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.     

Corrections for Trading Frictions in Multivariate Returns

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 non-positive 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 dynamic model of active portfolio management with benchmark orientation

This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factor model similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset pricing model. Journal of Economic Theory, 13, 342–360] and that portfolio managers adopt a mean tracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 13–22], we develop a dynamic model of active portfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike the case of constant proportional portfolios for standard utility maximization, our optimal portfolio policy is state dependent, being a function of time to investment horizon, the return on the benchmark portfolio, and the return on the investment portfolio. We define a dynamic performance measure which relates portfolio’s return to its risk sensitivity. Abnormal returns at each point in time are quantified as the difference between the realized and the model-fitted returns. Risk sensitivity is estimated through a dynamic matching that minimizes the total fitted error of portfolio returns. For illustration, we analyze eight representative mutual funds in the U.S. market and show how this model can be used in practice.     

Estimation error in the average correlation of security returns and shrinkage estimation of covariance and correlation matrices

The correlation matrix of security returns is an important input component for mean–variance portfolio analysis. This study uses the average of sample correlations to estimate the correlation matrix and derives an expression of its estimation error in terms of sampling variance. This study then considers the impact of such estimation error on shrinkage estimation, where a weighted average is sought between the sample covariance matrix and an average correlation target, and between the sample correlation matrix and the target. An illustrative example using monthly returns of the current Dow Jones stocks is provided.     

Characteristic-based mean-variance portfolio choice

We study empirical mean-variance optimization when the portfolio weights are restricted to be direct functions of underlying stock characteristics such as value and momentum. The closed-form solution to the portfolio weights estimator shows that the portfolio problem in this case reduces to a mean-variance analysis of assets with returns given by single-characteristic strategies (e.g., momentum or value). In an empirical application to international stock return indexes, we show that the direct approach to estimating portfolio weights clearly beats a naive regression-based approach that models the conditional mean. However, a portfolio based on equal weights of the single-characteristic strategies performs about as well, and sometimes better, than the direct estimation approach, highlighting again the difficulties in beating the equal-weighted case in mean-variance analysis. The empirical results also highlight the potential for ‘stock-picking’ in international indexes using characteristics such as value and momentum with the characteristic-based portfolios obtaining Sharpe ratios approximately three times larger than the world market.     

Estimating daily volatility in financial markets utilizing intraday data

This study proposes a new approach to the estimation of daily realised volatility in financial markets from intraday data. Initially, an examination of intraday returns on S&P 500 Index Futures reveals that returns can be characterised by heteroscedasticity and time-varying autocorrelation. After reviewing a number of daily realised volatility estimators cited in the literature, it is concluded that these estimators are based upon a number of restrictive assumptions in regard to the data generating process for intraday returns. We use a weak set of assumptions about the data generating process for intraday returns, including transaction returns, given in den Haan and Levin [den Haan, W.J., Levin, A., 1996. Inferences from parametric and non-parametric covariance matrix estimation procedures, Working paper, NBER, 195.], which allows for heteroscedasticity and time-varying autocorrelation in intraday returns. These assumptions allow the VARHAC estimator to be employed in the estimation of daily realised volatility. An empirical analysis of the VARHAC daily volatility estimator employing intraday transaction returns concludes that this estimator performs well in comparison to other estimators cited in the literature.     

Nonlinear mean reversion in stock prices

This paper provides new evidence on the time-series predictability of stock market returns by introducing a test of nonlinear mean reversion. The performance of extreme daily returns is evaluated in terms of their power to predict short- and long-horizon returns on various stock market indices and size portfolios. The paper shows that the speed of mean reversion is significantly higher during the large falls of the market. The parameter estimates indicate a negative and significant relation between the monthly portfolio returns and the extreme daily returns observed over the past one to eight months. Specifically, in a quarter in which the minimum daily return is −2% the expected excess return is 37 basis points higher than in a month in which the minimum return is only −1%. This result holds for the value-weighted and equal-weighted stock market indices and for each of the size decile portfolios. The findings are also robust to different sample periods, different indices, and investment horizons.     

