Variance Portfolio Research Articles

The Out-of-Sample Forecasting of Hedged Portfolio Variances using Bivariate Mixed Normal GARCH Models

In this study we compare two standard bivariate GARCH models with new bivariate mixed normal GARCH models in terms of the percentage variance reduction of the out-of-sample hedged portfolio and the statistical significance test of the performance improvements using Hansen’s (2001) Superior Predictive Ability statistics. All competing models are applied to estimate time-varying hedge ratios for corn and wheat, traded on the Chicago Board of Trade. The empirical results demonstrate that the standard BEKK-GARCH model significantly outperforms the other competing GARCH models at shorter horizons. However, as the hedge horizon is extended to longer than 10 days, it is evident that the mixed normal BEKK-GARCH model is the best at the usual significance level of 5%.

The Intertemporal Relation between Risk and Returns in Australia

We explore the intertemporal relation between the conditional mean and the conditional variance of the industry portfolio returns and Fama-French 25 Size/Book-to-Market portfolio returns in Australia using a bivariate generalized autoregressive conditional heteroskedasticity (GARCH) model. We estimate a portfolio's conditional covariance with the market and test whether the conditional covariance can help predict the time-variation in the portfolio's expected returns. Our results show that there is no significant relation between asset returns and market returns. We also examine asset returns with size and value factors, and we find that size and value factors only matter for the Size/Book-to-Market portfolios, but not for the industry portfolios. Lastly, we examine the role of the macroeconomic variables in determining the time-variation of asset returns, and there is no evidence of the relationship between asset returns and the macroeconomic variables.

ESTIMATION OF OPTIMAL PORTFOLIO WEIGHTS

The paper discusses finite sample properties of optimal portfolio weights, estimated expected portfolio return, and portfolio variance. The first estimator assumes the asset returns to be independent, while the second takes them to be predictable using a linear regression model. The third and the fourth approaches are based on a shrinkage technique and a Bayesian methodology, respectively. In the first two cases, we establish the moments of the weights and the portfolio returns. A consistent estimator of the shrinkage parameter for the third estimator is then derived. The advantages of the shrinkage approach are assessed in an empirical study.

Sectoral vs. Country Diversification Benefits and Downside Risk

Recently, the advantage of country diversification relative to sector diversification has been questioned especially against the background of the European monetary and financial integration. Correct estimates of the correlation matrix are central for the evaluation of the relative diversification gains. These estimates should take into account the time-varying and asymmetric behaviour of the correlation process particularly in the context of major changes in volatility and market trends. In this paper, the ADCC (Asymmetric Dynamic Conditional Correlation) model developed by Cappiello et al. (2003) is used to estimate the conditional correlation and volatility of weekly country, sector and country/sector return indexes over 1990-2003 . This model offers a relatively flexible specification for the conditional correlation process that is still computationally feasible for estimation on large portfolios. The estimation results point to an increase in the average correlation between country indexes during the last five years, but at the same time there is an important decline in the correlation between sector indexes. This trend is observed in both the euro area and the world-wide portfolios and is therefore not specific to the European integration process. At the same time, the volatility in the sector indexes has increased remarkably and in a relatively stronger way compared to the volatility in the country indexes. Both trends tend to cancel out in the calculations of optimal portfolio variance: lower sector correlation is offset by higher sector volatility and higher country correlation is neutralised by the relative lower volatility in country indexes. Therefore no clear trend appears from comparing the relative conditional variances of sector and country portfolios. After taking into account the effect of average returns, it turns out that country diversification is still the dominant strategy for world portfolios, whereas sector diversification is more interesting for euro area portfolios.

Statistical inference of the efficient frontier for dependent asset returns

In the paper we consider the three characteristics of the efficient frontier. These characteristics are estimated by substituting the unknown parameters by the sample counterparts. Assuming that the asset returns follow a stationary Gaussian process it is shown that the estimated characteristics are asymptotically normally distributed. This result is used to determine the joint asymptotic distribution of the estimated portfolio return and the estimated portfolio variance in the case of the expected utility portfolio and the tangency portfolio, respectively.

The Limits to Minimum-Variance Hedging

In this paper, we compare the estimated minimum-variance hedge ratios from a range of conditional hedging models with the ‘realized’ minimum variance hedge ratio constructed using intraday data. We show that the reduction in conditionally hedged portfolio variance falls far short of the ex post maximal reduction in variance obtained using the realized minimum variance hedge ratio. While this is partly due to systematic bias, correcting for this bias does little to improve hedging effectiveness. The poor performance of conditional hedging models is therefore more likely to be attributable to the unpredictability of the integrated hedge ratio.

The Distribution of the Sample Minimum-Variance Frontier

In this paper, we present a finite sample analysis of the sample minimum-variance frontier under the assumption that the returns are independent and multivariate normally distributed. We show that the sample minimum-variance frontier is a highly biased estimator of the population frontier and we propose an improved estimator of the population frontier. In addition, we provide the exact distribution of the out-of-sample mean and variance of sample minimum-variance portfolios. This allows us to understand the impact of estimation error on the performance of in-sample optimal portfolios.

