The Low Volatility Effect: Evidence from Europe
ANDREA ZAMPALONI*ABSTRACT
This paper investigates the presence of the low volatility effect in the European stock market for the period 2003-2015. I rank stocks based on their 24 months standard deviation and I form five equal-weight portfolios. I do not find consistent evidence to support the low volatility effect, as it is only in a bear market period that low volatility portfolio has higher returns than high volatility portfolio. On a risk-adjusted basis, the low risk portfolio outperforms the other riskier portfolios over the entire sample and for each market condition, providing a better risk-return relationship.
1. Introduction
The capital asset pricing model proposed by Sharpe (1964), Lintner (1965) and Mossin (1966) is a well-known equilibrium theory in finance. The model assumes that all investors are rational and want to maximize their utilities on the basis of two characteristics: return and risk. Risk, an uncertain quantity to the investor, is a measure of the probability of losses from future events. Investment opportunities in the market provide different trade-offs between risk and return, and each agent will choose the most suitable investment given his assessment of utility and tolerance of risk. The capital asset pricing model (CAPM) states that the relationship between risk and return is linear and positive. Therefore, rational agents are willing to take on additional risk only if it is expected to provide a higher return. However, from the early 1970s, several researchers find that low-risk assets provided returns that were too high relative to their risk, whereas higher-risk assets provided relatively low returns (see, among others, Black, Jensen, Scholes (1972) and Haugen & Heins (1975)). This, in academic terms, has been called “The low volatility effect” and has aroused interest among academics and practitioners in the last decade, as well as providing a significant investment strategy for several mutual funds.
paper, Blitz and van Vliet rank common stocks based on their historical volatility over the
period 1986-2006. They find that portfolios with lower volatility have higher Sharpe ratios1.
Most of the related studies classify portfolios based on the type of risk being measured, calculating performance and controlling results for different factors like market, value, size and momentum. To do so, it’s necessary to run cross-sectional regression models using respectively CAPM (market), Fama-French (value and size) and Carhart (momentum). However, apart from these features, the methods and data to be used in tests of the low volatility effect remain poorly defined. For example, one of the main critiques comes from Bali and Cakici (2008) where they argue that the low volatility effect is only driven by illiquid small-caps. Also, Scherer (2010) shows that this puzzle can be mainly explained by factors such as size and value.
In sum, a debate continues in the low volatility literature, due to different empirical results. Academics give different explanations of this puzzle, ascribing the results to four main causes: data mining (Black (1993)), leverage constraints (Blitz & van Vliet (2007)), agency problems (Baker & Haugen (2012)) and behavioral explanations (e.g. Kumar (2009)). Furthermore, the main literature on the low volatility puzzle is focused on the U.S. stock market, with only a few researchers showing evidence of this phenomenon using non-U.S. data like Ang et al. (2009) or Blitz and van Vliet (2007) which provide European and Japan results but even so, only at an aggregate level.
In this study, I attempt to identify the low volatility effect using the total historical volatility of stock returns as a measure of risk and focusing only on the European stock market, using an up-to-date sample. My first aim is to find evidence of the low volatility effect in the European stock market. If found, my second goal is to analyze any correlation between this phenomenon and different market trends or conditions; whether it’s driven by upward rather than downward market movements or if it’s more likely to show up in periods of expansion rather than recession. Then, in order to address the critique of Bali and Cakici (2008), where they assert that small stocks are the main drivers of this phenomenon, I use only large-caps in my data sample. Further, I compute cross-sectional regressions to examine whether the Jensen, Fama-French or Carhart models can explain the the low volatility effect
1 Sharpe ratio is defined as the ratio between the expected return in excess of the risk free rate and the
and compare my results with Scherer (2010) who argues that factors such as size and value can clarify this puzzle.
2. Literature review
In the context of the present analysis, financial risk is the uncertainty that an investor faces that actual returns can differ from expected returns. Published papers that seek evidence of the low volatility effect consider three types of risk: systematic risk, idiosyncratic risk and total volatility. In Section 2.1 below I address papers focusing on systematic risk, in Section 2.2, idiosyncratic risk and in Section 2.3, total volatility. Finally, in Section 2.4 I discuss the various explanations proposed.
2.1 Systematic risk
Black, Jensen and Scholes (1972) test the relationship between systematic risk (i.e. beta) and stock returns. They perform two econometric processes for estimating the relationship between different variables: time-series regression, which aims to define how one or more variables change over time; and cross-sectional regression, which considers changes in a single period of time. Firstly, by using the time series regression of excess returns on the portfolio against excess returns in the market, they show that high-beta securities had significantly negative intercepts and low-beta securities had significantly positive intercepts, contrary to the predictions of the traditional form of the model. Secondly, they perform a cross-sectional regression between the portfolio’s excess returns and betas, finding positive linearity. However, in different sub-periods and where an agent’s ability to borrow is limited, based on wealth and investment return, the return on low beta stocks was higher than would be expected under the the traditional CAPM model. Similar results come from Haugen and Heins (1975) over the period 1946 to 1971; they analyze the relationship between risk/return and market conditions, finding a positive relationship during a bull market and a negative relationship during a bear market.
positive return for the BAB factor, confirming the results of the previous literature. However, during periods when agents reach a maximum level of leverage in their investments and funding liquidity tightens, the return of the BAB factor is lower.
