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University of Amsterdam Amsterdam Business School

BSc Economics & Business Economics

Performance of low volatility investment portfolios

during the 2008 financial crisis

An empirical study on the low volatility anomaly during the 2008 financial

crisis

Author: Julia Antonina Rawecka Student number: 11588411

Economics and Business Economics Track: Finance

Supervisor: Philippe Versijp Date: July 2020

ABSTRACT

Low-volatility anomaly, that contradicts the basics of asset pricing, has been widely observed in financial markets. This research paper focuses on this anomaly during the 2008 financial crisis. The main research question is whether the low-volatility anomaly persists during the 2008 financial crisis. The research focuses on the US financial market in the period of time from 1992 till 2012. This existence of the anomaly and influence of the crisis are examined through regression and Sharpe ratio analysis. No statistically significant evidence for the influence of the crisis on the low-volatility anomaly is found.

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STATEMENT OF ORIGINALITY

This document is written by Student Julia Antonina Rawecka who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents

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Table of Contents

ABSTRACT ... 1

STATEMENT OF ORIGINALITY ... 2

INTRODUCTION ... 4

THEORETICAL FRAMEWORK ... 7

DATA AND METHODOLOGY ... 14

RESULTS... 16

FURTHER INVESTIGATION ... 18

CONCLUSIONS ... 20

LIMITATIONS AND FURTHER RESEARCH ... 21

REFERENCES ... 22

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INTRODUCTION

The theory of asset pricing is based on the notion of investors being rewarded for the risk they are taking. This relationship is described by CAPM, Capital Asset Pricing Model, which was proposed by Sharpe (1964), Litner (1965) and Mossin (1966). CAPM states that investors are rewarded for taking on non-diversifiable, systematic risk, due to the assumption that idiosyncratic, firm-specific risk is eliminated through portfolio diversification. A common measure of risk is the volatility of stock or portfolio returns. (Fama, French, 2003). This would imply that holding a portfolio with high systematic volatility should result in realizing higher returns. The CAPM, however, is based on numerous assumptions that may not hold perfectly in the real world. It is also very simplistic to connect the portfolio returns to only one systematic factor. After the model was proposed many researchers tried empirically testing it and they found out the relation to be stronger or weaker (Black, Jensen and Scholes, 1972) and at times even inverted (Haugen and Heins, 1975). Due to this empirical failure, many researchers tried expanding it to add additional explanatory factors; for example, Fama and French (1992) or Carhart (1997). Nowadays, we realize that when analysing the market solely through the lens of CAPM, we discover many anomalies, one of them is “low-volatility anomaly”.

The low-volatility anomaly is present when portfolios with low volatility outperform portfolios with high volatility, which is a contradiction to the central notion of CAPM. It was first described in 1991 by Haugen and Baker who tested the performance of cap-weighted and minimum-variance portfolios in the US in 1972-1989 and showed that the minimum-variance strategy yielded higher returns. The superiority of minimum-variance portfolios was confirmed in 2006 by Clark, Thorley and Silva who extended the research to a period of time from 1968 to 2005. The low-volatility anomaly has been later documented by many other researchers. In 2007 Blitz and van Vliet proved its existence in the US and global market in 1986-2006 period. In 2011 Baker, Bradley and Wurgler produced a paper where they tested the anomaly in the US market between 1968 and December 2008. They discovered that the anomaly holds for both risk definitions, systematic and total portfolio volatility and for all stock sizes, large and low capitalization

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stocks. In 2009 Ang, Hodrick, Xiang and Zhang proved the low-volatility effect in 23 countries, as measured by local and regional versions of the Fama French model. They also found that the anomaly simultaneously appeared in many parts of the world and was robust to controlling for additional factor loadings and firm characteristics. Researchers extended their research to include emerging financial markets, and in 2012 Haugen and Baker proved the existence of low-volatility anomaly in 21 developed countries and 12 emerging markets over a period of time from 1990 and 2011. In the same year, Blitz and van Vliet published a paper where they focused solely on the emerging equity markets and proved the effect there and showed that their result was robust to controlling for size, value, momentum, for omitting the least liquid stocks and extending the holding period.

The low-volatility anomaly is an extremely interesting research topic, and expanding the available knowledge-base is crucial for a number of reasons. First of all, the purpose of research is to increase the knowledge base and continuously challenge the existing theory to depict and understand reality better. This anomaly is a contradiction to what has always been taught about asset pricing and it could give rise to a new paradigm in finance. Second of all, understanding this anomaly and how it works in different states of the economy could give rise to new investment strategies or maybe even provide partial insulation from market shocks if it turns out to hold in times of financial crisis. Investors are assumed to be risk-averse (Bodie, Kane, Marcus, 2010) so this low-risk strategy could give rise to a new, desirable investment opportunity. Another reason is that many researchers tried to explain this anomaly with behavioural biases of the investors and also related it to how portfolio managers are incentivized (Blitz, Falkenstein, van Vliet). Extensive research of this anomaly can help understand it better and possibly redesign these incentive schemes. It is possible that this anomaly is just a market inefficiency that has not been traded away yet, but in such a case trading it away would improve the market efficiency, which is beneficial for the equity markets.

All of these studies focused on examining the effect over a long period of time. However, they did not distinguish between normal times and times of crisis, Some of them in their papers mentioned how the risk-return relationship acts in different states of the market, for example, Blitz van Vliet (2007) mentioned that low-volatility strategy outperforms in down

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markets and underperforms in up markets and has anti-bubble behaviour. Another example is Clark, Thorley and Silva (2006), who showed that market drop in 1982 and the market crash of October 1987 were muted in the minimum variance portfolios. Then Baker, Bradley and Wurgler in their paper from 2012 showed that the high-volatility portfolio provided relatively low returns in market downturns (1973–1974 and 2000–2002, the crash of 1987, and the financial crisis of 2008). However, none of these papers focused solely on examining the difference in this effect between the time of crisis and normal times.

