Axioma's Alpha Alignment Factor Demystified

Transcription

1 Axioma's Alpha Alignment Factor Demystified by Anureet Saxena, Ph.D. and Robert A. Stubbs, Ph.D. The construction of optimized portfolios entails complex interaction among three key entities, namely, the risk factors from the risk model, the alpha factors from the returns model and the constraints from the strategy. The problems that arise due to mutual misalignment among these three entities are collectively referred to as Factor Alignment Problems (FAP). Examples of FAP include risk-underestimation of optimized portfolios, undesirable exposures to factors with hidden and unaccounted systematic risk, consistent failure in achieving ex-ante performance targets, inability to harvest high quality alphas into above-average IR, and so on. Axioma's new research paper (Saxena and Stubbs) provides a detailed analysis of FAP and discusses solutions based on augmenting the user risk model with a single additional factor. The authors show how Axioma's Alpha Alignment Factor (AAF) approach (US Patent 7,698,202) emerges as a natural and effective remedy to FAP. The AAF not only corrects for risk underestimation bias in optimal portfolios but also pushes the ex-post efficient frontier upward thereby empowering a portfolio manager to access portfolios that lie above the traditional risk-return frontier. In this article we excerpt an illustration from this research to highlight the key findings of the paper and also to whet the reader's appetite for this ongoing exciting innovation topic at Axioma. Role of Constraints Research on factor alignment problems has gained some momentum in the past few years. For the sake of analytical tractability, researchers have mostly focused on the unconstrained MVO model trying to decipher the fundamental contributors to FAP. It is now well acknowledged that in an unconstrained setting, the optimized portfolio overloads on the orthogonal component of alpha thereby taking disproportionately large exposure to factors with hidden and unaccounted systematic risk. A natural remedy to this problem is to augment the user risk model with an additional factor, namely α. Alternatively, one can completely eliminate the exposure of the optimal portfolio to α by adding a factor neutrality constraint htα = 0. While a fine solution for the unconstrained case, this approach has some serious shortcomings when applied in the

2 presence of constraints. In this section, we highlight these shortcomings, discuss how the AAF circumvents them, and conclude with an illustrative example. First and foremost, we remind the reader that optimal holdings in a constrained setting are driven by implied alpha, γ, and not alpha. In other words, by aggregating the effect of constraints with the user alpha, the implied alpha takes a holistic view of the portfolio construction process and incorporates the effects of both the user alphas and constraints in determining the optimal portfolio holdings. Second, eliminating the exposure of the optimal portfolio to α by adding a constraint of the form htα = 0 does partially address the factor alignment problem but at a huge cost. Since α often has positive IC, by eliminating the exposure to α we are skewing the risk-return tradeoff by over-emphasizing the risk-aspect of α and completely ignoring its returnaspect. Indeed, the ideal approach is not to eliminate the exposure to α, rather, to actively manage it depending on the systematic risk and IC of α. Axioma's Alpha Alignment Factor (AAF) approach explicitly recognizes the role of implied alpha and offers the end user the means to actively manage the exposure of the portfolio to γ. In fact, under some minor assumptions, the AAF is the normalized form of the orthogonal component of γ. A keen reader might find it intriguing that the orthogonal component of γ differs significantly from α. Where does this disparity between α and γ arise? The answer to this question lies in the construction of implied alpha and its subtle yet significant relationship with constraint attribution (see Stubbs and Vandenbussche, 2010). Recall that the implied alpha is obtained by tilting the user alphas in the direction of constraints by taking weighted combination of the constraint gradients wherein the weighing scheme is dictated by constraint attribution. Consequently, the extent of (mis)alignment between implied alpha and the user's risk factors also depends on whether the constraint gradients have a significant orthogonal component or not. In fact, many of the constraints in real-world strategies that are binding in the optimal portfolios do have significant orthogonal components. For instance, most strategies impose bounds on maximum asset holdings for each asset, and the corresponding constraint is usually never spanned by the user risk factors. Note that it is extremely unlikely that the user risk model will have a factor with exposure to only one asset. The same argument applies to long-only constraints, turnover constraints, active asset bounds, etc. To summarize, there are essentially two schools of thought on addressing the FAP. Both rely on augmenting the user risk model with an additional factor, but they differ in the choice of the augmenting factor. While one uses α as the augmenting factor, the other uses γ. Next, we discuss an example to illustrate how these two approaches α versus γ (AAF) give different results depending on the nature of the constraints. Consider the following 130/30 strategy: Maximize Expected Return 5 * (Active Variance w.r.t S&P 500) s.t /30 Leverage constraint

