Factor

By: Rob Bush | April 11, 2017

For any multifactor investor, it's critical to really put the microscope on the excess returns of a fund to try and identify the main return drivers. And a good starting point is probably to regress a fund’s excess returns on a number of single factors to determine two things: the right factors that explain the excess returns, and, what those sensitivities look like. An example will help.

Using single and multifactor index returns from FTSE Russell, we regressed the daily excess returns of the FTSE Russell Comprehensive Factor Index on the excess returns to the five single factors that the index incorporates - value, size, momentum, low volatility and quality. The results are shown in Figure One.

Source: Deutsche Asset Management 6/30/2001—6/30/2016. The T-statistic is used to test hypotheses. It is a ratio of the departure of an estimated parameter from its notional value (total value of a leveraged position’s assets) and its standard deviation (volatility). The p-value is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. Past performance does not guarantee future results.

There are a few important points to consider with a process like this: understanding the numbers, interpreting them, and caveating them.

First, the r-square. This number provides the proportion of variability in excess return explained by the five factors. Here, that number is 77% which is quite high. It means that the majority of the variability is indeed accounted for by the reasons that you'd hope (overweighting stocks that have these characteristics).

The coefficients to each of the factors are really just "betas", and can be interpreted in the same way that we'd interpret the beta of any single stock - a gauge of sensitivity to another variable. So, just as we'd be used to saying that Stock A's beta of 1.2 means that, on average, if the market is up by 10%, then Stock A should be up by 12%, we can say that the 0.38 beta for Quality means that if the excess return to the Quality factor is 1% then that ought to contribute 0.38% to the excess return on average.

Finally, standard error, t-statistics and p-values give an idea of the statistical significance of the betas and should not be ignored. Significant betas should have low standard errors, high t-statistics and low p-values. A rule of thumb is that a t-statistic above two means that the beta is significant (i.e. different from zero). Note again that for all five of these factors that is the case.

One way to usefully interpret these numbers is to consider the fact that, if you multiply the betas (coefficients) by their respective excess returns, then that should describe the total excess return. So, for example, on a day when each of the five excess returns was 1%, then one would expect the excess return to the multifactor index to be 2.38% (1% * 0.38 + 1% * 0.70 + 1% * 0.17 + 1% * 0.61 + 1% * 0.52). This is the so-called fitted-value and it can be helpful to plot the fitted value against the actual returns to confirm visually that the relationship broadly holds.

There are a number of intricacies and assumptions within regression analysis that are beyond the scope of this discussion, but we would like to highlight just three critical issues around betas that investors should keep in mind.

First, running this regression on fewer than five factors, instead of all five, will produce a different beta - sometimes significantly different. And, this makes sense once one remembers that ultimately a regression is answering a very specific research question. An analogy may help here. Suppose you are given the free-throw percentages of a random sample of 1000 people and asked to work out what factors lead to better percentages. You might start by asking for the height of each of the 1000 people, and there's a good chance you'd find that height was an advantage (i.e. taller people had better percentages). In that instance, you would have a "beta" to height that explained hoop hitting ability. But, overall, the model would be a fairly basic one.

Now suppose that, in addition to height, you were also given each person's number of years of basketball experience. Now you would have two factors that explained their free throw shooting - height and experience. Clearly, the model has become more sophisticated and, in all likelihood, much better. You could now better explain why some tall people are terrible, and some short people are good (assuming that coaching paid off). In addition, a couple of things will almost certainly happen. The r-square will go up, reflecting this richer, more-nuanced model, and the beta to height will change, reflecting the fact that experience is also now being taken into account.

The second caveat, on the topic of changing betas, is that a regression will, rightly, be sensitive to the definition of the explanatory factors. So, returning to our basketball example, you will get different sensitivity to the coaching decision, depending on how you define it. Using "more than 200 hours per year" for example, instead of "more than 50 hours per year" will give different answers. Probably directionally similar, but different. Similar to coaching, investment factors can be defined in different ways and, given that specific funds will use their own specific definitions, they will tend to have higher betas to factors defined according to their own criteria, rather than factors defined according to other criteria.

Unfortunately, there's no easy solution to that problem. Using the definitions of a reliable third party system is one option, as is simply regressing on the proprietary definition and being aware of the limitation.

The final caveat is that the absolute beta size, while important, is only half of the story. The likely values of the factor itself are also important. In other words, I should care about my sensitivity to a factor, of course, but I should also care about the excess return to the factor itself. An example will clarify. A fund that has a beta of one to a factor that returns one basis point per year will, on average, have the same pick up to that factor. Clearly, it has a very high degree of sensitivity to something that effectively doesn't matter. Similarly, a fund that has a beta of 0.5 to a factor with an excess return of 10% per year will add 5%. The sensitivity, or beta, has been halved, but the factor now matters.

To round out our basketball analogy, it would be like asking your 1000 players to specify if they have played in the NBA. The chances are that if they answer yes, then that will have a very powerful impact on their free throw percentage (high beta), but if very few answer yes that was not a great factor to have added to your model.

To get under the hood of a multifactor fund’s return drivers, and gauge their sensitivities, start by regressing the multifactor excess returns on the single factor excess returns, but be aware of the subtleties around this approach.

To gauge sensitivities around multifactor fund’s return drivers, start by regressing the multifactor excessreturns on the single factor excess returns,

but be aware of the subtleties around this approach.

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