Thus this paper will examine whether the low volatility anomaly persists during crisis. The main research question is whether the low-volatility anomaly persists during the 2008 financial crisis. The research will focus on the American financial market as the US is where the crisis originated. It obviously had a spill-over effect on other countries, but these countries might have simultaneously experienced effects of different nature thus, results could be biased. That is why this paper focuses on the American market, where the effect was the strongest. First, portfolios sorted on volatility will be constructed, which is similar to the methodology used by Blitz and van Vliet (2007) and Baker and Haugen (2012). Later their returns will be examined based on the 4-factor Carhart model to which a binary variable indicating the influence of crisis will be added. The returns and excess returns represented by the alpha will be analysed (similar to Blitz, Pang and van Vliet, 2013) and the influence of the crisis on the excess returns based on the binary variable will be determined. The time range of the research is from 1992 till 2012, 20 years of monthly data. In that period of time, two crisis periods are identified: the dot com crash in 2000-2002 and the 2008 financial crisis.

The next section focuses on the literature review of existing theory and research on this anomaly. It elaborates on the CAPM, Fama-French and Carhart model used to explain the basics of asset pricing. Later data and methodology, that describe how data was gathered and how the research was conducted are presented. The last two sections present results and analyse them with respect to the existing literature and research.

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THEORETICAL FRAMEWORK

As mentioned in the introduction, the CAPM, Capital Asset Pricing Model, was first introduced in the 1960s. The proponents of the model were William F. Sharpe (1964), John Litner (1965) and Jan Mossin (1966). The main idea underlying the CAPM is that investors are rewarded for taking on risk. It assumes that firm-specific, also known as idiosyncratic risk is eliminated in a well-diversified portfolio and what remains is only the systematic risk, that is the degree of volatility of the portfolio returns due to a common market factor. This implies that the higher the systematic risk of a portfolio, the higher realized returns should be. A common measure for the total risk of a portfolio is volatility and a common measure for systematic risk is beta, which represents the co-movement of the stock with the market. The CAPM equation looks as follows:

𝐸(𝑟𝑖) − 𝑟𝑓 = 𝛼 + 𝛽(𝑟𝑚− 𝑟𝑓)

Where 𝛽 represents the co-movement of a stock with the market, 𝑟𝑚 represents the market return, 𝑟𝑓 is the risk-free rate, 𝐸(𝑟𝑖) is the expected return on an asset. 𝛼 represents abnormal returns, that if the CAPM equation is an accurate representation of reality, should be equal to zero.

CAPM is based on a set of strict assumptions such as that all investors are rational and are mean-variance optimizers. They should all have homogenous expectations and one period investment horizon. Furthermore, CAPM also assumes that all assets are publicly traded, they are no taxes and transaction costs, investors can borrow and lend without a limit at the risk-free rate and that there are no short-selling restrictions (Bodie, Kane, Marcus, 2010). As mentioned before, most of these assumptions are not realistic as in real life there are taxes, transaction costs, limits and costs of borrowing, short-selling restrictions and not all assets are available for trading. Furthermore, violations of these assumptions may alter investment decisions, for example, different tax exposure of two different investors will result in different profits for the same investment and can thus influence portfolio composition decisions (Bodie, Kane, Marcus, 2010).

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Soon after the model was published, many researchers started empirically testing its predictions. In 1972 Black, Jensen and Scholes published a paper “The Capital Asset Pricing Model: Some Empirical Tests.”. They used a sample of all stocks listed at New York Stock Exchange in a period of time from 1926 to 1966 and determined that expected return on a stock in excess of the risk-free rate is in fact not directly proportional to the co-movement with the market, so the beta and that 𝛼 is different from zero. For different periods of time, they have found the slope (the beta) to be either steeper, flatter or even inverted in relation to what CAPM predicted. In 1975 Haugen and Heins published a study where they found positive abnormal returns, alphas, for low-beta and low-volatility stocks which also suggested an inverted CAPM slope.

In 2013 Blitz, Falkenstein, van Vliet published a paper Explanations for the Volatility Effect: An Overview Based on the CAPM Assumptions in which they connected the low-volatility anomaly and empirical failure of CAPM to the failure of the CAPM assumptions. They extensively discussed the assumptions such as lack of leverage and short-selling constraints, one period investment horizon, risk-averse utility-maximizing investors caring only about mean and variance of returns, and completely rational processing of all information by all investors. The first three assumptions and their violation in real life are straightforward, but the last one is not. The first case they describe is the fact that investors tend to prefer attention-grabbing stocks. One of the authors, Falkenstein in 1996 showed that mutual funds tend to invest in securities that appeared on the news more. Then they mention the representativeness bias ,which implies people tend to make decisions based on anecdotes instead of pure data and statistics. Highly volatile stocks generate a lot of these, thus attracting investors and this high demand lowers the price and diminishes the returns. The last factor they describe is overconfidence that may nudge investors to pursue more risky and volatile stocks to “discover alpha (abnormal returns)” and prove their superior skills. All these violations of CAPM assumptions and behavioural biases may give rise to the anomaly described in this paper.

