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Statistical Graphs for Portfolio Returns
When You Have Questions the Statements Don't Answer

by Steve Miller, President, OpenBIFriday, November 16, 2007

Around the second week in January this year, I received annual statements for my meager mutual funds and managed accounts. The mailings detailed portfolio performance for the year, for 3 years, and for 5 years. 2006 was pretty good for the markets, as evidenced by lots of paper and hefty market write-ups. But, for me, a BI analyst, there was precious little supporting performance data. I had more questions than the statements had answers.

The January through December returns weren't entirely applicable to me, since I wasn't completely “in” on January 2, the starting point for calculated returns, and thus didn't participate fully for the year. And the 1, 3, and 5 year returns looked quite good until I realized we were now 4 years post bubble-burst recession. The 1 and 3 year figures were now completely post-recovery, and, for the 5 year return, we had just traded a bad year (2001), for a good one (2006).  I was frustrated that I had the1 year performance from January 1, 2006 through December 31, 2006, but not September 1, 2005 through August 31, 2006, or February 15, 2005 through February 14, 2006, when I might have invested my money. I was also frustrated my time slices were limited to 1, 3, and 5 years, when in fact my savings horizon was more like 10 to 15 years. So I decided to find the data to answer some of the questions the materials raised.

As luck would have it, Ken French, a professor at Dartmouth who's affiliated with the Center for Research on Security Prices (CRSP) at the University of Chicago, makes available 44 years of monthly, weekly, and daily returns for various  portfolios on his website (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). Choosing the daily returns, I first created a dataset consisting of  8 academic portfolios, 7 stock and 1 T-Bill (Risk Free, Market, Small Growth, Small Neutral, Small Value, Large Growth, Large Neutral, and Large Value). I then loaded the 11,000+ records (approximately 252 trading days per year times 44 years) into the R statistical package for analysis and graphics. With 44 years of daily data for 8 portfolios, I was now in a position to examine returns quite flexibly.

My very simple first questions of the data were as follows: 1) What was the Market growth of an initial $1 invested for 2006 (Jan 1-Dec 31)? and 2) What are the ranges of such Market returns for all one year periods ending in 2006 (Sept 30, 2005 to Sept 29, 2006, for example)? Corresponding answers: The Market growth of $1 invested for Jan 1-Dec 31, 2006 was $1.16; the minimum Market growth of $1  for a year ending in 2006 was $1.04, the maximum $1.22. If we consider the same question for bubble-bursting 2000, we get : $.89, $.87, and $1.33, respectively. Finally, 2002 yields: $.79, $.70, and $1.06. The range of returns for the years examined appears to be quite wide, and can vary considerably from the Jan 1-Dec 31 figure, suggesting the reported numbers might in many cases be a poor proxies for the actual returns real investors experience. I decided to examine further.

With daily data, it is feasible to calculate all possible 1 year returns by day. Thus, we can compute a growth of $1 return for Jul 1, 1963 to Jun 30, 1964, another for Jul 2, 1963 through Jul 1, 1964, another for Jul 3, 1963 through July 2, 1964, and so on – all the way to Oct 1, 2006 through Sept 30, 2007. In sum, there are over 10,000 of these day to day, one year returns in the 44 year period. I calculated all such returns for each of the 8 portfolios, then tabulated the corresponding percentile distributions, finally displaying all data in a graph. Figure 1A, shows the results. For each portfolio, the growth of $1 is shown on the y axis and the percentile distribution on the x. The portfolios are ordered from left to right by the median growth of $1 return, with Risk Free the smallest and Small Value largest. The graph is a Trellis, so the scales for each of the portfolio panels are identical, facilitating comparison. Note that all possible Risk Free returns are in the black, as would be expected. Note also that each portfolio has many years of losses, as evidenced by the points below the break even horizontal line at $1. At the same time, most of the points show a positive return. Still, as indicated by these findings, investors with just a 1 year horizon should avoid stock portfolios for the safety of Risk Free assets like a T-Bills.