2.2 Idiosyncratic risk
In seeking evidence for the low idiosyncratic volatility anomaly, the Ang, Hodrick, Xing, and Zhang (2006) paper is the pioneer. They use a cross-sectional regression of idiosyncratic volatility and stock returns. The reason for choosing this method, rather than a time-series regression, is that idiosyncratic volatility is the risk of a specific company, so that the company’s size or book value might play an important role in defining the security’s risk. For example, small companies are generally considered riskier than large companies since they are less liquid and have less access to capital and financial resources. As a result, this type of regression allows the authors to control for different companies’ factors and see if those factors have a positive or negative impact in explaining the low volatility effect. Before Ang et al. (2006), most of the empirical literature shows a positive relation between idiosyncratic volatility and average returns: for example Lintner (1965), Lehmann (1990) and Tinic & West (1986). More recent studies like Malkiel and Xu (2002), one of the first papers to argue the important role that idiosyncratic risk can play in asset pricing, find a positive relationship between unsystematic risk and stock returns.
idiosyncratic volatility in the U.S. market and abroad, showing that the results can be applied in other countries as well as the U.S.
After Ang et al. (2006), other studies also adopt a cross-section of stock returns to check the robustness of these results, but find positive or insignificant links with idiosyncratic risk. Bali & Cakici (2008) for instance develop their analysis based on the Fama and French (1992) model comparing periods of different lengths (daily and monthly) for volatility, and three different weighting schemes (value-weighted, equal-weighted and inverse volatility weighted). They find a significant negative relationship with small-cap stocks, but no significant relationship with large-cap stocks. Fu (2009) differs from previous studies by focusing on expected idiosyncratic volatility instead of historical. In this paper he studies the
relationship using an EGARCH 2 model, finding a significant positive relationship between
expected idiosyncratic volatility and expected returns.
2.3 Total volatility
Several papers test whenever the low volatility effect is a total risk (systematic and idiosyncratic) phenomenon. As I mentioned before, Ang et al. (2006) provide empirical evidence of the total volatility effect, resulting in a positive return of 1.06% and a standard deviation of 3.71% for the lowest quintile portfolio against a positive return of 0.09% with a standard deviation of 8.30% for the top quintile portfolio. Blitz and van Vliet (2007) find strong evidence that the total volatility effect is driven by the extremely high risk-adjusted returns of low volatility stocks, compared to the return of the overall market.
Clarke, de Silva & Thorley (2006) study the high performance of minimum variance portfolios, ranking them simply by their historical volatility. Their results show an increasing Sharpe ratio and alphas for portfolios with low historical volatility. In a sample of large caps over the period 1986-2006 they find an alpha spread of 12% annual return. They find significant evidence that high-risk stocks underperform low-risk stocks in the U.S. market and non-U.S. markets as well. For a robust result they control for factors such as size, value and momentum, and find no links with the volatility effect. This empirical study differs from Ang et al. (2006) for two reasons: first, on the adoption of the historical volatility length. While
Ang et al. (2006) uses a shorter period (one month) Blitz and van Vliet (2007) adopt a longer period (past three years). Second, Blitz & van Vliet (2007) find that low-risk stocks perform better than high-risk stocks.
Scherer (2010) argues that 83% of the excess returns for low volatility portfolios can be explained by size and value. He constructs minimum variance portfolios using the MSCI U.S. equity index from 1998 to 2009; then performs six regressions on the excess returns of the minimum variance portfolios against Fama-French factors. The paper shows that after adding value (HML) and size (SMB) factors, the model has a better fit with the data showing an
increase of the adjusted R2 (3) from 51% to 83%.
Baker and Haugen 2012 is one of the latest empirical papers on the evidence for the total low volatility effect. They cover the period from 1990 to 2011 ranking stock returns from all over the world by their 24 months volatility. They find that the high level of Sharpe ratio for the lowest risk portfolio leads to a positive difference between the lowest risk decile and the highest risk decile. Therefore not only high volatility stocks perform badly, but low volatility stocks perform well, all over the world.
2.4 Explanations of the phenomenon
In the literature several different explanations have been given for the low volatility effect. The various explanations fall into two main groups: rational and behavioral. In the following section, I provide an overview of the most shared and accepted ones.
2.4.1 Data mining
Black (1993) states that one of the most common causes of anomalies in the market is data mining and most of the time, anomalies will disappear as soon as it is discovered. However, most academics argue that the low-risk effect is fairly robust and consistent over time. Ang et al. (2006) for example, show that the effect appears during recessions and expansions and during stable and volatile periods. Ang et al. (2009) show that this phenomenon also takes place in international stock markets. Frazzini and Pedersen (2011) show that low-beta portfolios have high Sharpe ratios in U.S. stocks, non-U.S. stocks, bonds and other asset
3 The adjusted R2 is the coefficient of determination and by definition indicates the proportion of the
classes like commodities. Thus there is a consensus that the low-risk effect is pervasive in varied environments, regardless of data mining or its discovery.
2.4.2 Leverage constraints
Investment managers of mutual funds, pension funds, and households normally face restrictions on the use of leverage or of short-selling. This limits attractive investments, a condition called “leverage/short-selling constraint”. Investors can solve this problem by holding more risky stocks like high-beta securities, thus obtaining “built-in” leverage. However, if investors buy a greater proportion of high-beta stocks, the consequent increase of demand and market price for these stocks causes a divergence in their fair value and low returns. Blitz & van Vliet (2007) and Frazzini and Pedersen (2011) strongly support this position, arguing that the low returns of high-risk stocks are principally driven by the leverage restriction.
2.4.3 Agency problem
Another possible explanation of the low volatility puzzle can be found in the agency problem and portfolio compensation. The agency issue can arise whenever there is a conflict of interest between an investor and his delegated portfolio manager: the manager can aim to maximize his own return rather than pursue the investor’s interests. Investment managers usually earn a base salary plus a bonus which is linked to investment performance. Baker and Haugen (2012) provide a clear explanation of the agency issue, describing the option-like compensation that most managers enjoy.