The first researchers to identify the superior performance of low-volatility strategies were Haugen and Baker. In 1991 they published a study where they investigated cap-weighted and minimum-variance investment strategies from 1972 till 1998. They constructed the

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portfolios from the universe of 1000 largest by market capitalization US stocks using the covariance matrix of returns. The portfolios were rebalanced quarterly and their results included 2% transaction costs. They showed that cap-weighted portfolios were likely to be a sub-optimal investment strategy and that minimum variance portfolios consistently outperformed cap-weighted portfolios. They explained these results by differing investors’ expectations, restrictions on short-selling, differing tax exposure of investors, human capital (which is Present Value of manager’s future income) and foreign investors who choose stocks that combine well with their domestic portfolio so their preferences will differ from these of a US investor.

In 2006 Clark, Thorley, Silva confirmed the results of Haugen, Baker (1991). They used a longer and more recent dataset from January 1968 till December 2005 and estimated the covariance matrix based on 1000 largest market capitalization US stocks. They found out that minimum variance portfolios add value over the market-capitalization-weighted benchmark, while the standard deviation was lowered by about one fourth and risk measured by beta was lowered by about one third compared to the capitalization-weighted market benchmark. They also tested that inclusion of factor neutrality did not distort the results. What is interesting for this research is that they showed that the market drop in 1982 and the market crash of October 1987 were muted in the minimum variance portfolios.

In 2011 Baker, Bradley and Wurgler published a paper where they proved the existence of the anomaly in the US in the time range from January 1968 till December 2008. From the universe of 1000 biggest US stocks, they constructed 5 portfolios sorted on their total volatility and on trailing beta and rebalanced these portfolios monthly. They found out that for both risk definitions and all stock sizes (large caps and low caps) the low-risk portfolios performed better than the high-risk portfolios. The explanations they provided were irrational investor demand for high volatility stocks, investment firms with fixed benchmarks that create demand for high volatility stocks and leverage constraints. In their paper, it was also mentioned that high volatility stocks provided relatively low returns in market downturns (1973-1974, 2000-2002, 1978 and 2008) which is interesting with respect to the topic of this research paper.

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The research on this phenomenon was extended to other countries. Blitz and van Vliet (2007) proved the existence of the low-volatility effect in the US and global markets in the 1986-2006 period. Similar to other research, they constructed decile portfolios from the universe of the FTSE World Developed Index. Their segmentation was based on their past three-year return volatility and they were rebalanced them every 3 months, so turnover was lower than in most of the other research. They showed that stocks with historically low volatility delivered better risk-adjusted returns than high volatility stocks when measured by CAPM alphas and Sharpe ratios. They controlled for classic size, value and momentum strategies and they found that volatility effect is a separate effect, whose magnitude is comparable to that of size, value and momentum effects. They also stated that low-volatility strategy tends to outperform in down markets and underperforms in up markets and has an anti-bubble effect on the returns. This again relates directly to the topic of this research.

In 2009 Ang Hodrick Xiang Zhang proved the low-volatility effect in 23 countries, as measured by local and regional versions of the Fama-French model, where they adjusted for the market, size and book-to-market factors. They discovered a large and statistically significant difference of 1.31% in average monthly returns between the highest and lowest volatility portfolios, after controlling for the world market, size and value factors. What is interesting is that their results were robust when they controlled for factor loadings and firm characteristics and also appeared in many parts of the world.

Previously mentioned Haugen and Baker published another paper in 2012 where they proved the existence of low-volatility anomaly in 21 developed countries and 12 emerging markets over a period of time from 1990 and 2011. Their methodology included ranking stocks on volatility and rebalancing the portfolios monthly, which results in high portfolio turnover. As an explanation of the discovered volatility effect, agency issues and management compensation issues that increase demand for highly volatile stocks and decrease their returns were provided. They found out that financial institutions hold more volatile stocks, that analysts tend to cover more volatile stocks in their analysis, as these are more likely to impress the CIO and that more volatile stocks are frequently mentioned in the news, thus increasing the demand. In the conclusions, it was also mentioned that

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academics for various reasons try to preserve the existing paradigm, but Haugen and Baker believe we are experiencing a change of this old paradigm now.

Another study published in 2012 is that of Dutt and Humphrey-Jenner, who examined the low-volatility effect outside of the US market. They focused on equity markets of emerging Asian and EMEA (Europe, Middle East and Africa) countries, Latin America and other developed markets but without the US and Canada. They found out that low-volatility stocks indeed have higher returns than highly volatile stocks and argued that the reason for this could be a higher operating performance of less volatile firms.

In 2012 Blitz and van Vliet proved the low-volatility anomaly purely in the emerging equity markets. They related the effect in the emerging and in developed markets to check if they are driven by a common factor. They found low correlations which exclude a common global systematic driver. The low-volatility effect has received criticism saying that this relation is driven by small-caps, especially the very negative return of the highly-idiosyncratic volatile stocks, but Blitz and van Vliet obtained results that were robust to this critique. Furthermore their results, the alpha spread, spread remained large after controlling for size, value, momentum; after removing 50% of the smallest least liquid stocks and after extending the holding period from 1 year to 5 years. They also stated that the effect has strengthened over time and that it is stronger when total volatility is used instead of betas. As s factor that is responsible for strengthening the effect, they again proposed the agency issues, specifically delegated portfolio management and their incentive schemes.

