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Research & Analysis

Drawdowns Looming? It’s Not in the Numbers...

A quick glance at a chart of predicted risk over the past 30 years suggests that risk in the past has typically risen in advance of major market declines. We set out to quantify the likelihood of a major drawdown in the near future based on current risk model readings. Data from our US risk model suggests a low likelihood of a major drawdown, at least in the near term.

Anthony A. Renshaw, Ph.D.
Melissa Brown, CFA

Research Paper No. 052

Drawdowns Looming? It’s Not in the Numbers...
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Axioma’s Macroeconomic Model: Insights into equity portfolios from a new perspective

Axioma’s recently released macroeconomic‐based model provides a way to measure and manage financial risk in a U.S. portfolio by considering macroeconomic variables and events. The model’s horizon is three to six months—what we deem to be “medium” horizon. The Macro model joins four other model variants already offered by Axioma: two fundamental models and two statistical models, each offered for both a medium and a short (one‐to‐two month) horizon.

By Melissa Brown, CFA
Senior Director, Applied Research

Research Paper No. 051

Axioma’s Macroeconomic Model: Insights into equity portfolios from a new perspective
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Risk-Parity Strategies For Equity Portfolio Management

Risk-based strategies have gained popularity amid market uncertainty, and many are now being touted as "smart beta," providing a systematic way to outperform traditional capitalization-weighted benchmarks. Here we examine the notion of risk parity, taking what has almost exclusively been discussed in an asset allocation context and applying its concepts to equity-only portfolio construction.

Author: Frank Siu

Journal of Indexes, May/June 2014

Risk-Parity Strategies For Equity Portfolio Management
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Turmoil in Ukraine: Contained...So Far

Events in Ukraine have had a clear impact on risk there and in Russia, but thus far the spillover into neighboring countries has been largely contained, at least in terms of the risk forecasts for countries' equity markets and currencies.

By Melissa Brown, CFA
Senior Director, Applied Research

Research Paper No. 050

Turmoil in Ukraine: Contained...So Far
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Emerging Markets: Coming Full Circle?

At the start of 2013, it was hard to tell emerging markets from developed markets—on the basis of risk-return tradeoffs, anyway. But all that changed as the year unfolded. The latest data now suggest a reversal back toward where things were a year ago, with volatility easing in emerging markets and increasing in developed markets. If the trend continues, the big (risk) story in 2014 could be a shift back to developed markets.

By Melissa Brown, CFA
Senior Director, Applied Research

Research Paper No. 049

Emerging Markets: Coming Full Circle?
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Equity Markets: Correction or Contagion?

Turmoil in currency markets is being driven by, among others, Argentina and Turkey. But what about equities? Is the recent downturn simply the long-anticipated market correction following last year's bull market, or is the currency contagion infecting equities? Is last week's upturn cause for optimism or only a pause? Here Axioma examines some of the recent equity market results in an attempt to provide some answers.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 048

Equity Markets: Correction or Contagion?
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Is Low Volatility Changing Its Stripes?

Low volatility investing—once dominated by Consumer Staples and Utilities—took a distinct turn in 2013 and is now diversifying across other sectors.

Our analysis shows that the sector diversification that occurred in 2013 was driven primarily by historically low volatilities for all the sectors, which resulted in relatively small volatility differences among the sectors.

Low or minimum volatility investors face a strategic allocation decision.

If the primary goal of such investing is downside protection during market downturns, how relevant is market information in a bull market? Is it better to be strongly invested in Consumer Staples and Utilities, as was the case during the Great Recession; or will the investor be better off using the diversified, bull market sector allocations that occurred during 2013?

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 047

Is Low Volatility Changing Its Stripes?
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Enhancing the Investment Process with a Custom Risk Model

A case study by Axioma and Credit Suisse HOLT examines the benefits of using custom risk models generated by Axioma’s Risk Model Machine, in combination with HOLT’s highly regarded proprietary return models. The results of the study show that Axioma’s custom risk models provide further validation of the efficacy of HOLT’s fundamental-factor data. Combining a custom risk model with HOLT’s factors produced more efficient portfolios with better risk forecasts, and resulted in enhanced performance attribution.

Authors: Chris Canova and Melissa Brown (Axioma Inc.)
Adam Steffanus (Credit Suisse HOLT)

Axioma Case Study No. 001

 Error?
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Who Shrunk My Tracking Error?

