In previous posts, I have presented histograms of the sample distribution of the monthly returns estimated for the dynamic trading risk factor. With the most recent data set, we noted an apparent cluster of months with returns in the region of 2%. When performing an ad hoc analysis of data, it is my normal practice to make a histogram of the data and to fit a parametric model for the underlying probability density function (usually the generalized error distribution).

Histogramming is a basic and popular statistical technique. Among its benefits are:

• the definition of an histogram is elementary;
• histograms can be rapidly composed by hand (i.e. in the field, without the aid of a computer);
• their interpretation is straight forward;
• their sampling properties are well understood; and,
• fitting a model to the data is a rapid and well understood procedure

• the analyst has to make an arbitary choice of binning parameters;
• the resulting estimate is a step function, which likely does not represent the true underlying continuous p.d.f. and cannot be used to estimate the gradient of that function.

A populate alternative is Kernel Density Estimation, which estimates the population p.d.f. from the data via a kernel smoothing algorithm applied to the raw, unbinned, data. This procedure is easily to implement by computer, but not really suitable for use by hand. It has the advantage of the estimator being smooth and fairly robust with respect to the choice of smoothing kernel. The choice of smoothing bandwidth, analogous to the bin width in a histogram, is arbitary — but there are many good choices suggested in the literature. The chart above shows the application of this procedure to our data set: the monthly returns of the dynamic trading risk factor. This analysis illustrates a feature that, I feel, is not particularly noticable in the histograms previously made — that the data has a pronounced negative skewness. It doesn't appear to back the hypothesis that there is an anomalous cluster at exactly 2%. On the subject of the origin of this skew, of course, this data is silent. A suspicious mind might suggest that there is a tendency to adopt cookie jar accounting in a non-public company with obscure accounts — but salacious speculation is all we can really propose on the basis of this data alone.

Although I publish daily my record of our intraday forecasts of the $5 mini DJIA futures (ECBOT:YM) and of the e-mini NASDAQ-100 futures (GLOBEX:NQ), I haven't updated our regression summary charts for a while — so I thought that it was time to do that.

The above chart shows our summary analysis for the performance of the $5 mini DJIA futures system, and the one below the same analysis for the e-mini NASDAQ-100 futures system (which is based on the same platform).

These charts may be a little dense for some — those who wish to perform their own homebrew analysis are welcome to pick up the raw data, delayed by one day, for the two systems at the links at the top of this post.

In his recent post The Curious Incident of Hedge Funds During the Financial Crisis, Tadas Viskanta, mentions our analysis of the returns of Goldman Sachs and that they are well explained by the Dynamic Trading Risk Factor. Our conclusion was that Goldman generates it's returns by engaging in typical hedge fund trading activity, albeit with three times more leverage.

In view of this link, it seemed apropos to update the published charts with the most up to date data I have. In the prior post, I also stated that I personally held no investment position in Goldman Sachs. This is no longer correct. Largely as a result of the analysis referred to here, I now have a long position in Goldman (and other hedge fund like stocks).

With an extra month's data for the Dynamic Trading Risk Factor, we can look to see whether there has been any re-ordering of the members of the XLF that are selected for membership of the Poor Man's Hedge Fund.

The above data shows no change to our prior computation, and the membership is still:

1. Invesco plc IVZ
2. Goldman Sachs Group Inc. GS
3. Morgan Stanley MS
4. T Rowe Price Group Inc. TROW
5. Janus Capital Group Inc. JNS
6. Ameriprise Financial Services Inc. AMP
7. Franklin Resources, Inc. BEN
8. Lincoln National Corp. Inc. LNC

Note that Ameriprise is no longer in the top five, having been replaced by Janus.

Citadel's Kensington Global Strategies Fund, a multi-strategy hedge fund that apparently does a lot of Convertible Arbitrage, is apparently having a banner year so far. Last year, they we're the poster child for the Hedge Fund Crash. I have updated our archive of the Kensington Global Strategies Fund monthly returns data to reflect the most recent updates. (These are available from many sources such as the Market Folly blog.) My record has a hole in the 2008 data; if anybody has that data, or thinks that some of my numbers are wrong, please contact me at blog@gillerinvestments.com with your corrections.

Our regressions of this funds returns onto the dynamic trading risk factor series give an insignificant α and a β which is not significantly different from unity. As such, we forecast a return for Citadel for July, 2009, of 1%. However, it is fair to note that the returns of this fund per articulus (for the data points that I have) seem to drawn from a distribution with a β of unity, as before, but with a very negative α. The group of points is quite distinct from the main scatter. (In my scatter plot, the regression line is blue and the green line represents an identity relationship.)

With about half the funds having reported, it's now time to update our data and forecasts for the dynamic trading risk factor. As is our practice, we're doing this after about 50% of the fund universe has reported. This gives us an early estimate of a number that is not much likely to change during the rest of the month. For a variation, I've fitted the histogram of monthly returns to an Student's t distribution, rather than my normal usage of the Generalized Error Distribution.

From the above chart, we are starting to see a deficit of events in the 0%/month bin and an excess in the 2%/month bin. They are counts of 1 and 11 with an expectation of approximately 5 in each case. With a Poisson Distribution, the probability of drawing 1 or fewer events with an expectation of 5 is 4.0%. The probability of drawing 11 or more is 0.55%. The deficit of may be dismissed as a binning anomaly; however, the surfeit at 2% is a little more interesting. We have a less than 1% probability of this occuring by chance. Furthermore, 2% is a suspiciously round number. Going forward, we shall have to watch this bin with interest.

Out-of-sample, our final forecast of the return for June, 2009, was 2.17% (our early forecast was 2.11%), and the realization was 1.03%. At this point, we are forcasting a 0.75% return for July. These forecasts represent an a priori expected monthly return for any fund or firm that makes it's living by trading.