Figure 1 compares the likely performance of a low volatility portfolio with a high volatility portfolio. If an investment manager wants to maximize his return, he will invest more in risky stocks which could earn him a higher bonus. Contracts of this type will tend to increase demand for stocks with high volatility.
Fig. 1. Baker and Haugen (2012), Option-like manager compensation
This figure shows Baker and Haugen’s (2012) graph of a portfolio manager’s compensation, where it consists of salary plus bonus for a certain level of performance. The figure also shows the probability distributions between a high volatility portfolio and a low volatility portfolio.
Another agency problem linked to the volatility effect refers to benchmarking and the information ratio. The latter is a measure of the manager’s performance and shows how the returns on a specific portfolio differ from his defined index. These differences are then divided by their variance or also known as tracking error. Investment managers use this tracking error as the main measure of risk. Baker, Bradley and Wurgler (2011) argue that since low volatility strategies carry a very high level of tracking error the information ratio or manager’s performance decreases. For this reason, portfolio managers lack incentive to adopt low volatility strategies.
2.4.4 Behavioral explanations
Rational actions are not the only explanation provided by the literature to clarify this puzzle. A different set of theories attempts to explain the low volatility anomaly by studying how agents behave. Three major behavioral biases have been addressed as possible explanations for the low volatility puzzle: overconfidence (Baker, Bradley and Wurgler (2011)), preference for lotteries (Kumar (2009)) and representativeness (Blitz and van Vliet (2007); Baker, Bradley and Wurgler (2011)). I deal with these three perceived biases below.
Drivers of overconfidence might be memory issues (related to output bias), biased self-attribution (related to cognitive dissonance), illusion of knowledge, illusion of control and confirmation bias.
In the low volatility literature, Baker, Bradley and Wurgler (2011) argue that investments are based on judgment and estimation of the future. Overconfident investors are more likely to disagree about future estimations made by others, sticking with their own beliefs. Baker et al. (2011) also say that this overconfidence leads to a distorted view of future outcomes of different types of securities like high volatility stocks.
Preference for lotteries: This bias explains the tendency of investors to seek gains where the probabilities are very low. This tendency is a part of the “Prospect Theory” of Kahneman and Tversky (1979) where they explain the “Fourfold Pattern” of the value function. As you can see from figure 2 below, when investors are making profits, the preference line becomes concave, meaning that he is positive risk averse. However, when he is making losses, function becomes steeper and slightly convex, meaning that he is negative risk averse.
Fig. 2. Kahneman and Tversky (1979), Prospect theory: “Four Fold Pattern”
Kumar (2009) finds that some individual investors show a clear preference for stocks with lottery-like payoffs. This preference is more likely to appear during bad economic times, resulting in a decreasing performance of high volatility stocks.
Representativeness: Rationally, investment managers would ignore small and speculative stocks since the chance of failure is very high. However, Baker, Bradley and Wurgler (2011) explain that an apparently irrational preference for high volatility stocks is due to the chance of extremely high expected returns in a short time period. By ignoring the high chance of failure of small and speculative investments, investment managers tend to invest too much in risky stocks, resulting in lower returns.
Some academics, however, have put forward different explanations of the low volatility effect by reference to agents’ behavior. Blitz and van Vliet (2007) for instance refer to the mental accounting process, under which the investor splits his portfolio into two parts, unrelated between each other and using different rules. The first one represents the “asset allocation decision” and the second the “allocation to a certain type of asset class”. While the first portfolio aims to reduce losses, the second portfolio allows the investor to take riskier bets and to overpay for stocks with an uncertain outcome.
3. Methodology
The research objective of the present paper is to identify the low volatility effect in the European stock market and evaluate any possible explanations. This section discusses the research methods and procedures applied as follows: Section 3.1 deals with volatility-based portfolios, Section 3.2 discusses different performance measurements and regression-models; and Section 3.3 explains the robustness tests of different sub-samples.
3.1 Volatility portfolios
For the portfolio’s construction, I follow a similar approach to that of Blitz and van Vliet (2007), sorting stocks into five (i.e. quintile) equal-weight portfolios ranked by their historical volatility. Stocks are aggregated in different groups so that the risk of each individual stock is canceled out, providing a more diversified portfolio. At the end of every month, I create equally-weighted quintile portfolios ranked by the previous 24 months’ standard deviation, which is defined as the amount of variation or dispersion of a set of data. I apply the following formula:
𝜎!,!! = !! 𝑟!,!!!− 𝜇 !+ 𝑟!,!!!− 𝜇 !+ ⋯ + 𝑟!,!!!− 𝜇 ! , (1) where
N = number of months used to estimate volatility, i = stock for which the volatility is estimated, t = month over which the stock’s return is measured, r = monthly return for stock i at time T and
𝜇 = !
! 𝑟!,!!!+ 𝑟!,!!!+ ⋯ + 𝑟!,!!! .
returns in excess of the risk-free rate and standard deviation for each quintile portfolio. A problem of this sorting methodology is that the volatility observations are not independent of each other, so that similar results of risk are provided in the short run since they share 23 months of data. This is a problem in modeling aggregate past volatility because it is a variable that generates itself during a time series. Therefore, having dependent observations could lead to inconsistent results. Lastly, transaction costs are ignored in the monthly rebalancing.
