1
The volatility effect in the U.K.
Evidence from the FTSE 100
Master Thesis by Rob van Werven* June 21, 2013
ABSTRACT
This paper examines the empirical risk-return relation for U.K. common stocks. At the end of each month, I sort stocks into quintile portfolios based on their past three-year monthly volatility. I find a difference in the next month’s annualized arithmetic and geometric average returns between the high and low volatility-sorted quintile portfolios of -2.9% and -6.2% respectively. This ‘volatility effect’ is caused by significant outperformance of low-volatility stocks and under-performance of high-volatility stocks in months in which the market went down. These findings are in line with the argument that disagreement with short-sales constraints contribute to the volatility effect.
JEL Classification: G11, G12, G14
Keywords: volatility effect, asset pricing, CAPM, alpha, low-volatility, anomaly
*
2
1. Introduction
Logically, lower risk without lower return sounds appealing to many investors. This appealing idea contradicts with the dominant capital asset pricing model (CAPM), which predicts a positive linear relation between risk and return. What if, in practice, low-volatility stocks outperform high-volatility stocks?
Low-volatility investing has received much attention recently. Over the past couple of decades, low-volatility or low-beta portfolios appear to offer abnormal high returns and small drawdowns (see, e.g., Ang, Hodrick, Xing, and Zhang, 2006; Blitz and van Vliet, 2007; and Baker, Bradley, and Wurgler, 2011). In their empirical work, Blitz and van Vliet report significant outperformance of low-risk stocks and under-performance of high-low-risk stocks on a low-risk-adjusted basis. They label their findings as the ‘volatility effect’. Their results contradict the fundamental principle in finance that risk-averse investors should be compensated for risk with higher expected returns.
Although the empirical evidence of the volatility effect is growing, there are some critiques on these papers. According to Bali and Cakici (2008), the negative empirical relation between risk and return is driven by illiquid small-cap stocks only. Furthermore, Scherer (2010) argues that the success of minimum variance portfolios can be attributed for a large part to the classic size and value effects. Other authors contribute the success of low-risk stocks to short-term mean reversals and state that the risk-return relation is positive over longer investment horizons (see, e.g., Huang, Liu, Rhee, and Zhang, 2007; Amenc, Martellini, Goltz and Sahoo, 2011).
Other studies try to explain the apparent anomaly. There are three main groups of rationalizations for the volatility effect: behavior biases among private investors (see, e.g., Barberis and Huang, 2008; Kumar, 2009; and Baker and Haugen, 2012); short-selling and leverage constraints (see e.g., Black, 1972; de Giorgi and Post, 2011, and Hong and Sraer, 2012); and agency issues involved with delegated portfolio management (see, e.g., Baker, Bradley, and Wurgler, 2011; and Baker and Haugen, 2012). So far, these rationalizations are merely hypotheses and there is no consensus on the matter. If the volatility effect is persistent, than finding an explanation can contribute to the development of improved models predicting the relation between risk and return.
In short, there is no consensus about the apparent volatility effect or relation between risk and return. Therefore, this paper presents a thorough analysis of the empirical out-of-sample relation between historical volatility and one-month future returns. I thereby aim to answer two questions: (1) Is there evidence of a so-called ‘volatility effect’ for common stocks? (2) If there is evidence of a volatility effect, what causes this effect?
3 rebalance the portfolios at the end of each month. In the methodology that I use for answering the first question, I address several critiques previous studies have received. First, the data sample is a fresh data set with data of the U.K. stock market from January 1996 to December 2012. Most research thus far has focused on U.S. stocks only. If the results of my out-of-sample tests confirm the volatility effect documented by previous studies, then previous results are less likely to be the result of data mining. Furthermore, this data set includes the past couple of years and provides new insight into the risk-return relation during the latest credit crisis. Second, I address the critique of Bali and Cakici (2008) that the volatility effect is driven by illiquid small-cap stocks, by only including stocks listed on the FTSE 100, which are liquid large-cap stocks. Third, I control for the classic size, value, and momentum factors by both a regression-based methodology and a double-sorting methodology. Fourth, I analyze the risk-return characteristics for longer holding periods to address the critique that the volatility effect is driven by short-term mean reversals.