Investors have struggled to reach their target levels of tracking error in the current low-risk environment. In a search for solutions, we put tracking error under the microscope, dissecting its components and their relationship with market risk. We also reveal the potential risk-related pitfalls of different portfolio construction methodologies. But the real moral to the story is that investors cannot rely on tracking error alone. The focus must be on the overall quality of the portfolio.

Authors: Scott Hamilton and Melissa Brown

Research Paper No. 046

Who Shrunk My Tracking Error?
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Alpha Construction in a Consistent Investment Process

The three important ingredients in an MVO model are the alpha vector representing expected returns, the risk model that is used to measure the variance of the portfolio, and a set of constraints representing the portfolio managers' mandates and choices. In the traditional quantitative investment process, these three inputs are usually developed independently of each other, without much regard to the interaction between them. As a result, the optimal portfolio generated by the traditional approach may not consistently represent the views of the portfolio manager that are expressed in the expected returns.

Authors: Sebastian Ceria, Kartik Sivaramakrishnan, and Robert A. Stubbs

Research Paper No. 045

Alpha Construction in a Consistent Investment Process
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Demystifying Low-Volatility Strategies

Although the core ideas and research behind low-volatility investing have been in academia for several decades, only in the aftermath of the global financial crisis did this category of investment products begin to gain popularity.

Author: Frank Siu

Journal of Indexes, July/August 2013

Does Style Factor Alignment Foretell Market Crashes?
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Consistent Portfolio Management: Alpha Construction

The Fundamental Law of Active Management tells us that good forecasts should directly translate to outperforming portfolios. Why, then, do we so often hear the frustrated lament that they do not? Can this discrepancy between the clear theoretical rigor and the negative practical experience be explained?

Author: Robert A. Stubbs

Research Paper No. 044

Does Style Factor Alignment Foretell Market Crashes?
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Axioma Research Brief:
Does Style Factor Alignment Foretell Market Crashes?

In the US, Axioma’s style risk factors experienced two periods of high alignment: 1999 through 2001 and from 2008 to 2009. In each instance, the increased alignment of the style risk factors preceded subsequent steep market downturns by months. The recent decrease in US absolute correlation is likely a positive economic indicator.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 043

Does Style Factor Alignment Foretell Market Crashes?
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Equity Risk and the Presidential Election:
What Does History Tell Us?

Is there a relationship between US presidential elections and US equity volatility?

Historical data suggests there is, with volatility typically decreasing from June to November.

In 2012, volatility has decreased in every month to date, except for July and August, which recorded modest increases.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 042

Equity Risk and the Presidential Election
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Successfully Riding the Correlation Rollercoaster

Correlated markets dominated the investment landscape in 2011 and into 2012 — a situation that seems likely to persist. For investors, understanding the benefits and pitfalls of investing in different correlation environments is essential. Here we examine some strategies suited to a high correlation environment, and offer a series of "do's and don'ts" for investors as they ride the correlation rollercoaster.

Authors: Scott Hamilton, Sebastian Ceria, PhD, and Melissa Brown, CFA.

Research Paper No. 041

The Signpost Up Ahead: Risk Danger Zones
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The Signpost Up Ahead: Risk Danger Zones
What Multiple Risk Models Can Tell Us About Future Drawdowns

Do differences in risk predictions across a range of risk models provide market insight? Axioma produces four different risk models for each country and region, each with its own risk estimate calculated by looking at the data from a different perspective. In 2008 our short-horizon statistical models reacted more quickly and with more factor risk than our medium-horizon, fundamental models because the statistical models picked up a factor that was not “seen” by the fundamental models.

In this article, we show that multiple risk model predictions can potentially identify “danger zones” associated with future market drops. We use two risk metrics – average predicted risk and the maximum difference between Axioma’s four risk model predictions – to identify conditions when the maximum, forward, 90-day drawdown has been significant.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 040

The Signpost Up Ahead: Risk Danger Zones
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Neutralizing Unintended Bets in Factor ETFs
Purifying the Target Signal Improves Performance

In 2011, there was an explosion of ETFs offering a wide selection of affordable “factor” exposures, including the Russell-Axioma Factor ETFs and PowerShares ETFs. The factors selected – Volatility, Beta and Momentum, among others – are a subset of the “Style Risk Factors” used by equity fundamental factor risk models for the past three decades, so these factors clearly explain risk. Several of these factors are also closely associated with highly successful hedge funds, so the implication is that these factors are also potential alpha signals.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 039

A Matter of Perspective
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Aligning Alpha and Risk Factors, a Panacea to Factor Alignment Problems?

