Tukey’s transformational ladder for portfolio management

Abstract

Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted Standard and Poor (S&P 500) portfolio as well as the often overlooked disadvantages of the market capitalization weighted S&P 500’s portfolio (see Bloomfield et al. in J Financ Econ 5:201–218, 1977; DeMiguel et al. in Rev Financ Stud 22(5):1915–1953, 2009; Jacobs et al. in J Financ Mark 19:62–85, 2014; Treynor in Financ Anal J 61(5):65–69, 2005). However, portfolio allocation based on Tukey’s transformational ladder has, rather surprisingly, 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 (Tukey in Exploratory data analysis, Addison-Wesley, Boston, 1977): $1/x^2,\,\, 1/x,\,\, 1/\sqrt{x},\,\, \text {log}(x),\,\, \sqrt{x},\,\, x,\,\, \text {and} \,\, x^2$, where x is defined as the market capitalization weighted S&P 500 portfolio. Accounting for dividends and transaction fees, we find that the 1/$x^2$ weighting strategy produces cumulative returns that significantly dominate all other portfolio returns, achieving a compound annual growth rate of 18% 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. 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.

Appendices

Appendix A

Table 15 Annual returns (in %) for the EQU and the seven Tukey transformational ladder portfolios from 1958 to 1988

Table 16 Annual returns (in %) for the EQU and the seven Tukey transformational ladder portfolios from 1988 to 2015

Appendix B: Why rebalance monthly?

In Appendix B, we show that it is advantageous for investors holding the $1/x^2$ portfolio to rebalance their portfolios monthly.

In all of the calculations below, we begin with $100,000 in 1958 dollars. We assume transaction and administrative fees of $1 (in 2015 dollars) per trade and, additionally, a long-run average bid-ask spread of .1% of the closing value of the stock. Rebalancing daily, the portfolio goes broke. Having already considered monthly rebalancing shown in Fig. 1 in the main document, we now turn to an analysis of quarterly rebalancing and yearly rebalancing.

We first consider quarterly rebalancing. Figure 6 displays the cumulative returns calculated from 1958 to 2015 of the equally weighted S&P 500 portfolio (EQU) and the seven Tukey transformational ladder portfolios ($1/x^2,\,\, 1/x,\,\, 1/\sqrt{x},\,\, \text {log}(x),\,\, \sqrt{x},\,\,x,\,\,x^2$), where x is the market capitalization weighted portfolio, and the portfolios are rebalanced quarterly.

Fig. 6

Tukey transformational ladder portfolios with quarterly rebalancing from 1958 to 2015

The cumulative returns displayed in Fig. 6 are reproduced in Table 17.

Table 17 Ending balance on 12/31/15

We next consider annual rebalancing. Figure 7 displays the cumulative returns calculated from 1958 to 2015 of the equally weighted S&P 500 portfolio (EQU) and the seven portfolios given by the Tukey transformations ($1/x^2,\,\, 1/x,\,\, 1/\sqrt{x},\,\, \text {log}(x),\,\sqrt{x},\,\,x,\, x^2$), where x is the market capitalization weighted portfolio, and the portfolios are rebalanced annually.

Fig. 7

Tukey transformational ladder portfolios with yearly rebalancing from 1958 to 2015

The cumulative returns displayed in Fig. 7 are reproduced in Table 18.

Table 18 Ending balance on 12/31/15

We conclude by summarizing the findings of Figs. 6 and 7 for the $1/x^2$ portfolio. When rebalanced quarterly, the balance of the $1/x^2$ portfolio on 12/31/15 is $1.054 billion. When rebalanced annually, the value of the $1/x^2$ portfolio on 12/31/15 is $999.798 million. The $1.477 billion figure for the ending balance on 12/31/15 for the monthly rebalanced $1/x^2$ portfolio (Table 6) exceeds that of both quarterly rebalancing (Table 17) and annual rebalancing (Table 18).

