Factor state–space models for high-dimensional realized covariance matrices of asset returns

We propose a dynamic factor state–space model for high-dimensional covariance matrices of asset returns. It makes use of observed risk factors and assumes that the latent integrated joint covariance matrix of the assets and the factors is observed through their realized covariance matrix with a Wishart measurement density. For the latent integrated covariance matrix of the assets we impose a strict factor structure allowing for dynamic variation in the covariance matrices of the factors and the residual components as well as in the factor loadings. This factor structure translates into a factorization of the Wishart measurement density which facilitates statistical inference based on simple Bayesian MCMC procedures making the approach scalable w.r.t. the number of assets. An empirical application to realized covariance matrices for 60 NYSE traded stocks using the Fama–French factors and sector-specific factors represented by Exchange Traded Funds (ETFs) shows that the model performs very well in- and out of sample.     

The selection of multinational equity portfolios: forecasting models and estimation risk

The problem considered is the selection of a portfolio of international assets, particularly the forecasting of the inputs to a selection algorithm. Four models of the asset return generating process are considered, two of which ignore the international nature of the universe of assets, two which exploit it in different ways. Several estimation methods are considered for each component: expected return, variance and covariance of returns. The combinations of model and estimation method are first evaluated in terms of their forecasting performance for the components mentioned for the individual assets. The universe used is the components of the Financial Times Eurotrack 100 Index. Significant differences were found between the forecasting accuracy of the methods considered for each component. In the final stage of the analysis, a comparison of the returns on portfolios chosen using each combination showed a significant difference. The analysis suggests that the choice of estimation method is more critical than the choice of pricing model.     

High-dimensional covariance forecasting for short intra-day horizons

Asset return covariances at intra-day horizons are known to tend towards zero due to market microstructure effects. Thus, traders who simply scale their daily covariance forecast to match their trading horizon are likely to over-estimate the actual experienced asset dependence. In this paper, some of the key challenges are discussed that are encountered when forecasting high-dimensional covariance matrices for short intra-day horizons. Based on a novel evaluation methodology, and extensive empirical analysis, specific recommendations are made regarding model design and data sampling.     

Online learning of time-varying stochastic factor structure by variational sequential Bayesian factor analysis

Investment tasks include forecasting volatilities and correlations of assets and portfolios. One of the tools widely utilized is stochastic factor analysis on a set of correlated time-series (e.g. asset returns). Published time-series factor models require either sufficiently wide time windows of observed data or numeric solutions by simulations. We developed a ‘variational sequential Bayesian factor analysis’ (VSBFA) algorithm to make online learning of time-varying stochastic factor structure. The VSBFA is an analytic filter to estimate unknown factor scores, factor loadings and residual variances. The covariance matrix of the time-series predicted by the VSBFA can be decomposed into loadings-based covariance and specific variances, and the former can be expressed by ‘explanatory factors’ such as systematic components of various financial market indices. We compared the VSBFA with the most practiced factor model relying on wide data windows, the rolling PCA (principal components analysis), by applying them to 9-year daily returns of 200 simulated stocks with the ‘true’ daily data-generating model completely known, and by using them to forecast volatilities of long-only and long/short global stock portfolios with 25-year monthly returns of more than 800 stocks worldwide. Accuracy of the forecast covariance matrices is measured by a (symmetrized) Kullback–Leibler distance, and accuracy of the forecast portfolio volatilities is measured by bias statistic, log-likelihood, Q-statistic, and portfolio volatility minimization. The factor-based covariance and specific variances predicted by the best VSBFA are significantly more accurate than those by the best rolling PCA.     

Covariance complexity and rates of return on assets

This paper considers the estimation of the expected rate of return on a set of risky assets. The approach to estimation focuses on the covariance matrix for the returns. The structure in the covariance matrix determines shared information which is useful in estimating the mean return for each asset. An empirical Bayes estimator is developed using the covariance structure of the returns distribution. The estimator is an improvement on the maximum likelihood and Bayes–Stein estimators in terms of mean squared error. The effect of reduced estimation error on accumulated wealth is analyzed for the portfolio choice model with constant relative risk aversion utility.     