3 2. Active asset bounds (1.5%) 3. Asset bounds (1%) 4. Active factor exposure bounds (5%) In our experiments, we used US2AxiomaMH as the risk model and defined the alphas as, α = 0.5 * Growth * S/P. There are two sources of misalignment in the above strategy. First, the alphas are not completely spanned by the risk factors in US2AxiomaMH. To see this, note that the suite of style factors in our fundamental risk model US2AxiomaMH does not include the S/P factor although it does include the B/P factor. This disparity between the factors used to construct the alpha and those used to define the risk model is common. It occurs for various reasons, such as different adjustments to the common underlying financial statement data, different ways of handling intangibles such as goodwill or deferred taxes, individual preferences for one metric over another (e.g., EBITDA/EV vs. E/P), and so on. The second source of misalignment arises from the set of constraints. Similar to the strategy presented above, most real-world strategies have tight bounds on the portfolio allocation to specific types of securities. These constraints have a material impact on the structure of the optimal holdings. The AAF approach captures this impact by focusing on the implied alpha rather than the raw alpha. Figures 1 and 2 shed light into the way each one of these approaches affects the optimal holdings structure. Figure 1 gives the decomposition of the optimal holdings into the spanned and orthogonal components of alpha with respect to the risk factors X in the user risk model. Notice the varying proportion of the spanned component in the optimal portfolios obtained by each of the approaches, with the AAF approach clearly yielding portfolios with the highest spanned component. Figure 2 gives the predicted and realized risk of the orthogonal component of the portfolios arising from each of these approaches. Note the severe risk under-estimation that results when the user risk model is used without any augmenting factor. While augmenting the user risk model with α improves the accuracy of risk prediction, it does not completely eliminate the risk under-estimation problem. Using the AAF, on the other hand, yields an almost unbiased risk prediction. This dichotomy can be explained by digging into the mechanics of the two approaches. By focusing only on the exposure to α, the alternative approach can mislead one into believing that the bias in risk prediction of the optimal portfolio is only due to its concentrated exposure to α. The AAF approach, on the other hand, recognizes that in a constrained setting, the optimal portfolio holdings are governed by implied alpha not alpha, and consequently the bias in risk prediction is due to the exposure of the optimal portfolio to γ and not α. When compared on the basis of the ex-post utility function, the AAF approach outperforms the approach that uses the unaugmented user risk model by 30 basis points; the same metric for the alternative approach that penalizes the exposure to α is just 15 basis points.

4 Figure 1: Holdings Decomposition The decomposition of the optimized holdings into the spanned and orthogonal components with respect to the risk factors X in the user model Figure 2: Predicted and realized risk of h of resulting portfolios We conclude this article by highlighting an important aspect of the AAF approach, and its relation to the FAP. The problem of risk under-estimation of optimized portfolios is well known among portfolio managers. However, risk under-estimation is only a symptom of a much bigger and more complex issue, namely the factor alignment problem. Just as eliminating a symptom of a disease is not necessarily the same as