To be able to study the existence of low-volatility anomaly, one has to choose a proper model predicting portfolio returns. A model with a proper choice of explanatory variables can help disentangle the low-volatility anomaly from different factors or anomalies influencing the returns. As it was mentioned before CAPM did not withstand all of the empirical tests, thus academics constantly kept on trying to improve the CAPM by adding additional explanatory variables that could capture the nature of stock/portfolio returns. These explanatory variables are related to various firm characteristics; for example, in 1991 Chan, Hamao and Lakonishok discovered that book to market ratio influences expected

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returns. In particular, the higher the book to market ratio is, the higher expected returns are, this effect was later called the value effect. Before that, in 1981 Rolf Banz showed that market capitalization of a firm also influences the expected returns, this is also known as the size effect and it implies that smaller firms tend to exhibit higher returns than larger firms. In 1992 Fama and French proposed a three-factor Fama-French model that included explanatory variables supposed to capture so-called value and size effects. The model looks as follows:

𝐸(𝑟𝑖) − 𝑟𝑓 = 𝛽𝑖(𝑟𝑚− 𝑟𝑓) + 𝛽𝑠∗ 𝑆𝑀𝐵 + 𝛽𝑣∗ 𝐻𝑀𝐿

Where 𝐸(𝑟𝑖) represents expected return on a stock, 𝑟𝑓 is the risk-free rate and 𝑟𝑚− 𝑟𝑓 is the market premium. The added factors are 𝑆𝑀𝐵 (small minus big) representing the size effect and 𝐻𝑀𝐿 (high minus low) representing the value effect.

In 1997 Mark Carhart extended this model by adding a third factor- momentum. He published a paper where he evaluated the performance of mutual funds and found out that part of abnormal returns (alpha) could have been attributed to a strategy of purchasing stocks that have done well in the past and selling stocks that have performed poorly. The 4 factor Carhart model is as follows:

𝐸(𝑟𝑖) − 𝑟𝑓 = 𝛽𝑖(𝑟𝑚− 𝑟𝑓) + 𝛽𝑠∗ 𝑆𝑀𝐵 + 𝛽𝑣∗ 𝐻𝑀𝐿 + 𝛽𝑚∗ 𝑀𝑂𝑀

Where 𝐸(𝑟𝑖) represents expected return on a stock, 𝑟𝑓 is the risk-free rate and 𝑟𝑚− 𝑟𝑓 is the market premium. The added factors are 𝑆𝑀𝐵 (small minus big) representing the size effect, 𝐻𝑀𝐿 (high minus low) representing the value effect and 𝑀𝑂𝑀 being the momentum factor.

Another way to investigate the low-volatility anomaly is with the use of the Sharpe ratio, which measures risk-adjusted returns. It was introduced by William Sharpe as a means to measure expected returns per unit of risk (Sharpe, 1994). The formula is as follows:

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𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 = 𝐸(𝑅𝑝) − 𝑟𝑓 𝜎𝑝

Where 𝐸(𝑅𝑝) is the expected return of the portfolio, 𝑟𝑓 is the risk-free rate and 𝜎𝑝 is the

standard deviation of the portfolio.

An interesting study on low-volatility anomaly was performed by Riley and Jordan (2012). They examined the US equity market in the period of time from 1980 till 2011 and found out that investing $1 in low volatility portfolio in July 1980 grew to $89.5 at the end of 2011 but for the high-volatility portfolio that value was only $4.84. The results and evidence are even stronger when the risk of the portfolios is taken into account as the Sharpe and Treynor ratios of the lowest volatility portfolio are five times larger than the ratios of the high volatility portfolio (Riley, Jordan, 2012). What is extremely interesting about their results is the fact that when they examined the anomaly using the 4 factor Carhart model, they found out that the anomaly disappears in the period from 1996 to 2011. They related this to the momentum factor and the fact that highly volatile stocks, especially in times of bubbles and market crashes, tend to move quickly from being past winners to being past losers.

The purpose of this paper is to examine this relationship during a crisis, specifically the 2008 financial crisis. The mentioned research only briefly stated the nature of this risk-return relationship in different states of the market. As mentioned by Clark, Thorley, Silva (2006) minimum variance portfolio had a muting effect on market drop 1982 and the market crash of 1987. Baker, Bradley, Wurgler (2011) stated that high volatility portfolios provided relatively low returns in times of market downturns (1973/4, 2000/2, 1978, 2008). Then Blitz, van Vliet (2007) stated that low-volatility strategy outperforms in down markets and underperforms in up markets and has anti-bubble behaviour. This lays some foundation for the hypothesis that in times of crisis, the low-volatility anomaly tends to hold and could have an insulating effect on the stock returns.

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DATA AND METHODOLOGY

The data used for the purpose of this research is obtained from Wharton Research Data Services. The list of constituents is obtained from the Compustat and returns of individual stocks from the CRSP database. The research is based on total returns that include dividend payouts. The four factors used in the Carhart model and risk-free rate are also obtained from WRDS. The total list of constituents is analysed with the use of STATA to extract the exact list of constituents at a given point in time.

The time range for which the anomaly will be tested is 20 years of monthly data, from January 1992 till the end of December 2011. In the testing time range, two crisis periods are identified. The first one is the dot com crash that lasted from March 2000 till the end of 2002. The second one is the 2008 financial crisis that started in September 2008 with the collapse of Lehman Brothers and ended in June 2009 when the National Bureau of Economic Research in the US announced the end of the recession.

The measure of volatility used is the total volatility of the monthly returns and decile portfolios are created based on past 5-year volatility of returns. The portfolios are equally-weighted and rebalanced every five years, that is in 1992, 1997, 2002 and 2007 to make sure that the volatility does not change throughout the testing period. The returns of the stocks are analysed from 1987 to compute the past 5-year volatility needed to create decile portfolios. Not all of the SP500 constituents have enough past returns to compute the past volatility; thus, some stocks are omitted from the analysis. In the first rebalancing, 17 companies are omitted, in the second one 22, in the third one 5 companies and in the last one 9 companies are omitted from the analysis.

