Measuring Dependence in a Set of Asset Returns

Abstract

An index measuring the degree of dependence in a set of asset returns is defined as the ratio of an equivalent number of independent assets to the number of assets. The equivalence is based on either attaining the same optimized value enhancement or spread reduction. The value enhancement is the difference in value of a value maximizing portfolio and the maximum value delivered by the components. The spread reduction is the percentage reduction attained by a spread minimizing portfolio relative to the smallest spread for the components. Asset values or bid and ask prices of portfolios, are modeled by conservative valuation operators from the theory of two price economies. The dependence indices fall with the number of assets in the portfolio and they are explained by a measure of concentration applied to normalized eigenvalues of the correlation matrix along with the average level of correlation, the level of the (Rudin and Morgan, 2006) portfolio diversification index and the number of assets in the portfolio. A time series of the indices constructed on the basis of the \( S \& P\) 500 index and the nine sector ETF’s reveals a collapse during the financial crisis with no recovery until 2016, with a peak in February 2020 and a COVID crash in March of 2020. Furthermore, factor dependence benefits are richer than those found in equity indices. Dependence benefits across global indices are not as strong as dependence benefits across an equal number of domestic assets, but they rise substantially for longer horizons of up to three years.