Improved estimation of the covariance matrix of stock returns with an application to portfolio selection

This paper proposes to estimate the covariance matrix of stock returns by an optimally weighted average of two existing estimators: the sample covariance matrix and single-index covariance matrix. This method is generally known as shrinkage, and it is standard in decision theory and in empirical Bayesian statistics. Our shrinkage estimator can be seen as a way to account for extra-market covariance without having to specify an arbitrary multifactor structure. For NYSE and AMEX stock returns from 1972 to 1995, it can be used to select portfolios with significantly lower out-of-sample variance than a set of existing estimators, including multifactor models.     

Nonsynchronous Data and the Covariance-Factor Structure of Returns

Evidence is presented that indicates that the standard estimator of the covariance matrix of daily returns provides a distorted view of the true covariance-factor structure. An alternative estimator, based on a model of the price-adjustment delay process, reveals roughly twice as much covariation in individual security returns. The number of factors identified also appears to increase when this estimator is employed. Since the linear space spanned by the estimated factor-loading vectors is quite sensitive to the estimator used, it is important that the consistent estimator be considered in the usual two-stage empirical investigations of the APT.     

Dynamic factor long memory volatility

In this paper, we develop a long memory orthogonal factor (LMOF) multivariate volatility model for forecasting the covariance matrix of financial asset returns. We evaluate the LMOF model using the volatility timing framework of Fleming et al. [J. Finance, 2001, 56, 329–352] and compare its performance with that of both a static investment strategy based on the unconditional covariance matrix and a range of dynamic investment strategies based on existing short memory and long memory multivariate conditional volatility models. We show that investors should be willing to pay to switch from the static strategy to a dynamic volatility timing strategy and that, among the dynamic strategies, the LMOF model consistently produces forecasts of the covariance matrix that are economically more useful than those produced by the other multivariate conditional volatility models, both short memory and long memory. Moreover, we show that combining long memory volatility with the factor structure yields better results than employing either long memory volatility or the factor structure alone. The factor structure also significantly reduces transaction costs, thus increasing the feasibility of dynamic volatility timing strategies in practice. Our results are robust to estimation error in expected returns, the choice of risk aversion coefficient, the estimation window length and sub-period analysis.     

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.     

SYMMETRIC VERSUS ASYMMETRIC CONDITIONAL COVARIANCE FORECASTS: DOES IT PAY TO SWITCH?

Volatilities and correlations for equity markets rise more after negative returns shocks than after positive shocks. Allowing for these asymmetries in covariance forecasts decreases mean‐variance portfolio risk and improves investor welfare. We compute optimal weights for international equity portfolios using predictions from asymmetric covariance forecasting models and a spectrum of expected returns. Investors who are moderately risk averse, have longer rebalancing horizons, and hold U.S. equities benefit most and may be willing to pay around 100 basis points annually to switch from symmetric to asymmetric forecasts. Accounting for asymmetry in both variances and correlations significantly lowers realized portfolio risk.     

Size matters: Optimal calibration of shrinkage estimators for portfolio selection

We carry out a comprehensive investigation of shrinkage estimators for asset allocation, and we find that size matters—the shrinkage intensity plays a significant role in the performance of the resulting estimated optimal portfolios. We study both portfolios computed from shrinkage estimators of the moments of asset returns (shrinkage moments), as well as shrinkage portfolios obtained by shrinking the portfolio weights directly. We make several contributions in this field. First, we propose two novel calibration criteria for the vector of means and the inverse covariance matrix. Second, for the covariance matrix we propose a novel calibration criterion that takes the condition number optimally into account. Third, for shrinkage portfolios we study two novel calibration criteria. Fourth, we propose a simple multivariate smoothed bootstrap approach to construct the optimal shrinkage intensity. Finally, we carry out an extensive out-of-sample analysis with simulated and empirical datasets, and we characterize the performance of the different shrinkage estimators for portfolio selection.     

Beyond the Sharpe ratio: An application of the Aumann–Serrano index to performance measurement

We propose a performance measure that generalizes the Sharpe ratio. The new performance measure is monotone with respect to stochastic dominance and consistently accounts for mean, variance and higher moments of the return distribution. It is equivalent to the Sharpe ratio if returns are normally distributed. Moreover, the two performance measures are asymptotically equivalent as the underlying distributions converge to the normal distribution. We suggest a parametric and a non-parametric estimator for the new performance measure and provide an empirical illustration using mutual funds and hedge funds data.     

