Overlapping Effect and the Directionality Maps

Untangling the Overlapping Signals

As we have previously described, analyzing the relation of only one factor with the actions of a fund could result in overlapping signals. In such a case we cannot be sure if the reason of the action was the given factor. Moreover, it is quite logical to expect that the modern funds use more than 1 factor in making their trading decisions. So, trading actions based on different factors would introduce a noise. The theoretical method of resolving the overlap is to consider the cases when all except one factor values are identical and then analyze the fund’s trading action in relation to that one factor. In practice, of course, it is

Figure 2.25: Action Factors (1994-2014)

very hard to find stocks with all except one value being identical.

To resolve the issue of the overlap, for each factor I compare only the cases when the other factors are ”close enough” to each other. I do that for each fund in the following way:

• We save the coordinates (v1, ..., vd) of each stock towards which a given

fund had a Buy/Sell action

• For each factor j ∈ 1, ..., d, we consider the d − 1 dimensional subspace of the rest of the factors

Figure 2.26: Action & Attention Factor Pairs (1994-2014)

As an example, I performed the aforementioned steps for the fund with ID k = 13 of our database. I took 3 factors: size, bm and pcf for clustering and analyzed the directionality of the mom factor in each cluster. As we can see in the figure 2.30, in different clusters the fund has different directionality graphs. The combined (without clustering) directionality graph of the momentum factor is given in the figure 2.31.

The phenomenon also holds for other randomly selected funds in our database. To conclude, we saw that a fund might have different preferences towards the same factor in different factor regions. Hence, it may not be possible to aggre- gate the preferences of multiple funds by simply combining their 1-dimensional absolute directionality graphs.

Figure 2.27: Results of the IVNM A|A fitting on the remaining 63 factors (1994-2014)

Figure 2.29: Accessor Funds, Inc: Small to Mid Cap Fund; Class A Shares

Figure 2.30: Different Directionality in Different Clusters

Directionality Maps:

In theory, if we have a very large amount of observations, we could construct analogues of the directionality curves, but in multiple dimensions. In such a case, the overlapping effect will be no more relevant. However, considering too many dimensions is not practical either. The more dimensions we have, the exponentially less observations we get per a discrete lattice of a fixed length. So, the number of factors I will typically choose is less than 5. Moreover, with the 1-dimensional analysis we were able to find the absolute directionalities of the funds towards single factors. For the majority of the funds the directionality is limited to less than 5 factors.

Next, using an analogous approach as in the one-dimensional case, I construct a multi-dimensional directionality curve, which I call the preference directional-

ity map. In multiple dimensions, I use the range-search algorithm (see Bentley (1979) or Robinson (1981)) for the error smoothing. The algorithm works in O(dnlog(n)) time and runs typically in less than a second (on a desktop PC) if applied to any single fund in our database. After applying the smoothing, I also use a triangulation-based linear interpolant to first interpolate and then extrap- olate the values of the map at different points. The directionality map of the k=11 entry of our database is presented in the figure 2.32.

We could see that, although the aforementioned fund prefers low bm as its absolute directionality, as a result of multi-dimensional preference analysis, the factor mom overlaps the effect of the bm factor. Hence, we conclude once more that combining mutual funds by their general trading philosophies could intro- duce a bias because funds trade differently in the different factor regions. To resolve the issue, I will introduce a theoretical framework which will help us to

Figure 2.32: Sample Directionality Map - Accessor Funds: Growth Portfo- lio

understand how to combine the directionality maps of the different funds.

Further, I apply the model to reconstruct the directionality maps of two value and two growth funds. I chose the funds from BlackRock and Fidelity to rep- resent popular US equity mutual funds (see figure: 2.33). We can see that the growth funds (the upper two maps) have completely opposite trading patterns compared to the value funds (lower two maps). Although the directionality maps from different funds look quite different, it is also easy to note the visual similarity within the value funds and the growth fund. The two value funds tend to buy high bm and low mom stocks, which they sell when the mom in- creases. The two growth funds, on the other hand side, buy high mom stocks and sell them when the mom becomes lower. Thus, using the directionality maps we can clearly see the similarity within the different beliefs on how to

generate profits. Moreover, we note that two different funds under a larger company could take completely opposite bets in the equity market (counter- party).

Figure 2.33: Example: Directionality Maps of Value and Growth Funds

Combining the Directionality Maps:

We have seen that there exists some persistence in the trading patterns of the mutual funds. Given the stochastic and sometimes discretionary nature of in- vesting, it is unlikely that one could predict the exact trading decisions of an actively managed fund. However, a natural extension might be to combine the directionality maps of multiple funds. By doing so, the stochastic aspects of in- vesting could ”cancel each other out” and predicting the aggregate fund actions could become realistic. I devote the next chapter to discussing this topic.