3.2 Performance measurement
Every month, I calculate average return, standard deviation and Sharpe Ratio. Then, for each quintile I compute the monthly excess returns subtracting the risk-free rate. All the calculations provided are based on returns in excess of the risk-free rate. The average return is defined as:
𝐴𝑅 = !! ! 𝑅!,!
!!! , (2)
where Ri,t is calculated as the percentage change of the share price return or return index (RIi,t)
for the stock i at month t:
𝑅!,! = (!"!,!!!"!,!!!)
!"!,!!!
.
(3)In addition for reporting purpose, I provide portfolio returns on an annual basis applying the following formula:
𝐴𝑛𝑛𝑢𝑎𝑙𝑖𝑧𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 = ! 1 + 𝑅!
!!!
!
!.− 1, (4)
where n is the number of years, I then compute the annualized standard deviation of each
quintile simply by multiplying the equation (1) by 12.
Sharpe Ratio: A frequently used measure in evaluating portfolio performance is the
Sharpe ratio by Sharpe, (1966). It’s defined as the return of the portfolio Rp in excess of the
risk-free rate Rf divided by the standard deviation of the returns 𝜎𝑅p, providing reward per
unit of risk:
𝑆𝑅 = !!! !!
To analyze the statistical significance of the difference between two different Sharpe ratios, I apply the Jobson and Korkie (1981) formula shown below:
𝑧 = !"!!!"! ! !! !!!!,! ! ! ! !"!!!!"!!!!"!!"! !!!!,!! , (6)
where SR is the Sharpe ratio for portfolio i, pi, j is the correlation of the returns between
portfolios i and j and T is the number of observations.
Jensen’s Alpha: The Jensen’s alpha is a one-factor regression model and is equal to the difference in return of the actual portfolio in excess of the risk-free rate and the market benchmark (Jensen (1968)). I apply the following regression equation:
𝑅!− 𝑅! = 𝛼!+ 𝛽! 𝑅!− 𝑅! + 𝜀, (7)
where Rm is the return of the market index, p is the slope between the portfolio and the related
market index and 𝛼! is the intercept, or in other words, the average return in excess of the benchmark. This method differs from the Sharpe ratio in presenting a return number, and is based on the systematic risk rather than total risk. The risk is represented by the beta of the portfolio and is equal to the covariance between the portfolio and the market index divided by the variance of the portfolio.
Fama-French and Carhart Model: The Fama-French model by Fama & French (1992), also known as the three-factor model, is an extension of the Capital Asset Pricing Model by the addition of two benchmarks on top of the market factor: size and value. Fama and French show that these two factors have a strong power in explaining stock returns. Scherer (2010)
shows that 83%4 of the variation of the minimum variance portfolio based on U.S. stock
returns can be attributed to the Fama-French factors. In order to test my data sample and see if additional benchmarks other than the market can explain the volatility effect in Europe, I
download the market, size and value factors from Kenneth French’s website5 for European
stocks and regress them over the portfolios excess returns using the Fama-French regression below:
4 In Scherer (2010), the 83% is related to the R2, which provides the “goodness of fit” between the
model and the set of data.
5 I use http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html for the collection of
MKT, SMB, HML, and MOM factors data.
𝑅!,! = 𝛼 + 𝛽!𝑅!,!+ 𝑠!𝑆𝑀𝐵!+ ℎ!𝐻𝑀𝐿!+ 𝜀! , (8) where SMB (Small Minus Big) is the size factor and is the return on a long portfolio of small-cap stocks and a short portfolio of large-small-cap stocks while HML (High (book/market) Minus Low) is the return on a long portfolio of high book-to-market stocks and a short portfolio of low book-to-market stocks.
In addition, to see if a different benchmark improves the model providing additional explanation of the low volatility effect (e.g. by increasing the adjusted R2), I regress the excess returns of the portfolios using the Carhart four-factor model by Carhart (1997):
𝑅!,! = 𝛼 + 𝛽!𝑅!,!+ 𝑠!𝑆𝑀𝐵!+ ℎ!𝐻𝑀𝐿!+ 𝑚!𝑀𝑂𝑀!+ 𝜀!. (9) The difference from the regression in equation (8) is that this model adds a fourth factor called momentum (MOM) which explains the tendency of rising stock prices to increase further, and falling stock prices to keep dropping. The momentum factor is the difference in return between a long portfolio of winner stocks and a short portfolio of loser stocks.
The risk-free rate together with the benchmarks listed above such as market, value, size and momentum factor for European stocks are taken from the Kenneth French’s website. This ensures consistency in analyzing results over the different cross-sectional regression models. 3.3 Additional measurements
I perform other measurements on top of the ones described above to test the robustness of my results.
Modigliani-Modigliani Risk-Adjusted: The Modigliani-Modigliani (M2) risk-adjusted performance, also known as RAP (Modigliani and Modigliani (1997)), is derived from the Sharpe Ratio and measures the return of the portfolio relative to a certain benchmark (i.e. market index). One of the greatest advantages is that the return output is in percentage term providing a straightforward measure per unit of risk, different from the abstract output of the Sharpe ratio. The following formula represents the Modigliani RAP:
𝑅𝐴𝑃 = 𝑅!− 𝑅! !!
!!+ 𝑅!, (10)
where 𝜎! is the standard deviation of the excess returns of the benchmark, and 𝜎! is the
in abstract term as the other metrics that I use (e.g. Sharpe ratio) and is equal to the RAP minus the risk-free rate:
𝑅𝐴𝑃𝐴 = 𝑅!− 𝑅! !!
!!. (11)
Therefore, the RAPA is equal to the Sharpe ratio times the standard deviation of the benchmark.