The practical issues that arise due to the interaction between three principal players in any quantitative strategy, namely, the alpha model, the risk model and the constraints are collectively referred to as Factor Alignment Problems (FAP). While the role of misaligned alpha factors in causing FAP is relatively easy to understand, incorporating the impact of constraints entails considerable analytical complexity that most consultants and researchers found diffcult to fathom. A few of them have even gone to the extent of suggesting that aligning alpha and risk factors should suffice in handling FAP. We provide a solid rebuttal to this line of thinking by demonstrating typical symptoms of FAP in optimal portfolios generated by using completely aligned alpha and risk models. Additionally, we provide theoretical guidance to clarify the role of constraints in influencing FAP and illustrate how the Alpha Alignment Factor (AAF) methodology can handle misalignment resulting from constraints, analytical complexities notwithstanding.

Authors: Anureet Saxena, Ph.D, CFA, Christopher Martin, CAIA, and Robert A. Stubbs, Ph.D

Research Paper No. 038

A Matter of Perspective
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Factor Alignment Problem in Quantitative Portfolio Management

Factor alignment problems (FAPs) result from misalignment of the expected returns model, the risk model and the constraints used to formulate a portfolio construction problem. The impact on portfolio performance is invariably negative. This article discusses the effects of FAPs on optimized portfolios and demonstrates how the Alpha Alignment Factor (AAF) and custom risk models not only offer a practical solution to alignment problems, but also give the portfolio manager access to portfolios that lie above the traditional risk-return frontier defined by the user risk model.

The Journal of Portfolio Management
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A Matter of Perspective
Equity Correlations in 2011

Equity correlations surged in 2011, spiking in August roughly a year after a similar spike in 2010. However, when viewed over the entire 2011 time window, however, virtually all markets exhibited a near-linear increase in asset-asset correlations over the course of the year, starting from a January low and ending in December with values just slightly below August’s peak. In fact, if the 2011 trend were to continue in 2012 (which it cannot), the average asset-asset correlation in both the FTSE Developed Europe Index and the Russell 1000 Index would reach one sometime in May or June 2012—a cataclysmic event, to be sure. From the perspective of 2011 in its entirety, the relevant questions are: “Why were correlations so high in August?” and “Why were correlations so low in January?”

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 037

A Matter of Perspective
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An Empirical Case Study of Factor Alignment Problems using the United States Expected Returns (USER) Model

The practical issues that arise due to the interaction between three principal players in any quantitative strategy, namely, the alpha model, the risk model and the constraints are collectively referred to as Factor Alignment Problems (FAP). This paper concerns empirical illustration of various facets of FAP using the United States Expected Return (USER) model. Unlike previous studies on FAP that are either based on simulated returns or a black-box expected returns model, we leverage the detailed knowledge of the USER model to create an insightful narrative. We show that optimal portfolios constructed using the USER model without taking into account the misalignment issues betray typical symptoms of FAP and have exposure to certain hidden systematic risk factors that are not accounted for during portfolio construction. We trace the origins of these latent systematic risk factors to the constituent factors of the USER model and the turnover constraint. Finally, we leverage our understanding of the alignment issues to propose an alternative portfolio construction methodology that directly addresses FAP. Using the proposed methodology not only gives unbiased risk forecasts but also improves the ex-post performance in a statistically significant manner.

Authors: Anureet Saxena, Ph.D, CFA and Robert A. Stubbs, Ph.D

Research Paper No. 036

What Goes Up...
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What Goes Up…
Have Correlation and Volatility Turned the Corner in the US?

After an awful August, volatility and correlations in the US appear to be headed towards more typical values, making US market conditions more favorable to investors. However, the same trends in Europe and Emerging markets just reversed themselves. There are other indications, too, that we are not yet out of the woods...

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 035

What Goes Up...
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Slow Burn or Powder Keg?
Current Insights from the Axioma Risk Models

Markets are off 15% to 20% from their peaks in mid July, and volatility has been, well, volatile. As investors consider their responses to current market conditions, we report on the recent changes in the Axioma Risk Models™ in both the US and Europe.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 034

Slow Burn or Powder Keg?
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Which Factors Didn’t Flinch?
Analyzing Recent Market Movements Using Risk-Adjusted Factor Returns

Markets worldwide tumbled last week and plummeted yesterday. A dreadful combination of debt concerns, dysfunctional politics and rising investor fear has led to the worst one-week market performance since 2008.

