The Contribution of a Constituent Time Period-Asset Pair: Longitudinal Decompositions

Cross-sectional decompositions are well described for some common risk measures, enabling the attribution of risk across assets. Similar decompositions in the time series or jointly in the cross section and time series remain unarticulated. The author presents a very straightforward longitudinal risk decomposition using portfolio volatility as the risk metric. This decomposition expresses portfolio risk as the sum of mutually exclusive and cumulatively exhaustive contributions from each constituent time period-asset pair. It allows the isolation of contributions conditioned on arbitrary criteria and the aggregation of the contributions of flexible groupings of constituent assets, time periods, or both, into their combined contributions additively. For completeness, the author also decomposes means and Sharpe ratios and then extends the analysis to benchmarked portfolios and factor models. The utility of these decompositions is discussed both broadly and in some specific applications using real-world examples. Asset allocators, consultants, and portfolio managers alike should find in these decompositions a host of powerful uses in portfolio analysis and decision-making.