*The Fall issue of the Journal of Alternative Investments contains a great 75 page section on hedge fund replication. Articles cover the latest developments in the two major techniques used to approximate hedge fund returns (factor and distributional replication), performance characteristics of actual hedge fund replication programs, and practical hurdles to implementing these programs.*

*These articles have begun to attract interest from the hedge fund and broader financial communities. One paper by Jean-Francois Bacmann, Ryan Held, Pierre Jeanneret and Stefan Scholz called “The impact of missing factors on replication quality” has caught the eye of AllAboutAlpha.com contributor Pierre Laroche, head of R&D and Innocap, **a joint venture between Canada’s National Bank and BNP Paribas (related post). Below, Laroche examines the delicate balance between adding too many factors and too few factors in a factor-replication model.*

**Special to AllAboutAlpha.com by: **Pierre Laroche, Managing Director, R&D, Innocap Investment Management.

The issue of “missing factors” was raised soon after several major financial institutions launched their HF index replicators last year. The use of traditional regressions by these products raised some questions about the number of factors required to fully capture the nuances of HF returns. Specifically, the more factors one adds, the more likely those factors are to be collinear (correlated), thus lowering the regressors’ efficiency. This property of regression-based HF replicators (along with other properties such as their inability to track abrupt changes in weights) pushed financial institutions to look for more appropriate tracking models. One such model is the “Kalman Filter” (KF).

KFs can contribute greatly to hedge fund replication models for at least two reasons:

- Their tracking algorithm explicitly takes into account that exposure to return-generating factors are
*dynamic*(they vary through time). - The quality of the estimated weights is impacted much less by the presence of highly correlated factors.

In other words, KFs are influenced less by using a small number of highly correlated factors. Unfortunately, however, they do not settle the central question of the ideal number of factors to use when trying to “replicate” HF returns.

It is well known that working with too many factors increases the danger of overfit which,* ceteris paribus*, usually results in a lower *in-sample* tracking error but a higher *out-of-sample *tracking error. To mitigate risk, the “optimal” subset of factors is usually just based on heuristics known as the “information criteria”. For example, it is often suggested that HF index replication use only 4 to 6 factors – sometimes even less.

However, it is a mistake to select the number of factors solely on these guidelines because:

- These models completely ignore the results of more than 25 years of sound empirical literature on financial markets. The arbitrage pricing theory (APT), for Instance, suggests there are some factors that are often totally overlooked. The interest rate slope is perhaps the best example. It requires a
*long-term interest rate factor*– a factor that is not generally included in the information criteria. - The standard list of factors for HF replication are well suited to
*linear*regression-type replication models. But it’s still not clear that we can extend their results to KF-based replication models. - Working with too few factors also has a fairly important hidden cost:
*model risk*. This is a lesser known, but still crucial, problem identified by econometric literature.

Let’s focus on the last point here. To illustrate the problem of *model risk* (too few factors), imagine a portfolio composed of three factors whose allocations vary randomly around 0.4, 0.4 and 0.2 weightings for a long period of time.

Then imagine that these factor weightings suddenly shift to 0.25, 0.25 and 0.4 following a change in market conditions. The following figure illustrates such a scenario (with the three factors represented by the green, red and blue lines):

Now let’s use two Kalman Filters to infer the allocations to these factors from the observed monthly returns of the portfolio.

The first filter only uses the first two factors (the ones that would have been chosen by the “information criteria”) and the second filter uses *all three* assets. The following table contain the out-of sample monthly tracking errors of these two models:

We see that before the market regime change, the two-factor model performs better even if we know that the third asset contributes to the portfolio return.

But after the change in market conditions, the effectiveness of the two-factor model crumbles. In fact, its monthly out-of-sample tracking error is more than twice as large. Across the whole period, the three-factor model performs better even if the after-change period is much shorter that the initial period. Clearly, the third factor adds significantly to our ability to “replicate” this portfolio.

But this is not the end of the story. As can be seen in the upper panel of the exhibit below, during the pre-change period, the fit of the two-factor model is not only very good, but the two estimated weights are remarkably stable at around 0.5 respectively (which is the anticipated result of the two-factor model since they both have a 0.4 weight in the portfolio).

After the market conditions change, however, the allocations still oscillate around 0.5, but they become extremely volatile, which results in significantly higher trading costs. The estimated weights in the three-factor model do not, however, exhibit such an erratic behaviour. The estimated weights remain not only accurate, but they are highly stable as well (as can be seen in the lower panel of the exhibit below).

In conclusion, in line with Bacmann *et al*‘s paper in the *Journal of Alternative Investments*, choosing the right set of factors is a delicate compromise between efficiency (the *fewer *factors the better) and the cost of discarding a factor in case market environment changes (the *more *factors the better).

*– P. Laroche, November 2008*

*The opinions expressed in this guest posting are those of the author and not necessarily those of AllAboutAlpha.com.*