First, to investigate the presence of low-volatility anomaly, a graph of cumulative returns for all five portfolios is presented. After that, the Carhart model is used and alphas representing the abnormal returns are analysed with respect to the low-volatility anomaly.

Then to test the influence of the 2008 financial crisis on the low-volatility anomaly, the previously described Carhart model is used. The model is augmented by adding a dummy variable for crisis and it looked as follows:

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𝐸(𝑅𝑝) − 𝑟𝑓 = 𝛼1+ 𝛼2∗ 𝐶𝑟𝑖𝑠𝑖𝑠 + 𝛽1(𝑟𝑚− 𝑟𝑓) + 𝛽𝑠∗ 𝑆𝑀𝐿 + 𝛽3∗ 𝐻𝑀𝐿 + 𝛽4∗ 𝑀𝑂𝑀 + 𝜀

Where 𝐸(𝑅𝑝) represents the return on the portfolio, 𝑟𝑓 is the risk-free rate and 𝑟𝑚− 𝑟𝑓 is the market premium, 𝑆𝑀𝐿 represents the size effect, 𝐻𝑀𝐿 represents the value effect and 𝑀𝑂𝑀 being the momentum factor. Crisis is a binary variable included to test the influence of the financial crisis, 𝛼1 are the abnormal returns, and 𝛼2 represents abnormal returns due to the crisis.

This regression is performed for each of the five decile portfolios. Later a t-test is performed on each of the 𝛼2 coefficients to determine if the crisis had a statistically significant effect on the performance of the decile portfolios.

In the further investigation section, a different model is used to estimate the abnormal returns. As Riley and Bradford (2012) suggest, momentum factor during the crisis may distort the results; thus a 3-factor Fama-French model, that omits this factor, is used. Later a different crisis period definition is used to test the influence of the crisis on the anomaly. That definition is based on bear and bull markets. Bear market periods connected to the dot com crash and 2008 financial crisis are identified based on SP500 historical patterns and used as the definition of crisis.

In addition to regression analysis, Sharpe Ratios for all five portfolios are calculated to examine differences in risk-adjusted portfolio returns. They are first calculated for the aggregate testing period and then specifically for the time of the 2008 financial crisis. Then the difference in Sharpe Ratios of high and low volatility portfolios is tested using Jobson and Korkie test for equality of Sharpe ratios to determine if the difference is statistically significant.

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RESULTS

All of the portfolios presented below are sorted on volatility, with Portfolio 1 having the lowest volatility and Portfolio 5 having the highest volatility. The full output of all regressions can be found in the appendix.

The graph below presents the outcome of investing 100$ in each of 5 portfolios over the period from January 1992 till December 2011. These portfolios are sorted on volatility, with Portfolio 1 having the lowest volatility and Portfolio 5 having the highest. It is clear that the highest volatility portfolio has lower returns than medium volatility portfolios, but the cumulative return is slightly higher than of the lowest volatility portfolio.

Based on this graph, it can be concluded that the traditional CAPM relation does not hold in this time for these companies. According to CAPM, Portfolio 5 should obtain the highest returns while it performs similarly to Portfolio 2. This graph, however, does not depict the low-volatility anomaly, as the lowest volatility portfolio’s cumulative returns are still slightly lower, than these of Portfolio 5.

Figure 1: Cumulative returns for 5 decile portfolios for 1992-2012

USD 0 USD 500 USD 1,000 USD 1,500 USD 2,000 USD 2,500 USD 3,000 1 9 9 2 0 4 3 0 1 9 9 2 1 2 3 1 1 9 9 3 0 8 3 1 1 9 9 4 0 4 2 9 1 9 9 4 1 2 3 0 1 9 9 5 0 8 3 1 1 9 9 6 0 4 3 0 1 9 9 6 1 2 3 1 1 9 9 7 0 8 2 9 1 9 9 8 0 4 3 0 1 9 9 8 1 2 3 1 1 9 9 9 0 8 3 1 2 0 0 0 0 4 2 8 2 0 0 0 1 2 2 9 2 0 0 1 0 8 3 1 2 0 0 2 0 4 3 0 2 0 0 2 1 2 3 1 2 0 0 3 0 8 2 9 2 0 0 4 0 4 3 0 2 0 0 4 1 2 3 1 2 0 0 5 0 8 3 1 2 0 0 6 0 4 2 8 2 0 0 6 1 2 2 9 2 0 0 7 0 8 3 1 2 0 0 8 0 4 3 0 2 0 0 8 1 2 3 1 2 0 0 9 0 8 3 1 2 0 1 0 0 4 3 0 2 0 1 0 1 2 3 1 2 0 1 1 0 8 3 1

Cumulative Returns

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Table 1: Abnormal returns using Carhart and augmented Carhart model regression

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Carhart 4 factor model

Alpha 0.007* 0.007* 0.012*** 0.013*** 0.009* p-value 0.012 0.007 0.000 0.000 0.017 Agumented Carhart Alpha 0.007** 0.008** 0.012*** 0.012*** 0.010** p-value 0.003 0.003 0.000 0.000 0.004 Alpha Crisis -0.001 -0.003 0.002 0.004 -0.07 p-value 0.888 0.722 0.851 0.688 0.554 *p<0.05, **p<0.01, ***p<0.001

The output for the Carhart model shows statistically significant abnormal returns for all five portfolios. The abnormal returns are increasing with the volatility for Portfolios 1 to 4, but they are decreasing for the highest volatility Portfolio 5. However, abnormal returns of the highest volatility portfolio, are higher than these of the lowest volatility portfolio, so it does not fully support the existence of the low volatility anomaly.