We analyze market trends over the last three months using Axioma’s daily fundamental factor risk model. We assess factor performance by creating long-only portfolios that either maximize or minimize their exposure to one risk factor while remaining neutral to all other factors in Axioma’s risk models. The relative performance of these portfolios compared with the benchmark produces a risk-adjusted factor return that indicates the relative performance of that factor.

Author: Anthony A. Renshaw, Ph.D.

Research Paper No. 033

Which Factors Didn’t Flinch?
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Using Multiple Risk Models for Superior Portfolio Management…
A Practice Not Just For Quants

The focus on risk management is today unprecedented. We believe the use of daily multiple risk models – both short and longer horizon, and both fundamental and statistical – can help managers to predict portfolio risk more accurately. Multiple risk estimates provide a more comprehensive view of portfolio risks, and the daily data that underlies these models can help managers to react faster and with more confidence. Indeed, for growing numbers of Axioma’s clients the use of multiple daily risk models is already becoming best practice. The case study presented here highlights the benefits of looking at portfolios through the lenses of multiple daily risk models.

Authors: Melissa Brown, CFA and Chris Canova, CFA

Research Paper No. 032

Using Multiple Risk Models for Superior Portfolio Management…
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Market Aftershocks? A Seven-Day Update
The Global Impact of the Japan Earthquake As Seen Through the Lens of Axioma’s Daily Risk Models

A seven-day update of Axioma’s first article1 on the impact of events in Japan on markets worldwide shows the initial trends continuing to play out. Non-Japanese markets continue to recover from their lows on March 14 and remain well de-correlated from the events in Japan2, while Japan’s domestic market remains challenging.

The initial response of the market was, of course, to reduce its exposure to Japan. As the situation began to improve or at least stabilize following March 14, investors who were short on volatility in Japan started to increase their exposure to and risk in Japan.

The situation merits continued close observation, at least until control of the Fukushima nuclear plant is regained. Positive news from Japan is likely to be followed by strong re-investment in that market.

Author: Anthony Renshaw, PhD

Research Paper No. 031

[1] Market Aftershocks? The Global Impact of the Japan Earthquake as Seen Through the Lens of Axioma’s Daily Risk Models

[2] The market movements in response to Japan were not driven by any particular industries, so there was little contagion to other markets as a result of the high level of industry-industry correlations. See Link(correlations piece) for details.

Market Aftershocks? A Seven Day Update
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Is Big Beautiful Again?

Over the last two to three years, assets under management left large cap equity markets in favor of small cap and emerging markets, where plenty of alpha opportunity remained. However, the recent rally of the US and other developed equity market suggests a reversal of that trend with rebalancing aimed at de-risking investments by moving out of small and emerging markets and into large, liquid stock in developed markets.

An examination of the factor returns over a range of Axioma’s fundamental factor risk models shows that four technical factors have dominated performance over the last four months through the end of February: beta or market sensitivity, volatility, short-term momentum, and liquidity.

In the EM and smaller markets such as Asia Pacific Ex Japan, all four of these factors have exhibited consistently negative returns. In the largest developed equities markets – US, Great Britain, and, prior to the earthquake, Japan – strategies with high beta exposure have shown the best factor returns, suggesting that the developed equity markets may now be a safe haven for investments compared with other world equity markets.

Author: Anthony Renshaw, PhD

Research Paper No. 030

Is Big Beautiful Again?
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Market Aftershocks?
The Global Impact of the Japan Earthquake as Seen Through the Lens of Axioma’s Daily Risk Models

Markets worldwide have been jolted by the effects of the devastating earthquake off the coast of Japan on Friday March 11. This article highlights the changes in Axioma’s equity risk models that have occurred over the last few days (through market close 3/17) and provides some interpretation of these changes. Because Axioma’s risk models are updated daily, we are able to observe market developments as they occur, in contrast to conventional risk models in which the covariance matrix is updated on a monthly basis. While markets outside of Japan flinched in response to initial reports of the disaster, the good news at this point in time is that those markets now seem to have stabilized. The situation for Japan, however, remains volatile.

