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Portfolio Returns and the Bootstrap

by Steve Miller, President, OpenBIMonday, December 7, 2009

It's been a tough couple of years in the stock market. In 2008, the Wilshire 5000 Total Market index declined a calamitous 39%. Through October, that same index has advanced 16.8% for 2009,  an overall 2008 through October decline of about 28%. Unfortunately, portfolio return arithmetic is pretty cruel. A drop of 50% in year one mandates an advance of not 50% but 100% in year two just to get back to even.

What will stock returns look like in 2010, 2011 and beyond? We certainly don't know, though there's no shortage of opinion in the business press and blogs, both bullish and bearish. I tend to listen to experts like Warren Buffet and John Bogel, who've stood the test of investment time. Alas, they're warning us to prepare for lower annual returns than we've experienced historically.

Increasingly, investment planners are using Monte Carlo simulation techniques to estimate future market performance for their customers. With basic assumptions about the nature and distribution of market returns, these simulations use computing power and random number generation to help predict the future.

Among the the more useful simulation techniques for investment planning is the bootstrap, formulated in 1979 by Brad Efron, a statistics professor at Stanford University. The idea behind the bootstrap is quite simple. Analysts are interested in knowing information about a population, but observe only a sample of data from that population. The bootstrap treats the available sample as the population, using the computer to generate repeated random samples with replacement (resamples)  from that sample data, in turn calculating statistics of interest from these resamples. Over thousands of iterations, the distribution of such statistics can be well behaved. Rather than wrestling with arduous and often presumptive mathematics to make inferences, the analyst finesses from the multitude of bootstrap samples using computer power.

Ken French, a financial economist from Dartmouth, provides access to over 46 years of investment returns data on his website. For the analyses that follow, I used daily returns for eight different portfolios from July 1963 through October 2009. These include a Risk Free index of 3-month Treasury Bills, a Total Market index similar to the Wilshire 5000, and six indexes comprised of two size categories by three value classifications – Small Growth, Small Neutral, Small Value, Large Growth, Large Neutral, and Large Value.

From 7/1963 through 10/2009, there were 11,665 daily returns for each of the 8 portfolios. $1 invested in the Risk Free portfolio at the outset would have been worth $12.20 at the end of October, 2009, an annualized return of about 5.55%. Similarly, $1 invested in the Total Market index portfolio in 7/1963 would have grown to $66.42, an annualized percentage of 9.49%. The most lucrative portfolio was Small Value: $1 in 1963 would have become $848 in 2009 – an annualized growth rate of 15.7%!

A bootstrap methodology applied to this data might look something like the following. First, construct vectors of length 11,665 for each of the eight portfolio daily returns. Next, simulate the 1 year, 5 year, 10 year and 20 year performance of each of the eight by calculating and storing hypothetical growth of $1 returns of multiple bootstrap samples. Use a bootstrap sample size computed as 252 days per year multiplied by the number of years. For example, for 5 year returns, the number of daily percentages randomly sampled with replacement is 252x5=1260. For each of the 4 time frames and each portfolio, resample a large number of times, say, 100,000. Finally, compute percentile distributions (quantiles) for the calculated bootstrap returns. The growth of $1 at say, the 75 percentile, would be the return greater than or equal to 75% of such calculations.

To perform the computations, I used the open source R Project for Statistical Computing. R is an integrated platform  consisting of a programming language, a data management capability, statistical/mathematical procedures and graphics. For the analyses here, I used basic R language programming in tandem with the lattice graphics package.

Below are graphs depicting the results of 100,000 bootstrap samples for each of the eight portfolios for 1 year, 5 year, 10 year and 20 year simulations. Consider first the 1 year results. The chart on the left shows the percentile returns organized by Size. Note that each of the two panels is comparable on x and y scales. The left panel displays the 3 distributions of Small portfolios, along with Market and Risk Free; the right the Large portfolios with replicated Market and Risk Free. For each portfolio's plot, the x axis lists the percentiles from 0-100; the y axis the portfolio returns on a log base 2 scale. Logs are critical for visualizing stock performance, since returns multiply and get large quickly. A base 2 log has a natural investment interpretation, indicating the number of times the portfolio return has doubled. Zero represents no change; 1 depicts a single doubling; -1 a halving, etc. As an illustration,  35% of the Small Growth portfolios are in the red after 1 year – a log 2 return value less than 0, which translates to a shrinkage of an initial $1 investment.

graph depicting the results of 100,000 bootstrap samples
click on image for a more detailed version

The essentially horizontal Risk Free curve demonstrates little difference in the percentile distribution of returns – the 10th percentile of Risk Free is roughly the same as the 90th. This is no surprise of course. That Value and Neutral are above and to the left of Market and Growth in the Small panel indicates they perform better in the short time frame with this methodology. The horizontal line at 0 denotes the break-even mark. The vertical line that intersects the Market portfolio at 0 shows that over 25% of Market returns are in the red for the 1 year time frame. Similarly, the vertical line through the intersection of the Market and Risk Free demonstrates that 40% of the time the Market portfolio performs worse than Risk Free over a 1 year period. No wonder advisers caution against stock investments for short time horizons. The chart on the right shows the same portfolio returns dimensioned by Value – Growth, Neutral and Value. Even at one year, there's some indication that Small performs better in Neutral and Value. Finally, the darker gray horizontal lines include 95% of Market returns. 95% of Market log base 2 returns are between roughly -.32  and .57, indicating a $1 initial Market investment growing to between about $.80 and $1.48 95% of the time the first  year.

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