The second part of the table shows abnormal returns for augmented Carhart regression. The alpha crisis coefficient shows the influence of the crisis on the anomaly. When looking strictly at alpha coefficients, it is clear that the highest volatility portfolio had suffered the most during the crisis which is consistent with the research of Baker, Bradley, Wurgler (2011) who stated, that high volatility portfolios provided relatively low returns in times of market downturns. Unfortunately, these results are statistically insignificant, which can be concluded by looking at the p-values of these coefficients, thus based on this output it can not be concluded that the 2008 financial crisis had an influence on the low-volatility anomaly.

According to Riley and Bradford (2012), using the Carhart model in the time range 1996-2011 makes the low-volatility anomaly disappear, due to the Momentum factor distorting the results in times of crisis. Even though in this research, a longer time range is used (1992-2012), the reason identified by Riley and Bradford could still interfere with observing the low-volatility anomaly. To overcome this problem, in the Further Investigation section, the 3 factor Fama-French model is used to estimate abnormal returns.

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Table 2: Sharpe Ratios

Sharpe Ratio Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 1992-2012 Monthly SR 0.153 0.159 0.271 0.239 0.124 Crisis Monthly SR -0.150 -0.224 -0.192 -0.102 -0.161 Z-statistic 1992-2012 0.558 P-value 0.58 Crisis 0.068 P-value 0.95

To test if the difference between Sharpe Ratio of the low-volatility Portfolio 1 and high-volatility Portfolio 5 is statistically significant, the Jobson and Korkie test is performed. The outcome of the test is presented in the last row in the form of the Z-statistic. Based on the statistic and the p-values corresponding to it, the difference in the Sharpe Ratios is statistically insignificant for both periods: 1992-2012 and the 2008 crisis period.

FURTHER INVESTIGATION

As stated by Riley and Bradford (2012) using the Carhart model in the time range from 1996-2011 may make the low-volatility anomaly disappear, due to the momentum factor. The table below presents abnormal returns for the 3-factor Fama-French model and the augmented Fama-French model.

Table 3: Abnormal returns using Fama-French and augmented Fama-French model regression

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 3 factor model Alpha 0.003 0.004 0.009*** 0.009*** 0.003 p-value 0.159 0.139 0.001 0.006 0.250 Agumented 3 factor Alpha 0.002 0.003 0.009*** 0.008** 0.002 p-value 0.316 0.254 0.000 0.003 0.512 Alpha Crisis 0.009 0.007 -0.000 0.003 0.012 p-value 0.443 0.528 0.964 0.836 0.384 *p<0.05, **p<0.01, ***p<0.001

For the 3 factor model, the abnormal returns are increasing with volatility for Portfolios 1 till 4 and are decreasing for Portfolio 5. The magnitude of the alpha coefficient of the highest volatility portfolio is the same as of the lowest volatility portfolio. Compared to Carhart model, where the abnormal returns of highest volatility portfolio are slightly higher

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than that of lowest volatility portfolio, this could be a stronger hint towards the existence of the anomaly, but unfortunately, both of the alphas are statistically insignificant, so no conclusions about the existence of the low-volatility anomaly can be made.

In the second part of the table, the Fama French model augmented with the binary variable is presented. Looking purely at the magnitude of Alpha Crisis, the results are contradictory to existing research, because the highest volatility Portfolio 5 seems to perform best during the crisis. Unfortunately, the Alpha Crisis coefficients are again statistically insignificant, which means that data according to this data, the 2008 crisis did not affect the low-volatility anomaly.

Another factor worth investigating is a different definition of the crisis. The definition used before was based on economic events (collapse of Lehman Brothers) that started the crisis and announcement of the US National Bureau of Economic Research about the end of the recession. Another possible definition is connected to bull and bear markets. Based on the historical SP500 patterns, the bear period corresponding to the dot com crash and 2008 financial crisis were used as the crisis period. The abnormal returns using this crisis definition, for Carhart and Fama-French model, are given below.

Table 4: Abnormal returns using augmented Carhart and Fama-French regression for bear market period Carhart model Alpha 0.006* 0.006** 0.011*** 0.011*** 0.010* p-value 0.020 0.023 0.000 0.001 0.009 Alpha Crisis -0.009 -0.007 -0.005 -0.006 -0.016 p-value 0.296 0.457 0.596 0.553 0.146 3 factor model Alpha 0.005* 0.005* 0.010*** 0.010** 0.007 p-value 0.038 0.068 0.000 0.002 0.057 Alpha Crisis -0.008 -0.005 -0.004 -0.004 -0.013 p-value 0.365 0.581 0.684 0.682 0.271 *p<0.05, **p<0.01, ***p<0.001

Using the augmented Carhart model, each of the portfolios has statistically significant abnormal returns. When testing the influence of the crisis, we see crisis had the biggest negative effect on the most volatile portfolio, which is consistent with previous research, but the results are statistically insignificant.

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Looking at the augmented 3-factor Fama-French model, the same pattern can be observed. The negative influence of crisis was the strongest for the most volatile portfolio, but the results are statistically significant.

CONCLUSIONS

The aim of this paper was to investigate the low-volatility anomaly in the US stock market in the period of time from 1992 to 2012 and whether the 2008 financial crisis had an influence on it. The investigation was based on Carhart and Fama-French model regression and Sharpe ratio analysis of volatility sorted portfolios. The portfolios were constructed from SP500 constituents.