Author: Anthony Renshaw, PhD

Research Paper No. 029

Market Aftershocks?
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Finance, Statistics, Accounting, Optimization and some Alignment Problems

Quantitative equity portfolio management has evolved into an inter-disciplinary activity that draws expertise from fields of finance, statistics, accounting and optimization. Each one of these streams is a matured discipline in itself, having its own body of knowledge and operates under assumptions that are usually well-accepted within the respective community. However, when concepts from these diverse fields are applied in a common setting, there is bound to be friction between various assumptions which get magnified due to the use of an optimizer.

This paper focuses on problems that arise due to interaction between three principal players in any quantitative strategy, namely, the alpha model, the risk model and the constraints. We present a detailed investigation of these misalignment problems, survey some of their common sources, analyze and document their effects on the ex-post performance of optimized portfolios and finally conclude with a practical and effective remedy in the form of augmented risk models to circumvent them.

Authors: Anureet Saxena, PhD, Robert A. Stubbs, PhD

Research Paper No. 028

Finance, Statistics, Accounting, Optimization and some Alignment Problems
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The Domino Effect in Industries:
Why Equity Correlation Surges Are Now So Severe

Equity correlations world-wide have dropped significantly after soaring to all-time highs between March and July 2010. The speed with which correlations surged and fell, as well as the fact that the swing could not be easily attributed to specific market events perplexed many market observers. The surge of 2010 prompted some to wonder if the new highs represented a permanent change in the market that might fundamentally alter approaches to portfolio management. Or was it merely a short-term “bubble” that would revert relatively quickly to lower correlation levels? The recent sharp decline back to precrisis correlation levels would seem to support the latter view.

For more detailed coverage of this topic please see Research paper No. 026

Authors: Anthony Renshaw, PhD, Anureet Saxena, PhD

Research Paper No. 027

The Domino Effect in Industries
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What’s Up with Equity Correlations?
They’re Down…and Factor Volatility Is to Blame
Using Axioma’s Risk Models to Explain the Recent Surge in Equity Correlation

Equity correlations have dropped significantly after soaring to all-time highs between March and July 2010. Predictions of equity correlations derived from Axioma’s Risk Models reflect these recent correlation changes accurately. Short-horizon risk models have tracked the recent decline in asset-asset correlations closely, while medium-horizon risk model correlations are declining according to a longer, less reactive time horizon.

For abbreviated coverage of this topic, please see Research paper No.027

Authors: Anthony Renshaw, PhD, Anureet Saxena, PhD

Research Paper No. 026

What’s Up with Equity Correlations?
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Yesterday’s News in Today’s Risk Model:
A Risk Model’s Perspective on the Chinese Currency Debate

The currency, country and market correlations in Axioma’s Global Fundamental Factor Risk Model have closely tracked the political and economic Chinese currency movements that occurred in 2010. This data illustrates how quickly and accurately daily risk models can capture and model market news.

Author: Anthony Renshaw, PhD

Research Paper No. 025

Yesterday’s News in Today’s Risk Model?
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Is This An Equity Correlation Bubble?

Is This An Equity Correlation Bubble? An historical examination of equity correlations suggests that equity correlations will remain at historically high levels. The correlations may not be as high as those achieved during the two recent correlation spikes, but they are likely to be higher than those experienced prior to 2007.

In addition, the recent return correlations of non-US equities in US dollar have been noticeably higher than the correlations of the returns in their local currencies. The indication is that currencies returns have become an increasingly large and increasingly common component of non-US equity performance, a fact that should be considered for those implementing currency hedges.

Author: Anthony Renshaw, PhD

Research Paper No. 024

Is This An Equity Correlation Bubble?
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Returns-Timing: A Solution to Market Asynchronicity (Short Version)

Until recently, risk models have tended to be built using low frequency (weekly or monthly) data. This has resulted in a necessary compromise between stability (for which one needs a long history of data) and responsiveness, for which, the shorter the history, the better. Stability plus responsiveness can be achieved if one uses daily data, which allows for a large number of observations to be used in model estimation without the historical baggage of long out-of-date and irrelevant events. Daily data have additional problems, however, as the differing closing times of markets worldwide may induce spurious relationships across model factors; in particular, correlations between markets may appear lower than they are in fact, due to the market lag effect. In this report, we offer a solution to this dilemma, that enables us to build a stable, daily data-based model, but which takes account of the differing market closing times and corrects the model factor correlations accordingly.

