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University Of Amsterdam, Amsterdam Business School Master in International Finance
Master Thesis
The Low-Risk Anomaly in European Equity Markets
August 2013
Student: Quinby, Yvette Supervisor: Benninga, Simon
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Abstract
I examine selected European equity markets for evidence consistent with the hypothesis that low risk assets have higher risk adjusted returns than high risk assets.
Consistent with the prediction that this low risk anomaly is a global phenomenon, I find evidence that low beta is associated with high alpha in the selected markets. I also find that a strategy designed to exploit the anomaly can yield significant alpha and a high Sharpe ratio relative to the value-weighted market index.
In contrast to previous research in U.S. markets, I do not find evidence that the low risk anomaly can be attributed to a specific investor group overweighting high risk assets relative to their peers in the selected European equity markets.
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Table of Contents
I. Introduction ... 4
a. The Low-Risk Anomaly ... 4
b. Research in European Equity Markets ... 6
II. Literature Review ... 8
a. Are mutual funds and their investors irrational? ... 8
b. Behavioural Finance As An Explanation To The Low-Risk Anomaly ... 10
c. Leverage Aversion As An Explanation To The Low-Risk Anomaly... 11
d. Speculation and Limits to Arbitrage ... 14
III. Data ... 15
IV. Methodology ... 16
a. Beta as a Measurement of Risk ... 16
b. Testing if High-Beta is Associated With Low-Alpha ... 16
c. Testing the Low-Risk Anomaly Arbitrage ... 17
d. Testing Portfolio Beta of Investor Groups ... 19
V. Results ... 21
a. Testing if High-Beta is Associated With Low-Alpha ... 21
b. Testing the Low-Risk Anomaly Arbitrage ... 24
c. Testing Portfolio Beta of Investor Groups ... 27
VI. Conclusion ... 30
VII. Appendix A – Additional Empirical Results ... 32
Figure A ... 32 Figure B ... 33 Figure C ... 34 Figure D ... 36 Figure E ... 38 Table D1 ... 39 VIII. References ... 40
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I.
Introduction
The low risk anomaly refers to evidence that low risk assets offer higher risk adjusted returns than high risk assets. This evidence contradicts a core principal of modern financial theory – that investors must be compensated for accepting higher risk with higher returns.
In this section I present (a) a brief overview of the low risk anomaly and (b) an introduction to my research on the anomaly in European equity markets.
a. The Low-Risk Anomaly
A central facet of modern financial theory is that investors are compensated for taking on risk with higher returns. Yet an ever growing body of research continues to present evidence that this may not be the case at all. In fact the opposite often appears to be the case: low risk assets are observed offering higher returns per unit of risk than high risk assets.
Modern Portfolio Theory (MPT) from Markowitz (1952) explains how an investor should select a portfolio of assets that maximises the ratio between reward and risk. The portfolio which achieves the maximum ratio is known as the tangency portfolio and the risk-reward ratio is known as the Sharpe ratio.
The Capital Asset Pricing Model (CAPM) from Sharpe (1964) and Lintner (1965) assumes that if all investors are rational and informed, they will select the tangency portfolio. They prove that if all agents invest this way, the tangency portfolio will equal to the value-weighted market portfolio. But as far back as Black, Jensen, and Scholes (1972), it was noted that the security market line (SML) for U.S. stocks is flatter than predicted by CAPM. In other words, the value-weighted market
portfolio of U.S. stocks did not appear to equal the tangency portfolio and consequently achieved a lower Sharpe ratio. The SML is better explained by CAPM with restricted borrowing than the standard CAPM (Black (1972)).
Since then, evidence consistent with the flatness of the SML has continued to be presented in published research, with authors noting one consistent symptom of the SML flatness: that low risk assets offer higher risk adjusted returns than high risk assets. A notable, but by no means
exhaustive, list of research covering this low risk anomaly includes Karceski (2002), Baker, Brendan & Wurgler (2010), Frazzini & Pedersen (2011), Asness, Frazzini & Pedersen (2012), Hong & Sraer (2012) and Baker & Haugen (2012).
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Evidence for the low risk anomaly has been found in U.S. stocks during the period from 1968 to 2008 (Baker, Brendan and Wurgler (2012) and from 1926 to 2009 (Frazzini & Pedersen (2011)). Evidence consistent with the low risk anomaly was also found in non-U.S. developed equity markets (Frazzini & Pedersen (2011) and Baker & Haugen (2012)), emerging equity markets (Baker & Haugen (2012)), as well as U.S. treasury and corporate bond markets (Frazzini & Pedersen (2011)).
Explanations for the low risk anomaly have so far been primarily focused on leverage aversion and other behavioural finance theories.
The theory of leverage aversion assumes that some investors are constrained from applying leverage to their investments. CAPM assumes that investors will select the portfolio of assets which offers the highest Sharpe ratio. If an investor requires higher returns than those offered by the tangency portfolio, they should simply apply leverage to the same portfolio in order to maintain the most efficient risk-reward ratio. But what are investors who cannot, or will not, apply leverage to do? Leverage aversion theory proposes that such investors choose to overweight high risk assets in order to achieve the desired higher returns at the expense of an efficient risk-reward ratio. The increased demand for high risk stocks inflates prices, suppressing future expected returns. The existence of such investors means that the value-weighted market value portfolio is no longer equal to the tangency portfolio, resulting in a flatter SML.
Frazzini & Pedersen (2011) argue that it is reasonable to assume that such leverage constrained investors do indeed exist. In the United States, mutual funds are restricted by regulation from applying leverage. They are also required to keep cash on hand in order to meet redemptions. Individual investors in all markets are likely to face high costs for borrowing and/or onerous margin constraints. The authors present evidence that U.S mutual funds and individual investors hold investment portfolios with a beta higher than one, where as firms that actively apply leverage (leveraged buyout firms and Berkshire Hathaway) hold investment portfolios with a beta lower than one. The evidence is consistent with the leverage aversion theory.
Several other interesting theories seeking to explain the low risk anomaly have been put forward by researchers. Most focus on different aspects of behavioural finance and limits to arbitrage due to constraints on investors that restrict them from short selling and/or applying leverage. These are covered in detail in the Literature Overview section, but I briefly discuss two of the theories here. Karceski (2000) argues that the behaviour of mutual funds may explain the low risk anomaly.
Following a dramatic upward movement in the stock market, mutual funds experience large investor cash-inflows. Investors will select whichever mutual fund has the best recent performance. Because
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high risk stocks outperform low risk stocks in a bull market, mutual funds are incentivised to overweight high risk stocks in order to capture a larger share of the cash inflow. The increased demand for high risk stocks inflates prices, suppressing future expected returns. Karceski (2000) finds that between 1984 and 1996, aggressive growth equity funds had more than twice as much (18% of all investment) as income equity funds (8% of all investments), with respective beta’s of 1.12 and 0.72.
Baker, Brendan and Wurgler (2010) present existing evidence that investor behaviour is irrational and tilted towards overweighting high risk assets. One example is evidence that investors have a preference for lottery-like bets. That is, they have a preference for scenarios that offer an unlikely probability for very large financial gain at the expense of near certain, but very limited, financial loss. High-risk assets emulate this lottery-like scenario. Another behavioural trait is representative bias. For example, investors may incorrectly identify all “tech” stocks as “winners” due to the few that did exceedingly well and received a lot of press coverage as a result. Investors will ignore or be unaware of the overwhelmingly large number of “tech” stocks that perform very poorly.
b. Research in European Equity Markets
The existing research on the low risk anomaly is primarily focused on U.S. markets. Some research, such as Frazzini & Pedersen (2011), Asness, Frazzini & Pedersen (2012) and Baker & Hagen (2012), does partially cover international equity markets. But the results for individual country markets are not subjected to detailed analysis.
