Whitepaper An introduction to performance attribution through a factor lens

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1 Whitepaper An introduction to performance attribution through a factor lens The document is intended only for Professional Clients and Financial Advisers in Continental Europe (as defined in the Important information); for Qualified Investors in Switzerland; for Professional Clients in Ireland and the UK; for Institutional Investors in Australia; for Professional Investors in Hong Kong; for Institutional Investors and/or Accredited Investors in Singapore; for certain specific Qualified Institutions/Sophisticated Investors only in Taiwan and for Institutional Investors in the USA. The document is intended only for accredited investors as defined under National Instrument in Canada. It is not intended for and should not be distributed to, or relied upon, by the public or retail investors. Sanne de Boer Senior Quantitative Analyst, Invesco Quantitative Strategies Julian Keuerleber Head of Global Portfolio Analytics, Invesco Quantitative Strategies Carsten Rother Research Analyst, Invesco Quantitative Strategies Guest contributor: Elroy Dimson Chair of the Centre for Endowment Asset Management, Cambridge Judge Business School Contents 1. Executive summary 2. Introduction 3. Performance attribution: an enduring puzzle 3.1. The eternal question 3.2. The riddle of residuals 4. Performance attribution through a factor lens 4.1. A novel and non-linear approach 4.2. Implementation and insight 5. Conclusion 1. Executive summary Factor investing has become an established cornerstone of asset management. Amid everaccelerating product innovation and expansion, the question of exactly how exposure to different factors contributes to overall investment success is perhaps more pressing than ever. We make the case for a novel factor lens approach to performance attribution. We argue that traditional linear models of stock returns cannot be relied upon to capture all the possible interactions within sophisticated, multi-factor strategies and that a nonlinear methodology is better equipped to reconcile portfolio performance with investment process. We suggest that our proposed model is easy to implement as an extension of standard factor attribution. It intuitively adopts the view of a portfolio manager rather than the view of an econometrician, as each stock s residual contribution is attributed according to the factors that drove its position. 2. Introduction Factor investing has evolved from an academic concept to a cornerstone of asset management in the space of just a few decades. Once largely confined to arcane journal papers, it has come to play a substantial role in the analysis and construction of hundreds of billions of dollars worth of portfolios. It is no exaggeration to say that it has transformed thinking around risk and return and how markets as a whole are viewed. Its origins can be traced to the 1960s and the introduction of the Capital Asset Pricing Model (CAPM), which posited that every stock is in some way sensitive to the movement of the broader market. It was this idea of a single factor market risk that gave rise to the formative notion of beta. We have certainly come a long way since then. Today, thanks to a wealth of research into nontraditional beta and the underlying drivers of returns, we know that there are many factors that can affect portfolio performance. The boundaries of what might be described as portfolio craftsmanship are constantly being pushed as appreciation of the science of factors continues to grow. In turn, attention is increasingly being drawn to the crucial question of precisely how exposure to different factors contributes to overall investment success. This question is assuming ever-greater significance as factor-based products become more numerous and more complex. Investors may welcome choice and sophistication, but they do not want these advances to hinder their grasp of how their investments actually work. Proliferation and intricacy should not serve as barriers to understanding Invesco An introduction to performance attribution through a factor lens 1

2 With this concern in mind, in this white paper we take a closer look at performance analysis through what we call a factor lens. Specifically, we examine an important technical issue that arises in the performance of factor strategies that are subject to investment constraints. The suggested factor lens approach, which may also be applied to asset classes other than equities, can help deliver clarity that embraces both craftsmanship and cost-effectiveness a potentially attractive combination in an investment environment defined by rapid product innovation and expansion 1. Attention is increasingly being drawn to the crucial question of precisely how exposure to different factors contributes to overall investment success. 3. Performance attribution: an enduring puzzle 3.1. The eternal question Performance attribution has always been a source of fascination and frustration in the sphere of factors. Although many studies have demonstrated how factor exposure can deliver a return premium over the broader market, the finer nuances remain a matter of debate and discovery among the academic and practitioner communities. The outperformance was originally considered to be a rational premium for systematic risk, but explanations have come to include the inefficiency of markets and the irrationality of those who participate in them 2. This might be sufficient for investors who are content to focus on single-factor strategies, for which the question of performance attribution is answered with comparative ease, but it falls a long way short of unravelling all the mysteries inherent in multi-factor methodologies. The Norwegian Government Pension Fund, the world s largest sovereign wealth fund, has perhaps provided the most