Treynor Ratio: The Treynor Ratio, developed by Treynor (1965), is the slope of the line connecting the portfolio’s return with the risk-free rate:
𝑇 =!!! !!
!! . (12)
This type of ratio, unlike those described before, is based on the systematic risk of the portfolio. Since systematic risk is a part of total risk, I expect to find similar results to those based on total volatility.
3.4 Testing sub-samples
To test whether the low volatility puzzle can be partly explained by different market conditions or is just a random walk, I compute all the risk-adjusted performance metrics and Jensen’s alpha for different sub-groups. In this case, as a proxy for the portfolio’s market benchmark, I use the STOXX 600 Europe Index. Therefore, I download monthly returns (RI) from January 2003 to December 2015. Next, as a surrogate for the risk-free rate I apply the monthly returns (RI) of one-month EURIBOR and I subtract it from the portfolio’s and market’s returns. The reader should note that these results are only comparable within this section and cannot be correlated with the results of the overall sample where I use a different risk-free rate and market return.
Fig. 3. STOXX 600 Europe Index
This figure shows the monthly return of the STOXX 600 Europe Index from January 2001 to December 2015. I divide the entire sample into three periods to analyze the volatility-sorted portfolios under different market conditions. Therefore, I calculate all the performance metrics and compute the Jensen’s alpha of the quintiles excess returns during rising market periods (i.e. 2003-2007 and 2011-2015) and a falling market period (i.e. 2007-2009).
4. Data
This section presents the data of my research analysis and is organized as follows: Section 4.1 discusses the data sample relative to the dependent variables (portfolios’ excess returns) and Section 4.2 explains the different independent variables: market return, risk-free rate, value, size and momentum.
4.1 Dependent variables – portfolios’ excess returns
I collect monthly returns6 of stocks listed in the STOXX 600 Europe from January 2001
to December 2015. The STOXX 600 Europe index contains 600 large stocks from 18 European developed countries measured and weighted by free float market capitalization. I choose this index so that I can test the argument in Bali and Cakici (2008) that the low
6 I use the data-type RI (return index), which includes the reinvestment of dividends. This shows a
volatility effect is only caused by small-caps. The country most represented in the STOXX
600 Europe is the United Kingdom (30-35%) followed by France and Switzerland (12-15%)7.
The data sample ranges from a maximum of 595 stocks to a minimum of 439, with an average of 523 active stocks. In the construction of quintile portfolios ranked by 24 months of historical volatility, the data sample starts from January 2003. For the sample in analysis, no stocks were delisted. However, the reported stocks’ returns increase over time. Therefore, for every month from January 2003 onwards I calculate the previous 24 months volatility and rank the stocks in ascending order based on the standard deviation. I exclude stocks with less than 12 months of returns. In other words, I include in the portfolio each stock that in the specific month provides 12 months of data. Every month I divide the stocks into equally-weighted quintile portfolios. The weights of the portfolios increase over time so the sample starts with 91 stocks per quintile at the beginning of January 2013 and ends with 117 stocks per quintile in December 2015. The rise of monthly observations can decrease the standard deviation, but over time the observations’ growth is weak and should not affect its calculation. All the calculations for the overall sample in analysis are based on the portfolios’ return in excess of the risk-free rate for European stocks available on Kenneth French’s website. Table 1 shows the descriptive statistics of the portfolios where the bottom (first) quintile represents the lowest risk stocks and the top (fifth) quintile represents the highest risk stocks. The standard deviation increases for each quintile with a final difference of 6.34% between high-risk and low-risk portfolios. The mean is in line with the core finance theory where higher risk provides higher return, though the mean of the third quintile is unexpectedly lower than that of the second quintile, even though its risk is 1% higher.
Table 1
Summary Statistics of Portfolios’ Excess Returns – Dependent Variable
Every month, I form equal-weighted quintile portfolios by ranking stocks in ascending order based on their previous 24 months’ standard deviation. I use monthly data from the constituent list of the STOXX 600 Europe. All the statistics are measured in monthly percentages and calculated over the portfolios’ excess returns, after subtracting the risk-free rate downloaded from Kenneth French’s website. The first portfolio on the left represents the lowest risk quintile while the fifth portfolio on the right is the highest risk quintile. The last column “5-1” refers to the difference in monthly returns between portfolio “5” and portfolio “1”. The “Jarque-Bera (p-value)”row reports the probability of a specific series to be normally distributed. The sample period runs from January 2003 to December 2015.
Summary Statistics Low Risk Quintile
High Risk
Quintile High-‐ Low
1 2 3 4 5 (5-‐1) Mean 1.04% 1.14% 1.07% 1.25% 2.15% 1.12% Standard Deviation 3.17% 4.35% 5.34% 6.56% 9.51% 6.33% Median 1.52% 1.74% 1.60% 2.15% 2.41% 0.89% Maximum 7.56% 13.28% 15.27% 21.82% 49.59% 42.02% Minimum -‐18.05% -‐23.49% -‐26.42% -‐31.24% -‐39.12% -‐21.06% Skewness -‐1.840 -‐1.460 -‐1.017 -‐0.742 0.508 2.348 Kurtosis 7.971 6.605 4.584 4.315 5.884 -‐2.087 Jarque-‐Bera (p-‐value) 0.000 0.000 0.000 0.000 0.000 -‐ Observations 156 156 156 156 156 -‐ 4.2 Independent variables
As a proxy for the risk-free rate and for the market benchmark of my portfolios, I use the data for European stocks from Kenneth French’s website for the period January 2003 to December 2015. To calculate the market benchmark, Fama and French first create European portfolios based by country. Then they collect returns for all the firms that provide four ratios: book-to-market (B/M); earnings-price (E/P); cash earnings to price (CE/P); and dividend yield (D/P). Finally, the market return is calculated as the value-weighted average return for each
country8. All the returns for European stocks are calculated in U.S. dollars. The market excess
return is the return on a region’s value-weight market portfolio minus the U.S. one-month Treasury bill rate. I apply the same risk-free rate in calculating the volatility portfolios’ excess returns.
