Using forward-looking option prices, the authors develop a family of estimators of the covariance matrix. These estimators rely on only forward-looking information in contrast to estimates that draw on historical returns. Applying the forward-looking estimators developed to a minimum-variance investment strategy, the authors find that their strategy performs better than various benchmark strategies, especially in periods of crisis. The strategies they compare include index investing, naive diversification with an equally weighted investment strategy (1/N investing), and those based on historical estimates.

Covariance matrices are of primary importance in asset allocation, portfolio management, pricing, modeling, and risk management.

The global minimum variance portfolio (GMVP), as the name suggests, is defined in the context of modern portfolio theory as the portfolio with the lowest possible risk (measured by variance) among all possible portfolios. Within a mean–variance framework, the GMVP can be obtained using the covariance matrix without the need to estimate expected returns.

The authors’ dataset consists of daily prices of the DJIA index and stocks. For the DJIA stocks, they use Black–Scholes implied volatilities for multiple strike prices and maturities to calculate the forward-looking moments. DJIA daily stock prices from 1998 to 2012 are used for the out-of-sample analysis. The authors do not allow for shorting and rebalance monthly.

Using their forward-looking estimators of the covariance matrix, the authors find that the minimum-variance strategy outflanks various investment strategies in out-of-sample tests and that the results are statistically significant. They compare the performance of their strategy with that of the GMVP strategy based on three historical returns of covariance estimates (including two shrinkage estimators), a naive diversification strategy that invests equally in each of the N assets (1/N diversification), and both price-weighted and capital-weighted benchmark strategies on the DJIA. The outperformance is stronger in periods of crisis, which the authors explain is the result of higher information flow during crises and the ability of informed investors to use the option markets to exploit information asymmetry.

The authors conclude that adopting forward-looking estimates may help investors better realize the potential gains from optimal portfolio diversification. They also suggest that investors using historical estimators use more current data and shrinkage estimators. They also note that because their study uses DJIA stocks only, the performance of the implied estimators may not show identical results for other investment universes.

Asset returns are modeled using the standard one-factor Sharpe-based market model, with coefficients permitted to vary in time. The authors’ strategy is to use forward-looking options on the DJIA stock index and options on the individual index constituents to determine the family of covariance matrix estimators.

To derive the family of estimators of the covariance matrix, the authors impose a cross-sectional constraint on a selected central moment of the return distribution. Depending on the moment chosen, a family of estimators can be generated. For example, to derive the first estimator of the covariance matrix, the authors use the second moment (i.e., variance) of the return distribution and assume that the same time-varying proportion of total variance is attributable to systematic factors in the market model. This constraint is in line with data from other research studies. Similarly, the authors derive covariance matrix estimators based on the third central moment (i.e., skewness) and the fourth central moment (i.e., kurtosis).

Critically, for the second central moment, the authors’ assumption ensures a positive-definite covariance matrix, which is important in finding the GMVP (or more generally, in solving any optimization problem). A symmetric, positive-definite matrix is one with positive eigenvalues.

Positive-definiteness is not assured for the skewness- and kurtosis-based estimators, so the authors empirically examine the data, which consist of DJIA index and constituent stock returns over a five-year period. They find that the kurtosis-based estimator shows better properties than the skewness-based estimator and do not use the latter.

Expected returns are an important variable in mean–variance optimization, and as the authors point out, including expected returns and expanding the sample to stocks beyond the DJIA would be logical future research directions.