Outliers can have a considerable influence on the conventional measure of covariance, which may lead to a misleading understanding of the comovement between two variables. Both an analytical derivation and Monte Carlo simulations show that the conventional measure of covariance can be heavily influenced in the presence of outliers. This paper proposes an intuitively appealing and easily computable robust measure of covariance based on the median and compares it with some existing robust covariance estimators in the statistics literature. It is demonstrated by simulations that all of the robust measures are fairly stable and insensitive to outliers. We apply robust covariance measures to construct two well-known portfolios, the minimum-variance portfolio and the optimal risky portfolio. The results of an out-of-sample experiment indicate that a potentially large investment gain can be realized using robust measures in place of the conventional measure.
Covariance, a common measure of the linear dependence of two random variables, has played an important role in many academic fields, including finance (particularly for the construction of optimal portfolios). Since the seminal work of Markowitz (1952), a number of studies have proposed a wide range of portfolios. The fundamental reason to construct a portfolio is to achieve diversification and one can obtain more benefit from diversification when the assets included in the portfolio are less correlated; that is, the lower the correlation, the greater benefit from diversification. Therefore, it is vital to carefully measure the correlation between asset returns before constructing portfolios. Usually, the correlation between asset returns is measured using the conventional sample covariance – apart from normalization by standard deviations.
Kim and White (2004) demonstrate that the conventional measures of skewness and kurtosis, which are based on sample averages, are highly sensitive to outliers. Hence, it can be conjectured that the conventional measure of covariance is also sensitive to outliers because it is computed as a sample average. In this paper, we will demonstrate both analytically and by Monte Carlo simulations that this conjecture indeed holds in the presence of outliers. Financial data, particularly data on asset returns, are widely known to include outliers, which typically result from financial crises such as the 9/11 attack and the 2008 global financial crisis. Therefore, simply applying the conventional measure of covariance to asset returns can lead to a misunderstanding of the comovement between assets. Since the covariance measure is one of the important inputs to the optimization of Markowitz portfolios, the use of the conventional measure can potentially cause a reduction in the performance of the resulting portfolios.
This paper proposes an intuitively appealing and easily computable robust measure of covariance, and we compare it with several existing robust covariance estimators from the statistics literature. To do this, we closely follow the main idea in Kim and White, 2004, Bonato, 2011, Ergun, 2011 and White et al., 2010 by constructing the proposed measure to be based on the median rather than on averages. After comparing the conventional and robust measures through Monte Carlo simulations, we employ the robust measures to construct two well-known portfolios, the minimum-variance portfolio and the optimal risky portfolio, using return data obtained from Professor Kenneth R. French’s website. Previous studies have noted the instability of portfolio optimization (e.g. Jobson and Korkie, 1980, Jobson and Korkie, 1981 and Michaud, 1989), but they typically focused on the sensitivity and uncertainty in the mean and variance measures used in the optimization process. As a result, such studies have proposed some techniques for stabilizing the mean and variance measures (e.g. Jobson et al., 1979, Adrian and Brunnermeier, 2008 and Kane et al., 2011). The present paper focuses mainly on the role of the covariance measure in the construction of optimal portfolios. The results of an out-of-sample experiment indicate that a large investment gain can be realized by using robust measures in place of the conventional measure.
Through both an analytical derivation and Monte Carlo simulations, this paper has demonstrated that outliers can have considerable influence on the conventional measure of covariance. It was also shown that outliers are most likely to influence the conventional measure when they occur at the same time in two variables. An implication is that the calculated conventional measure can be large even when the true covariance is zero, reflecting the phenomenon of a spurious correlation.
To address these problems, we have proposed an alternative robust measure based on the median. Simulations demonstrate that the proposed measures show good performance in finite samples and that its performance is comparable to existing robust covariance measures such as the Campbell, Sign, Rank, and MCD covariance measures. The proposed measure is more intuitively appealing and more easily computable than other robust measures. Hence, the proposed measure can be complementary to the conventional covariance measure.
The empirical application of the proposed measure as well as the other robust measures to portfolio optimization indicates the potential for a large investment gain by substituting robust measures for the conventional one. It should be noted that the out-of-sample exercise is just one example that shows the usefulness of the proposed measure. It is feasible for use in many other areas requiring covariance estimation.
