Our original Poor Man's Hedge Fund was built from five stocks. However, evidence presented in a recent post, studying the tracking of portfolios of various sizes, suggests that the right number might be larger. If one looks at the chart in detail, it's probably best to conclude that the optimal size is somewhere in the region 4–10 stocks. Since my prior bias is too include more stocks, thereby acting to reduce the exposure to idiosyncratic factors, I decided to go with the eight stocks that give the best tracking, as measured by the in sample regression R². The data for these eight are listed in the table below.

The added companies are:

1. Janus Capital Group Inc. JNS
2. Franklin Resources Inc. BEN
3. Lincoln National Corp. LNC

When implementing the Poor Man's Hedge Fund as a small portfolio of companies that, from statistical analysis of prior returns, appear to generate their profits by exposure to the Dynamic Trading Risk Factor, I arbitarily decided to choose an equally weighted portfolio of just five companies. This choice was made without deep thought other than the idea that managing a large, weighted, portfolio is difficult for a small investor, and the whole point of the idea was to provide a vehicle suitable for a small investor.

Not all financial stocks have returns that are explained by this risk factor and in fact, some members of the XLF are very poorly explained by it. So a legitimate question to ask is how the tracking of a composite portfolio behaves as a function of the number of assets included within it?

The chart shows the result of ranking individual regressions of all current XLF member companies onto the dynamic trading risk factor. We rank the 80 index members by their individual regression R²'s and then build a portfolio including an equal weighted combination of the first stock; the first two stocks; the first three stocks; etc., all the way to the entire index membership. We then regress all of those portfolios onto the same risk factor and plot the portfolio regression R² for all of these possible portfolios.

There are two competing forces that describe these portfolio's relative performance. The first is the fact that the larger the portfolio we build the less idiosyncratic risk it will contain and the more systematic risk, in proportion, will be left. This is basic factor theory at work, and it drives us to build as large a portfolio as possible, as that will produce the purest expression of the risk factors. The second is the fact that the lesser ranked stocks have proportionately less exposure to the dynamic trading risk factor, and more to other factors such as general market risk. Thus the more of these stocks we include in the portfolio the more we dilute its tracking of the risk factor we seek exposure to and replace it with general market performance. Between these two extremes should lie an optimum, which seems to be at about eight stocks — although, our arbitary choice of five is not so bad.

In an earlier post we introduced the idea of investing in hedge fund strategies, and thereby picking up the premium income that arises from exposure to the Dynamic Trading Risk Factor — which he hypothesize is the true reason why hedge funds outperform the general market — by investing in the publicly traded companies who's monthly returns are strongly influenced by the returns of that factor.

We used standard linear regression analysis to identify a small portfolio of the five stocks who's monthly total returns are best described by this factor. To select these members, we studied the current constituency of the S&P 500 Financials Select Sector Sub-Index, represented by the XLF exchange traded fund. As this universe selection imposes a style and survivorship bias on our universe, we need to exercise caution in our use of the data.

The chart below illustrates an attempt to peer into this data. Following the cross validation idea of comparing analysis on randomly selected sub-samples of the data, we took the current members of the XLF and, for each member, took a randomly chosen sample of 50% of our data, and it's complement, for regression of the member's montly returns onto our series of the dynamic trading risk factor. Each member's data had a freshly chosen set of random sub-samples, which should remove systematic correlations on a stock-by-stock basis. The data plotted is the signed square root of the sub-sample regression R² for each stock analyzed.

We see that, on average, there does seem to be meaningful similarity of the effectiveness of the risk factor between the two random sub-samples.

The second draft of my working paper on the analysis of the relationship between the number of Twitter followers and the success of our intraday futures trading system is now available online; both here and at my Social Science Research Network Author Page. Below, is a chart of the number of followers as a function of time, together with the model developed in the working paper above.

In the chart the black line represents the number of followers, as measured through the Twitter API; the blue line is our fitted model, using the procedures as described in the paper above; and the shading represents the success variable (dark for right; light for wrong).

I just completed the first draft of an analysis of the relationship between the number of Twitter followers and the success of our intraday futures trading system. You can download it from my site by following the link above, or from my Social Science Research Network Author Page. In summary, we find a statistically weak but positive correlation between forecasting success and the change in the number of followers. I will post more details of this research later.

In an earlier post we examined the covariance between the Poor Man's Hedge Fund and the SPY ETF. Using a cross-validation method, we demonstrated an apparently stable linear relationship between the total montly returns of the Poor Man's Hedge Fund and those of the SPY ETF. Given this relationship, it is natural to ask whether it can be exploited to remove systematic risk factors that are not idiosyncratic to the hedge fund trading style — which is a long winded way to say that we can hedge out market risk, for example by taking a long position in the SDS ultra-short inverse tracking ETF.

The chart above is visually dense (perhaps violating the suggestions of Edward Tufte's The Visual Display of Quantitative Information) but has a lot of information to convey, so please forgive me. Most strikingly, the vertical bars indicate which dates were in the training set (shaded grey) or validation set (left white). These set memberships were chosen at random with a 50:50 chance of a given date being in each set. I seeded the random number generator with a known seed (in this case the number 12345), so that I could reproduce the chart made and importantly to prevent selection bias that I might engage in by picking a chart that looks good.

The four lines are a value added monthly index which represents the cumulative total return of investing in each vehicle. Individually, they are:

1. the total return, including dividends, of the SPY ETF;
2. the cumulative return of the dynamic trading risk factor;
3. the total return, including dividends, of the Poor Man's Hedge Fund tracking portfolio;
4. the cumulative residual return of the Poor Man's Hedge Fund, using the β to SPY estimated with the training set data.

The stories of each time-series are interesting. The total return of the SPY, which is designed to track the S&P 500 share index, for the entire decade to the and of May, 2009 is a loss of 23%. This is a terrible return for a decade. Conventional wisdom is that a ten year horizon for an equity investment to pay off is ten years. (Even Warren Buffett suggests this horizon in interviews.) Over the same period, investment in the dynamic trading risk factor, which should represent the performance of a typical hedge fund, has delivered a return of 65%. On its own, the Poor Man's Hedge Fund delivered 27% — not as good as a pure hedge fund factor play, but impressive versus the benchmark. On a residual basis, this series is an outstanding performer, giving a total return of 111% over the period. However, one should not write off the hedge funds — as we have not presented an analysis of that series with its market risk exposure hedged out.

As a final datum, we can look at the simple sharpe ratio (the ratio of the annualized average monthly return to the annualized standard deviation) of the residual returns series; this is 1.1 for the training set and 0.6 for the validation set.