A bivariate Markov regime switching GARCH approach to estimate time varying minimum variance hedge ratios

This article develops a new bivariate Markov regime switching BEKK-Generalized Autoregressive Conditional Heteroscedasticity (GARCH) (RS-BEKK-GARCH) model. The model is a state-dependent bivariate BEKK-GARCH model and an extension of Gray's univariate generalized regime-switching (GRS) model to the bivariate case. To solve the path-dependency problem inherent in the bivariate regime switching BEKK-GARCH model, we propose a recombining method for the covariance term in the conditional variance-covariance matrix. The model is applied to estimate time-varying minimum variance hedge ratios for corn and nickel spot and futures prices. Out-of-sample point estimates of hedging portfolio variance show that compared to the state-independent BEKK-GARCH model, the RS-BEKK-GARCH model improves out-of-sample hedging effectiveness for both corn and nickel data. We perform White's (2000) data-snooping reality check to test for predictive superiority of RS-BEKK-GARCH over the benchmark model and find that the difference in variance reduction between BEKK-GARCH and RS-BEKK-GARCH is not statistically significant for either data set at conventional confidence levels.

Improving Performance By Constraining Portfolio Norms: A Generalized Approach to Portfolio Optimization

In this paper, we provide a general framework for determining the portfolio that has superior out-of-sample performance even in the presence of estimation error. This general framework relies on solving the traditional minimum-variance problem (based on the sample covariance matrix) but subject to the additional constraint that the norm of the portfolio weight vector must be smaller than a given threshold. We show that our general framework nests as special cases the shrinkage approaches of Jagannathan and Ma (2003) and Ledoit and Wolf (2004b), and the 1/N policy studied in DeMiguel, Garlappi, and Uppal (2007), and that all these policies can be interpreted as those of a Bayesian investor who has a certain prior belief on portfolio weights. We also use our general framework to propose several new portfolio strategies. Finally, we compare empirically (in terms of portfolio variance, Sharpe ratio, and turnover), the out-of-sample performance of the new polices we propose to ten strategies in the existing literature across ten datasets. We find that the new policies we propose can outperform the policies proposed in Jagannathan and Ma (2003) and Ledoit and Wolf (2004b), the 1/N policy evaluated in DeMiguel, Garlappi, and Uppal (2007), and also other strategies in the literature such as Brandt, Santa-Clara, and Valkanov (2005).

Sequential monitoring of minimum variance portfolio

This paper evaluates the economic effect of monitoring the minimum variance portfolio weights, which depend solely on the covariance matrix of returns. The investor decides whether the portfolio composition providing the smallest portfolio variance remains optimal at the beginning of every new investment period. For this purpose changes in the optimal weights are sequentially detected by means of EWMA control charts. Signals obtained from monitoring are used for improvement of the covariance matrix estimation procedure. The investment strategy exploiting signals from control charts is compared with a number of alternative approaches in the empirical study.

An Examination of Price Discovery and Hedging Efficiency of Indian Equity Futures Market

Present study investigates the price discovery and hedging efficiency of NIFTY and all those stock futures whose trading started on 9th November 2001 and are continuously traded till 30th June 2006. The study observes information asymmetry in both futures and cash market and significant Jarque-Bera test rejects the hypothesis that returns in both markets follow normal distribution. Both futures and cash market returns are found to be integrated of order 1, which implies that strong long-run relationship exists between two markets and these results are strongly supported by predictable and stationary basis. Presence of information asymmetry and cointegration implies that both markets are inefficient in weak form. Moreover, Granger causality and Vector Autoregression (VAR) results provides significant evidence that futures market leads cash market, which implies that futures market is an efficient price discovery vehicle. On the basis of price discovery efficiency of the futures market, hedge ratio through EGARCH (1,1) and VAR (based on Error correction Methodology) have been estimated, which suggests that efficient price discovery of futures market provides good opportunities for the traders to hedge their market risk because hedging through futures (except for RELIANCE) help the traders to reduce portfolio variance by approximately 90% and even more in some cases.

The distribution of the sample variance of the global minimum variance portfolio in elliptical models

In this article, we consider the variance of the global minimum variance portfolio. Assuming a matrix variate elliptically contoured distribution for the portfolio asset returns, we give the exact distribution and the density of an estimator of the portfolio variance. These results have many useful applications. They permit statements about the performance of the estimator. Moreover, it is possible to derive confidence intervals and to construct a test for the hypothesis that the global minimum variance is less than or equal to a certain value. We illustrate our results in an empirical study dealing with the daily returns of three developed countries.

On the existence of solutions to the quadratic mixed-integer mean–variance portfolio selection problem

In the standard mean–variance portfolio selection approach, several operative features are not taken into account. Among these neglected aspects, one of particular interest is the finite divisibility of the (stock) assets, i.e. the obligation to buy/sell only integer quantities of asset lots whose number is pre-established. In order to consider such a feature, we deal with a suitably defined quadratic mixed-integer programming problem. In particular, we formulate this problem in terms of quantities of asset lots (instead of, as usual, in terms of capital per cent quotas). Secondly, we provide necessary and sufficient conditions for the existence of a non-empty mixed-integer feasible set of the considered programming problem. Thirdly, we present some rounding procedures for finding, in a finite number of steps, a feasible mixed-integer solution which is better than the one detected by the necessary and sufficient conditions in terms of the value assumed by the portfolio variance. Finally, we perform an extensive computational experiment by means of which we verify the goodness of our approach.

