Quantile plots, partial orders, and financial risk

Quantile-quantile plots are most commonly used to compare the shapes of dis tributions, but they may also be used in conjunction with partial orders on distributions to compare the level and dispersion of distributions that ha ve different shapes. We discuss several easily recognized patterns in quant ile-quantile plots that suffice to demonstrate that one distribution is sma ller than another in terms of each of several partial orders. We illustrate with financial applications, proposing a quantile plot for comparing the r isks and returns of portfolios of investments. As competing portfolios have distributions that differ in level, dispersion, nan shape. it is not suffi cient to compare portfolios using measures of location and dispersion, such as expected returns and variances; howe:ver, quantile plots, with suitable scaling, do aid in such comparisons. In two plots, we compare specific por tfolios to the stock market as a whole, finding these portfolios to have hi gher returns, greater risks or dispersion, thicker tails than their greater dispersion alone would justify. Nonetheless, investors in these risky port folios are more than adequately compensated for the risks undertaken.