To answer the second question, I link my empirical findings to different hypotheses that have been put forward in the literature to contribute to the volatility effect. Specifically, I look at the volatility effect over time and the pattern of the risk-return relation to determine if the results are in line or contradict different hypotheses.
The main findings indicate a volatility effect although the evidence is not overwhelming. For the total sample period, low-risk stocks outperform high-risk stocks, especially on a risk-adjusted basis. The empirical relation between risk and return indicates an inverted U-shape. The highest volatility-sorted quintile portfolio shows considerable under-performance. This under-performance is larger for stocks sorted on historical volatility than on historical market beta. An equally weighted strategy that goes long in the 20% highest volatility stocks and short in the 20% lowest volatility stocks, i.e., the high-minus-low volatility-sorted portfolio, earned an annualized arithmetic and geometric average return of -2.9% and -6.2% respectively. However, these return differences are not statistically significant. A 95% confidence interval of the estimated mean excess return of the different quintile portfolios shows that a positive linear relation between risk and return cannot be rejected. The high-minus-low volatility-sorted portfolio has an insignificant but negative Jensen’s alpha of -6.1%. The low volatility-sorted portfolio has a marginally significant positive Jensen’s alpha of 3.3%. The low volatility-sorted portfolio has a significantly higher Sharpe ratio than the market while the high volatility-sorted portfolio has a significantly lower Sharpe ratio than the market.
4 The pattern of the volatility effect is robust to different formation, waiting, and holding periods. The results show that low volatility-sorted portfolios continue to outperform high volatility-sorted portfolios even for longer holding periods. Thus, the volatility effect cannot be explained by short-term mean reversals.
The sub-sample results in this study show some support for the explanation that disagreement with short-sales constraints contribute to the volatility effect. The results for bull and bear markets show that the volatility effect is present in bear markets, while the results for bull markets are ambiguous. Furthermore, in up-months, I find a perfect linear relation between risk and return and there is no evidence for a volatility effect. In down-months, I find clear evidence of a volatility effect with both outperformance of low-risk stocks and under-performance of high-risk stocks. The findings for bull/bear markets and up/down months are in line with the model of Hong and Sraer (2012). Their model predicts an inverted U-shape when disagreement about the macro-economy between investors is high and investors face short-sales constraints. Disagreement is typically larger in bear markets (which contain a lot of down months), which would explain the found volatility effect in bear markets and down-months.
The remainder of this paper is organized as follows. Section 2 discusses the main literature on the relation between risk and return. Section 3 discusses the used methodology to construct volatility-sorted portfolios. Furthermore, I address the relevant risk and return measures used to evaluate the ex-post risk-return relation of different portfolios. In Section 4, I discuss the data sources and summary statistics of both dependent and independent variables in this research. Section 5 presents and evaluates the ex-post risk-return characteristics of volatility-sorted portfolios. I offer concluding remarks in Section 6.
2. Literature review
This section discusses the main literature on the relation between risk and return. Section 2.1 discusses the classical finance theory. In Section 2.2, I discuss the main empirical findings of key papers on the relation between risk and return. Section 2.3 discusses possible explanations that have been put forward in the literature to explain the volatility effect.
2.1. Theory
The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965), and Black (1972) or Sharpe-Lintner-Black (SLB) model has been the dominant classical finance model for decades. According to the CAPM, there is a linear relation between ‘systematic’ risk and expected return of a stock. This relation is
5 where the tildes correspond to random variables, ( ̃) is the expected return of stock i in excess of the risk-free rate, ̃ is the expected return on the ‘market portfolio’ in excess of the risk-free rate, and is the systematic risk of stock i defined as ( ̃ ̃ ) ̃ . If we define alpha, , as
( ̃) ̃ , (2)
then equation (1) implies that the alpha of each stock is equal to zero.
Fama and French (1992) extend the one-factor model of (1) with two variables; size and book-to-market equity, which help to explain the cross-sectional variation in average stock returns. A further extension is the four-factor model of Carhart (1997). This four-factor model includes an additional momentum factor compared to the Fama and French model. Although these extensions appear to do a good job explaining the cross-sectional variation in average stock returns in empirical studies, there is little theoretical justification.
