You may have seen in different papers or websites, analysis of how a specific active portfolio is exposed to different financial factors (Value, Growth, Size, Quality, etc).

This insight is very interesting in order to know what to expect from a strategy and to explain and understand its behaviour, compared with a benchmark (which defines the field of play).

The analysis could be done by several means, such as:

• A regression analysis over historical returns of the active portfolio and the factor scores to calculate the attributed performance to that factor.

• Calculate the spread between the weighted average factor score of the active portfolio and the benchmark.

Increasing factor portfolios

The idea presented in this post consists of defining a series of portfolios with progressively increasing exposure to the factor we want to study, so we can have a reference scale of factor values.

Let’s assume our investment universe is the components of an index (e.g. S&P 500), so our benchmark will be this index. Consider that our objective is to analyse the exposure of a portfolio to the Value Factor in comparison with the benchmark S&P 500. To do so, we are going to define a set of portfolios based on the index components, which will establish ascending scale values, that would act as a reference.

The mechanism to create the set of portfolios is:

• Sort the components of the index by their Value Factor scores, tracking the index weights.
• Define the granularity of the scale we want: each step of the scale is going to be calculated according to the stocks weights in the index aggregated. In this case, we will define each value of the scale with 1% weight steps.

• For the first value of the scale (lowest value), take from the sorted list the assets with the lowest Value Factor score until summing up a weight of 1%. So the first value of the scale would be:

Where s_(j) are the sorted factor scores and w_(j) the weights of the corresponding assets in the index. We separate w’_(k1), as the weight in the index will not add up to complete exactly 1%, so it can not be used directly and it should be tidied up.

The weights of the first portfolio would be:

• For the second scale value, take all the stocks necessary to reach a weight of 2% of the assets with the lowest Value score. Then the second value of the scale would be:

• The calculations for the rest of the values until 100% would be equivalent. Portfolio 100 would correspond to the entire index.

• To obtain higher values than the index, we should take away 1%, 2%, .., 99% weights of the assets with less factor value score.

For example, value 102 would exclude the assets of the lowest value scores that add up a weight of 2% or, in other words, it would include only the assets with more value factor that add up 98% of weight. The calculation would be:

So, with step 1% we obtain a scale of 199 possible values corresponding to the 199 portfolios created, where portfolio 100 corresponds to the index. If we wanted more accuracy we could use a step of 0.5% and get 399 portfolios.

At this point it is important to note:

• If the stock of lowest score weighs more than 1% in the index due to its high capitalization, the first portfolio would only have one component. The same would occur with the portfolio 199 if the stock with the highest score would weigh more than 1%.
• The first portfolios (or last portfolios) will present the same values if the stock of lowest score (or highest score) weighs more than 2%. For example, if Apple, that weighs more than 4%, had the lowest Value Score, the first 4 reference portfolios would contain only Apple and obviously would show the same value.

Why is this method so interesting?

Because it allows joining the different scale values to the concept of Active Share, which is a very common way of expressing how far a portfolio is from a benchmark in a normalized way.

Let’s remember Active Share formula:

Where w _j^portfolio and w _j^benchmark represent the weights of the portfolio and the benchmark in all the components of the universe.

Digging a bit in the definitions of reference portfolios and active share formula, we realise that reference portfolio 100 has active share 0%, portfolios 101 and 99 have active share 1%, portfolios 102 and 98 have active share 2%, and so on. Let’s proof this statement with portfolio 98:

The stocks with weight in the portfolio 98 sum up to 100% and all of them have more weight than in the index (since they have been normalized, as we saw) while these same stocks sum up to 98% in the index/benchmark. On the other hand, the rest of the stocks out of the portfolio sum up to 0% (obviously) and 2% in the benchmark. So, from the active share formula and grouping weights appropriately, we obtain the expected 2% active share:

So we have a series of 199 portfolios increasing in Value Factor and associated with Active Shares. Then, any portfolio constructed over S&P 500 components can be analysed according to its weighted value factor score, which can be linked to the closest reference value among the 199 defined previously. A simple approximation to obtain the (signed) active share associated is just by calculating the percentage of scale values lower than the score of the portfolio at study:

We could create several scales with different factors to have a multidimensional analysis. Again, we could classify the portfolio in terms of active share regarding the different factors, with equal scales from 0% to 100%, which simplifies a lot the visualization.

Then, it is possible to represent visually the exposures in terms of active shares by means of a classical spyder chart, where the center and the outer edge would mean active share 100% of the reference portfolios (theoretical boundaries); further from the center would mean more exposure to the factor. The middle dotted line represents the benchmark 0% active share.

I hope this post has been of interest to you. Do not hesitate to share with us your thoughts about this method or others of your liking.