Before revision:  In this work, we consider the S\&P 500 portfolio over the 1958-2015 time horizon weighted using John Tukey's transformational ladder (Tukey2): 1/x^2,\,\, 1/x,\,\, 1/x,\,\, \text{log}(x),\,\, \sqrt{x},\,\, x, \\ \,\,  \,\, } x^2, where x is the market capitalization weighted  portfolio.
After revision:  In this work, we consider the S\&P 500 portfolio over the 1958-2015 time horizon weighted by Tukey's transformational ladder (Tukey2): 1/x^2,\,\, 1/x,\,\, 1/x,\,\, \text{log}(x),\,\, \sqrt{x},\,\, x, \\ \,\, \,\, } x^2, where x is the market capitalization weighted portfolio.