Covariance and correlation stationarity: Experiences from seven Asian emerging markets

Because of the rising interest and the growing importance of the Asian emerging markets in international diversification, this paper examines the covariance and correlation stationarity in stock return relationships among seven Asian emerging markets. This paper also covers the issue of seasonality in stock return co-movements. Empirical results show that because the correlations among them and those with other developed markets are very small, huge gains from diversifying into the seven Asian emerging markets are possible. Results on stationarity indicate that correlation matrices of stock returns are much more stable then the corresponding variance-covariance matrices and that the length of the estimation period seems to have no impact on the stationarity of the correlation matrix. We also found that virtually no seasonality in the correlations exists among the seven Asian emerging markets. However, we did find that during our sample period covariance among stock returns is nonstationary in January. The author thanks an anonymous referee ofFinancial Engineering and the Japanese Markets for the valuable comments on the earlier version of this article.     

Hedge fund portfolio construction: A comparison of static and dynamic approaches

This article studies the impact of modeling time-varying covariances/correlations of hedge fund returns in terms of hedge fund portfolio construction and risk measurement. We use a variety of static and dynamic covariance/correlation prediction models and compare the optimized portfolios’ out-of-sample performance. We find that dynamic covariance/correlation models construct portfolios with lower risk and higher out-of-sample risk-adjusted realized return. The tail-risk of the constructed portfolios is also lower. Using a mean-conditional-value-at-risk framework we show that dynamic covariance/correlation models are also successful in constructing portfolios with minimum tail-risk.     

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.     

Mean–variance and mean–semivariance portfolio selection: a multivariate nonparametric approach

While univariate nonparametric estimation methods have been developed for estimating returns in mean-downside risk portfolio optimization, the problem of handling possible cross-correlations in a vector of asset returns has not been addressed in portfolio selection. We present a novel multivariate nonparametric portfolio optimization procedure using kernel-based estimators of the conditional mean and the conditional median. The method accounts for the covariance structure information from the full set of returns. We also provide two computational algorithms to implement the estimators. Via the analysis of 24 French stock market returns, we evaluate the in-sample and out-of-sample performance of both portfolio selection algorithms against optimal portfolios selected by classical and univariate nonparametric methods for three highly different time periods and different levels of expected return. By allowing for cross-correlations among returns, our results suggest that the proposed multivariate nonparametric method is a useful extension of standard univariate nonparametric portfolio selection approaches.     

Asset pricing and foreign exchange risk

According to the International Capital Asset Pricing Model (ICAPM), the covariance of assets with foreign exchange currency returns should be a risk factor that must be priced when the purchasing power parity is violated. The goal of this study is to re-examine the relationship between stock returns and foreign exchange risk. The novelties of this work are: (a) a data set that makes use of daily observations for the measurement of the foreign exchange exposure and volatility of the sample firms and (b) data from a Eurozone country.The methodology we make use in reference to the estimation of the sensitivity of each stock to exchange rate movements is that it allows regressing stock returns against factors controlling for market risk, size, value, momentum, foreign exchange exposure and foreign exchange volatility. Stocks are then classified according to their foreign exchange sensitivity portfolios and the return of a hedge (zero-investment) portfolio is calculated. Next, the abnormal returns of the hedge portfolio are regressed against the return of the factors. Finally, we construct a foreign exchange risk factor in such manner as to obtain a monotonic relation between foreign exchange risk and expected returns.The empirical findings show that the foreign exchange risk is priced in the cross section of the German stock returns over the period 2000-2008. Furthermore, they show that the relationship between returns and foreign exchange sensitivity is nonlinear, but it takes an inverse U-shape and that foreign exchange sensitivity is larger for small size firms and value stocks.     

Does the choice of estimator matter when forecasting returns?

While the literature concerned with the predictability of stock returns is huge, surprisingly little is known when it comes to role of the choice of estimator of the predictive regression. Ideally, the choice of estimator should be rooted in the salient features of the data. In case of predictive regressions of returns there are at least three such features; (i) returns are heteroskedastic, (ii) predictors are persistent, and (iii) regression errors are correlated with predictor innovations. In this paper we examine if the accounting of these features in the estimation process has any bearing on our ability to forecast future returns. The results suggest that it does.