5 curing the disease, simply circumventing the risk under-estimation problem does not necessarily shield the portfolio from the ill-effects of the factor alignment problem. Many practitioners fail to acknowledge this aspect of portfolio optimization and resort to various kinds of ad hoc techniques (such as reducing the risk threshold or increasing the risk aversion parameter) to resolve the risk under-estimation problem, i.e., treating the symptom without addressing the disease the factor alignment problem. While ad hoc approaches may be adequate to solve the risk under-estimation problem, they fail to add any value to the portfolio construction process per se. In other words, they merely move the portfolio on the frontier defined by the user risk model. The AAF, on the other hand, materially alters the mechanism of portfolio construction itself. The AAF not only corrects for risk underestimation bias of optimal portfolios but also pushes the ex-post efficient frontier upward, thereby empowering a PM to access portfolios that lie beyond the traditional risk-return frontier. Figure 3 from our research paper demonstrates this facet of the AAF. We refer the reader to this document for related discussions. To conclude, while the alternative approaches provide a symptomatic cure to the FAP, the AAF attacks the problem at its very core and therein lies its value. Figure 3: Pushing Frontiers, literally! References Anureet Saxena and Robert A. Stubbs. Pushing frontiers (literally) using alpha alignment factor. Technical report, Axioma, Inc. Research Report #022, February Robert A. Stubbs and Dieter Vandenbussche. Constraint attribution. The Journal of Portfolio Management 36 (4): 48-59, Summer 2010.

6 New ETF Companion File for Axioma Risk Models Starting this October, Axioma's worldwide and U.S. risk models will include new companion files that enable clients to easily incorporate ETFs into their portfolio construction and rebalancing processes. "The new ETF package simplifies ETF integration with Axioma Portfolio with a plug and play' capability that automatically provides look-through functionality right on the desktop," said Ron Perez, Vice President for Product Management and Strategy. "To use an ETF in your portfolio construction process, you simply choose your ETF they are all prepackaged in the file we provide and our software loads in all the constituents with the specified weights." While the ETF is treated in the portfolio as a single asset, Axioma's "look-through" capability gives clients the ability to fully decompose the basket that comprises the ETF and offer asset level transparency. "It's not enough to know the constituents of a given ETF you need to understand how that ETF is going to interact with your portfolio holdings," added Perez. "And that's what our look through' capability does, by exposing the specific risk factors for each asset in the ETF. By fully decomposing all the assets and weights within the ETF, clients can see the effect of the union of the ETF constituents with the other individual assets in the portfolio." The look-through capability also allows clients to compare similar ETFs to identify those that deliver the lowest tracking error, or perhaps fail to meet the user's objectives in some way, such as by violating a constraint. While it was possible to use ETFs with Axioma's tools in the past, the process was manual and time consuming. "With this new companion file, we've completely streamlined and automated the end-toend workflow so that clients can quickly obtain the insights they need to understand the potential opportunities and risks of a given ETF," said Perez. ETFs are typically used by portfolio managers to obtain exposure to a particular sector, country or style that may be outside their realm of expertise. For example, for the manager who wants exposure to a region or country but has concerns about local market liquidity and transaction costs may find that ETFs offer a quick and reliable means of obtaining that exposure. ETFs also can be used by managers for hedging purposes. Unlike options, which have specified holding periods and easy-to-overlook expiration dates, ETFs offer added flexibility because they can be traded, long or short, like a single asset.

7 The new ETF package is included with Axioma's U.S. and global risk models at no additional charge and without any additional end user license. Concluded Perez: "Clients have been asking for this capability and I think we've responded with an integrated, easy-to-use tool that really provides them with the insights to capitalize on the features that make ETFs such attractive and convenient investment vehicles."

