Abstract:

For a single asset or portfolio of assets, we are used to looking at the skewness and kurtosis of returns to understand aspects of risk not captured by standard deviation alone. For multiple assets, we often rely on the covariance matrix to describe the dependence relationship among assets. However, just as standard deviation is an incomplete description of the riskiness of individual assets, the covariance matrix is an incomplete description of the dependence structure for multiple assets. Analogously to the (2-way) covariance matrix, the multivariate version of skewness and kurtosis are 3-way and 4-way objects called cumulant tensors. They can be used to model higher-order dependence and perform portfolio optimization that accounts for skewness and kurtosis as well as mean and variance. One can also build factor models using these objects. We describe a new approach for building these factor models to analyze cumulant tensors.