Sampling error and double shrinkage estimation of minimum variance portfolios

Shrinkage estimators of the covariance matrix are known to improve the stability over time of the Global Minimum Variance Portfolio (GMVP), as they are less error-prone. However, the improvement over the empirical covariance matrix is not optimal for small values of n, the estimation sample size. For typical asset allocation problems, with n small, this paper aims at proposing a new method to further reduce sampling error by shrinking once again traditional shrinkage estimators of the GMVP. First, we show analytically that the weights of any GMVP can be shrunk – within the framework of the ridge regression – towards the ones of the equally-weighted portfolio in order to reduce sampling error. Second, Monte Carlo simulations and empirical applications show that applying our methodology to the GMVP based on shrinkage estimators of the covariance matrix, leads to more stable portfolio weights, sharp decreases in portfolio turnovers, and often statistically lower (resp. higher) out-of-sample variances (resp. Sharpe ratios). These results illustrate that double shrinkage estimation of the GMVP can be beneficial for realistic small estimation sample sizes.     

Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps

Green and Hollifield (1992) argue that the presence of a dominant factor would result in extreme negative weights in mean‐variance efficient portfolios even in the absence of estimation errors. In that case, imposing no‐short‐sale constraints should hurt, whereas empirical evidence is often to the contrary. We reconcile this apparent contradiction. We explain why constraining portfolio weights to be nonnegative can reduce the risk in estimated optimal portfolios even when the constraints are wrong. Surprisingly, with no‐short‐sale constraints in place, the sample covariance matrix performs as well as covariance matrix estimates based on factor models, shrinkage estimators, and daily data.     

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.     

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.     

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.     

Optimal asset allocation using a combination of implied and historical information

This paper investigates the contribution of option-implied information for strategic asset allocation for individuals with minimum-variance preferences and portfolios with a variety of assets. We propose a covariance matrix that exploits a mixture of historical and option-implied information. Implied variance measures are proposed for those assets for which option-implied information is available. Historical variance and correlation measures are applied to the remaining assets. The performance of this novel approach for constructing optimal investment portfolios is assessed out-of-sample using statistical and economic measures. An empirical application to a sophisticated portfolio comprised by a combination of equities, fixed income, alternative securities and cash deposits shows that implied variance measures with risk premium correction outperform variance measures constructed from historical data and implied variance without correction. This result is robust across investment portfolios, volatility and portfolio performance metrics, and rebalancing schemes.     

Mitigating Estimation Risk in Asset Allocation: Diagonal Models Versus 1/N Diversification

Recent literature suggests that optimal asset‐allocation models struggle to consistently outperform the 1/N naïve diversification strategy, which highlights estimation‐risk concerns. We propose a dichotomous classification of asset‐allocation models based on which elements of the inverse covariance matrix that a model uses: diagonal only versus full matrix. We argue that parsimonious diagonal‐only strategies, which use limited information such as volatility or idiosyncratic volatility, are likely to offer a good tradeoff between incorporating limited information while mitigating estimation risk. Evaluating five sets of portfolios over 1926–2012, we find that 1/N is generally not optimal when compared with these diagonal strategies.     

On the distribution and estimation of trading costs

This paper investigates the uncertainty about the trading costs associated with a given portfolio strategy. I derive accurate approximations of the ex ante probability distributions of proportional trading costs and portfolio turnover under the conventional assumption of normal asset returns. Based on these approximations, I express the expected trading costs as a function of asset and portfolio characteristics. All else equal, the expected trading costs increase with: i) the deviations of the expected asset returns from the expected portfolio return, ii) the assets' volatility and iii) the portfolio volatility. At the same time, they decrease with the covariance between the assets and the portfolio. Furthermore, I propose novel estimators of the expected turnover and trading costs and show that they offer small bias and low variance, even when the sample size is small. Finally, I incorporate my results into a portfolio selection framework to produce portfolios with low levels of risk and trading costs. Several experiments with real and simulated data confirm the practical value of the results.     

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.     

The Optimal Log-Utility Asset Management under Incomplete Information

In this paper, we propose a theory for deriving the optimal portfolio that assures the log-utility investors of maximizing their expected utility. Restricting investors' information at defined levels, we propose the sample path-wise optimal portfolio (SPOP), which is consistent with the back-test framework used in actualinvestment. It is proven that, at any finite terminal time, this SPOP is asymptotically optimal among all the portfolios which are predictable under investors' incompleteinformation. The optimality is guaranteed by the continuous Bayesian updating formula. Finally, we discuss an algorithm for searching the SPOP, based on asset prices at discrete time intervals.     

Using industry momentum to improve portfolio performance

Minimum-variance portfolios, which ignore the mean and focus on the (co)variances of asset returns, outperform mean–variance approaches in out-of-sample tests. Despite these promising results, minimum-variance policies fail to deliver a superior performance compared with the simple 1/N rule. In this paper, we propose a parametric portfolio policy that uses industry return momentum to improve portfolio performance. Our portfolio policies outperform a broad selection of established portfolio strategies in terms of Sharpe ratio and certainty equivalent returns. The proposed policies are particularly suitable for investors because portfolio turnover is only moderately increased compared to standard minimum-variance portfolios.     

