene and from statistically transformed object signatures and covariance matrices from another time. This study found a substantial reduction in the number of false alarms (factor of 10 to 1000) using WED, ACE, RMLC relative to MF searches for the two independent data collects. The regularization of in-scene and transformed covariance matrices substantially reduced false alarms relative to using unprocessed covariance matrices. This study adds simple, high performing algorithms to the object search arsenal.