Evidence in support of theories that seek to explain the low risk anomaly is overwhelmingly focused on the U.S. market. It typically involves an analysis of U.S. investor portfolios in domestic equities. In order to gain new insights, I have chosen to direct my research at non-U.S. markets. I have selected five European equity markets in which I research the low risk anomaly: Germany, France, Spain, Switzerland and The Netherlands.
Aside from offering a more global perspective of the anomaly, these markets are also of interest because they are so much smaller than the U.S. equity market. The limited existing research on European equity markets covers (nearly) all listed equities in each market. Yet in many European markets, more than 90% of the market capitalization is held in only a handful of stocks. For example in Switzerland, where more than 90% of the market capitalization is held by less than 20 stocks. This raises the question that even if the low risk anomaly exists in these countries, could it be exploited
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without relying on the questionable assumption that small-cap, illiquid stocks be part of any exploitative strategy?
For this reason, I narrow my research to those stocks with the highest market capitalization and trading volumes in each country. In Germany, this is the constituents of the DAX index; in France, constituents of the CAC index; in Spain, constituents of the IBEX index; in Switzerland, constituents of the SMI index; and in The Netherlands, constituents of the AEX index.
For each market, I aim to answer three key questions: (1) is there empirical evidence consistent with the proposition that high beta is associated with low alpha? (2) Is there empirical evidence
consistent with the proposition that the low risk anomaly can be arbitraged for profit? (3) Is there empirical evidence consistent with the proposition that a specific investors group over or under weights risky assets relative to other market participants?
To answer these questions, I examine the returns for stocks that are constituents of the DAX, CAC, IBEX, SMI and AEX indices between January 2001 and January 2013. I use ex-ante beta, calculated relative to the returns on the market index, as a measurement for risk.
In all five markets, I find evidence consistent with the proposition that high beta is associated with low alpha. Between January 2002 and January 2013, the 20% of stocks with the lowest beta per market record a positive expected excess return, a positive alpha and a positive Sharpe ratio. For the same time period, the 20% of stocks with the highest beta per market record a negative expected excess return, a negative alpha and a negative Sharpe ratio.
To test the question of arbitrage profitability, I apply the Bet Against Beta (BAB) portfolio strategy from Frazzini & Pedersen (2011) to my five selected European markets. The BAB portfolio is a beta-neutral, zero-cost, arbitrage strategy which is long low beta stocks and short high beta stocks. The strategy is rebalanced each month, with the stocks weighted according to their beta ranking. I find evidence that the BAB portfolio strategy produces significant alpha in the DAX, IBEX and AEX markets. In all five markets, I find that a unit of currency invested in the BAB portfolio results in a profit. The profit is greater than a unit of currency invested in the respective market index for the same period.
To address question (3), I examine the equity portfolios for different groups of investors in each market index. The result is an unclear picture in which no evidence is found to support the theory that specific investor groups allocate risk significantly different from other groups, or that their specific risk allocation behaviour might result in the low risk anomaly.
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The rest of the paper is organized as follows. Section II provides a comprehensive literature review of recent papers covering the low risk anomaly. Section III describes the data used in my research. Section IV described the empirical methodology of my research. Section V presents my results and Section VI concludes. Appendix A provides additional empirical results.
II.
Literature Review
a. Are mutual funds and their investors irrational?
Karceski (2000) considers how cash inflows to equity mutual funds in bull markets influence the investment behaviour of the fund managers. The paper presents evidence based on U.S. equity mutual fund investment portfolios to support the case that equity mutual funds overweight high beta stocks.
The paper considers the lack of empirical evidence for CAPM, citing evidence from past papers dating back as far back as Black (1972). Noting that the actual security market line is flatter than that predicted by CAPM, the author proposes the explanation that investors in mutual funds are chasing returns, resulting in increased demand for high beta stocks and reduced future returns on those stocks.
The author cites three empirically verifiable facts that support his proposal. (1) When investors decide in which mutual fund to invest, they tend to select those funds with superior past
performance. (2) There tend to be large cash inflows to the aggregate equity market mutual fund industry just after a dramatic upward movement in the stock market. (3) High-beta stocks outperform low beta stocks in a bull market.
If these facts are accepted, then mutual fund managers care more about performance in a bull market than they do in a bear market. In a bear market, cash inflows will dry up regardless of performance. In a bull market, it makes sense for a manager to tilt their portfolio in favour of high beta stocks in order to maximise returns and capture a greater portion of the cash inflows from investors.
To investigate this theory, the paper examines the shareholdings of U.S. mutual funds and compares the beta of their portfolios to that of the S&P 500. The author finds that between 1984 and 1996, aggressive growth equity funds had more than twice as much (18% of all investment) as income equity funds (8% of all investments), with respective beta’s of 1.12 and 0.72. For the same period, the beta of all equity portfolio held by all equity mutual funds was 1.05 (when controlling for cash held). However, the statistical significance of this result is not reported.
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Baker & Haugen (2012) provides broad research supporting the hypothesis that low risk stocks outperform high risk stocks in 21 developed countries and 12 emerging markets during the period 1990 to 2011. The authors propose and present evidence to consistent with the explanation that institutional investors overweight high risk stocks, driving up prices and suppressing future returns. The authors calculate the volatility of stock returns in worldwide markets. For each market, at the start of each month, they rank the stocks by volatility and sort the stocks into deciles. At the end of the month they calculate the returns for each decile portfolio.
The paper compares the difference in (1) returns, (2) volatility and (3) Sharpe ratio between the lowest and highest decile portfolios. In all markets, the lowest volatility decile portfolio outperforms the highest volatility decile portfolio.
Whilst the results are impressive, the authors do not provide tests for statistical significance. The likelihood that the highest volatility decile portfolio would contain a high proportion of illiquid small-cap stocks could also make it hard to arbitrage the difference in returns in practice.
The authors propose an explanation for the low risk anomaly based on compensation structure and agency issues with institutional investment managers.
The paper argues that when an investment manager is paid a bonus based on performance over a certain benchmark, the investment manager is incentivised to select a more volatile portfolio. The authors also argue that high-volatility stocks have a higher level of analyst and news coverage. Investment managers are therefore driven to select these stocks because the increased coverage allows the manager to more easily explain his stock selection in monthly portfolio meetings and to explain changes in the portfolio to clients.
The result is that institutional investors overweight their portfolios with high risk stocks. This drives up the price of high-volatility stocks, reducing their future expected return.
The authors provide evidence to support to proposition that high-volatility stocks have increased analyst and news coverage by examining the relationship between the number of analyst reports and the returns of U.S. stocks between 2000 and 2011. They find a statistically significant positive relationship between the two variables. The authors also examine the relationship between volatility and the number of stories on the Dow Jones News Wire for each stock. Again, they find that more volatile stocks have higher news coverage.