One important assumption underlying CAPM is that all investors hold the market portfolio in equilibrium and that all idiosyncratic risk is diversified. Hence, investors are only compensated for systematic risk. In practice, however, many investors may not hold well-diversified portfolios. Only 10% of household investors hold portfolios of more than 10 stocks (Goetzmann and Kumar, 2008). A portfolio should consist of about 50 stocks to achieve relatively complete portfolio diversification (Campbell, Lettau, Malkiel, and Xu, 2001). Assuming under-diversified portfolios, there are different theories predicting a positive relation between idiosyncratic risk and expected stock return (see, e.g., Levy, 1978; Merton, 1987, Malkiel and Xu, 2002).
The total volatility of a stock consists of both a systematic en idiosyncratic part. Theory predicts a positive relation between both parts of total volatility and the expected return of a stock. Thus based on these theories, I would expect to find a positive linear relation between risk and return.
2.2. Empirical findings
Since the development of the CAPM, empirical tests show that the relation between risk and expected return is not linear as predicted by theory. Early empirical work by Black, Jensen, and Scholes (1972) shows that the expected excess returns on assets are not strictly proportional to the market betas of the assets. They find negative alphas for high beta stocks and positive alphas for low- beta stocks. These findings are confirmed by other authors (see, e.g., Miller and Scholes, 1972; Fama and MacBeth, 1973; Haugen and Heins, 1975).
6 Lanstein (1985) find a positive relation between a firms’ book-to-market-value of common equity and average return.
An important paper in history of the CAPM is the paper of Fama and French in 1992. They find a negative relation between risk and return which implies that the CAPM is not even a good approximation of the risk-return relation. They find a raw return spread of -2.4% for the high-minus-low beta-sorted decile portfolio. Furthermore, they find that two variables (size and market-to-book value) help to explain the cross-section of expected stock returns. This was the birth of the three factor model of Fama and French. The negative relation that Fama and French find is confirmed by Black (1993), although he questions the size and value factors. Black argues that borrowing restrictions might cause low-beta stocks to have higher expected returns than theory predicts. Haugen and Baker (1991, 1996) and Falkenstein (1994) provide more evidence for a flat or even negative relation between risk and return of stocks.
In the past couple of years, new evidence also shows that the relation between risk and return is flat or negative and that this effect is even stronger when risk is measured as the past volatility of stocks. An important paper in this regard is the paper of Ang, Hodrick, Xing, and Zhang (2006). They sort stock on their one-month idiosyncratic volatility and find a raw return spread of -12.7% for the high-minus-low quartile value-weighted portfolio for stocks in the Center for Research in Security Prices (CRSP) database. Blitz and van Vliet (2007) provide a detailed analysis by sorting stocks on total volatility. They find a raw return spread of -5.9% for the high-minus-low volatility-sorted decile portfolio on a compounded basis. Furthermore, they conclude that their results are robust after controlling for size, value and momentum effects. Clarke, de Silva and Thorley (2010) show further evidence of outperformance of minimum variance portfolios. They show that this outperformance is largely a function of low-beta stocks in the minimum variance portfolio. Baker, Bradley, and Wurgler (2011) show further empirical evidence of what they call ‘the low-risk anomaly’.
Other studies report opposite findings or doubt the robustness of earlier studies. Bali and Cakici (2008) argue that the findings of Ang, Hodrick, Xing, and Zhang (2006) are caused by small-cap stocks and that if these stocks are excluded their results become insignificant. Martellini (2008) includes only surviving stocks and finds a positive relation between risk and return. Fu (2009) argues that the focus should be on expected instead of historical volatility and finds a large positive relation between risk and return when using EGARCH models. In addition to these findings, the volatility effect can be the result of data mining, driven by classic effects, or driven by short-term mean reversals.
Data mining: The volatility effect can be the result of data mining and in this case it will disappear
7 consistently worked in the past. However, in many cases, such an ‘anomaly’ will disappear as soon as it is discovered. If the volatility effect is the result of data mining than using a fresh data set can lead to new insights.