8 Two Years Later... What are Axioma's clients talking about and doing two years after the global meltdown? After countless conversations, certain threads have emerged that speak to the ways in which Axioma's clients are dealing with the ongoing effects of the crisis. Axioma Advisor recently spoke with three of Axioma's senior managers, to collect the insights they have gathered "The crisis still weighs heavily on peoples' minds," said Chris Canova, Director of Sales. "One of the things we're seeing is that there's a lot of concern out there that the alpha is good but it's just not making its way into the realized results of the portfolio. So clients are looking very hard not only at their alpha process, but how it relates to their risk model. And as a direct consequence of that, we're seeing a lot of backtesting. Constraint attribution is a big issue, too. For constraints that are self imposed, for example, clients obviously want to know which ones are helping and which ones are hurting. And they're taking nothing for granted in working through that process." "That's a valid point," added Olivier d'assier, Managing Director for Europe and Asia. "The best managers, in my opinion, do not think of their alphas, risk models or constraints as independent of each other. And the need for that kind of holistic view has become increasingly evident. In 2008, post crisis, no one wanted to hear about our optimizer because quants were telling us that they had to focus on rebuilding their alpha models. But the fact is and more and more people are now recognizing this looking at the alpha model alone is not the answer. Unless you're looking at the trilogy alphas, risk models and constraints you're missing the point. Alphas are not what is packaged into your final portfolio; implied alphas are, and they are a creation of the interaction of these three parts." Indeed, clients are working harder than ever and making use of more tools than ever to improve their investment process. Noted Canova: "When we launched our multiple risk models back in October of 2008, the response was, frankly, a little lukewarm. That has now changed. The use of multiple risk models is really starting to gain traction. Why? Because managers that have traditionally only used fundamental models are curious to see the alternative, and vice versa. It's just another tool, another way of building better portfolios. You know, we said at the outset that multiple models can provide a better view of your risk. And now, two years later, our clients are coming around to the same point of view." It's also worth noting what clients are not doing. Said Rob Stubbs, Vice President of Research, "After the collapse, clients wanted to know what we were doing to help them avoid another catastrophe in the future. So we came back with a model that would respond to that kind of downside risk. And the response from clients was all but unanimous: How well does it work and how much does it eat into returns?' When it was all said and done, no one was interested in it

9 because while the model did indeed reduce drawdown in a crisis, it also sharply reduced returns under normal market conditions. And, simply put, no one wanted to pay that kind of a price for downside protection." What's next on the horizon? "Let's put it this way: People are not about to give up on seeking returns," said Stubbs. "Though more people do seem willing to place bets on alternative investments fixed income, gold, options a whole range of different asset classes. And if that's where our clients are going, I wouldn't be surprised to see Axioma delivering at least some of the risk models needed to help get them there."

10 Don't Believe Everything You Read in the Newspapers... As this issue of Axioma Advisor was being assembled, The New York Times published a story on the front-page of its business section entitled, "The Quants Are Reeling; Funds That Relied on Computer Models Try to Get Off the Mat.1" The article spoke and perhaps just a bit too eagerly of "a stinging comedown for the wizards of high finance." Once "revered as the brightest minds in finance" the "so-called quantitative investment managers no longer look like geniuses," observed the reporter. To be sure, quant funds have witnessed a steep decline in assets. But times are hard for all fund managers, regardless of whether their approach is fundamental or quantitative. A fog of uncertainty reigns in the macroeconomic outlook, in the minds of investors, and in the confidence investors have in the strategies of their fund managers. The reality is that making money today is as tough as it has ever been. And in these tough times, solutions are being sought wherever they can be found. At Axioma, for example, we have witnessed no small number of former clients who, after being downsized from quant funds, have since found work in of all places fundamental shops. Recognizing the value of quant strategies, some fundamental firms are working to develop quantitative dimensions, or overlays, to their fundamental approaches. Our former clients, now embedded in fundamental firms, are working with us to develop tools and approaches tailored to this new hybrid environment. And as far as our quant-fund clients are concerned, far from being down for the count, they have redoubled their efforts to find an edge. Clients are studying new factors and the ways in which those factors interact with both their alpha processes and their risk models. They are questioning assumptions about when and how often to rebalance. They are looking at risk from multiple perspectives, and, in many cases, utilizing more than one risk model. And they are examining ways to capitalize on our daily updates of those risk models. And that's not all. Portfolio managers are looking to diversify with new geographic strategies. As emerging markets continue to decouple from those of the developed world, clients see an opportunity to exploit quantitative phenomena based on the experiences that worked so well in the most developed markets. They are now also looking beyond equities to new asset classes and they are demanding new tools to invest in those asset classes. Quants are also looking to differentiate themselves by customizing risk models with proprietary factors. Imagine a tool that would allow clients to create customized risk models, while still being able to benchmark them against an existing out-of-the-box risk model. Those who succeed in squeezing out returns in these challenging times will do so by

