Here we go the other way around.\u00a0 We take the decision as fact and think of the returns over the period as just one realization of a random process.<\/p>\n
Supposings<\/h2>\n
Suppose we are holding a portfolio at the end of 2010, and it has the constraints:<\/p>\n
- \n
- constituents from the S&P 500<\/li>\n
- long-only<\/li>\n
- at most 50 names<\/li>\n
- no asset may account for more than 4% of the portfolio variance<\/li>\n<\/ul>\n
Suppose further that the value of the portfolio is $10,000,000 and we are willing to trade $100,000 (buys plus sells).<\/p>\n
The trade we did resulted in the return being larger by 16.28351 basis points during 2011 (not counting trading costs).\u00a0 But of course that is only how it happened to have turned out — history could have been different.\u00a0 A gain of 16.28351 basis points was not an inevitable result of that decision.<\/p>\n
(It doesn’t really matter, but in the example presented here the initial portfolio is merely a random portfolio obeying the constraints<\/a>, and the decision is to move towards smaller predicted variance.)<\/p>\n
Bootstrapping<\/h2>\n
One way to assess variability is to use the statistical bootstrap<\/a>.\u00a0 Figure 1 shows the bootstrap distribution of the return difference using daily data for the year.<\/p>\n
Figure 1: Distribution of the return difference for the decision from bootstrapping daily returns in 2011.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackboot.png\" "\\"jackboot\\"")
<\/a>26% of the distribution in Figure 1 is below zero.<\/p>\nThe bootstrap samples the days with replacement.\u00a0 It is (essentially) assuming that the returns are independent.\u00a0 That’s a quite strong assumption — it probably produces a distribution that is too variable.<\/p>\n
We should be able to consider 26% as an upper limit on the p-value<\/a> for the hypothesis that the difference is less than or equal to zero.<\/p>\n
Figure 2 shows a typical case of the number of replicates for each value in a sample with 252 observations.\u00a0 Many observations (about a third) do not appear at all, and a few observations appear more than three times.<\/p>\n
Figure 2: Number of replications for each observation in a bootstrap sample for 252 observations.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/bootcount.png\" "\\"bootcount\\"")
<\/a><\/p>\nJackknifing<\/h2>\n
The jackknife is generally considered to be a primitive version of the bootstrap.\u00a0 Another name for it is “leave-k-out”.\u00a0 The “k” is often 1.\u00a0 The equivalent of Figure 2 for the jackknife would be that all observations are at one replicate except for k at zero.<\/p>\n
The jackknife disturbs the distribution much less — we’ll get less variability.<\/p>\n
Figure 3 shows the return difference distribution from leaving out one daily return during the year.<\/p>\n
Figure 3: Distribution of the return difference in 2011 for the decision from leaving out 1 daily return.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackrd1.png\" "\\"jackrd1\\"")
<\/a><\/p>\nFigures 4 through 8 show the distributions from leaving out various fractions of the returns.<\/p>\n
Figure 4: Distribution of the return difference in 2011 for the decision from leaving out 1% (that is, 3) of the daily returns.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackrd01f.png\" "\\"jackrd01f\\"")
<\/a><\/p>\nFigure 5: Distribution of the return difference in 2011 for the decision from leaving out 5% of the daily returns.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackrd05f.png\" "\\"jackrd05f\\"")
<\/a><\/p>\nFigure 6: Distribution of the return difference in 2011 for the decision from leaving out 10% of the daily returns.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackrd10f.png\" "\\"jackrd10f\\"")
<\/a><\/p>\nFigure 7: Distribution of the return difference in 2011 for the decision from leaving out 20% of the daily returns.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackrd20f.png\" "\\"jackrd20f\\"")
<\/a><\/p>\nFigure 8: Distribution of the return difference in 2011 for the decision from leaving out 50% of the daily returns.
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/jackrd50f.png\" "\\"jackrd50f\\"")
<\/a>The fractions of the distributions below zero are: 0.5% for leave out 5%, 3% for leave out 10%, 9% for leave out 20%, and 26% for leave out 50%.<\/p>\nQuestions<\/h2>\n
A quite interesting feature of Figure 1 is the long negative tail.\u00a0 Any ideas what is happening there?\u00a0 The trade closes MHS and opens WMT, but their return distributions for the year are both reasonably symmetric.\u00a0 The less symmetric MHS has a fairly trivial weight.<\/p>\n
How can we decide what fraction of observations is best to leave out?<\/p>\n
Have others done analyses similar to this?<\/p>\n
Summary<\/h2>\n
If we could figure out how to determine the right amount of variability, this would be an interesting approach.<\/p>\n
Epilogue<\/h2>\n
You’re old enough to make a choice
\nIt’s just that in between all the words on the screen
\nI doubt you’ll ever hear a human voice<\/p><\/blockquote>\nfrom “Walk Between the Raindrops” by James McMurtry
\n