In the previous post, we investigated whether outperforming funds report their results earlier in the month, and whether that induces a systematic bias into the early reported hedge fund indices relative to the finalized values at the end of the month. The table below shows my measurements of the Dynamic Trading Risk Factor, as estimated through the procedures outlined on this blog, as determined according to a fairly ad-hoc updating schedule determined by my commitments to other projects.

For each month we present the first recorded estimate of the factor return, the final recorded estimate of the factor return, and a “corrected estimate” or forecast final value based upon our bias expression from the prior post and the initial estimate. We compute residuals for two hypotheses:

1. the null, H₀, that the early measurement is an unbiased estimate of the final value; and,
2. the alternate, H₁, that the corrected estimate is a better forecast of this final value.

We can compare these hypothesis by evaluating their forecasting skill, which gives a 23% edge to the corrected estimate. This skill is usually defined by the following equation, where MSE means mean square error.

In an earlier post we addressed the question as to whether there is any bias in the delay between a fund's month end and the performance of that month. Our cynical theory is that managers with good news to report report it early and those with bad news to report report it late.

Since September, 2009, I've been sampling the reported monthly return of the BarclayHedge Hedge Fund Index. This is a simple average of the monthly returns of all funds that have reported to the group at that time. From time-to-time during each month I've sampled the main index's reported average monthly rate of return and the number of funds that have reported. You can find this data on my blog at the page Return Index Accumulation Report. The chart below is an analysis of the error, meaning the difference between the average monthly rate of return for the entire universe reporting on the sample date and that value finally reported at the end of the month. To gauge the average scale of the bias we fit a simple model by least squares:

Here B is the bias, or the average error between the sampled monthly rate of return and the final monthly rate of return; S is the scale we seek to estimate; and, p is the proportion of funds reported (i.e. the number in the sample divided by the final number of funds reporting in that month). Our estimate of the average scale of the error is (51 ± 8) bp/month.

Professional managers are fully awhere of the transient and random nature of the returns they create, whether actively or passively, and are real human beings with the behavioural biases and oddities that characterize us as a group. Thus, when we are presented with a month in which we do very well, we are aware that the future will likely hold periods of underperformance. Furthermore, it is likely that the month following a good month, the month during which we are preparing a formal summary of the prior returns that we know were good, we are more likely to underperform that recent history than outperform it. Nobody wants to write the letter:

Dear Investor, last month we did very well. However, as I write this I know that we're doing less well, so don't get too carried away with your newfound wealth that I've already lost.

Furthermore, a manager who is confessing to a particularly dire prior period of returns would greatly like to write:

Dear Investor, last month we did badly. However, as I write this I know that we're doing very well, so please do not distress too much over your losses, which have already been erased.

For an example of this latter tendency, I can simply refer to my prior post on the September, 2009, performance of our NASDAQ-100 futures trading system. Both these forces together, provide the incentive for outperforming managers to report their returns promply and for underperforming managers to linger a while before sending the letters out of the door. Thus, we can explain the tendency observed in our analysis of the incremental updates of the BarclayHedge data.

In my prior post, describing the August, 2009, performance of the Dynamic Trading Risk Factor, I alluded to the fact that the source data, the Barclay Hedge hedge fund indices, does not accrue atomically — in fact it is gradually updated during the course of each month.

In prior posts, I pointed out that the final month's factor return is mostly little changed from that computed from earlier, incomplete, data. As it should be, if the Law of Large Numbers is working it's usual magic. However, nature is often a little more subtle than that, and mankind often deliberately so. It therefore falls to us to ask whether there is any bias between the latency of a given fund's publication of its performance numbers and that same fund's performance relative to the mean of all reporting funds? Our strong prior is that human frailty would lead the bearers of good news to rush it out as soon as possible; and, in contrast, the bearers of bad news to wait a little longer than they should before revealing their shame.

The above chart illustrates my attempt to quantify this effect on the September, 2009, data, which represents the aggregated reported August, 2009, returns of around 2,500 funds as reported during the month. Not possessing an inside track at the data vendor, I had to resort to archiving and timestamping sampled copies of the cumulatively reported data. Thus, it was tedious to collect. Exhibited above is the mean relationship between the relative error in the Hedge Fund Index's reported average return and the relative proportion of the total number of funds that had reported at that time. (In both cases, relative means relative to the final sample value I have for that month's datum.) From this sample there does appear to be a very strong negative correlation, confirming our cynical prior from the earlier paragraph. Unfortunately, the serial correlation of the errors complicates a statistical analysis of the goodness of fit — but by eye it appears excellent.