The main findings were that the 2008 financial crisis did not have a statistically significant effect on the low-volatility anomaly. This result is based on augmented Carhart and Fama-French regressions and confirmed with the crisis period Sharpe ratios. The influence of the crisis on the anomaly remains insignificant when different crisis definitions are used, the first one based on the of collapse Lehman Brothers and the second one on the bear market periods. These results are contradictory to the research of Baker, Bradley, Wurgler (2011) who stated that high volatility portfolios provided relatively low returns in times of 2008 market downturn and to the research of Blitz, van Vliet (2007), who stated that low-volatility strategy has anti-bubble behaviour.

When it comes to the presence of the low-volatility anomaly in the 1992-2012 period, it was not explicitly observed. The abnormal returns identified using the Carhart model suggest some anomaly, as abnormal returns for the most volatile portfolio are declining and are only slightly higher than these of the lowest-volatility portfolio. Using the Fama-French model did not produce significant abnormal returns for the highest and lowest volatility portfolios, so no conclusions about the presence of the anomaly can be made. This is inconsistent with Ang et al. (2011), who proved the existence of the anomaly in 23 countries, including the US. This difference could be due to different portfolio rebalancing or due to different time periods examined, as their research focused on the time between 1963 and 2003.

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The Sharpe Ratio analysis also did not yield significant results, as the difference between ratios for the highest and the lowest volatility portfolios was statistically insignificant. This is contradictory to Blitz and van Vliet (2007) and Riley and Jordan (2012) who found the anomaly using the Sharpe ratios. The contradictory results could be due to different portfolio formation and time range used. Riley and Jordan (2012) examined a shorter period of time (1996-2011) and Blitz and van Vliet used FTSE World Developed Index for a period of time from 1985 till 2006.

LIMITATIONS AND FURTHER RESEARCH

The first shortcoming of this paper is the rebalancing frequency. For the purpose of this research, the portfolios were rebalanced every five years to account for the changing volatility of the stocks and changing composition of the SP500 index. This rebalancing could be done more frequently to improve the portfolio composition and possibly deliver stronger results.

Another limitation is the model used for estimating abnormal returns. As stated by Riley and Bradford, the Momentum factor in times of crisis can distort the result of the regressions. In their paper, they found that in a period of time from 1996 till 2011 using the Carhart model, the low-volatility anomaly disappeared. This research uses Carhart and 3 factor Fama-French model (in the further investigation section), but additional models could be used to predict the abnormal returns better.

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REFERENCES

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Economics, 91(1), 1-23. doi:10.1016/j.jfineco.2007.12.005

Baker, M., Bradley, B., & Wurgler, J. (2011). Benchmarks as Limits to Arbitrage:

Understanding the Low-Volatility Anomaly. Financial Analysts Journal, 67(1), 40-54. doi:10.2469/faj.v67.n1.4

Baker, N. L., & Haugen, R. A. (2012). Low Risk Stocks Outperform within All Observable Markets of the World. SSRN Electronic Journal. doi:10.2139/ssrn.2055431

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Berk, J. B., & DeMarzo, P. M. (2017). Corporate finance. Boston: Pearson.

Blitz, D. C., & Van Vliet, P. (2007). The Volatility Effect. The Journal of Portfolio Management, 34(1), 102-113. doi:10.3905/jpm.2007.698039

Blitz, D., Falkenstein, E. G., & Vliet, P. V. (2013). Explanations for the Volatility Effect: An Overview Based on the CAPM Assumptions. SSRN Electronic Journal.

doi:10.2139/ssrn.2270973

Blitz, D., Pang, J., & Vliet, P. V. (2012). The Volatility Effect in Emerging Markets. SSRN Electronic Journal. doi:10.2139/ssrn.2050863

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Fama, E. F., & French, K. R. (2003). The Capital Asset Pricing Model: Theory and Evidence The Capital Asset Pricing Model: Theory and Evidence (CRSP Working Paper No. 550) Retrieved from SSRN website:

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Haugen, R. A., & Baker, N. L. (1991). The efficient market inefficiency of capitalization– weighted stock portfolios. The Journal of Portfolio Management, 17(3), 35-40. doi:10.3905/jpm.1991.409335

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APPENDIX

The figures in the brackets are the standard errors of the coefficients.

Table 5: Carhart model regression

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Market Premium 0.442*** (0.08) 0.482*** (0.09) 0.419*** (0.09) 0.462*** (0.10) 0.754*** (0.11) SMB -0.220** (0.07) -0.118 (0.07) -0.120 (0.08) 0.029 (0.09) 0.331** (0.11) HML 0.256* (0.10) 0.249* (0.11) 0.146 (0.11) 0.116 (0.13) -0.193 (0.12) MOM -0.076 (0.06) -0.128* (0.06) -0.082 (0.05) -0.141* (0.07) -0.282** (0.09) Constant 0.007* (0.00) 0.007** (0.00) 0.012*** (0.00) 0.013*** (0.00) 0.009* (0.00) R-squared 0.28 0.29 0.23 0.22 0.42 *p<0.05, **p<0.1, ***p<0.001