Classic effect: The volatility effect can be a ‘classic’ effect in disguise. The outperformance of
low-volatility stocks can be the effect of implicit loadings on size or value factors. If this is true, the three-factor model of Fama and French (1992) should capture the abnormal returns of the one-factor regression. In the line of classic effects, the highest volatility portfolio can be dominated by losing stocks in terms of the momentum factor of Jegadeesh and Titman (1993). Following the theory of the ‘leverage effect’, rising asset prices are accompanied by declining volatility, and vice versa (see, e.g., Black, 1976; Christie, 1982). In addition, declines in asset prices are generally accompanied by a larger increase in volatility than the decline in volatility that accompanies a rising asset price (see, e.g., Nelson, 1991; Engle and Ng, 1993). If losing stocks dominate the highest volatility portfolio, momentum can result in a short-term negative relation between high volatility and return. In this case, the four-factor model of Carhart (1997) should explain the abnormal returns of the one-factor regression.
Short-term return reversals: The relation between risk and return is positive, but short-term return
reversals cause the relation to be negative. Huang, Liu, Rhee and Zhang (2010) argue that the highest volatility portfolio is dominated by winning stocks in the months in which the portfolio is formed. Thus, the negative risk return relation should be caused by short-term return reversals of winner stocks instead of high volatility. If short-term effects cause the volatility effect, the volatility effect should disappear for longer investment horizons.
9
2.3. Possible explanations
If the volatility effect is robust to critiques mentioned in Section 2.2, there are three types of explanations that have been put forward in the literature to contribute to the volatility effect.1
Behavioral biases: The first explanation is that less than fully rational private investor behavior
leads to excess demand for high volatile stocks which lowers their expected return. There are at least three behavioral biases that afflict the individual investor:
- Preference for lottery tickets: Investors see stocks as lottery-tickets. High risk or high beta stocks are an easy way to get rich quickly although their expected return is negative (see e.g., Barberis and Huang, 2008; and Kumar, 2009).
- Representativeness: For example, investors believe that the road to riches is paved with speculative investments in new technologies. They thereby ignore the high rate of failure of small, speculative investments and overpay for volatile stocks (Baker and Haugen, 2012). - Overconfidence or disagreement: Overconfident investors are more likely to disagree about
the future prospects of a firm. This disagreement is larger for more volatile stocks. If there is a market with optimists acting more aggressive than pessimists, this will lead to excess demand for volatile stocks by overconfident optimists (Baker and Haugen, 2012).
Above behavioral biases can lead to excess demand for high volatile stocks. For these arguments to cause a volatility effect, they must be complemented by an argument explaining why institutional investors do not capitalize on the volatility anomaly. That is, why do large institutions not short high-risk stocks?
Leverage and short-selling constraints: The second explanation is that the volatility effect can be
attributed to leverage and short-selling constraints. This argument states that investors are not allowed or unwilling to apply leverage. To take fully advantage of the returns of low-risk stocks, investors must apply leverage. Without applying leverage, return seeking investors prefer high-beta stocks to obtain high returns (see e.g., Black, 1972; and de Giorgi and Post, 2011). Furthermore, investors face short-sales constraints and are not able to short high-volatility stocks. An interesting study in this regard is the work of Hong and Sraer (2012). They provide a model based on disagreement (also see ‘overconfidence’ in the former paragraph) and short-sales constraints. They find that high risk stocks are overpriced compared to low-risk stocks when disagreement about the market or common factor of firms’ cash flows is high. In times of high aggregate disagreement, the risk return relation can take on an inverted U-shape. They state that high-beta stocks such as retailers load more on the macro-factor than low-beta stocks. If investors disagree about the macro-economy, then their return forecasts of high-beta stocks will diverge much more than their forecasts of low-beta stocks. Optimistic investors will want to buy high beta stocks while pessimistic investors will want to sell high-beta stocks. However, because of short-sales constraints, the demand for high-beta stocks is larger than the supply
1
10 of high-beta stocks and consequently high-beta stocks become overpriced. According to Hong and Sraer, arbitrageurs are to risk-averse to completely eliminate the over-pricing of high beta stocks. If disagreement and short-sales constraint contribute to the volatility effect, we would expect to find the relation between risk and return to take on an inverted U-shape in times of high aggregate disagreement. Since disagreement is typically high in times of high economic uncertainty, we would expect to find the volatility effect to be present in bear markets.