Authors: Simon Bell, Frank Siu, Stefan Schmieta

Research Paper No. 023


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Pushing the Frontier (literally) with the Alpha Alignment Factor

Construction of optimized portfolios entails complex interaction between three key entities, namely, the risk factors, the alpha factors and the constraints. The problems that arise due to mutual misalignment between 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, and inability to harvest high quality alphas into above-average IR. In this paper, we give a detailed analysis of FAP and discuss solution approaches based on augmenting the user risk model with a single additional factor y. For the case of unconstrained MVO problems, we develop a generic analytical framework to analyze the ex-post utility function of the corresponding optimal portfolios, derive a closed form expression of the optimal factor volatility value and compare the solutions for various choices of y culminating with a closed form expression for the optimal choice of y. Ultimately, we show how the Alpha Alignment Factor (AAF) approach emerges as a natural and effective remedy to FAP. AAF not only corrects for risk underestimation bias of optimal portfolios but also pushes the ex-post efficient frontier upwards thereby empowering a PM to access portfolios that lie above the traditional risk-return frontier.

Authors: Anureet Saxena, Robert A. Stubbs

Research Paper No. 022


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Alpha Alignment Factor: A Solution to the Underestimation of Risk for Optimized Active Portfolios

The underestimation of risk of optimized portfolios is a consistent criticism about risk models. Quantitative portfolio managers have historically used a variety of ad hoc techniques to overcome this issue in their investment processes. In this paper, we construct a theory explaining why risk models underestimate the risk of optimized portfolios. We show that the problem is not necessarily with a risk model, but is rather the interaction of expected returns, constraints, and a risk model in an optimizer. We develop an optimization technique that incorporates a dynamic Alpha Alignment Factor (AAF) into the factor risk model during the optimization process. Using actual portfolio manager backtests, we illustrate both how pervasive the underestimation problem can be and the effectiveness of the proposed AAF in correcting the bias of the risk estimates of optimized portfolios.

Authors: Anureet Saxena, Robert A. Stubbs

Research Paper No. 015


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Constraint Attribution

Constraints are now an integral part of the portfolio construction process. With constraints comes the challenge of understanding how they cause the optimal portfolios to deviate from a trade-off dictated by the forecasts of risk and return. We describe the theory and application of a technique able to quantify the impact of individual constraints in several different ways. This includes decomposing the difference between the optimal constrained and unconstrained portfolios and the difference between alphas and implied alphas as described in earlier work by Grinold and others. Furthermore, we introduce a new technique that applies these decompositions on an ex-post basis, providing on understanding of how constraints actually impact realized risk and return.

Authors: Robert A. Stubbs, Dieter Vandenbussche

Research Paper No. 014


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Multi-Portfolio Optimization and Fairness in Allocation of Trades

When trades from separately managed accounts are pooled for execution, the realized market-impact cost can be far greater than the sum of the predicted cost over all accounts. Multi-portfolio optimization is a technique for rebalancing multiple portfolios at the same time, considering their joint effects while adhering to account-specific constraints. The interaction of accounts in a multi-portfolio setting can bias particular accounts if fairness is not considered in the solution methodology. With respect to the trading of multiple accounts, fairness is not well-defined. Definitions vary among portfolio managers often based on their particular investment offering. For this reason, we do not prescribe a single best approach for multi-portfolio optimization. Instead, we discuss the pros and cons of two approaches that each has foundations in economic theory, the Cournot-Nash equilibrium and the collusive solution. We present a unified framework capable of solving either problem.

Authors: Robert A. Stubbs, Dieter Vandenbussche

Research Paper No. 013


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How to Evaluate a Risk Model

Risk model providers commonly report the average R2 value of the asset returns model. Some models, such as statistical models, will consistently have greater R2 values than others. However, strong explanatory power from a returns model does not necessarily translate into an accurate risk model. The ultimate test of a risk model lies in the testing of its risk forecast against realized values.

Research Report No. 012

How to Evaluate a Risk Model
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How Stale is your Risk Model?
Daily Risk Changes in September 2008

In this short article, we report some of the dramatic risk changes that occurred during September 2008, and quantify the inaccuracies that occurred with a monthly risk model. The results show that stale risk models seriously misestimated risk during the second half of September. In times like these, there is no justification for using anything other than a risk model that is updated daily.

Author: Anthony Renshaw - PhD

Research Report No. 011

How Stale is your Risk Model? Daily Risk Changes in September 2008
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September 15, 2008: A Market Analysis

This short article analyzes the market situation on September 15, 2008, using Axioma US fundamental risk model.