Orthogonal GARCH and covariance matrix forecasting: The Nordic stock markets during the Asian financial crisis 1997–1998

In risk management, modelling large numbers of assets and their variances and covariances in a unified framework is often important. In such multivariate frameworks, it is difficult to incorporate GARCH models and thus a new member of the ARCH-family, Orthogonal GARCH, has been suggested as a remedy to inherent estimation problems in multivariate ARCH modelling. Orthogonal GARCH creates positive definite covariance matrices of any size but builds on assumptions that partly break down during stress scenarios. This article therefore assesses the stress performance of the model by looking at four Nordic stock indices and covariance matrix forecasts during the highly volatile years of 1997 and 1998. Overall, Orthogonal GARCH is found to perform significantly better than traditional historical variance and moving average methods. Out-of-sample evaluation measures include symmetric loss functions (RMSE), asymmetric loss functions, operational methods suggested by the Basle Committee on Banking Supervision, as well as a forecast evaluation methodology based on pricing of simulated ‘rainbow options’.     

Expected stock returns,common idiosyncratic volatility and average idiosyncratic correlation

Motivated by Herskovic et al. (2016), we examine the role of the average idiosyncratic correlation (ICOR) in two types of markets: an emerging market and a developed market. Examining daily stock data from the Chinese stock market for the period 1995 to 2020 and from the US for the period 1926 to 2019, we adopt high-dimensional principal component analysis (PCA) and thresholding methods to re-estimate ICOR. We find that ICOR plays an important role in explaining the expected stock returns, as the common idiosyncratic volatility (CIV) does in Herskovic et al. (2016). ICOR has been neglected in the literature due to large estimation error in the idiosyncratic covariance matrix and our analysis provides evidence that ICOR is nonnegligible in both markets when we control for several common market factors. We show that the average idiosyncratic covariance, which is the numerator of ICOR, exhibits the same pattern as CIV. Furthermore, our regression analyses of expected stock returns in response to ICOR change in both markets show that, in contrast to the negative result for CIV, the stocks’ high risk exposure to ICOR change comes with a higher risk premium, perhaps because of the synchronized but disproportionate changes in the monthly idiosyncratic covariance and idiosyncratic volatility.     

Estimating a covariance matrix for market risk management and the case of credit default swaps

We analyze covariance matrix estimation from the perspective of market risk management, where the goal is to obtain accurate estimates of portfolio risk across essentially all portfolios—even those with small standard deviations. We propose a simple but effective visualisation tool to assess bias across a wide range of portfolios. We employ a portfolio perspective to determine covariance matrix loss functions particularly suitable for market risk management. Proper regularisation of the covariance matrix estimate significantly improves performance. These methods are applied to credit default swaps, for which covariance matrices are used to set portfolio margin requirements for central clearing. Among the methods we test, the graphical lasso estimator performs particularly well. The graphical lasso and a hierarchical clustering estimator also yield economically meaningful representations of market structure through a graphical model and a hierarchy, respectively.     

Estimating correlation and covariance matrices by weighting of market similarity

We discuss a weighted estimation of correlation and covariance matrices from historical financial data. To this end, we introduce a weighting scheme that accounts for the similarity of previous market conditions to the present situation. The resulting estimators are less biased and show lower variance than either unweighted or exponentially weighted estimators. The weighting scheme is based on a similarity measure that compares the current correlation structure of the market to the structures at past times. Similarity is then measured by the matrix 2-norm of the difference of probe correlation matrices estimated for two different points in time. The method is validated in a simulation study and tested empirically in the context of mean–variance portfolio optimization. In the latter case we find an enhanced realized portfolio return as well as a reduced portfolio risk compared with alternative approaches based on different strategies and estimators.     