Agency issues involved with delegated portfolio management: The third explanation is that agency
issues involved with delegated portfolio management for institutional investors lead to excess demand for volatile stocks. This excess demand lowers their expected return. There are two main arguments that support this explanation:
- Benchmarking as limit on arbitrage: Institutional investors focus on tracking error instead of total risk. Low-volatility stocks exhibit high tracking errors and are unattractive from this perspective which tends to flatten the risk-return relation. Baker, Bradley, and Wurgler (2011) argue that this argument, in combination with irrational investor behavior, can explain a more flat relation between risk and return. However, this argument cannot explain an inverted relation between risk and return.
- Option-like manager compensation and agency issues (Baker and Haugen, 2012): If managers have option-like compensation, they would prefer high volatile portfolios and tend to overpay for high-risk stocks. Furthermore, investment firms typically make portfolios for their clients based on a model portfolio. Managers are attracted to stocks for which they can make a compelling case to be included in this model portfolio. They are likely to pick more noteworthy stocks that have received attention in the media. These stocks are likely to be more volatile stocks.
In general, if agency issues involved with delegated portfolio management contribute to the volatility effect, we would expect to find the volatility effect to strengthen over time along with the increase in institutional ownership.
3. Methodology
11
3.1. Portfolio construction
I create equally weighted (EW) quintile portfolios at the end of every month based on ranking stocks on their past 36-months volatility. The first quintile portfolio (henceforth the low volatility-sorted portfolio) consists of stocks with the lowest historic volatility while the last quintile portfolio (henceforth the high-volatility-sorted portfolio) consists of stocks with the highest historic volatility. I calculate the past volatility of each stock by the standard deviation of total monthly returns over the last 36 months. I only include stocks with at least 18 months of return data in portfolio formation to ensure a reliable estimate of the historic risk of a stock.
3.2. Risk and return measures
I calculate the return of each quintile portfolio by taking an average of the returns of the stocks in the quintile in the month following portfolio formation. I subtract the risk-free rate to calculate returns in excess of the risk-free rate. Thus, all regressions and calculations are based on simple excess returns.
I report the annualized geometric and arithmetic mean returns, annualized standard deviation (monthly standard deviation times √ , Sharpe ratio, CAPM beta, CAPM alpha, Fama-French alpha, four factor alpha and t-statistics for each quintile portfolio. All t-statistics are Newey-West adjusted to have heteroskedasticity and autocorrelation consistent standard errors (see Appendix A).
Arithmetic versus geometric returns: I report both annualized arithmetic and geometric mean
returns in excess of the risk-free rate. I calculate the simple or arithmetic return for stock i at month t,
Ri,t, by:
, (3)
where RIi,t is the total return index for stock i at month t. The arithmetic return of portfolio p in month t
in excess of the risk-free rate, Rp,t, is:
∑ , (4)
where Np is the number of stocks in portfolio p, and Rf,t is the risk-free rate at month t. The annualized
arithmetic return is the average monthly excess return multiplied by 12.
The annualized geometric excess return for portfolio p over months t=1 to t=T, GRp, is:
[∏ ] . (5)
12 implicitly assumes the total sample period is used to evaluate a certain type of strategy. In reality, different types of investors have different types of investment horizons, both short- and long-term. For this reason, both measures are important and included in reporting. All regressions are based on arithmetic returns.
Sharpe ratio: The Sharpe ratio of portfolio p, Sp, measures the ex-post excess return per unit of
risk:
( ̅ )
(√ ) , (6)
where ̅ is the average return of portfolio p in excess of the risk-free rate, and √ is the standard deviation of the excess returns of portfolio p.
I apply the Jobson and Korkie (1981) test with Memmel (2003) correction to test if there is a statistical significance difference between two Sharpe ratios. This test is as follows:
√ [ [ ] ]
, (7)
where Si is the Sharpe ratio of portfolio i, ρi,j is the correlation of the returns between portfolios i and j,
and T is the number of observations.
Alpha: I perform a number of regressions to test if the different portfolios or investment strategies
earn abnormal returns relative to the CAPM one-factor model, Fama-French three-factor model and four-factor model of Carhart. An investment strategy earns abnormal profits if the intercept in the regression is significantly different from zero.