In addition to identifying the factors that drove returns that day, the results highlight recent changes in market conditions that may be important to portfolio managers.

Author: Anthony Renshaw - PhD

Research Report No. 010

September 15, 2008: A Market Analysis
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Using Axioma's Alpha Factor Method To Correct the Misalignment
of Alpha Model and Risk Model Factors

Typically, the number of factors used in an alpha or risk model is much lower than the number of assets in the portfolio, or in the investable universe. This short note studies the consequence of this "dimensionality gap" and shows how Axioma's Alpha Factor™ method can limit the potential impact of the difference between the asset space and the factor space to produce improved realized portfolio performance.

Author: Anthony Renshaw - PhD

Research Report No. 009

Using Axioma's Alpha Factor Method To Correct the Misalignment of Alpha Model and Risk Model Factors
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Portfolio Construction Strategies Using More Than One Risk Model

Using more than one risk model in a portfolio construction strategy allows a portfolio manager to exploit the fact that different risk models measure and capture risk differently. Having both a fundamental and statistical risk model simultaneously in the strategy ensures that the optimized portfolio reflects both points of view. Two risk model strategies can produce just as conservative portfolios and better overall performance than one risk model alone provided that the strategies are calibrated so that both risk models affect the optimal portfolio solution.

Author: Anthony Renshaw - PhD

Research Report No. 008

Portfolio Construction Strategies Using More Than One Risk Model
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Real World Case Studies in Portfolio Construction
Using Robust Optimization

Robust portfolio optimization has been a core feature of Axioma's portfolio construction tools for years, and many of our clients use robust optimization in their portfolio construction processes to deliver higher value added. This study reports a series of real world portfolio construction case studies documenting different approaches for implementing robust portfolio optimization and their benefits. The results provide guidance for designing robust portfolio construction strategies.

Author: Anthony Renshaw - PhD

Research Report No. 007

Real World Case Studies in Portfolio Construction Using Robust Optimization
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Risk Model Reliability: Daily vs. Monthly Re-Estimation

Research demonstrates significant risk model errors are likely when risk models are not up-to-date.

Author: Anthony Renshaw - PhD

Research Report No. 006

Risk Model Reliability: Daily vs. Monthly Re-Estimation
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Diagnosing When Leverage Will Benefit Active Equity Funds

Axioma's Applied Research team offers a practical method for predicting when leverage is likely to improve the performance of a long-only portfolio.

Author: Anthony Renshaw - PhD

Research Report No. 005

Diagnosing When Leverage Will Benefit Active Equity Funds
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Constraint Attribution

Understanding the impact of constraints imposed during portfolio construction and being able to quantify that impact to the asset owners is critical.

This research paper demonstrates how to measure the effects that individual constraints have on a portfolio, compared with a portfolio dictated exclusively by a trade-off of forecast risks and returns.

Author: Robert A. Stubbs - PhD, Dieter Vandenbussche - PhD

Research Report No. 004

Constraint Attribution
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Incorporating Estimation Errors Into Portfolio Selection: Robust Portfolio Construction

Portfolio managers who rely on mean-variance efficiency often find that their portfolios are unintuitive or do not behave as expected. In this paper we discuss how estimation error can affect the quality of your portfolio and what you can do about it.

Authors: Sebastian Ceria - PhD, Robert A. Stubbs - PhD

Research Report No. 003

Incorporating Estimation Errors Into Portfolio Selection: Robust Portfolio Construction
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Axioma's Alpha Factor Method: Improving Risk Estimation by Reducing Risk Model Portfolio Selection Bias

Axioma's patent-pending Alpha Factor™ method can be used to improve your risk estimation by reducing risk model portfolio selection bias.

Authors: Anthony Renshaw - PhD, Robert A. Stubbs - PhD, Stefan Schmieta - PhD, Sebastian Ceria - PhD

Research Report No. 002

Axioma's Alpha Factor Method: Improving Risk Estimation by Reducing Risk Model Portfolio Selection Bias
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Computing Return Estimation Error Matrices for Robust Optimization

A practical look at methods for computing effective error estimates to be used as inputs for robust optimization.

Authors: Robert A. Stubbs - PhD, Pamela Vance - PhD

Research Report No. 001

Computing Return Estimation Error Matrices for Robust Optimization
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