11 developing better strategies based on educated insights and thorough research, and by making the best use of the most powerful existing and emerging new tools. There is still alpha in the markets, it is just much harder to find it than it used to be. This challenge is right up our alley, because working side by side with clients to develop the next new tool, the next breakthrough solution, is what we do best. According to headlines in the newspaper, quants are hoping for a revival. But according to what Axioma's clients are saying and doing, it's not about hope. We are entering what I believe is likely to be one of the most exciting and creative periods of invention and innovation that the quant community has ever witnessed. 1 Please note that the article received a different headline on The New York Times' website: "Shrinking Quant' Funds Struggle to Revive Boom"

12 Using the Time Series Workspace Converter In Axioma Portfolio version 6.4, we introduced the Time Series Converter. This feature allows you to easily convert a point-in-time Axioma Portfolio workspace into a time series workspace, without the need to recreate the data import details for each of your rebalancing inputs. This exporter is a time-saver: it automates the process of setting up the data library in a new time series workspace with the data used in the originating point-in-time workspace. This includes maintaining links to external (user) data sources as well as dynamically created data structures. At the same time as it populates the data library, it also creates a new time series workspace that is ready to open and manipulate further. The conversion process behaves differently depending upon the way that the data is saved in the point-in-time workspace and relies on certain assumptions based on the typical requirements of a backtest. For example, a benchmark can be imported through a single delimited file for the date in the point-in-time workspace. If we make sure it is imported with the "Save Reference" feature activated, the Times Series Converter will create a benchmark object in the time series workspace that looks for a new benchmark file for every period of the backtest. As a result, it is also important to have a naming convention for your time series files that incorporates YYYMMDD into the name, e.g. mybenchmark_ csv. If your benchmark is not saved with a date in the file name, the Converter will assume that this data is a "single period" object and the file will be used for each date of the backtest. That being said, the converted data library can be changed after the conversion and updated with new file names or different parameters. In addition to benchmark files, other common time-series data examples include alpha files, transaction cost files, and other attributes that may change over time. On the other hand, certain objects such as a Strategy are not imported but rather created within the workspace. These items are saved as XML files and are linked to the newly created time series workspace automatically. This may also apply to any dynamic attributes, sets or classifications that are created within the point-in-time workspace. Here is an example of how one might create a simple point-in-time workspace and convert it to a times series workspace: First, we import an initial account with the Save Reference box checked. Note that in a backtest only one initial account is needed, therefore we can import this with a date in the file name, such as a benchmark, or without it. Without the date in the file name, the Time Series Converter will create a "single period" data object.

13 We can also import a custom benchmark that will have a separate file for each period and is saved by reference. Our file has the dating convention YYYYMMDD in the name. A different date format can be used as long as it is done consistently throughout the entire workspace. The desired date format can be chosen in the Time Series workspace options.

14 We can save alpha files and other attributes in the same way, using dated file names and saving by reference. Also, we will create the Dynamic Attribute below to show how classifications, sets and dynamic attributes are converted. The attribute below filters for any asset in the workspace with a price under $5.

15 Finally, we will create a sample strategy that could be evaluated in the backtest. We could also import a strategy that was saved in XML in a separate workspace but the result would be identical. Once the workspace is set up and saved, go to Tools > Export > TimeSeriesConverter > Convert Workspace to Times Series By default, a time series workspace will be saved in the same directory as the point-intime workspace, with the name ConvertedTimesSeriesWorkspace.tsw. Both the location and the name of the file can be changed, as long as the extension remains.tsw. Those using an initial account other than Cash for the initial rebalance must select the Include Accounts check-box. Doing so will ensure that our Sample Account is included in the times series workspace.

16 After clicking on Ok, the times-series workspace is created along with certain data folders containing the XML for data, such as dynamic attributes and the strategy. Open the Time Series application by going to Tools > Open Times Series Analysis and open the workspace by going to File > Open and select the workspace. The Data Library will show all of the data objects that were converted, and it is here that objects can be deleted or added as necessary.

17 It is now possible to setup the times series of dates and continue with the backtest setup without worrying about reimporting data. For more information about the Time Series Converter please see Chapter 15. Importing and Exporting Data in the Axioma User's Guide. For more general information about backtesting and Time Series Application setup, please refer to the Axioma Portfolio Time Series Software Manual. If you have any additional questions or comments please contact us at or Axioma Portfolio is a trademark of Axioma, Inc.