The evidence to support the claim that institutional investors hold more volatile portfolios is less clear. The authors examine the institutional investor ownership of the top 1000 U.S. stocks by
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market capitalization between 2000 and 2009. The stocks are sorted into deciles according to market capitalisation. Then within each decile, three equally populated groups are formed according to percentage ownership by institutional investors. When comparing the highest ownership group with the lowest ownership group, the highest ownership group portfolio has a higher volatility than the lowest ownership group. However, no statistical significance tests are provided.
b. Behavioural Finance As An Explanation To The Low-Risk Anomaly
Baker, Brendan and Wurgler (2010) present further evidence that low risk stocks have outperformed high risk stocks in the U.S. market. They propose an explanation incorporating key elements of behavioural finance: irrational behaviour by investors and limits to arbitrage.
The paper cites strong evidence that high risk stocks have substantially underperformed low risk stocks in U.S. markets for the previous 41 years, whether risk is measured by beta or volatility. The authors present research on U.S. stock return data between 1968 and 2008. Their methodology sorts stocks in five groups each month according to volatility or beta. The returns of each group are then tracked. The process is repeated for the top 1000 stocks by market capitalization.
The results show that the portfolio group with the lowest risk stocks, whether measured by volatility or beta, outperform the portfolio group with the highest risk stocks for the period. For beta sorted portfolios, the lowest risk portfolio group has significant positive alpha and the highest risk portfolio group has significant negative alpha. This is applies to both the top 1000 stocks and all stocks. For volatility sorted portfolios, the lowest risk portfolio group has significant positive alpha. The highest risk portfolio group had significant negative alpha when applied to all stocks, but not when restricted to the top 1000 stocks.
The authors also note that growth in the gap between returns in the lowest and highest risk portfolios accelerated after 1983 (with the exception of the technology bubble).
They propose and provide two explanations for this anomaly: individual investor’s preference for risk and institutional investors mandate to maximise their information ratio.
The proposed explanations are part of existing behavioural finance models that assume that some market participants are irrational in some way and that there are limits to arbitrage. For individual investors, the authors emphasise preference for lotteries, representativeness bias and
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The lottery preference refers to evidence that people are often will to accept near-certain losses in exchange for a slight chance of winning big. A lottery, for example. The concept can be applied to low-priced volatile stock where there is a small chance of double or tripling invested money, but a much larger chance of losing some of the investment.
The representativeness bias refers to evidence that people routinely mistake narrow characteristics as representative of a group. For example, investors may consider high-tech shares as representative of stocks with above average stock market performance. But this would ignore the much larger number of high-tech stocks that perform badly.
The authors propose that overconfidence in individual investors is relevant due to the relative scarcity of short selling in this group. As a result, pessimists do not act as aggressively as optimists. Therefore prices are generally set by optimists.
Institutional investors supposedly do not step in to arbitrage this irrational behaviour due to their mandate to maximise the information ratio. The information ratio is the difference between the returns of the managed portfolio and some benchmark (usually a market index), divided by the tracking error (volatility of the difference). The authors provide numerous examples and
mathematical proof that seeking to maximise the information ratio prevents institutional investors from exploiting the low risk anomaly. They argue that as long as fixed benchmarks remain the predominate measure of performance, the low risk anomaly is likely to remain.
c. Leverage Aversion As An Explanation To The Low-Risk Anomaly
Frazzini & Pedersen (2011) contributes more evidence in support of the low risk anomaly by presenting strong empirical results in multiple assets classes and multiple countries that demonstrate the strong performance of low beta asset returns. The paper also examines the investment portfolios of U.S. based mutual funds, finding evidence consistent with the theory that leverage constrained investors overweight high risk assets.
The authors provide evidence to support the propositions that high beta is associated with low alpha and that a portfolio that bets against beta produces significant risk adjusted returns. They also find evidence consistent with the theory that leverage aversion drives the low risk anomaly.
The authors examine U.S. equity returns between 1926 and 2009, international equity returns from 19 MSCI developed markets between 1984 and 2009, U.S. treasury bonds between 1952 and 2009 and U.S. corporate bonds between 1973 and 2009.
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To test the theory that high beta is associated with low alpha, the authors construct asset portfolios sorted by beta. Each month, per asset class, the beta-sorted portfolios were constructed. The Sharpe ratios for each portfolio clearly show that the low beta portfolios outperformed the high beta portfolios. For U.S. stocks, treasury bonds and credit indices, the Sharpe ratios and alphas declined monotonously with beta. Similar results were found for international stocks, although the trend was not as strong.
To test the theory that betting against beta can produce significant risk adjusted returns, the authors propose a dual-portfolio that is long low beta assets and short high beta assets. In order to construct the portfolio, assets are ranked by beta. The bottom half of the assets with lower betas become part of the “long portfolio” and the top half of assets with higher betas become part of the “short
portfolio”. Within each portfolio, assets are weighted according to their beta, so that assets with the lowest betas have more weight in the “long portfolio” and assets with the highest beta have more weight in the “short portfolio”. At the start of each month, the “long portfolio” is purchased using leverage to bring the beta up to 1. The “short portfolio” is sold, but at a de-levered quantity in order to bring the beta down to 1. The result is a beta-neutral portfolio that is long low beta assets and short high beta assets.
The authors tested this bet against beta (BAB) portfolio in U.S. equities, 19 individual MSCI developed equity markets, combined equity markets, U.S. treasuries, U.S. corporate bonds and combined assets classes. For robustness, they tested different time periods, applied different risk free rates and different beta-calculating regressions (one, three, four and five factor CAPM). In an overwhelming number of cases, the BAB portfolio was found to generate significant alpha. For example, in U.S. equities the BAB portfolio returned a monthly excess return of 0.71% (t-stat=6.76) and a significant alpha of between 0.69% - 0.46%, depending on which factor is applied. Combined international equities achieved a similar result.
The results for individual international equity markets were not so clear. Only 7 out of the 19 markets achieved significant positive alpha with the BAB portfolio. These markets included Spain, Italy and The Netherlands with monthly alphas of 0.52%, 0.49% and 0.76% respectively.
The authors suggest that leverage aversion is the driver for this low risk anomaly, as leverage constrained investors overweight high risk assets in order to achieve higher returns. The increased demand inflates prices and depresses future expected returns. Even investors who are able to apply leverage are forced to de-lever during times of tightening funding liquidity constraints. Consistent
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with this theory, the authors use the TED spread to measure funding liquidity and find that during times of tightening funding liquidity constraints, the BAB portfolio realizes negative returns.
Seeking further evidence to shed some light on the theory of leverage aversion, the authors studied the investment portfolios of actively managed open-end U.S. domestic mutual funds between 1980 and 2009, individual U.S. investors between 1991 and 1996, firms targeted by leveraged buyout funds between 1963 and 2009 and Berkshire Hathaway between 1980 and 2009.
Mutual funds and individual investors can reasonably be expected to experience leverage
constraints. Mutual funds are prevented by regulation from applying leverage and must also keep cash on hand in order to meet redemption requirements. Individual investors are likely to face high costs for borrowing and/or onerous margin constraints. Leveraged buyout firms are well known for applying leverage in their investments and Bershire Hathaway, owned by Warren Buffet, is a well-known hedge fund that uses leverage (by issuing debt) to invest.
Mutual funds and individual investors were found to hold investment portfolios with betas significantly higher than 1. Private equity LBO firms and Berkshire Hathaway were found to hold investment portfolios with betas significantly lower than 1. The results are consistent with the leverage aversion theory.
In a follow up paper, Asness, Frazzini & Pedersen (2012) contribute more empirical evidence that low risk assets outperform high risk assets by considering the returns of U.S. stocks and bond and global stocks and bonds over varying time horizons. They propose that leverage aversion results in a market portfolio with a lower Sharpe ratio to the tangency portfolio and is therefore responsible for the anomaly.