I obtain the one-factor or Jensen’s alpha by regressing the excess returns of each quintile portfolio against the excess returns of the market by equation:
, (8)
where Rp,t is the return of portfolio p in period t, in excess of the risk-free return, αp is the alpha of
portfolio p, βp,MKT is the beta of portfolio p with respect to the market portfolio, RMKT,t is the return on
the market portfolio in excess of the risk-free return, and εp,t is the error term of portfolio p in period t.
13 alpha for the four factor model, I also add an up-minus-down (UMD) momentum factor. Including these factors results in the following regressions:
; (9)
, (10) where βp,SMB, βp,HML, βp,UMD are the betas of portfolio p with respect to the size, value, and momentum factors, and RSMB,t, RHML,t, RUMD,t are the return factors on the size, value and momentum effect. I
calculate the return factors by ranking stocks on ascending order of the scores and subtracting the return of the equally weighted bottom quintile portfolio from the top quintile portfolio. I only include stocks with positive book-to-market values for HML quintile portfolios. I calculate the UMD scores by calculating the past 12-minus-1 month returns for each stock. Therefore, I only include stocks with past 12 months return data for UMD quintile portfolios. In addition to creating the size, value, and momentum return factors myself, I re-run equations (8), (9), and (10) with the Europe factors from the website of Kenneth French.2
For the double-sorting methodology, I first sort stocks into five groups on ascending order of size, value, or momentum. Then, I sort each of these five groups into five sub-groups on ascending order of their past three-year monthly volatility. Finally, I merge the 25 sub-groups into five quintile portfolios by combining the five lowest volatility sub-groups, the five next lowest volatility sub-groups, etc. I regress the excess returns of the double-sorted quintiles against the excess returns of the market using regressions (8), (9), and (10). By using a double-sorting methodology, I control for implicit loadings of size, value, or momentum factors ex-ante.
I also test the hypothesis that all five alphas of the quintile portfolios are jointly equal to zero with the Gibbons, Ross, and Shanken (GRS) (1989) test. I report the relevant p-values for rejecting this hypothesis.
3.3. Robustness tests
I perform a number of robustness tests to test the robustness of the volatility effect and to control for other effects like the beta effect and short-term mean reversion. First, I compare the volatility effect with the beta effect. Since volatility and beta are related risk measures, the volatility effect could just capture the long-standing critique of the CAPM that the empirical relation between average return and risk is flat (see, e.g., Black et al., 1972; Fama and French, 1992). I compare the volatility effect with the beta effect by both creating quintile portfolios based on sorting stocks on their market beta and using a double-sorting methodology. I create the beta-sorted quintile portfolios following the same methodology of creating the volatility-sorted quintile portfolios. I calculate the beta of each stock by
2 I use http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html for the collection of MKT, SMB, HML, and UMD return
14 dividing the covariance of the stock’s returns with the market’s return by the variance of the market’s returns based on monthly return data. To ensure a reliable estimate of the historic risk of a stock, I only include stocks with at least half of the 36 months of return data in portfolio formation. I calculate the betas of each stock by using the FTSE 100 as a proxy for the market portfolio. The double-sorting methodology is similar to the methodology described in Section 3.2.
Second, I test the effects of different L/M/N strategies similar to Ang, Hodrick, Xing, and Zhang (2006) and Huang, Liu, Rhee and Zhang (2010). This strategy can be described by an L-month initial formation period, an M-month waiting period, and an N-month holding period. Specifically, I form portfolios based on total volatility at month t over an L-month period from the end of month t – L –M to the end of month t - M. I hold these portfolios from month t to month t + N for N months. For the main results, I follow a 36/0/1 strategy. That is, I use the past 36 months to form portfolios, and calculate the return of each portfolio directly in the month following portfolio formation. To test the robustness of the volatility effect for different formation periods, I also use a 12/0/1 and a 24/0/1 strategy. By skipping M months, I can control for possible short-term mean reversals or microstructure noises (Huang, Liu, Rhee and Zhang, 2010). Therefore, I also test the effect of skipping a few months by using a 36/1/1, 36/2/1, and a 36/3/1 strategy. At last, I test the effects of different holding periods. Instead of calculating the one-month returns, I calculate the 6, 12, 24, and 36 months returns after portfolio formation. I construct overlapping portfolios to increase the power of the test following the methodology of Jegadeesh and Titman (1993, 2001), Ang, Hodrick, Xing, and Zhang (2006) and Blitz, Pang, and van Vliet (2012). Following this methodology, the return of a quintile portfolio in a given month is an equally weighted average of the returns of N past quintile portfolios, where N is the holding period in months. Therefore, a portfolio changes 1/N part of its composition each month. If the volatility effect is the result of short-term return reversals, the alphas should converge to zero for longer holding periods.