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.     

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.     

Correlation Risk and Optimal Portfolio Choice

We develop a new framework for multivariate intertemporal portfolio choice that allows us to derive optimal portfolio implications for economies in which the degree of correlation across industries, countries, or asset classes is stochastic. Optimal portfolios include distinct hedging components against both stochastic volatility and correlation risk. We find that the hedging demand is typically larger than in univariate models, and it includes an economically significant covariance hedging component, which tends to increase with the persistence of variance–covariance shocks, the strength of leverage effects, the dimension of the investment opportunity set, and the presence of portfolio constraints.     

Shrinkage estimation of Kelly portfolios

Although the Kelly portfolio is theoretically optimal in maximizing the long-term log-growth rate, in practice this is not always so. In this paper, we first show that the sample plug-in estimator of the Kelly portfolio weights is actually biased, and we then propose an unbiased estimator as an alternative. We further derive a shrinkage estimator under the objective of minimizing the expected growth loss of the actual growth relative to the true growth. An explicit formula for the shrinkage coefficient is established. Statistical properties for the shrinkage coefficient are studied through extensive Monte Carlo simulations, and conditions for obtaining accurate estimates for the shrinkage coefficient are also discussed. The effectiveness of the proposed unbiased and shrinkage Kelly portfolios in reducing the expected growth loss are validated by various simulation studies. It is found that our proposed shrinkage Kelly portfolio has superior performances in growth loss reduction, followed by the unbiased Kelly portfolio, and the sample plug-in Kelly portfolio. The advantages of our proposed unbiased and shrinkage Kelly portfolios for long-term investments are additionally confirmed by stock investment in the U.S. market.     

Portfolio selection with commodities under conditional copulas and skew preferences

This article investigates the portfolio selection problem of an investor with three-moment preferences taking positions in commodity futures. To model the asset returns, we propose a conditional asymmetric t copula with skewed and fat-tailed marginal distributions, such that we can capture the impact on optimal portfolios of time-varying moments, state-dependent correlations, and tail and asymmetric dependence. In the empirical application with oil, gold and equity data from 1990 to 2010, the conditional t copulas portfolios achieve better performance than those based on more conventional strategies. The specification of higher moments in the marginal distributions and the type of tail dependence in the copula has significant implications for the out-of-sample portfolio performance.     

Cardinality versus q-norm constraints for index tracking

Index tracking aims at replicating a given benchmark with a smaller number of its constituents. Different quantitative models can be set up to determine the optimal index replicating portfolio. In this paper, we propose an alternative based on imposing a constraint on the q-norm (0?<?q?<?1) of the replicating portfolios’ asset weights: the q-norm constraint regularises the problem and identifies a sparse model. Both approaches are challenging from an optimization viewpoint due to either the presence of the cardinality constraint or a non-convex constraint on the q-norm. The problem can become even more complex when non-convex distance measures or other real-world constraints are considered. We employ a hybrid heuristic as a flexible tool to tackle both optimization problems. The empirical analysis of real-world financial data allows us to compare the two index tracking approaches. Moreover, we propose a strategy to determine the optimal number of constituents and the corresponding optimal portfolio asset weights.     

Portfolio optimization under a generalized hyperbolic skewed t distribution and exponential utility

In this paper, we show that if asset returns follow a generalized hyperbolic skewed t distribution, the investor has an exponential utility function and a riskless asset is available, the optimal portfolio weights can be found either in closed form or using a successive approximation scheme. We also derive lower bounds for the certainty equivalent return generated by the optimal portfolios. Finally, we present a study of the performance of mean–variance analysis and Taylor’s series expected utility expansion (up to the fourth moment) to compute optimal portfolios in this framework.     

Volatility timing: How best to forecast portfolio exposures

This paper investigates how best to forecast optimal portfolio weights in the context of a volatility timing strategy. It measures the economic value of a number of methods for forming optimal portfolios on the basis of realized volatility. These include the traditional econometric approach of forming portfolios from forecasts of the covariance matrix. Both naïve forecasts using simple historical averages, and those generated from econometric models are considered. A novel method, where a time series of optimal portfolio weights are constructed from observed realized volatility and direct forecast is also proposed. A number of naïve forecasts and the approach of directly forecasting portfolio weights show a great deal of merit. Resulting portfolios are of similar economic benefit to a number of competing approaches and are more stable across time. These findings have obvious implications for the manner in which volatility timing is undertaken in a portfolio allocation context.     

Volatility and covariation of financial assets: A high-frequency analysis

Using high frequency data for the price dynamics of equities we measure the impact that market microstructure noise has on estimates of the: (i) volatility of returns; and (ii) variance–covariance matrix of n assets. We propose a Kalman-filter-based methodology that allows us to deconstruct price series into the true efficient price and the microstructure noise. This approach allows us to employ volatility estimators that achieve very low Root Mean Squared Errors (RMSEs) compared to other estimators that have been proposed to deal with market microstructure noise at high frequencies. Furthermore, this price series decomposition allows us to estimate the variance covariance matrix of n assets in a more efficient way than the methods so far proposed in the literature. We illustrate our results by calculating how microstructure noise affects portfolio decisions and calculations of the equity beta in a CAPM setting.