Multi-dimensional Risk and its Diversification

In this paper, we propose defining the ¡®risk¡¯ of a portfolio as a multi-dimensional concept characterized by three components: variance, skewness and kurtosis. Unlike most previous papers studying how the first component of risk,i.e., variance, is diversified, we use both analysis and simulations to investigate how the other two components (skewness and kurtosis) are diversified when the number of stocks in a well-diversified portfolio increases. We find a couple of interesting results. When a portfolio is skewed and fat-tailed, then not only its variance, but also its skewness and kurtosis are simultaneously reduced as the number of risky assets in the portfolio increases. When the risky assets in a portfolio are moderately correlated, the three components tend to decrease and eventually converge to non-zero values. These non-zero limit values can be used to define the true multi-dimensional systematic risk of the portfolio. Hence, it can be argued that multi-dimensional non-systematic risk can be diversified by constructing a well-balanced portfolio. Another interesting result is that the skewness risk of a portfolio tends to decrease more slowly than the other two types of risk, the variance risk and the kurtosis risk, which indicates that the skewness risk is the most difficult to diversify among the three components.