The size, value and momentum factors for European markets are also available on Kenneth French’s website. The first two monthly factors’ returns are constructed sorting stocks into two market-cap and three book-to-market equity (B/M) groups at the end of each June. Therefore, the size (SMB) factor is the equal-weight average of the returns on three small European stock portfolios minus the average of the returns on the three large stock portfolios. Likewise, the value (HML) factor is calculated as the equal-weight average returns for the two high B/M European portfolios minus the average of the returns for the two low B/M portfolios. The monthly momentum (MOM) return factor is constructed sorting European stocks by size and previous 10 months cumulative returns. Then, these stocks are allocated into six different portfolios from SL (Small, Losers) to BW (Big, Winners). Lastly, the MOM factor is the equal-weight average of the returns for the two winner portfolios minus
the average of the returns for the two loser portfolios9.
To see if market conditions can explain the low volatility effect, I divide the entire sample into sub-periods and analyze the volatility-sorted portfolios’ returns with the market trend. In this analysis I use a different surrogate for the risk-free rate and market benchmark: I download monthly returns (RI) of the STOXX 600 Europe Index from January 2003 to December 2015 from DataStream and use it as the market benchmark. Next, as a proxy for the risk-free rate I apply the monthly returns (RI) of one-month EURIBOR over the same period and I subtract it from the portfolios’ returns and market returns. All the results of this paper are based on stock returns in excess of the risk-free rate. In table 2, I provide summary statistics of the independent variables.
Table 2
Summary Statistics of Independent Variables
The following table presents the summary statistics of the independent variables. The column “MKT-RF” refers to the return of the STOXX 600 Europe Index used as the market benchmark in the sup-period analysis. The other columns on the right refer to respectively market, size, value and momentum benchmarks collected from Kenneth French’s website and used to analyze the overall sample period. The “Jarque-Bera (p-value)” row reports the probability of a specific series to be normally distributed. The sample period runs from January 2003 to December 2015.
Summary Statistics Market Index Fama-‐French/Carhart factors
MKT-‐RF MKT-‐RF (3FF) SMB HML MOM Mean 0.63% 0.76% 0.21% 0.08% 0.74% Standard Deviation 5.30% 5.50% 1.90% 2.16% 3.97% Median 1.22% 1.00% 0.25% 0.16% 1.18% Maximum 17.00% 13.86% 4.99% 8.31% 10.20% Minimum -‐27.17% -‐22.17% -‐4.67% -‐4.60% -‐26.15% Skewness -‐0.941 -‐0.629 -‐0.108 0.405 -‐2.508 Kurtosis 5.045 1.835 0.031 1.016 14.441 Jarque-‐Bera (p-‐value) 0.000 0.000 0.861 0.007 0.000 Observations 156 156 156 156 156 5. Results
This section presents the results of the research analyses. Section 5.1 discusses the results of the portfolios’ performance, Section 5.2 analyzes the regression results and Section 5.3 presents the results over the different sub-sample periods.
5.1 Portfolios’ performance
Table 3 shows the annualized and average performance metrics of the full sample of the quintile portfolios ranked on previous 24 months’ volatility. Looking at the returns and standard deviations of the quintile portfolios, risk and return have a positive relationship where higher risks produce higher returns. The high minus low column shows that the difference in return between the highest and lowest quintiles is 10.12% annualized and 1.12% averaged. The results of the third portfolio are anomalous: though the risk is consistently higher than the first quintile, it underperforms in annual terms by -0.82%.
Table 3
Performance metrics of the quintile portfolios sorted by volatility
After the construction of quintile portfolios ranked by their historical volatility, I compute all the metrics listed in the first column of this table. The first three rows represent annualized performances while the rest of the table shows the results on average. The calculations are based on the portfolios’ excess returns over the Fama-French risk-free rate for European markets and refer to the entire time period from January 2003 to December 2015. Variables that are significant at the 1%, 5% and 10% level are marked by ***,** and *.
Performance Measurements
Low Risk
Quintile High Risk Quintile High-‐ Low
1 2 3 4 5 (5-‐1) Annualized return 12.51% 13.23% 11.69% 13.08% 22.63% 10.12% Standard Deviation 11.00% 15.00% 18.42% 22.67% 32.89% 21.90% Sharpe Ratio 1.14 0.88 0.63 0.58 0.69 -‐0.45 Average Return 1.04% 1.14% 1.07% 1.25% 2.15% 1.12% Standard Deviation 3.17% 4.35% 5.34% 6.56% 9.51% 6.33% Sharpe Ratio 0.33 0.26 0.20 0.19 0.23 -‐0.10 (z/t-‐test) 2.734** 3.026*** 0.676*** -‐1.508 1.856* -‐ RAPA 0.018 0.014 0.011 0.010 0.012 -‐0.006 Treynor Ratio 0.035 0.027 0.020 0.019 0.023 -‐0.012
Fig. 4. Risk-Return trade-off between Highest and Lowest Portfolio
I plot the monthly average returns and standard deviations of the low-risk quintile and high-risk quintile. However, this figure does not represent the trend over time, but simply shows the stocks’ average risk-return trade-off for the sample period 2013-2015.
