Quantile Plots,Partial Orders,and Financial Risk

Quantile-quantile plots are most commonly used to compare the shapes of distributions, but they may also be used in conjunction with partial orders on distributions to compare the level and dispersion of distributions that have different shapes. We discuss several easily recognized patterns in quantile-quantile plots that suffice to demonstrate that one distribution is smaller than another in terms of each of several partial orders. We illustrate with financial applications, proposing a quantile plot for comparing the risks and returns of portfolios of investments. As competing portfolios have distributions that differ in level, dispersion, and shape, it is not sufficient to compare portfolios using measures of location and dispersion, such as expected returns and variances; however, quantile plots, with suitable scaling, do aid in such comparisons. In two plots, we compare specific portfolios to the stock market as a whole, finding these portfolios to have higher returns, greater risks or dispersion, thicker tails than their greater dispersion alone would justify. Nonetheless, investors in these risky portfolios are more than adequately compensated for the risks undertaken.