The authors claim that risk parity (RP) investing has been gaining popularity in recent years. RP investing starts with considering traditional asset allocations such as a stock/bond 40/60 portfolio. The performance of such a portfolio is dominated by the performance of the stock assets due to the higher volatility of this asset class.
RP investing advocates that investments be diversified by risk, not by value. This generally requires the investor to invest more capital in low risk assets than in high risk assets. As a result, the RP portfolio has a lower expected return than a classic stock/bond 40/60 portfolio, but a higher Sharpe ratio. RP investors must apply leverage in order to achieve higher expected returns.
The authors cite numerous existing papers that provide evidence supporting the claim that low risk stocks outperform high risk stocks. They add to this evidence by considering three data samples: (1)
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U.S. stocks and bonds from 1926 to 2010; (2) global stocks, U.S. bonds, credit and commodities from 1973 – 2010; (3) global stocks and bonds from 11 countries between 1986 and 2010. For each asset class, they created a value-weighted index. They then constructed a value-weighted market index across asset classes.
The author’s found that the market value-weighted index and the classic stock/bond 40/60 portfolio had higher expected returns than the unlevered RP portfolio. But the RP portfolio achieved a higher Sharpe ratio and could therefore be levered to achieve high returns for less risk. The RP portfolio also achieved highly significant alpha when regressed on the market value-weighted portfolio returns.
The paper proposes that leverage aversion is responsible for this low risk anomaly. Revisiting the theory of leverage aversion, they explain its impact on Markowitz’s (1952) Modern portfolio theory (MPT) and the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965). MPT considers how investors select asset portfolios in order to achieve the maximum ratio of expected return to risk, otherwise known as the Sharpe ratio. The portfolio which achieves this maximum ratio is known as the tangency portfolio. CAPM goes one step further by showing that if all investors are rational and informed, they will select the tangency portfolio. In this case, the tangency portfolio will equal the market value-weighted portfolio. If an investor wishes to achieve a higher return, they can simply apply leverage to the portfolio.
However, what can investor do to achieve higher expected returns if they cannot or are unwilling to apply leverage? They must overweight riskier assets, which reduces their Sharpe ratio, but increases the expected returns.
Leverage aversion theory assumes that many such investors exist. As a result, the CAPM assumption that the market value-weighted portfolio is equal to the tangency portfolio does not hold. Because some investors overweight riskier assets, the price of these assets is elevated and suppresses future expected returns. Consequently, the market value-weighted portfolio achieves a lower Sharpe ratio than the tangency portfolio.
The authors provide evidence for this assertion by calculating the tangency portfolio for U.S. stocks and bonds between 1926 and 2010. The market portfolio achieved a significantly lower Sharpe ratio than the tangency portfolio.
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Hong & Sraer (2012) present evidence to support an explanation for the low risk anomaly based on speculative overpricing. Specifically, that in times of high market uncertainty, pessimists are sidelined and optimists over-price high beta stocks, reducing future expected returns.
The paper cites existing research that demonstrates how low risk stocks have outperformed high risk stocks for the last 30 years. The authors submit that the reason for this anomaly is due to the
speculative nature of high beta stocks. The theory allows investors to disagree about the market outlook and the common factor of a firm’s cash flows. At the same time, some investors are constrained from short-selling due to costs, regulatory requirements or institutional policy. In such a world, low beta stocks would have a relatively muted reaction to high aggregate disagreement. By definition, low beta stocks have lower sensitivity to market disagreement. By contrast, high beta stocks have a higher sensitivity to disagreement. Beta amplifies sensitivity to disagreement.
In the case of high beta stocks and high levels of disagreement, optimists set the market price because pessimists are sidelined due to short selling constraints. This speculative overpricing can inflate prices for high beta stocks and depress future returns.
The authors predict that the relationship between beta and return should take an inverted U shape when aggregate disagreement is significantly high.
The paper presents evidence to support this theory. The authors use monthly data of analysts’ earnings forecasts for U.S. stocks between 1981 and 2010. Disagreement is measured by the standard deviation of those forecasts. Aggregate disagreement is measured as a beta-weighted average of disagreement.
They find that in months when aggregate disagreement is low, expected returns increase with beta. In months where aggregate disagreement is high, the relationship between beta and returns takes on the predicted inverted U shape.
III.
Data
In this section I describe the data used in my research on the low risk anomaly in European equity markets.
Data identifying the historical constituents of the DAX, IBEX, SMI, AEX and CAC indices is sourced from Wharton Data Research Services COMPUSTAT global database. Historical constituent data covers the period from January 2001 to January 2013.
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Historical price data for the DAX, IBEX, SMI, AEX and CAC indices and their stock constituents is sourced from Thomson Datastream. I use the total return index (RI), which assumes any cash
dividends are re-invested. Historical price data covers the period from January 2001 to January 2013. Risk free rate data is sourced from Thomson Datastream. I use the 3 month EURIBOR interbank rate for EU based assets and the central bank Lombard rate for Swiss based assets.
Mutual fund and hedge fund investment data is sourced from Thomson One, covering the period from December 2005 to December 2012. Any investments in DAX, IBEX, SMI, AEX or CAC index constituents held by actively managed mutual funds and hedge funds are included in the data. The investment data is the U.S. dollar value each fund has invested per stock on 31st of December.
IV.
Methodology
In this section I describe the methodology of my research on the low risk anomaly in European equity markets.
a. Beta as a Measurement of Risk
I use rolling regressions of daily stock returns against the daily market index returns to calculate individual stock ex-ante beta. I use a standard single-factor CAPM regression:
𝐸 𝑅 = 𝑟𝑓+ 𝛽( 𝑟𝑚 − 𝑟𝑓 )
where rf is the risk free rate (3 month Euribor or Swiss Lombard rate) and rm is the daily return of the relevant market index (DAX, IBEX, SMI, AEX or CAC).
The beta calculation is based on 250 historical daily returns. Where 250 days are not available, a minimum of 30 days are required and each daily beta calculation increments the number of historical observations until 250 is reached. Stocks with less than 30 days of data are ignored.
b. Testing if High-Beta is Associated With Low-Alpha
To test if high beta is associated with low alpha in the selected market indices, I examine expected excess returns and alphas of stock portfolios formed according to beta. The methodology is applied for each market index (DAX, IBEX, SMI, AEX and CAC).
At the start of each month, the market index constituents are listed. The stocks are then ranked according to their beta. The stocks are sorted into quintile portfolios according to beta. Within each quintile portfolio, the stocks are evenly weighted.
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At the end of each month, the excess return for each portfolio is calculated and new quintile
portfolios are formed. The process is repeated each month for the period between January 2002 and January 2013. The expected excess return for each quintle portfolio is simply the mean of all excess returns: 𝑟𝑞𝑝 = 1 𝑛 (𝑟 − 𝑟𝑓) 𝑛 𝑖=0
I also calculate the alpha for each quintile portfolio based on the monthly excess returns of the portfolio against the monthly excess returns of the relevant market index:
𝑟𝑞𝑝 = 𝛼 + 𝛽 ( 𝑟𝑚− 𝑟𝑓 )
If high beta is associated with low alpha, the low beta portfolio(s) should record higher alpha than the high beta portfolio(s).
c. Testing the Low-Risk Anomaly Arbitrage
To test if it is possible to arbitrage the low risk anomaly in European markets, I apply the “Bet Against Beta” (BAB) portfolio methodology from Frazzini & Pedersen (2011). I briefly describe this methodology below, but readers should refer to Frazzini & Pedersen (2011) for more details on the approach.