celebrated and influential illustration of the latter approach. A landmark analysis of the fund s drivers of returns in the 10 years leading up the global financial crisis might be seen as marking a dividing line between the before and after eras in the history of factor performance attribution. The fund reported a loss of 23.3% amid the turmoil of 2008, yet its equity/fixed-income benchmark was down only 19.9% during the same period. Commissioned by the Norwegian Ministry of Finance to investigate the disparity, researchers from Columbia Business School, Yale School of Management and London Business School famously concluded that 70% of all the fund s active returns from 1998 to 2008 were due to exposure to systematic factors including the widely acknowledged likes of size, value and momentum and the less recognised likes of credit spread, liquidity and foreign exchange. The findings were outlined in Evaluation of Active Management of the Norwegian Government Pension Fund Global. Published in 2009, the study created unprecedented awareness of the argument that diversification purely by asset class could conceal risk concentrations and that factors might therefore serve as the building blocks for portfolio construction. Major impetus was thus given not only to the cause of factor investing itself but to an associated field of inquiry that is still exploring the puzzle of performance attribution today. I believe factor rewards can be harvested successfully over the long term by taking exposures that are appropriate for the client The riddle of residuals The basic aim of factor performance attribution is to decompose a portfolio s realised returns into an array of clear-cut contributions. This should reveal how much performance is due to factors based on, say, style, geographic and industry exposures. Stock returns are assumed to follow a fundamental linear factor model, with exposures based on valuation ratios, price momentum and other characteristics that have been found to explain return dispersion. This being the case, factor returns are estimated through the linear regression of stock returns on factor exposures. Each factor s contribution is then approximated by the portfolio s exposure to the factor multiplied by its estimated return. This is a widely used approach to performance attribution offered by the main providers of off-the-shelf factor models. A problem with this approach is the treatment of residuals that are left unexplained. These are commonly attributed to what is known as stockspecific risk, a catch-all term that encompasses events with narrow impacts for instance, merger announcements, product launches, industrial accidents, earnings hits and misses, accounting issues and so on. Much the same convenient fallback was employed in the days of the Capital Asset Pricing Model to cover anything that could not be ascribed to the only factor believed to exist at the time, market risk. While it is true that stock-specific risk may have some effect, investors keen to understand the performance drivers within a multi-factor portfolio are unlikely to be satisfied if residuals are shown to dominate attribution. One reason for this is that meaningful levels of such returns are inconsistent with a diversified factor strategy; another is that asset owners have rightly come to expect a more insightful explanation of how their investments function. The issue, we say, lies in the assumption of linearity. Traditional linear factor models of stock returns cannot be relied upon to capture all the possible interactions within a sophisticated, multi-factor investment process that entails proprietary definitions, seeks to avoid unrewarded risks and attempts to mitigate implementation costs. As the example below shows, the picture that emerges is often inaccurate. We set about addressing this shortcoming in the following chapter Invesco An introduction to performance attribution through a factor lens 2

3 Appearances can be deceptive Suppose that the returns from a price momentum factor were especially strong in its tails. In other words, imagine that a small number of stocks with the very best trailing returns continued their winning streak while those with the very worst returns continued to underperform. Something similar happened during the global financial crisis. Defensive industries held up relatively well, although only a handful of counter-cyclical stocks managed to rally; cyclical industries suffered heavy losses; and financials fell off a cliff following the failure of Lehman Brothers. A price momentum factor s contribution to returns in such circumstances might be estimated to be highly positive. However, a trend-following long-short investor may have been unable to capitalise, since position limits for diversification and liquidity purposes would have shifted market exposure to more moderate momentum stocks which might have performed in line with the market. Here, then, we are left with a misleading picture of performance attribution. Stock-specific risk does not accurately explain the residuals: instead a richer and more satisfying explanation is likely to be found in the non-linearity of the factor returns to which the strategy was exposed as a result of its investment constraints. Traditional linear factor models of stock returns cannot be relied upon to capture all the possible interactions within a sophisticated, multi-factor investment process. Q&A with Elroy Dimson, former chair of the Strategy Council of the Norwegian Government Pension Fund Global Elroy Dimson chairs the Centre for Endowment Asset Management at Cambridge Judge Business School. He chaired the Strategy Council of the Norwegian Government Pension Fund Global (NGPFG) from 2010 until 2016, during which time the fund cemented its reputation as a pioneer in several areas of investment. Today NGPFG consistently ranks as the world s largest sovereign