Table 6: Agumented Carhart model regression

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Crisis -0.001 (0.01) -0.003 (0.01) 0.002 (0.01) 0.004 (0.01) -0.07 (0.01) Market Premium 0.439 *** (0.08) 0.475*** (0.09) 0.423*** (0.09) 0.472*** (0.10) 0.739*** (0.12) SMB -0.218 (0.07) -0.115 (0.08) -0.122 (0.08) 0.025 (0.09) 0.336* (0.11) HML 0.258* (0.11) 0.253* (0.11) 0.144 (0.11) 0.111 (0.13) -0.185 (0.13) MOM -0.077 (0.05) -0.131* (0.05) -0.080 (0.05) -0.136* (0.07) -0.290* (0.08) Constant 0.007** (0.00) 0.008** (0.00) 0.012*** (0.00) 0.012*** (0.00) 0.010** (0.00) R-squared 0.29 0.29 0.23 0.22 0.42 *p<0.05, **p<0.1, ***p<0.001

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Table 7: Fama-French model regression

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Market Premium 0.469*** (0.09) 0.528*** (0.09) 0.448*** (0.09) 0.513*** (0.10) 0.858*** (0.12) SMB -0.228** (0.08) -0.136 (0.09) -0.130 (0.09) 0.009 (0.10) 0.286* (0.12) HML 0.274* (0.11) 0.281* (0.11) 0.165 (0.11) 0.152 (0.13) -0.118 (0.13) Constant 0.003 (0.00) 0.004 (0.00) 0.009*** (0.00) 0.009*** (0.00) 0.003 (0.00) R-squared 0.27 0.27 0.21 0.20 0.38 *p<0.05, **p<0.1, ***p<0.001

Table 8: Agumented Fama-French model regression

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Market Premium 0.476*** (0.09) 0.534*** (0.09) 0.448*** (0.09) 0.516*** (0.11) 0.868*** (0.12) SMB -0.231** (0.07) -0.138 (0.07) -0.130 (0.08) 0.008 (0.09) 0.282* (0.11) HML 0.262* (0.11) 0.271* (0.11) 0.166 (0.12) 0.148 (0.14) -0.136 (0.13) Crisis 0.009 (0.01) 0.007 (0.01) -0.000 (0.01) 0.003 (0.01) 0.012 (0.01) Constant 0.002 (0.00) 0.003 (0.00) 0.009*** (0.00) 0.008** (0.00) 0.002 (0.00) R-squared 0.28 0.27 0.21 0.20 0.38 *p<0.05, **p<0.1, ***p<0.001

Table 9: Agumented Carhart model regression for bear market period

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Market Premium 0.414*** (0.08) 0.461*** (0.09) 0.403*** (0.09) 0.443*** (0.11) 0.706*** (0.12) SMB -0.204** (0.07) -0.105 (0.07) -0.110 (0.08) 0.041 (0.09) 0.354** (0.11) HML 0.263* (0.11) 0.253* (0.11) 0.147 (0.11) 0.120 (0.13) -0.176 (0.13) MOM -0.087 (0.06) -0.137* (0.06) -0.091 (0.05) -0.150* (0.07) -0.298*** (0.09) Crisis -0.009 (0.01) -0.007 (0.01) -0.005 (0.01) -0.006 (0.01) -0.016 (0.01) Constant 0.006* (0.00) 0.006** (0.00) 0.011*** (0.00) 0.011*** (0.00) 0.010* (0.00) R-squared 0.29 0.29 0.23 0.22 0.43 *p<0.05, **p<0.1, ***p<0.001

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Table 10: Agumented Fama-French model regression for bear market period

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Market Premium 0.449*** (0.09) 0.516*** (0.10) 0.439*** (0.10) 0.503*** (0.11) 0.826*** (0.12) SMB -0.221** (0.07) -0.131 (0.07) -0.127 (0.08) 0.013 (0.09) 0.298** (0.11) HML 0.286* (0.11) 0.289* (0.11) 0.171 (0.11) 0.158 (0.14) -0.099 (0.14) Crisis -0.008 (0.01) -0.005 (0.01) -0.004 (0.01) -0.004 (0.01) -0.013 (0.01) Constant 0.005* (0.00) 0.005* (0.00) 0.010*** (0.00) 0.010** (0.00) 0.007 (0.00) R-squared 0.28 0.27 0.21 0.20 0.38 *p<0.05, **p<0.1, ***p<0.001

Table 11: Summary of abnormal returns

Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Carhart 4 factor model

Alpha 0.007* 0.007* 0.012*** 0.013*** 0.009* p-value 0.012 0.007 0.000 0.000 0.017 3 factor model Alpha 0.003 0.004 0.009*** 0.009*** 0.003 p-value 0.159 0.139 0.001 0.006 0.250 Agumented Carhart Alpha 0.007** 0.008** 0.012*** 0.012*** 0.010** p-value 0.003 0.003 0.000 0.000 0.004 Alpha Crisis -0.001 -0.003 0.002 0.004 -0.07 p-value 0.888 0.722 0.851 0.688 0.554 Agumented 3 factor Alpha 0.002 0.003 0.009*** 0.008** 0.002 p-value 0.316 0.254 0.000 0.003 0.512 Alpha Crisis 0.009 0.007 -0.000 0.003 0.012 p-value 0.443 0.528 0.964 0.836 0.384

Different crisis definition Carhart model Alpha 0.006* 0.006** 0.011*** 0.011*** 0.010* p-value 0.020 0.023 0.000 0.001 0.009 Alpha Crisis -0.009 -0.007 -0.005 -0.006 -0.016 p-value 0.296 0.457 0.596 0.553 0.146 3 factor model Alpha 0.005* 0.005* 0.010*** 0.010** 0.007 p-value 0.038 0.068 0.000 0.002 0.057 Alpha Crisis -0.008 -0.005 -0.004 -0.004 -0.013 p-value 0.365 0.581 0.684 0.682 0.271 *p<0.05, **p<0.01, ***p<0.001

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