At the start of each month, the market index constituents are listed. The stocks are then ranked according to their beta. Two portfolios are formed: a low beta portfolio and a high beta portfolio. One half of the index constituents go into the low beta portfolio and the other half go into the high beta portfolio, depending on their beta ranking. Within each portfolio, the stocks are weighted according to their beta so that the lowest beta stocks receive the most weighting in the low beta portfolio and the highest beta stocks receive the most weighting in the high beta portfolio. The following calculations are used to determine weightings:
𝑧𝑖 = 𝑟𝑎𝑛𝑘(𝛽) where zi is a 1 x n vector of beta rankings;
𝑧 = 1’𝑛 𝑧/𝑛 where 𝑧 is the average beta ranking;
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𝑤𝐿 = 𝑘 (𝑧 – 𝑧 )+
where wL is the weight given to members of the low beta portfolio; 𝑤𝐻 = 𝑘 (𝑧 – 𝑧 )−
where wH is the weight given to members of the high beta portfolio;
The BAB portfolio is a self-financing zero-beta portfolio that is long in the low beta portfolio and short in the high beta portfolio. The low beta portfolio is leveraged to have the same beta as the market index, while the high beta portfolio is deleveraged to also have a unit beta. This achieves are beta-neutral portfolio. The return of the BAB portfolio is then calculated as:
𝑟𝑡+1𝐵𝐴𝐵 = 1 𝛽𝑡𝐿 (𝑟𝑡+1 𝐿 – 𝑟𝑓) − 1 𝛽𝑡𝐻 (𝑟𝑡+1 𝐻 – 𝑟𝑓) where: 𝑟𝑡+1𝐿 = 𝑟′𝑡+1 𝑤𝐿 𝛽𝑡𝐿 = 𝛽′𝑡 𝑤𝐿 𝑟𝑡+1𝐻 = 𝑟′ 𝑡+1 𝑤𝐻 𝛽𝑡𝐻 = 𝛽′𝑡 𝑤𝐻
Intuitively, it is the case that rLt+1 is the weighted monthly return of the stocks in the low beta portfolio and rHt+1 is the weighted monthly return of the stocks in the low beta portfolio. Likewise, βLt is the weighted beta of the stocks in the low beta portfolio and βHt is the weighted beta of the stocks in the high beta portfolio
Each month, the return for the previous month’s BAB portfolio is calculated. A new BAB portfolio is then formed. The process is repeated each month for the period between January 2002 and January 2013.
In order to test the performance of the BAB portfolio, I calculate the annualized expected excess return, the annualized alpha, and the annualized Sharpe ratio.
The annualized expected excess return is simply the annualized mean of monthly BAB portfolio returns. The alpha the annualized intercept from a regression of the realized monthly BAB portfolio returns against the realized monthly market index risk premium:
19
The Sharpe ratio is an annualized figure based on the annualized BAB portfolio returns and the annualized volatility of those returns:
𝑆 =( 1 + 1 𝑛 𝑛𝑖=0(𝑟 𝐵𝐴𝐵 ) )12− 1 𝜎𝐵𝐴𝐵 12
For comparative purposes, I also calculate the annualized expected excess returns and annualized Sharpe ratio for each market index.
d. Testing Portfolio Beta of Investor Groups
Several theories have been proposed research papers as to the cause of the low risk anomaly. Leverage aversion is a popular explanation put forward in papers including Black (1972), Frazzini & Pedersen (2011) and Asness, Frazzini & Pedersen (2012). Baker & Haugen (2012) propose that compensation and agency issues with institutional investors drive them to overweight risky assets causing the anomaly. Karceski (2000) argues that individuals directing cash into mutual funds exhibit returns chasing behaviour that incentivises mutual funds to overweight risky assets. Baker, Brendan and Wurgler (2010) present existing evidence that investor behaviour is irrational and tilted towards overweighting high risk assets.
If correct, any of these theories would result in a value-weighted market portfolio that is not equal to the tangency portfolio. The investor groups identified in the theories would hold asset portfolios with a beta higher than that of the tangency portfolio. They would also hold asset portfolios with a beta higher than that of the market portfolio if their risk allocation behaviour was significantly different the other supposedly rational investors in the market.
My research examines the beta of equity portfolios for three different groups of investors: leverage constrained investors, unconstrained investors and mutual funds.
Frazzini & Pedersen (2011) and Asness, Frazzini & Pedersen (2012) argue that it is reasonable to assume that U.S. mutual funds are leverage averse because they are prevented by regulation from applying leverage and are also required to keep significant amounts of cash on hand in order to meet redemptions. Regulation regarding leverage varies from country to country and European UCITS regulation does permit mutual funds to apply limited leverage. I therefore choose U.S. based mutual funds to be the proxy for the leverage constrained investor group.
Hedge funds which have significantly less leverage constraints are chosen as the proxy for non-leverage constrained investors.
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I will also examine the beta of portfolios held by all mutual funds to test for evidence consistent with the theories of returns chasing behaviour and agency issues.
The data used in this test is an annual snapshot of investments held by actively managed mutual funds and hedge funds on December 31st. Eight annual snapshots are available during the period 2005 to 2012.
Per investment group and market, I calculate the portfolio ex-ante beta. This is calculated as the value weighted beta of the portfolio:
𝛽𝑝 = ( 𝑉𝑠𝑖 𝑉𝑝 ∗ 𝛽𝑠𝑖 ) 𝑛 𝑖=0
Where Vs is the dollar value invested in the stock, Vp is the total dollar value invested in all index constituent stocks and βs is the ex-ante beta of the stock.
The beta per investment group is simply the mean of all investor portfolios belonging to the group during the period from 2005 to 2012.
I also consider the beta of overall investment in each market per investor group. This is calculated as the value weighted beta of all investments by all investors in the market index:
𝛽𝑎𝑝 = ( 𝑉𝑠 𝑖 𝑉𝑎𝑝 ∗ 𝛽𝑠𝑖 ) 𝑛 𝑖=0
Where Vs is the dollar value invested in the stock, Vap is the total dollar value invested in all index constituent stocks by all investors and βs is the ex-ante beta of the stock.
The beta of the investor group is simply the mean of overall investment beta during the period from 2005 to 2012.
As previously mentioned, all of the proposed explanatory theories for the low risk anomaly would result in an asset portfolio beta higher than that of the tangency portfolio. An asset portfolio beta that is not significantly different from the market portfolio would not necessarily contradict any of the theories. But an asset portfolio beta that is significantly different from the market portfolio would be consistent with the theory that one investor group is allocating their risk differently from the rest of the market.
Therefore, I test the null hypothesis that each investor group has an equity portfolio beta of 1. In other words, that each investor group does not allocate risk differently from the market as a whole.
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If I find that any of the investor groups has an equity portfolio beta significantly different from 1, I will reject the null hypothesis.
In the case of the leverage constrained investor group, an equity portfolio beta significantly higher than 1 would be consistent with the theory that this particular investor group overweights risky assets relative to other market participants.
In the case of the unconstrained investor group, an equity portfolio beta significantly lower than 1 would be consistent with the theory that this particular investor group underweights risky assets relative to other market participants.