Market-wide illiquidity and the distribution of non-parametric stochastic discount factors

Employing out-of-sample non-parametric estimation techniques, we show that market-wide liquidity risk matters for asset pricing independently of the specific functional form of the stochastic discount factor (SDF) and, therefore, of the asset pricing model specification. Market-wide illiquidity significantly affects the distribution of the SDF. Specifically, it boosts up the volatility of the SDF, causing minor effects on higher moments of its distribution. This finding is robust to the use of different sets of test assets in the estimation of the SDF, including equity and corporate bond portfolios, and the use of a high-dimensional data estimation procedure.     

Forecasting time-varying covariance with a range-based dynamic conditional correlation model

This paper proposes a range-based dynamic conditional correlation (DCC) model combined by the return-based DCC model and the conditional autoregressive range (CARR) model. The substantial gain in efficiency of volatility estimation can boost the accuracy for estimating time-varying covariances. As to the empirical study, we use the S&P 500 stock index and the 10-year treasury bond futures to examine both in-sample and out-of-sample results for six models, including MA100, EWMA_, CCC_, BEKK, return-based DCC, and range-based DCC. Of all the models considered, the range-based DCC model is largely supported in estimating and forecasting the covariance matrices.     

The Effect of Serial Dependence on Multiperiod Holding Period Return Performance

The impact of the investment time horizon on risk‐return properties of asset returns depends on the presence of serial correlation and higher order serial dependencies. We present a methodology for decomposing multiperiod holding period return covariance into serial and cross‐sectional components using a recursive multiplicative model that captures the effects of serial and cross‐sectional dependencies and their joint effects without requiring a distributional form assumption. Applying this model to historical monthly return series for commonly held financial assets and portfolios of assets, we investigate the significance of the investment time horizon, the existence and relevance of time diversification, the inflation‐hedging effectiveness of different assets, and the appropriateness of applying traditional capital market theory in a multiperiod framework.     

Parameter uncertainty in portfolio selection: Shrinking the inverse covariance matrix

The estimation of the inverse covariance matrix plays a crucial role in optimal portfolio choice. We propose a new estimation framework that focuses on enhancing portfolio performance. The framework applies the statistical methodology of shrinkage directly to the inverse covariance matrix using two non-parametric methods. The first minimises the out-of-sample portfolio variance while the second aims to increase out-of-sample risk-adjusted returns. We apply the resulting estimators to compute the minimum variance portfolio weights and obtain a set of new portfolio strategies. These strategies have an intuitive form which allows us to extend our framework to account for short-sale constraints, transaction costs and singular covariance matrices. A comparative empirical analysis against several strategies from the literature shows that the new strategies often offer higher risk-adjusted returns and lower levels of risk.     

Bayesian realized-GARCH models for financial tail risk forecasting incorporating the two-sided Weibull distribution

The realized-GARCH framework is extended to incorporate the two-sided Weibull distribution, for the purpose of volatility and tail risk forecasting in a financial time series. Further, the realized range, as a competitor for realized variance or daily returns, is employed as the realized measure in the realized-GARCH framework. Sub-sampling and scaling methods are applied to both the realized range and realized variance, to help deal with inherent micro-structure noise and inefficiency. A Bayesian Markov Chain Monte Carlo (MCMC) method is adapted and employed for estimation and forecasting, while various MCMC efficiency and convergence measures are employed to assess the validity of the method. In addition, the properties of the MCMC estimator are assessed and compared with maximum likelihood, via a simulation study. Compared to a range of well-known parametric GARCH and realized-GARCH models, tail risk forecasting results across seven market indices, as well as two individual assets, clearly favour the proposed realized-GARCH model incorporating the two-sided Weibull distribution; especially those employing the sub-sampled realized variance and sub-sampled realized range.     