Portfolio Dot Plot Graph
       Figure 1A

Figure 1B  provides a summary view of the same data using a modification of the handy boxplot. The portfolios are ordered from top to bottom by median return. The portfolio “cartridges” are interpreted as follows: the thickest rectangle houses the middle 25% of returns; the next thickest rectangle the middle 50%, followed by 75%, 90%, and 100%, respectively. The vertical bar denotes the median growth of $1 return, the dot the mean. The vertical line at $1 is the break even point.
 
Portfolio Boxplot Graph
Figure 1B

Not surprisingly, this graph corroborates Figure 1A, showing the much wider range of 1 year returns, including negative, for stock portfolios in contrast to Risk Free. Note the extremes of Small Growth 1 year performance: $1 “growing” to $.50 in some cases while doubling in others.

Portfolio Dotplat Graph
      Figure 5A

The same calculations and graphics used to showcase 1 year growth of $1 returns are also pertinent for 5 years. We calculate all possible 5 year returns, starting with July 1, 1963 to June 30, 1968 through Oct 1, 2002 to Sept 30, 2007. Figure 5A presents the percentile distributions; Figure 5B the corresponding boxplots.

Portfolio Boxplot Graph
Figure 5B

Over 5 year time periods, the performance advantages of stock portfolios over Risk Free become more prominent. While investors can still earn less with stock portfolios than with Risk Free, and in fact can lose principal, stock portfolio returns show much higher upside at 5 year consideration. The median 5 year growth of $1 for Small Value exceeds the maximum return for Risk Free.

Dotplot Graph
      Figure 15A

Finally, we calculate and display the 15 year returns in Figure 15A and Figure 15B. None of the 7 stock portfolios has a decreasing 15 year growth of $1 return. And while there are overlaps between Risk Free and stock , the lowest returns of 3 portfolios: Large Value, Small Neutral, and Small Value, exceed the highest of Risk Free. It seems the longer the investment horizon, the better stock portfolios perform relative to Risk Free. Go figure!

Boxplot Graph
Figure 15B

Lest anyone become too giddy about the results presented, let me offer words of caution. The returns of 1963 through 2007 depicted here may, in fact, not be representative of returns going forward. Like financial services firms, we note that the past is no indication of the future – but then ignore our own warning! The portfolios discussed are academic and hypothetical, though approximations exist with mutual fund purveyor, Dimensional Fund Advisors (http://www.dfaus.com/). Readers should be wary of equating the Small and Value maintained by CRSP to those with the same names from other vendors, since each has unique definitions, making the portfolios not fully comparable. Also, in contrast to commercial mutual funds or ETFs, there are no fees associated with these portfolios. Over time, the impact of 0.25% to 1.50% annual charges takes a significant toll on returns. Finally, the differences in returns of the portfolio styles examined is quite stark, but investors should be aware that style performance runs in cycles. Large and Growth dominated in the mid to late nineties, while Small and Value prevailed from 2000-2005. There is evidence that style popularity is changing again -- back to Large and Growth. So now might not be the time to drop your savings in a Small Value portfolio.

One result that presents unequivocally in these graphs is the salutary effect of time on returns.  For holding periods of 5 years or less, there's a clear risk that stock portfolios will under-perform the Risk Free asset and maybe even lose principal. At 15 years, each portfolio shows positive returns for all instances, and three -- Large Value, Small Neutral, and Small Value -- dominate. For longer periods, that effect is even more pronounced, showcasing the magic of compounding. Investors who buy solid portfolios and hold them for long periods of time will most likely be amply rewarded. Doesn't this sound like something Warren Buffet might say?

About the Author:

Steve Miller is President of OpenBI, LLC, a Chicago-based services firm focused on delivering business intelligence solutions with open source software. A statistician/quantitative analyst by education, Steve has 30 years BI experience. His charter – and OpenBI's – is to help customers manage performance through optimal deployment of analytics. Steve is a columnist for DMReview and writes also for BIReview and the B-Eye-Network. In addition to R, OpenBI specializes in the Pentaho and JasperSoft open source BI platforms and Weka data mining. Steve can be reached at steve.miller@openbi.com.

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