In the case of the mutual fund investor group, an equity portfolio beta significantly higher than 1 would be consistent with the theory that this particular investor group exhibits irrational risky behaviour relative to other market participants.
V.
Results
In this section I present the results of my research on the low risk anomaly in European equity markets.
a. Testing if High-Beta is Associated With Low-Alpha
The expected excess returns per quintile portfolio (Table A1) are consistent with the hypothesis that the high beta is associated with low alpha in the selected markets.
In all five market indices, the lowest beta quintile portfolio recorded higher expected excess returns than the highest beta quintile portfolio. In all five market indices, the lowest beta quintile portfolio recorded positive expected excess returns. In four out of five of the market indices (the exception being the CAC market index), the excess returns were statistically significant at 5%. In all five market indices, the highest beta quintile portfolio achieved statistically significant (5%) negative excess returns.
22 Table A1
Excess Returns of Quintile Portfolios
The table below shows the expected excess return per quintile portfolio for the period between January 2002 to January 2013. Returns are annualized percentages and 5% statistical significance is highlighted in bold.
P1 P2 P3 P4 P5 DAX 4.98 1.20 1.32 -1.49 -10.92 IBEX 3.67 1.81 -0.70 -6.25 -4.88 AEX 3.77 -2.42 -1.56 -8.57 -20.73 CAC 1.93 -1.60 -1.40 -6.38 -6.97 SMI 2.80 -4.17 1.59 3.07 -7.17
The alphas per quintile portfolio (Table A2) are consistent with the hypothesis that high beta is associated with low alpha in the selected markets.
In all five market indices, the lowest beta quintile portfolio was found to have higher alpha than the highest beta quintile portfolio. In all five market indices, the lowest beta quintile portfolio achieved positive alpha, although no results were statistically significant at 5%. In all five market indices, the highest beta quintile portfolio achieved negative alpha. For the DAX, IBEX and AEX market indices, these results were statistically significant at 5%. Alpha is generally found to decline with beta, although not monotonically.
Table A2
Alpha per Quintile Portfolio
The table below shows the alpha per quintile portfolio for the period between January 2002 to January 2013. Alpha is the intercept in a regression of monthly portfolio excess returns against the relevant monthly market index excess returns.
Alphas are annualized percentages and 5% statistical significance is highlighted in bold.
P1 P2 P3 P4 P5 DAX 4.28 0.13 0.05 -3.03 -12.72 IBEX 2.76 0.48 -2.27 -7.99 -6.90 AEX 5.15 -0.72 0.77 -6.07 -17.21 CAC 2.65 -0.62 -0.33 -5.09 -5.32 SMI 2.29 -4.99 0.46 1.55 -8.93%
The Sharpe ratios per quintile portfolio (Table A3) are consistent with the hypothesis that high beta is associated with low alpha in the selected markets.
In all five market indices, the lowest beta quintile portfolio was found to have a positive Sharpe ratio and the highest beta quintile portfolio was found to have a negative Sharpe ratio. Sharpe ratio is generally found to decline with beta, although not monotonically.
23 Table A3
Sharpe Ratio per Quintile Portfolio
The table below shows the Sharpe ratio per quintile portfolio for the period between January 2002 to January 2013. The columns P1 - P5 reference each quintile portfolio, with P1 containing the lowest beta stocks and P5 containing the highest
beta stocks. The Sharpe ratio is based on annualized monthly portfolio excess returns.
P1 P2 P3 P4 P5 DAX 0.30 0.05 0.05 -0.04 -0.25 IBEX 0.22 0.09 -0.03 -0.23 -0.16 AEX 0.22 -0.11 -0.06 -0.25 -0.41 CAC 0.12 -0.08 -0.06 -0.22 -0.19 SMI 0.23 -0.23 0.07 0.10 -0.19
A time series for a unit of currency invested in the low beta quintile and a unit of currency invested in the high beta quintile portfolio further clarifies the results. The final values from the time series are show below in Table A4. A chart for the full time series is shown in Appendix A, Figure C. In all five markets, the low beta quintile portfolio yielded a profit. The high beta quintile yielded a loss.
Table A4
Quintile Portfolio Investment Time Series
The table below shows the final value of a unit of currency invested in a quintile portfolio for the period between January 2002 to January 2013. At the start of each month, market index constituent stocks are listed and ranked by beta. Stocks are
sorted into five quintile portfolios according to their beta. Within each quintile portfolio, stocks are evenly weighted. Quintile portfolios are rebalanced every month. The rows P1 - P5 reference each quintile portfolio, with P1 containing the
lowest beta stocks and P5 containing the highest beta stocks.
DAX IBEX AEX CAC SMI
P1 1.45 1.26 1.26 1.08 1.25
P5 0.09 0.35 0.01 0.20 0.19
Evidence that high beta is associated with low alpha is strongest in the DAX, AEX and IBEX market indices. All three markets recorded statistically significant positive excess returns in the lowest beta quintile and statistically significant negative excess returns in the highest beta quintile. All three markets recorded statistically significant negative alpha in the highest beta quintile.
The evidence for the SMI and CAC is less certain due to the lack of statistical significance in most results. However, the difference in Sharpe ratios and the investment time series are consistent with the hypothesis that the low risk anomaly is present.
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Only the DAX market records monotonic decline of excess returns and alpha with beta. It is difficult to say if this indicates that the low risk anomaly is not as strong in other markets. Due to the small number of stocks in each index – the CAC has the most (40) and the SMI has the least (20) – it is reasonable to expect a significant amount of noise in the results due to the lack of diversity in each quintile portfolio.
b. Testing the Low-Risk Anomaly Arbitrage
The results of the BAB portfolio arbitrage strategy (Table B1) show that it is possible to arbitrage the low risk anomaly in the DAX, IBEX and AEX markets. Results for the CAC and SMI markets were consistent with the hypothesis that arbitrage is possible, but the results were not statistically significant.
All five markets recorded positive expected excess returns and positive alpha. The results were statistically significant at 5% for the DAX, IBEX and AEX markets. The BAB portfolio recorded a higher Sharpe ratio and expected excess return than the respective market index in all five markets. More detailed results are shown in Appendix A, Table D1.
Table B1
BAB portfolio Performance
The table below shows the performance of the BAB portfolio for the period between January 2002 to January 2013. BAB XRet is the annualized expected excess return percentage of the BAB portfolio. BAB Alpha is the intercept in a regression of
monthly BAB portfolio excess returns against the relevant monthly market index excess returns. BAB S-Ratio is the Sharpe ratio is based on annualized monthly BAB portfolio excess returns. Index XRet is the annualized expected excess return percentage of the market index. Index S-Ratio is the Sharpe ratio is based on annualized monthly market index portfolio
excess returns. XRet and alphas with 5% statistical significance are highlighted in bold.
BAB XRet BAB Alpha BAB S-Ratio Index XRet Index S-Ratio
DAX 12.78 12.87 0.67 1.20 0.05
IBEX 13.22 13.16 0.58 1.76 0.08
AEX 16.75 16.63 0.69 -2.29 -0.10
CAC 4.04 4.16 0.26 -1.01 -0.05
SMI 4.19 4.37 0.28 0.94 0.06
It is clear that in the DAX, IBEX and AEX markets, the BAB arbitrage is possible and highly profitable. This is emphasised when comparing the expected excess return and Sharpe ratio of the BAB
portfolio to that of the respective market index. Importantly, the BAB portfolio was able to achieve the positive alpha with a higher reward to risk ratio than that of the market index.