wealth fund, yet a little over 20 years ago it was the smallest. Here Professor Dimson discusses the role of factors in the fund s success and in influencing investment thinking more generally. How significant was the 2009 study of NGPFG both for the fund itself and in terms of raising wider awareness of factors potential contribution to portfolio construction and performance? People had bought into the fundamental law of active management, which asserts that you can make money by taking lots of small, insightful bets and spreading them across a very broad portfolio. Particularly in the fixed income space, the outcome of this approach was to have lots of little exposures that appeared to provide small amounts of alpha. What the study s assessment of NGPFG s performance revealed was that this apparent alpha was actually factor-driven. In light of this, the study s authors recommended that the fund shift its active management from bottom-up security selection to top-down factor allocation. So this was certainly a watershed in the history of factors for Norway, because factor exposures were very influential in understanding where NGPFG s performance had gone wrong in the early stages of the global financial crisis. And it also helped to change how a lot of other investment organisations thought, because it showed them how performance could be attributable to factors other than market risk. In 2011, the year after you became chair of NGPFG s Strategy Council, you published a report that re-emphasised the importance of harvesting risk premiums but which noted the absence of consensus over which factors might actually drive asset prices. Are we any nearer a consensus today? If you ask people now about the single-factor model that is, just looking at the overall equity market I think most would concede that even in that instance there s a huge amount of statistical uncertainty about the market risk premium. So the question of how confident we can be that there s a factor premium or discount for other sorts of risk exposures is a difficult one. It s the kind of thing we try to look at with long-term data, and it s important to remember that the effects can be quite subtle. Sometimes you might get what appears to be a risk premium, and sometimes that premium might disappear. I think the notion that there might be hundreds of rewarded factors is nonsense that s just people coming up with things that only work in-sample. But I do think there s compelling evidence that there has been an expected reward for a number of risk exposures. A key point is that those risk premiums can t be bigger than the discount investors accept for avoiding such exposures. In other words, the sort of factors that make sense have to be ones where you can identify a group of investors who will be on the other side of the transaction and who will have a good reason for accepting the premium with a minus sign in front of it. Regardless of how many there might be, are factors here to stay? I think there s reason to expect interest in factors to continue. Although we should anticipate factor risk premiums that are modest in magnitude, I believe factor rewards can be harvested successfully over the long term by taking exposures that are appropriate for the client Invesco An introduction to performance attribution through a factor lens 3

4 4. Performance attribution through a factor lens 4.1. A novel and non-linear approach In this chapter we offer a brief overview of a nonlinear approach that aims to provide a more accurate analysis of residuals. It is important to note that our principal objective here is simply to outline some basic elements. We focus less on the mathematics and mechanics behind the idea and more on why it is needed and what benefits might be derived from it 3. In short, our model assigns each stock s contribution to residuals to the factors by which it is most characterised, in proportion to its squared standardised factor exposures. The aggregation over all stocks then identifies which factors were most responsible for residuals. This approach to attribution, which we might call non-linear residual reallocation, enables us to develop a more comprehensive and factor-centric understanding of performance. In the example in section 3.2, for instance, most of the disappointing shortfall would be traced to the curtailed bets on the momentum tails, whose estimated contribution would be cut accordingly. Let us apply the model to a hypothetical portfolio consisting of just four stocks and two commonly employed factors: value and momentum. The table below lists each stock and its standardised exposures. Remember that we look to attribute a stock s residual return in proportion to its squared factor exposures, so the residual return of Stock 1 which has standardised exposures of 2 and -1 respectively would be attributed as follows: ( 2 + ( 1) ) =. = % momentum; ( 2 + ( 1) ) =. = % value Mathematically, the correlation between stocks factor exposures and their contribution to residuals is zero. However, the stocks residual returns correlation with our non-linear classification scores allows us to decompose the attribution residual across factors, as illustrated in the table. This approach, which we might call non-linear residual reallocation, enables us to develop a more comprehensive and factorcentric understanding of performance Implementation and insight Our proposed factor lens approach is easy to implement as an extension of standard factor attribution. An interesting feature is that it intuitively adopts the view of a portfolio manager rather than the view of an econometrician, as each stock s residual contribution is attributed according to the factors that drove its position. Depending on the situation and the insights required, an analyst has a choice as to which potential nonlinearities in factor returns to explore. As in our example of a