5.2 Regression models
Table 4
Jensen, Fama-French and Carhart Regression of Portfolios’ Excess Returns
Panel A shows the alpha of the Jensen, Fama-French and Carhart regression models while Panel B reports the coefficients of the independent variables. All the results are calculated on the portfolios’ monthly returns over the Fama-French’s risk free rate for European stocks. Variables that are significant at the 1%, 5% and 10% level are marked by ***,** and *.
The coefficients of the regressions are listed in Panel B of table 4 above and most of them are statistically significant. However, the value factor (HML) provides no significant results for the Fama-French and Carhart models. The market betas for all the regressions are positive and increasing with risk. I find positive and increasing size (SMB) for the volatility portfolios,
all significant at the 1% level. The use of the size and value factors increases the adjusted R2
by 6.22% for the low portfolio and by almost 13% for the high portfolio. This slight improvement of the low portfolio cannot be seen as a proof of the Scherer (2010) findings of a much higher improvement for a minimum variance portfolio of U.S. stocks after adding the size and value factor (from 51% to 73%). However, compelling findings come from the
high-Coefficients Low Risk Quintile High Risk Quintile High-‐ Low
PANEL A: Alphas 1 2 3 4 5 (5-‐1)
risk portfolio. After adding the momentum factor, the last quintile has an adjusted R2 equal to 50.46% with a considerable improvement from the Jensen regression of about 21%. This means that 21% of the variation of the high-portfolio’s excess returns can be explained by the value, size and momentum factors. The coefficients for the momentum factor are negative, but not significant for the low and second quintile.
5.3 Sub-samples performance
The main goal in analyzing sub-periods is to detect any possible relationship between the portfolio’s performance and the market trend. To do so, I re-run all the performance metrics of section 5.1 and the Jensen’s alphas for three unequal sub-groups based on their market trend: 2003/2007 (rising), 2007/2009 (falling) and 2011/2015 (rising).
In this section, I compute the portfolios’ excess returns using as a proxy for the risk-free rate the monthly return of the one-month EURIBOR. For the Jensen’s regressions I use a different benchmark to compare the portfolio’s returns with their market Index. Therefore, I download the monthly STOXX 600 Europe Index returns (RI) over the same periods.
2003/2007: The STOXX 600 Europe Index returns for the period 2003-2007 show a rising trend as can be seen from figure 3. For the first period in analysis, the market has an average return of 1.32% and standard deviation of 4.24%, underperforming all the volatility-sorted portfolios. The average returns for the quintile portfolios are in line with the overall sample. Indeed, as seen in table 5, the difference between high and low risk portfolios is a positive 1.37%. Also, the standard deviation is increasing, providing evidence of linearity between risk and return. However, on a risk-adjusted basis the low volatility portfolio carries a Sharpe ratio of 0.73 versus the 0.45 of the highest risk quintile. These results are confirmed also from the RAPA measure which shows a difference of -0.012 between the high and low portfolios. Again, it seems that the low-risk portfolio has a much higher trade-off between risk and return than the high-risk portfolio confirming the findings of Blitz and van Vliet (2007). The Treynor ratio follows the same path as the other risk-adjusted measures providing a negative difference between the high and low portfolios of -0.013.
risk portfolio is 0.94 higher than the lowest risk portfolio. All the results are statistically significant at the 1% level except for the highest portfolio’s alpha where it is still significant, but at the 5% level.
Table 5
Portfolios’ performance 2003-2007
The table below refers to the first sub-sample, which is from February 2003 to June 2007. During this period the monthly returns of the STOXX 600 Europe Index increase over time. All the results of the quintiles are based on the portfolios’ excess returns using the monthly return of the 1-month EURIBOR as a risk-free rate. Panel B presents the results of the Jensen’s alpha. In this case, I use as a benchmark the monthly return of the STOXX 600 Europe Index. Variables that are significant at the 1%, 5% and 10% level are marked by ***,** and *.
In figure 5, I plot the standard deviation and average returns of the lowest and highest portfolio for the period 2003-2007. The portfolios’ trend lines have similar orientation with the overall sample. The stocks in the high portfolio have a positive and linear relationship between risk and return while the stocks in the low portfolio have an opposite trend where higher return provides lower risk. An important premise is that the stocks’ returns and standard deviation in figure 5 are not plotted following a time series.
Performance
2003/2007 Low Risk Quintile High Risk Quintile High-‐ Low Market Index
Fig. 5. Risk-Return trade-off between Highest and Lowest Portfolio
I plot the monthly average returns and standard deviations of the low-risk quintile and high-risk quintile. However, this figure does not represent the trend over time, but simply shows the stocks’ average risk-return trade-off for the sample period 2003-2007.
2007/2009: The second sub-sample runs from July 2007 to March 2009. During this period the return of the STOXX 600 Europe index decreases over time. Table 6 provides information of the portfolios and market performance where the average return of the index is equal to -3.67% with a very high standard deviation (7.82%). During negative market condition, the low volatility portfolio has the best performance. Indeed, the average return of the first quintile is equal to -2.03% against the -3.61% of the high volatility portfolio. Different from the previous sub-period of positive market trend, the low portfolio shows higher returns than the higher volatility portfolios not only on the risk-adjusted measures but also on average return. All the performance metrics such as the Sharpe, RAP and Treynor ratios perform poorly in respect of the riskier portfolios. One possible explanation can be ascribed to the preferences for lotteries during bad times argued by Kumar (2009), where some individual investors show a clear preference for stocks with lottery-like payoffs during bad times, resulting in a decreasing performance of high volatility stocks.