The results for the CAC and SMI markets are less clear. Although the BAB portfolio recorded a higher Sharpe ratio than the respective market index, the alphas were not statistically significant. This is
25
probably explained by the fact that these two markets recorded the weakest evidence associating high beta with low alpha in the quintile portfolio tests. However, it should be noted that the BAB portfolio did not perform worse than the respective market index.
I next examine a time series which compares a unit of currency invested in the BAB portfolio compared with a unit of currency invested in the market index. The final results of the time series are shown in Table B2. The full time series is shown in Appendix A, Figure D.
The BAB portfolio time series emphasises the existing results. The DAX, IBEX and AEX perform strongest, with the AEX being the stand-out performer for a 4.05 EUR result in the BAB portfolio versus 0.57 EUR in the AEX index.
Table B2
BAB and Market Index Investment Time Series
The table below shows the final value of a unit of currency invested in a BAB portfolio and a unit of currency invested in the market index for the period between January 2002 to January 2013.
DAX IBEX AEX CAC SMI
BAB 3.03 2.89 4.05 1.34 1.38
Market 0.84 0.92 0.57 0.72 0.96
Given that two markets (SMI and CAC) appear to be lagging behind the others in terms of evidence supporting the presence of the low risk anomaly, I analyse some properties of the BAB portfolio in each market to see if the SMI and CAC are in some way different.
First, I examine the realized beta of the BAB portfolios (Table B3). The BAB portfolios are designed to be beta-neutral by selecting and weighting stocks in such a way that the portfolio has an ex-ante beta of zero. This assumes that past beta is a good indicator of future beta. I regress the monthly excess returns of the BAB portfolio against the monthly market index excess returns to test the null hypothesis that the BAB portfolio beta is equal to zero. I find that the SMI BAB portfolio has a realized beta significantly different from zero. This may explain the relatively poor performance of the SMI BAB portfolio.
26 Table B3
BAB portfolio Realized Beta
The table below shows the realized beta of the BAB portfolio for the period between January 2002 to January 2013. Realized beta is the slope of the regression of monthly BAB portfolio excess returns against the relevant monthly market
index excess returns. Realized beta with 5% statistical significance is highlighted in bold.
DAX IBEX AEX CAC SMI
Beta -0.07 0.03 -0.05 0.11 -0.19
T-Stat -0.94 0.34 -0.05 1.63 -2.28
Next, I next examine the average hedge ratios and beta spread in each market (Table B4). If the difference in ex-ante beta between the lowest beta stocks and the highest beta stocks is large, this should improve the performance of the BAB portfolio arbitrage.
The BAB portfolio consists of two sub-portfolios: one long low beta stocks and one short high beta stocks. The long portfolio is leveraged up to a beta of one, while the short portfolio is deleveraged to the same beta in order to create a hedged beta-neutral position. For example, the BAB portfolio in the DAX market had on average 1.82 EUR invested in long low beta stocks for every 0.85 EUR in short high beta stocks.
Table B4
BAB Hedge Ratio & Beta Spread
The table below shows the ratio of investment in long low beta shares to short high beta shares in the BAB portfolio at formation. Beta Spread is defined as (HBeta- LBeta) / (HBeta* LBeta) where HBeta (LBeta) are the betas of the short (long)
leg of the BAB portfolio at portfolio formation.
DAX IBEX AEX CAC SMI
$ (long) low beta 1.82 1.95 1.76 1.53 1.58
$ (short) high beta 0.85 0.85 0.69 0.73 0.68
Beta Spread 0.97 1.10 1.07 0.80 0.89
The CAC and SMI record the lowest beta spread out of all markets. This may explain the relatively poor performance in the BAB portfolio arbitrage.
A full time series for the beta spreads is shown in Appendix A, Figure E. Interestingly, each market appears to follow the same trend, with beta spread compressing in 2004, expanding in 2009 and then compressing again in 2010. Beta compression as a result of tightening funding liquidity was discussed in Frazzini & Pedersen (2011).
27
c. Testing Portfolio Beta of Investor Groups
The first null hypothesis to be tested is that actively managed U.S. mutual funds – my proxy for leverage constrained investors – do not allocate risk differently from the market as a whole. In no market were actively managed U.S. mutual funds found to hold portfolios with a beta significantly different than one (Table C1). In fact in all markets, except the SMI, this group of
investors recorded a portfolio ex-ante beta of slightly less than one. I therefore do not reject the null hypothesis.
Table C1
Portfolio Beta of U.S. Actively Managed Mutual Funds
The table below shows the mean ex-ante beta of equity portfolios held by actively managed U.S. mutual funds on December 31st between the years 2005 to 2012. T-stats report the statistical significance that the beta is different from 1.
DAX IBEX AEX CAC SMI
Beta 0.93 0.94 0.95 0.99 1.07
T-Stat -0.16 -0.12 -0.08 -0.02 0.13
For robustness, I also examine the ex-ante beta of overall investment in each market by this group of investors. Once again, in no market were actively managed actively managed U.S. mutual funds found to hold a portfolio with a beta significantly different than one. In fact in all markets, except the SMI and CAC, this group of investors recorded a beta slightly less than one. I therefore do not reject the null hypothesis.
Table C2
Total Investment Beta of U.S. Actively Managed Mutual Funds
The table below shows the mean ex-ante beta of all equity investments held by actively managed U.S. mutual funds on December 31st between the years 2005 to 2012. T-stats report the statistical significance that the beta is different from 1.
DAX IBEX AEX CAC SMI
Beta 0.92 0.98 0.97 1.00 1.01
T-Stat -0.45 -0.07 -0.12 -0.02 0.03
The results do not directly contradict the theory that leverage aversion (at least in part) explains the low risk anomaly. Leverage constrained investors overweight risk stocks relative to the tangency portfolio, but not necessarily the market portfolio. Because this test calculates ex-ante beta relative to the market portfolio, a unit beta is not proof that the investor group are risk-weighting their
28
portfolios efficiently. But it does indicate that actively managed U.S. mutual funds are not allocating their risk in a manner that is significantly different from the market as a whole.
The next null hypothesis to be tested is that all actively managed mutual funds do not allocate risk differently from the market as a whole.
If mutual funds are driven to overweight risky stocks due to “returns chasing behaviour” or other agency issues, we might expect to see that they hold stock portfolios with a beta greater than one. However, in no market were actively managed actively managed mutual funds found to hold portfolios with a beta significantly higher than one (Table C3). The AEX, CAC and SMI recorded portfolio betas slightly higher than one, but the numbers are not significant. I therefore do not reject the null hypothesis.
Table C3
Portfolio Beta of All Actively Managed Mutual Funds
The table below shows the mean ex-ante beta of equity portfolios held by actively managed mutual funds on December 31st between the years 2005 to 2012. T-stats report the statistical significance that the beta is different from 1.
DAX IBEX AEX CAC SMI
Beta 0.91 0.98 1.05 1.02 1.02
T-Stat -0.23 -0.04 0.09 0.04 0.03
For robustness, I also examine the ex-ante beta of overall investment in each market by this group of investors (Table C4). Once again, in no market were actively managed mutual funds found to hold a portfolio with a beta significantly higher than one. I therefore do not reject the null hypothesis.
Table C4
Total Investment Beta of Actively Managed Mutual Funds
The table below shows the mean ex-ante beta of all equity investments held by actively managed mutual funds on December 31st between the years 2005 to 2012. T-stats report the statistical significance that the beta is different from 1.