momentum investor, including only the main intended return drivers of a factor strategy would result in reducing the estimated contributions from those that failed to deliver; by contrast, including only portfolio-construction-related riskcontrol or liquidity factors might identify which are most responsible for any shortfall in reaping returns. By way of further illustration, consider the active positions of a high-capacity strategy benchmarked against a capitalisation-weighted index. These might inherently be concentrated among mega-caps, where factor efficacy could be lower. Similarly, including country or industry indicators might identify segments of the investable universe where factor performance was differentiated. We mentioned earlier the venerable fallback of stockspecific risk. This still has some relevance. While the method we have outlined here can be used to fully cast realised performance in factor terms, it may be more realistic to allow for some contribution from stock-specific risk. This is because the unexplained contribution of positions not driven by any of the included interaction factors cannot reasonably be reattributed in any other way. The residual return contribution of stocks whose outlying performance indicates some stock-specific event also falls into this category, and this intuition can easily be captured in the final attribution. Non-linear classification in action This table shows the results of applying a non-linear residual reallocation rule to a simple hypothetical portfolio consisting of four stocks with exposure to two factors. Exposures (linear factor model) Classification (non-linear) Momentum Value Momentum Value Stock % 20% Stock % 50% Stock % 69% Stock % 0% Source: Invesco. For illustrative purposes only Invesco An introduction to performance attribution through a factor lens 4

5 In summary: reasons for and benefits of a factor lens approach Standard factor performance attribution assumes a linear relationship between factor exposures and returns. This overlooks the fact that investment constraints can affect the linearity between factor exposures and optimal portfolio weights. A factor lens approach aims to classify each stock according to its predominant factor exposures that is, the drivers behind its active position in a portfolio. This should make it possible to attribute previously unexplained returns and so link realised performance directly to the factor-driven investment process. Such an approach can better satisfy sophisticated investors desire for a comprehensive explanation of how their multi-factor strategies work. The proposed approach intuitively adopts the view of a portfolio manager rather than the view of an econometrician. 5. Conclusion Much has changed in the world of factors since the Capital Asset Pricing Model first introduced the concept of market risk more than half a century ago. As in so many areas of science, advances are now being made at an ever-accelerating rate. It is perhaps worth remembering that as recently as the early 1990s the idea of a single factor held sway in academia, whereas now some estimates hold that there are several hundred. While this is a questionably high number, as we remarked earlier, we have certainly come a long way. One thing that has remained reasonably constant throughout, however, is the quest to shine a truly meaningful light on performance attribution. This was an issue when market risk was supposedly the only factor in town and stock-specific risk was reckoned to solve all attendant mysteries; and it is still an issue today maybe even more so in the age of multiple factors and high-level portfolio craftsmanship. The goal of fully reconciling portfolio performance with investment process is not easily achieved. We have seen, for instance, how assumptions of linearity can lead to an inaccurate and unsatisfying picture with regard to residuals when investment constraints must be observed. In this white paper we have argued that a non-linear approach can offer greater precision in attributing previously unexplained returns. Cutting-edge multi-factor products increasingly entail proprietary definitions, strive to avoid unrewarded risks and seek to mitigate implementation costs. Our proposed model is intended to complement such strategies, which involve myriad inputs and interactions and whose complexities are unlikely to be captured by traditional linear models of stock returns. We therefore believe that a factor lens approach to performance attribution can appeal to asset owners and asset managers alike, particularly at a time of unprecedented innovation and expansion in this arena. Irrespective of how sophisticated or large the factor phenomenon becomes, the fundamental importance of understanding exactly how investments work should never be overlooked or forgotten. The fundamental importance of understanding exactly how investments work should never be overlooked or forgotten. Notes 1 The purpose of this white paper is to provide a relatively simple introduction to a novel factor lens approach to performance attribution. Readers interested in a more in-depth explanation may wish to refer to our article in the Q edition of Invesco s Risk & Reward. We explore the issue in full in Non- Linear Factor Attribution (2018), which is available on request and is perhaps best suited to devotees of the arcane journal papers that we mentioned at the outset. 2 More background on factor investing can be found in Factor investing: the third pillar of investing alongside active and passive, an article in the Q edition of Risk & Reward. 3 As stated previously, readers who require a more detailed explanation may wish to consult the Q edition of Risk & Reward or our new paper, Non-Linear Factor Attribution (2018) Invesco An introduction to performance attribution through a factor lens 5

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