# Factor contribution

### J. González

#### 22/09/2021

In this post we are going to examine two alternative methods of calculating the factor contribution to the performance of an equity portfolio.

To evaluate the performance of an equity portfolio regarding the exposure to risk factors, it is common to calculate the contribution of each factor to the performance.

When we use the term contribution we are speaking about the absolute return of something (sector, region, factor), that is, with respect to nothing.

Another term commonly used is “factor attribution”, which refers to the excess return against a benchmark as a consequence of a different exposure to factors than the benchmark.

The allocation of the performance to the different risk factors poses some difficulties concerning:

• Factors definition: there are infinite forms of defining the factor scores of each asset, depending on the signals used (eg. ROE, P/B, etc), weights given to the signals, window of calculation, etc.
• Method to measure the exposure to the factors: regressions (Fama-French) or models that measure the degree of “belonging” to a factor, asset-grouping models (Brinson model) or models that divide the asset universe in groups, which is quite straightforward when dealing with sectors or regions, but not so much when dealing with factors.
• Active portfolios: when the allocation of the portfolio changes throughout time, the performance attribution complicates as multi-period returns must be calculated and arithmetic returns can not be applied.

Next we will present two methods of calculating the factor contribution of a portfolio based on the factor exposure calculation method (Active Share method) presented in this previous post. We will use our own definition of the different factors, aggregating several scores, of which we will not go into detail (you can use your own factors scores to carry out the calculations).

## Method 1

This first method allocates the whole return of the portfolio among the factors of interest. The algorithm is as follows:

1. Choose the factors of interest to be analysed.
2. Establish the benchmark to compare with for the factor exposure scale.
3. Calculate the factor exposures of each asset of the portfolio, defined according to the exposure method already mentioned, normalizing the exposure among the factors in each date.
4. Calculate each factor return contribution for date j as:

where w is the weight of the asset n in the portfolio in date j, w^i is the factor i exposure normalized in day j, and r is the return of the n asset, having a total of k assets allocated.

• To obtain the contribution evolution we just have to accumulate the amounts obtained by multiplying the return contribution by the gav and refer it to the initial gav value.

### Method 1 example

Let us show an example of this calculation on a long only UK Low Volatility portfolio over the components of the FTSE 100:

In the next graph you can see that the exposure to low volatility anomaly is the strongest, much higher than the benchmark FTSE 100 (remember that the benchmark is on the 50% level for all the factors in this exposure method), followed by momentum and then quality, which present higher exposure than FTSE in most periods.

Higher exposure to low volatility provokes a higher contribution of this anomaly in the whole period analysed, as can be seen in the Factor Contribution graph. However, the higher exposure to momentum than quality factor during the 2000s does not result in a higher contribution to performance. At the same time, Value factor, with much less exposure, shows one of the highest contributions up to 2010. All this agrees with the relative behaviour of risk factors during 2000s.

The main advantage of this first method is the clear link of the contribution with the degree of exposure to factors according to the Active Share method representation.

However, as there is not a selection contribution term, since the returns are distributed completely on the factors chosen, we can not assess the contribution of other causes (e.g. selection contribution), which may be important to extract conclusions about a fund manager selection ability, for example.

## Method 2

With this method we try to solve the disadvantage shown by the previous method. The algorithm is very similar to the one of method 1, but instead of using the returns of the components of the portfolio to obtain the factor contribution, we are going to use the returns of the reference portfolios built in the factor exposure.

Remember that to obtain the factor exposure we make use of several reference portfolios with an active share progressively higher against the benchmark chosen, reference portfolios that define a score scale for each factor. As the reference portfolio associated with our portfolio (for each factor and date) represents the portfolio factor exposure, the return of this reference portfolio can be considered our pattern for the factor scores of the portfolio.

Then, we could calculate each factor contribution just weighting this reference return with the normalized portfolio exposure.

Obviously, with this calculation the factor contribution will not add up to the portfolio return, so the return due must be considered the selection contribution of the portfolio.

In the factor contributions for the UK Low Volatility Portfolio, the quality factor is the most affected by the new method of calculation. Now, its contribution is very similar to the one of the momentum factor, while the selection contribution assumes more weight mostly during the last four o five years.

## Summary

We have presented two methods to calculate the factor contribution to the performance of an Equity Portfolio. The first of them is very intuitive as is a direct consequence of the factor exposure. but lacks the selection term that may be very clarifying in some contexts.

The second method depends on the reference factor portfolio return associated with the degree of exposure and includes a selection contribution, giving importance to the specific asset selection of the portfolio different from the benchmark, not only the asset allocation.

If you use some contribution/attribution analysis do not hesitate to sharing it here.