DAX IBEX AEX CAC SMI
Beta 0.92 1.00 1.01 1.02 1.01
T-Stat -0.54 -0.02 0.04 0.11 0.04
The results are not consistent with the theory that mutual funds behave in a manner different to other investors by overweighting riskier stocks. If the manner in which mutual funds select their stocks – be it portfolio managers selecting high-profile risky stocks in order to impress
colleagues/clients, or selecting high risk stocks to maximise the share of industry cash-inflow during bull-markets – we might reasonably expect to see this reflected in the beta of their portfolio. But this
29
assumes that all other investor groups are not behaving in a similar manner and that a significant number of rational investors exist who invest in the tangency portfolio.
The final null hypothesis to be tested is that actively managed hedge funds do not allocate risk differently from the market as a whole.
Hedge funds have more flexibility in the types of strategies that they may implement, such as the use of leverage. Hedge funds are well placed to apply their available leverage and exploit the low risk anomaly. If they were to do so, we would reasonably expect to see this reflected in the beta of their portfolios.
However, I find that in no market is the mean beta of hedge fund portfolios significantly less than one (Table C5). The evidence is not consistent with the theory that unconstrained investors are exploiting the low risk anomaly in significant numbers. I therefore do not reject the null hypothesis.
Table C5
Portfolio Beta of All Actively Managed Hedge Funds
The table below shows the mean ex-ante beta of equity portfolios held by actively managed hedge funds on December 31st between the years 2005 to 2012. T-stats report the statistical significance that the beta is different from 1.
DAX IBEX AEX CAC SMI
Beta 0.91 0.97 1.05 1.01 1.06
T-Stat -0.21 -0.07 0.09 0.01 0.10
For robustness, I also examine the ex-ante beta of overall investment in each market by this group of investors (Table C6). The results confirm the previous finding and the null hypothesis is not rejected.
Table C6
Total Investment Beta of Actively Managed Hedge Funds
The table below shows the mean ex-ante beta of all equity investments held by actively managed hedge funds on December 31st between the years 2005 to 2012. T-stats report the statistical significance that the beta is different from 1.
DAX IBEX AEX CAC SMI
Beta 0.94 1.03 1.01 0.99 1.02
T-Stat -0.33 0.06 0.03 -0.03 0.09
Although I have found evidence consistent with the theory that the low risk anomaly exists in the selected market indices, I find no evidence that any of the selected groups of investors is either specifically responsible for the anomaly or specifically exploiting the anomaly.
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VI.
Conclusion
The low risk anomaly is identified where low risk assets outperform high risk assets on a risk adjusted basis. The anomaly was noted as far back as Black (1972) who attributed the anomaly to leverage constrained investors. Researchers have since continued uncovering evidence consistent with the anomaly, particularly in U.S. markets.
I find evidence consistent with the hypothesis that investors prefer high risk assets to low risk assets in five actively traded European equity markets: DAX, IBEX, AEX, CAC and SMI.
For each market I find that a monthly-rebalanced portfolio containing the 20% of index constituents with the lowest betas records a positive expected excess return, positive alpha and positive Sharpe ratio. A monthly-rebalanced portfolio containing the 20% of index constituents with the highest betas records a negative expected excess return, negative alpha and negative Sharpe ratio. A time-series analysis for a unit of currency invested in each portfolio from January 2002 to January 2013 paints a clearer picture by visualizing the stark difference in profits each portfolio achieved during the selected period.
The results are consistent with the hypothesis that high beta is associated with low alpha in the selected equity market indices. The evidence in strongest in the DAX, IBEX and AEX due to the statistical significance of some results.
I find that a strategy to arbitrage the low risk anomaly – the BAB portfolio - results in highly
significant excess returns and alpha in the DAX, IBEC and AEX market indices. The results in the CAC and SMI were not statistically significant. When examining the properties of the arbitrage strategy in each market, I find that both the CAC and SMI have constituent stocks with a narrower range of betas than the markets where the strategy performed successfully. I also find that the ex-ante beta for constituent stocks of the SMI is a poor predictor for future beta. This confirms what intuition should already tell us - that a broad range of beta and predictable beta is important to the success of a strategy seeking to exploit the low risk anomaly.
The results offer new insight to the low risk anomaly. A detailed analysis of equity markets outside of the U.S. confirms that the low risk anomaly is indeed global. A focus on only the most liquid stocks in each country confirms that the low risk anomaly is not a product of small-cap, illiquid outliers. And the relatively small size of the selected European markets also confirms that the anomaly can still be found and exploited in smaller markets.
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My analysis of different investor groups and their equity investment portfolios raises more questions than it answers. Previous studies, notably Frazzini & Pedersen (2011) have found U.S. mutual funds to hold equity investment portfolios with a beta higher than 1. This is not only consistent with the theory that mutual funds overweight risky assets, but also that they overweight risky assets relative to all other market participants. This in turn implies that there are other investors actively exploiting the low risk anomaly by overweighting low risk assets relative the value-weighted market index. I do not find evidence consistent with these results. None of the investor groups I identify – U.S. mutual funds, all mutual funds or all hedge funds – hold equity portfolios with a beta significant different from that of the value-weighted market index.
Given that there is clear evidence (at least in the DAX, IBEX and AEX) that high risk assets are
overpriced relative to low risk assets, this would appear to indicate that the preference for high beta assets is shared across different investor groups active in European markets. The practice is not limited to any specific investor group. If this is indeed the case, it could present an interesting opportunity to institutional investors seeking to outperform their peers in European markets.
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VII. Appendix A – Additional Empirical Results
Figure ASharpe Ratio per Quintile Portfolio
The figure below shows the Sharpe ratio per quintile portfolio for the period between January 2002 to January 2013. At the start of each month, market index constituent stocks are listed and ranked by beta. Stocks are sorted into five quintile
portfolios according to their beta. Within each quintile portfolio, stocks are evenly weighted. Quintile portfolios are rebalanced every month. The columns P1 - P5 reference each quintile portfolio, with P1 containing the lowest beta stocks
and P5 containing the highest beta stocks. The Sharpe ratio is based on annualized monthly portfolio excess returns.
0.22 0.09 -0.03 -0.23 -0.16 P1 P2 P3 P4 P5
IBEX
0.30 0.05 0.05 -0.04 -0.25 P1 P2 P3 P4 P5DAX
0.22 -0.11 -0.06 -0.25 -0.41 P1 P2 P3 P4 P5AEX
0.12 -0.08 -0.06 -0.22 -0.19 P1 P2 P3 P4 P5CAC
0.23 -0.23 0.07 0.10 -0.19 P1 P2 P3 P4 P5SMI
33 Figure B
Alpha per Quintile Portfolio
The figure below shows the alpha per quintile portfolio for the period between January 2002 to January 2013. At the start of each month, market index constituent stocks are listed and ranked by beta. Stocks are sorted into five quintile portfolios according to their beta. Within each quintile portfolio, stocks are evenly weighted. Quintile portfolios are rebalanced every month. The columns P1 - P5 reference each quintile portfolio, with P1 containing the lowest beta stocks and P5 containing
the highest beta stocks. Alpha is the intercept in a regression of monthly quintile portfolio excess returns against the relevant monthly market index excess returns.
2.76% 0.48% -2.27% -7.99% -6.90% P1 P2 P3 P4 P5