Appendix C: Results of bootstrap for random N

Appendix C displays the bootstrapped distributions for fixed N for seven different values of $N(N=20,50,100,200,300,400,500)$. The results herein are presented to support our findings in Sect. 6.2 of the main manuscript.

Fig. 8

Bootstrap distribution for $N=20$

Table 19 Sample statistics for the cumulative return on 12/31/15 for $N=20$, calculated from 20,000 simulations

Fig. 9

Bootstrap distribution for $N=50$

Table 20 Sample statistics for the cumulative return on 12/31/15 for $N=50$, calculated from 20,000 simulations

Fig. 10

Bootstrap distribution for $N=100$

Table 21 Sample statistics for the cumulative return on 12/31/15 for $N=100$, calculated from 20,000 simulations

Fig. 11

Bootstrap distribution for $N=200$

Table 22 Sample statistics for the cumulative return on 12/31/15 for $N=200$, calculated from 20,000 simulations

Fig. 12

Bootstrap distribution for $N=300$

Table 23 Sample statistics for the cumulative return on 12/31/15 for $N=300$, calculated from 20,000 simulations

Fig. 13

Bootstrap distribution for $N=400$

Table 24 Sample statistics for the cumulative return on 12/31/15 for $N=400$, calculated from 20,000 simulations

Fig. 14

Bootstrap distribution for $N=500$

Table 25 Sample statistics for the cumulative return on 12/31/15 for $N=500$, calculated from 20,000 simulations

Appendix D: Value of S&P 500 portfolio over the 1980–2015, 1990–2015, and 2000–2015 horizons

In Appendix D, we show that the returns of the eight portfolios under consideration precisely follow the order of the Tukey transformational ladder for three additional time horizons: 1980–2015, 1990–2015, and 2000–2015.

We first consider the 1980–2015 horizon. We invest $272,028 on 1/2/80 (the equivalent of $100,000 in 1958 dollars) and let each portfolio grow until 12/31/15. The cumulative returns are displayed in Fig. 15 and Table 26.

Fig. 15

Cumulative $\text {log}_{10}$ returns (from 1980–2015) for the EQU portfolio and the seven Tukey transformational ladder portfolios. The calculation assumes that $272,028 is invested on 1/2/80 and left to grow until 12/31/15

Table 26 Cumulative returns for the EQU portfolio and the seven Tukey transformational ladder portfolios. The calculation assumes that $272,028 is invested on 1/2/80 and left to grow until 12/31/15

We now consider the 1990–2015 time horizon. We invest $445,455 on 1/2/90 (the equivalent of $100,000 in 1958 dollars) and let the portfolios grow until 12/31/15. The results are displayed in Fig. 16. The cumulative returns displayed in Fig. 16 are reproduced in Table 27.

Fig. 16

Cumulative $\text {log}_{10}$ returns (from 1990–2015) for the EQU portfolio and the seven Tukey transformational ladder portfolios. The calculation assumes that $445,455 is invested on 1/2/90 and left to grow until 12/31/15

Table 27 Cumulative returns for the EQU portfolio and the seven Tukey transformational ladder portfolios. The calculation assumes that $445,455 is invested on 1/2/90 and left to grow until 12/31/15

Finally, we consider the 2000–2015 time horizon. We invest $590,210 on 1/2/00 (the equivalent of $100,000 in 1958 dollars) and let the portfolios grow until 12/31/15. We display the results in Fig. 17. The cumulative returns displayed in Fig. 17 are reproduced in Table 28.

In conclusion, Tables 26, 27 and 28 each show that the portfolio returns precisely follow the order of the Tukey transformational ladder.

Fig. 17

Cumulative $\text {log}_{10}$ returns (from 2000–2015) for the EQU portfolio and the Tukey transformational ladder portfolios. The calculation assumes that $590,210 is invested on 1/2/00 and left to grow until 12/31/15

Table 28 Cumulative returns for the EQU portfolio and the seven Tukey transformational ladder portfolios. The calculation assumes that $590,210 is invested on 1/2/00 and left to grow until 12/31/15