Our latest paper on equity factor investing addresses the question of performance attribution of multi-factor equity strategies.
Multifactor strategies are based on investable portfolios, benchmarked against market capitalisation indices and tilted towards stocks with given characteristics (factors): they should be cheaper (value), more profitable (quality), less risky (low risk), and have outperformed in the past (momentum).
The task at hand is to explain the returns of multi-factor portfolios in terms of the tilts created by each of the factors used to construct those portfolios.
This is essentially an ex-post process that seeks to explain realised performance by assessing the performance
- From the stock tilts created by the factors
- From the impact that portfolio constraints may have in pushing the portfolio away from those stock tilts.
This is important because it allows investors to understand, and not just to blindly profit or suffer from, the performance of the investment strategy.
Why is factor attribution of multi-factor portfolios difficult?
The traditional performance attribution used for active benchmarked funds decomposes the excess returns into a stock-picking contribution and a sector or regional contribution. While informative, this fails to account for the contribution of the factors driving the investment decisions in multi-factor strategies.
Constructing an attribution based on the factors is, however, not a straightforward exercise. The main difficulty is to attribute the extent by which a stock tilt in the portfolio is just driven by the factors or also driven by portfolio constraints that necessarily have an impact on the final allocation. Being able to tell why a tilt in favour or against a stock is found in the portfolio is thus key to this attribution challenge.
What do we propose?
In our paper, we propose a model for the return attribution of multi-factor portfolios that takes into account how the portfolio is constructed, which stocks should be overweight and underweight, and the impact of portfolio constraints. In this model, we can actually represent the impact from constraints as additional non-targeted factor tilts that are created just for the sake of complying with the constraints.
Thus, the model first produces a linear combination of the factor tilts coming directly from the desire to overweight value, quality, low risk and momentum stocks, and from a non-linear combination of additional factor tilts resulting from the impact of constraints. Then, as wanted, this attribution is expressed only on the targeted factors.
Indeed, we show how the final optimal constrained multi-factor portfolio should be well represented by a linear mixture of the stock tilts coming from each of the targeted factors and two additional terms due to constraints. These terms measure by how much the most typical constraints push the portfolio away from the original factor-driven stock tilts.
The terms are additional non-linear combinations of stock tilts that can be related to the original factors. For example, it may happen that a given constraint reduces stock tilts associated with value stocks, while another increases stock tilts associated with quality stocks. If that is the case, our model will capture it.
By explicitly considering the impact of constraints, the return attribution model reduces the unexplained part of the multi-factor portfolio returns. By representing the impact of constraints as functions of the original factors, the contribution of constraints to the performance can be easily be reallocated to the original factors.
The estimation of the model, which allows us to produce the return attribution, is based on cross-sectional regression. This essentially regresses the stock returns on factor data by using the model described in the paper. Then, by using the model output and the multi-factor portfolio stock weights, we can proceed with the construction of the final attribution.
Illustration: Multi-factor strategy for European equities
In this example, our multi-factor strategy for European stocks uses value, quality, low risk and momentum factors to drive the portfolio allocation. The factors and the portfolio constraints are the only drivers of the stock allocation. The portfolios, rebalanced every month, are constrained to invest only in stocks in the benchmark MSCI Europe index. The maximum allocation to stocks is not allowed to deviate by more than 1.5% around their respective weight in the index. Short-selling is not allowed.
The results at the end of August 2021 are in the table below. Year-to-date, the strategy outperformed the benchmark by 3.27%, gross of fees in EUR. Low risk factors contributed the most: 0.79% of the excess returns. All factors had a positive contribution.
The total contribution of the factors to the excess returns was 2.36%, which leaves 0.91% explained by effects of portfolio constraints that cannot be mapped onto an allocation to the factors. Optimal portfolio construction should minimise the unexplained contribution over medium to long-term investment horizons, typically spanning several years.
We propose a framework for return attribution of multi-factor portfolio strategies that has several advantages over other methods:
- It faithfully reflects how factors impact the stock tilts in the current portfolio
- The number of explanatory variables is limited
- It leads to a good approximation of multi-factor portfolio returns with a relatively small residual error
- It provides a straightforward interpretation of portfolio performance in terms of the factors used to construct it.
We believe this approach should become a standard tool in the performance attribution of multi-factor investment strategies as it provides a faithful and intuitive description of their performance that directly relates to the building blocks used in the construction process.