Table 6
Portfolios’ performance 2007-2009
The table below refers to the second sub-sample, which is from July 2007 to March 2009. During this period the monthly returns of the STOXX 600 Europe Index decrease over time. All the results of the quintiles are based on the portfolios’ excess returns using the monthly return of the 1-month EURIBOR as a risk-free rate. Panel B presents the results of the Jensen’s alpha. In this case, I use as a benchmark the monthly return of the STOXX 600 Europe Index. Variables that are significant at the 1%, 5% and 10% level are marked by ***,** and *.
During the bear period, the high and low portfolios maintain the same risk-return relationship. However, as figure 6 shows, the low-risk quintile has a flattening trend while the stocks in the high-risk portfolio maintain almost the same angle as the previous period. Even if the trend line of the low portfolio slightly increases, it maintains a negative relationship between risk and return, while the high portfolio looks to be identical providing a positive trade-off. In addition, the low portfolio during the falling period shows acceptable returns compared to the high portfolios where the risks are more spread out and highly volatile. An important premise is that the stocks’ returns and standard deviation in figure 6 are not plotted following a time series.
Performance
2007/2009 Low Risk Quintile High Risk Quintile High-‐ Low t Index Marke
Fig. 6. Risk-Return trade-off between Highest and Lowest Portfolio
I plot the monthly average returns and standard deviations of the low-risk quintile and high-risk quintile. However, this figure does not represent the trend over time, but simply shows the stocks’ average risk-return trade-off for the sample period 2007-2009.
2011/2015: During the last sub-period the market has a positive trend with an average return of 1.21% and standard deviation equal to 4.30%. In table 7, the low-risk portfolio outperforms the high-risk portfolio only in risk-adjusted terms with a difference of Sharpe ratio around -0.19. Overall, the performance measurement shows similar results as the first sub-period analyzed (2003-2007) but in this case, the differences between the sorted portfolios are smaller, showing a more stable performance over the volatility-sorted quintiles.
Table 7
Portfolios’ performance 2011-2015
The table below refers to the third sub-sample, which is from August 2011 to May 2015. During this period the monthly returns of the STOXX 600 Europe Index increase over time. All the results of the quintiles are based on the portfolios’ excess returns using the monthly return of the 1-month EURIBOR as a risk-free rate. Panel B presents the results of the Jensen’s alpha. In this case I use as a benchmark the monthly return of the STOXX 600 Europe Index. Variables that are significant at the 1%, 5% and 10% level are marked by ***, ** and *.
The risk-return relationship for this period looks inverted for the stocks in the low volatility portfolio. Figure 7 shows that for this sample period the stocks in the low quintile exhibit a positive and linear relationship between standard deviations and average returns meaning that higher risk is rewarded with higher return. This positive trend for the low volatility portfolio was not detected in the previous sub-samples showing a lack of “low volatility benefit” where lower risk follows higher return. The line of the high volatility stocks is much steeper than the previous years, but maintaining a positive and linear relationship also during this sub-period. An important premise is that the stocks’ returns and standard deviation in figure 7 are not plotted following a time series.
Performance
2011/2015 Low Risk Quintile High Risk Quintile High-‐ Low Market Index
Fig. 7. Risk-Return trade-off between Highest and Lowest Portfolio
I plot the monthly average returns and standard deviations of the low-risk quintile and high-risk quintile. However, this figure does not represent the trend over time, but simply shows the stocks’ average risk-return trade-off for the sample period 2011-2015.
6. Conclusion
maintain greater returns only when the index has a positive trend with a difference between the high and low portfolios of 1.37% and 0.92% respectively during the first and second sub-period.
Interesting results come from the non-linearity of the trade-off between risk and return. For all of the samples analyzed, the low volatility portfolio provides much higher risk-adjusted returns than all the other quintiles like in Blitz and van Vliet (2007). Compelling outcomes show that when the returns over the high minus low portfolio increases twice, the standard deviation is three times higher. This means that the low-risk stocks’ returns are too high given their risk or that high-risk stocks’ returns are too low compared to their risk.
Moreover, all the performance metrics such as Sharpe, M2 and Treynor give the same
outcomes, proving differences in risk-return trade-off over the volatility-sorted portfolios. Morevoer, it can be seen from table 5 to 7 that this diversity is higher during the boom period of the early 2000s. For example, in 2003-2007 the Sharpe ratio of the high minus low portfolio is -0.29 and reducing to a difference of -0.19 over the last 5 years. In addition, Figures 5, 6 and 7 provide a closer look to the changes in trade-off of the stocks in the high and low quintile. The angle of the trend line connecting risk to return becomes flatter over time, switching its direction from negative to positive for the low volatility portfolio.
The fact that I do not find consistent evidence of the low volatility effect using only large-caps could support the argument of Bali and Cakici (2008) where they argue that the low volatility effect refers only to small-caps. It would therefore be interesting to restrict the data sample only to small European stocks and see if applying the same methodology leads to opposite results. Then, I address the criticism of Scherer (2010) who argues that Fama-French factors can explain the low volatility puzzle, by controlling the results for size, value and
momentum factors. For the low volatility portfolio I find increasing R2 after controlling the
results for Fama-French and Carhart factors. In other words, the R2 moves from 26% of the
Jensen’s regression to around 32% (adjusted R2) when additional factors are included.
minus low alpha was close to zero (0.13%), the last sub-sample shows a negative, but insignificant difference (-0.29%). During down-months the Jensen’s results are in line with the overall sample where higher risk follows higher return.
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