A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms

In this paper, we provide a general nonlinear programming framework for identifying portfolios that have superior out-of-sample performance in the presence of estimation error. This general framework relies on solving the traditional minimum-variance problem but subject to the additional constraint that the norm of the portfolio-weight vector be smaller than a given threshold. We show that our general framework nests as special cases several wellknown (shrinkage and constrained) approaches considered in the literature. We also use our general framework to propose several new portfolio strategies that we term partial minimum-variance portfolios. These portfolios are obtained by applying the classical conjugate-gradient method to solve the minimum-variance problem. We compare empirically (in terms of portfolio variance, Sharpe ratio, and turnover), the out-of-sample performance of the new portfolios to various wellknown strategies across several datasets. We find that the norm-constrained portfolios we propose outperform shortsale-constrained portfolio approaches, shrinkage approaches, the 1/N portfolio, factor portfolios, and also other strategies considered in the literature.

Private Equity versus Stocks

In this article the risk and return characteristics and diversification benefits when private equity is used as a portfolio component are investigated. The analysis is based on actual cash flows to and from portfolio companies. Using a dataset that provides complete information for 3620 investments made by 123 funds from 37 investment managers the authors find a dramatic increase in the extent to which private equity outperforms stock investment during the late 1990s. Within the overall class of private equity, returns on earlier private equity investment categories, including venture capital, show on average higher variations and even higher rates of failure. However, by selecting a few extremely well-performing companies, one can achieve high average returns, thus compensating for lost investments. An analysis of various private equity subcategories shows that there is a wide range of optimal allocations to this asset class that minimizes mixed-asset portfolio variance or maximizes performance ratios. <b>TOPICS:</b>Private equity, security analysis and valuation, portfolio construction, performance measurement

Visual Portfolio Statistics

This paper develops visual aids for the understanding of two asset portfolio mathematics. Specifically, visual aids are utilized in teaching portfolio variance and correlation coefficient concepts. The presentation is simple, yet powerful, and is useful for an audience with varying levels of statistical sophistication. Consequently, the visual aids can replace or complement standard presentations of basic portfolio theory.

Visual Portfolio Statistics

This paper develops visual aids for the understanding of two asset portfolio mathematics. Specifically, visual aids are utilized in teaching portfolio variance and correlation coefficient concepts. The presentation is simple, yet powerful, and is useful for an audience with varying levels of statistical sophistication. Consequently, the visual aids can replace or complement standard presentations of basic portfolio theory.

Static Mean-Variance Analysis with Uncertain Time Horizon

We generalize Markowitz analysis to the situations involving an uncertain exit time. Our approach preserves the form of the original problem in that an investor minimizes portfolio variance for a given level of the expected return. However, inputs are now given by the generalized expressions for mean and variance-covariance matrix involving moments of the random exit time in addition to the conditional moments of asset returns. Although efficient frontiers in the generalized and the standard Markowitz case may coincide under certain conditions, we demonstrate that, by means of an example, in general that is not true. In particular, portfolios efficient in the standard Markowitz sense can be inefficient in the generalized sense and vice versa. As a result, an investor facing an uncertain time horizon and investing as if her time of exit is certain would in general make suboptimal portfolio allocation decisions. Numerical simulations show that a significant efficiency loss can be induced by an improper use of standard mean-variance analysis when time horizon is uncertain.

The generalized value at risk admissible set: constraint consistency and portfolio outcomes

Generalized value at risk (GVaR) adds a conditional value at risk or censored mean lower bound to the standard value at risk and considers portfolio optimization problems in the presence of both constraints. For normal distributions the censored mean is synonymous with the statistical hazard function, but this is not true for fat-tailed distributions. The latter turn out to imply much tighter bounds for the admissible portfolio set and indeed for the logistic, an upper bound for the portfolio variance that yields a simple portfolio choice rule. The choice theory in GVaR is in general not consistent with classic Von Neumann–Morgenstern utility functions for money. A re-specification is suggested to make it so that gives a clearer picture of the economic role of the respective constraints. This can be used analytically to explore the choice of portfolio hedges.

Modern Portfolio Management with Conditioning Information

This paper studies models in which active portfolio managers utilize conditioning information unavailable to their clients to optimize performance relative to a benchmark. We derive explicit solutions for the optimal strategies with multiple risky assets, with or without a risk-free asset, and consider various constraints on portfolio risks or weights. The optimal strategies feature a mean-variance efficient component (to minimize portfolio variance), and a hedging demand for the benchmark portfolio (to maximize correlation with the benchmark). A currency portfolio example shows that the optimal strategies improve the measured performance by 53% out of sample, compared with portfolios ignoring conditioning information.