Over the past half century, portfolio managers have carefully documented the advantages of the equally weighted S\&P 500 portfolio as well as the often overlooked disadvantages of the market capitalization weighted  S\&P 500  portfolio (see Bloom, Uppal, Jacobs, Treynor). However, rather surprisingly, portfolio allocation based on the seven simple transformations of John Tukey's ladder are nowhere to be found in the literature. 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. We find that the 1/x^2 weighting strategy produces cumulative returns which significantly dominates all other portfolios, achieving an annual geometric mean return of 20.889\\%DIF < . Further, the 1/x^2 weighting strategy is superior to a 1/x weighting strategy, which is in turn superior to a 1/\sqrt{x} weighted portfolio, and so forth, culminating with the x^2 transformation, whose cumulative returns are the lowest of the seven transformations of John Tukey's transformational ladder. Rather shockingly, the order of cumulative returns precisely follows that of John Tukey's transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon.\end{abstract} %DIF > over the 1958-2015 horizon. Our story is furthered by a startling phenomenon: both the cumulative and annual returns of the 1/x^2 weighting strategy are superior to those of the 1/x weighting strategy, which are in turn superior to those of the 1/\sqrt{x} weighted portfolio, and so forth, ending with the x^2 transformation, whose cumulative returns are the lowest of the seven transformations of Tukey's transformational ladder. Indeed, the order of cumulative returns precisely follows that of Tukey's transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon, and we take care to differentiate it from the well-known "small-firm effect."
Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted S\&P 500 portfolio as well as the often overlooked disadvantages of the market capitalization weighted Standard and Poor's ( S\&P 500 ) portfolio (see Bloom, Uppal, Jacobs, Treynor). However, portfolio allocation based on the transformations of John Tukey's ladder , rather surprisingly, have remained absent from the literature. 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, \\ \,\, \text{and \,\, } x^2, where x is the market capitalization weighted S\&P 500 portfolio. We find that the 1/x^2 weighting strategy produces cumulative returns that significantly dominates all other portfolios, achieving a compound annual growth rate of 20.889\\%DIF < . Further, the 1/x^2 weighting strategy is superior to a 1/x weighting strategy, which is in turn superior to a 1/\sqrt{x} weighted portfolio, and so forth, culminating with the x^2 transformation, whose cumulative returns are the lowest of the seven transformations of John Tukey's transformational ladder. Rather shockingly, the order of cumulative returns precisely follows that of John Tukey's transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon.\end{abstract} %DIF > over the 1958-2015 horizon. Our story is furthered by a startling phenomenon: both the cumulative and annual returns of the 1/x^2 weighting strategy are superior to those of the 1/x weighting strategy, which are in turn superior to those of the 1/\sqrt{x} weighted portfolio, and so forth, ending with the x^2 transformation, whose cumulative returns are the lowest of the seven transformations of Tukey's transformational ladder. Indeed, the order of cumulative returns precisely follows that of Tukey's transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon, and we take care to differentiate it from the well-known "small-firm effect."
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