Disentangling the role of variance and covariance information in portfolio selection problems

The covariation among financial asset returns is often a key ingredient used in the construction of optimal portfolios. Estimating covariances from data, however, is challenging due to the potential influence of estimation error, specially in high-dimensional problems, which can impact negatively the performance of the resulting portfolios. We address this question by putting forward a simple approach to disentangle the role of variance and covariance information in the case of mean-variance efficient portfolios. Specifically, mean-variance portfolios can be represented as a two-fund rule: one fund is a fully invested portfolio that depends on diagonal covariance elements, whereas the other is a long-short, self financed portfolio associated with the presence of non-zero off-diagonal covariance elements. We characterize the contribution of each of these two components to the overall performance in terms of out-of-sample returns, risk, risk-adjusted returns and turnover. Finally, we provide an empirical illustration of the proposed portfolio decomposition using both simulated and real market data.     

When do improved covariance matrix estimators enhance portfolio optimization? An empirical comparative study of nine estimators

The use of improved covariance matrix estimators as an alternative to the sample estimator is considered an important approach for enhancing portfolio optimization. Here we empirically compare the performance of nine improved covariance estimation procedures using daily returns of 90 highly capitalized US stocks for the period 1997–2007. We find that the usefulness of covariance matrix estimators strongly depends on the ratio between the estimation period T and the number of stocks N, on the presence or absence of short selling, and on the performance metric considered. When short selling is allowed, several estimation methods achieve a realized risk that is significantly smaller than that obtained with the sample covariance method. This is particularly true when T/N is close to one. Moreover, many estimators reduce the fraction of negative portfolio weights, while little improvement is achieved in the degree of diversification. On the contrary, when short selling is not allowed and T?>?N, the considered methods are unable to outperform the sample covariance in terms of realized risk, but can give much more diversified portfolios than that obtained with the sample covariance. When T?<?N, the use of the sample covariance matrix and of the pseudo-inverse gives portfolios with very poor performance.     

Menu simplification for portfolio selection under short-sales constraints

We introduce a risk-reduction-based procedure to identify a subset of funds with a resulting opportunity set that is at least as good as the original menu when short-sales are imposed. Relying on Wald tests for mean-variance spanning, we show that the better results for the subset can be explained by a higher concentration of covariance entries between its assets, ultimately leading to smaller Frobenius norms of the associated matrices. With data on US-defined contribution plans, where participants have limited financial literacy, tend to be overwhelmed and prefer to make decisions among fewer choices, we obtain a 75% average reduction.     

A semi-analytic method for valuing high-dimensional options on the maximum and minimum of multiple assets

Valuing high-dimensional options has many important applications in finance but when the true distributions are unknown or complex, numerical approximations must be used. Approximation methods based on Monte-Carlo simulation show a steep trade-off between estimation accuracy and computational efficiency. This article presents an alternative semi-analytic approximation method for pricing options on the maximum or minimum of multiple assets with unknown distributions. Computational efficiency is shown to improve significantly without sacrificing estimation accuracy. The method is illustrated with applications to options on underlying assets with mean-reverting prices, time-dependent correlations, and stochastic volatility The authors would like to thank the two anonymous referees, the associate editor, and Dr. Jess H. Chua at the University of Calgary for valuable comments and insights on this research. This research was partly supported by NUS grant R-146-000-059-112     

ABC of SV: Limited information likelihood inference in stochastic volatility jump-diffusion models

We develop novel methods for estimation and filtering of continuous-time models with stochastic volatility and jumps using so-called Approximate Bayesian Computation which build likelihoods based on limited information. The proposed estimators and filters are computationally attractive relative to standard likelihood-based versions since they rely on low-dimensional auxiliary statistics and so avoid computation of high-dimensional integrals. Despite their computational simplicity, we find that estimators and filters perform well in practice and lead to precise estimates of model parameters and latent variables. We show how the methods can incorporate intra-daily information to improve on the estimation and filtering. In particular, the availability of realized volatility measures help us in learning about parameters and latent states. The method is employed in the estimation of a flexible stochastic volatility model for the dynamics of the S&P 500 equity index. We find evidence of the presence of a dynamic jump rate and in favor of a structural break in parameters at the time of the recent financial crisis. We find evidence that possible measurement error in log price is small and has little effect on parameter estimates. Smoothing shows that, recently, volatility and the jump rate have returned to the low levels of 2004–2006.