Third, I look at different types of sub-sample results. I do this for multiple reasons:
- The volatility effect may have strengthened over time. A strengthening over time would support the hypothesis that agency issues and delegated portfolio management contribute to the volatility effect since institutional ownership has increased over time.
- The volatility effect may be more apparent in certain type of market trends. If the volatility effect is mainly present in bear markets this would support the hypothesis that disagreement with short-sales restrictions contribute to the volatility effect as argued by Hong and Sraer (2012).
- The factor exposures (market beta, size, value, and momentum) in the regressions may not be constant over time.
15 period into five unequal sub-periods by determining the ex-post turning points of the FTSE 100. At last, I compare the volatility effect for months in which the return of the FTSE 100 in excess of the risk-free rate was positive and for months in which it was negative. I use the same methodology as described above to determine the one-factor, three-factor, and four-factor alphas.
4. Data
This section is organized as follows: Section 4.1 discusses the total data sample. Section 4.2 describes the independent variables and section 4.3 describes the dependent variables in equations (6) to (8).
4.1. Data sample
The data sample consists of all common stocks listed on the FTSE 100 from the moment Thomson Datastream has available constituents list (January 1996) until December 2012. The FTSE 100 is a highly tradable index, designed to represent the performance of the hundred most highly capitalized blue chip companies of the U.K. market. By choosing all constituents listed on this index, I only include relatively liquid large-cap stocks in the data sample. This choice ensures that the results are not driven by illiquid small-cap stocks.
I identify all constituents of the FTSE 100 at the end of each month in the sample period. Identifying all stocks that were actually listed in a particular month greatly reduces survivorship bias. For all constituents, I gather the total return series, market capitalization, and book-to-market values from Thomson Datastream. The sample includes a total number of included stocks of 249 (96 dead stocks). The sample consists of a minimum of 97 and a maximum of 102 active stocks included in the FTSE 100 for a particular month. The median number of active stocks included in the sample is 101 stocks.
4.2. Independent variables
This paragraph describes the independent variables are the excess return on the market, the size factor, the value factor, and the momentum factor. Appendix B reports the correlations between the different return factors.
Market returns: I take the FTSE 100 as a proxy for the market index. This index is a value-weighted (VW) index. The average monthly excess return on the market in the sample period, as reported in Table 2 Panel B column eight, is 0.3%. The return distribution is negatively skewed (-0.615) and the Jarque-Bera test rejects normality.
Risk-free rate: All calculations are based on simple returns in excess of the risk-free rate. A proxy
16 calculate the return of each portfolio in the month following portfolio formation, I require a risk-free proxy with a one-month period. The one-month U.K. Treasury Bill Tender rate would be most suited as it is line with the approach of other authors (see e.g., Fama and French, 1993). However, due to limited availability of these rates, I use the one-month GBP London-Interbank Offered Rate (LIBOR) as a proxy of the risk-free rate. Although this rate is a good proxy of the risk-free rate, it is likely to be higher than the one-month U.K. Treasury Bill rate, especially during a financial crisis because it exhibits some default risk. This choice will only have a minor impact on the results. I take the total return index of the one-month GBP LIBOR from Thomson Datastream.
Size factor returns: The average monthly return of size (SMB) factor in the sample period, as
reported in column nine of Table 2 Panel B, is -0.2%. The under-performance of small firms in the sample, as reported in Table C1 Panel A of Appendix C, causes this negative return difference. This positive relation between size and return contradicts the findings of other authors (see e.g., Fama and French, 1993) who document a negative relation between size and average return. However, the dataset consists of large-cap stocks only and the return difference is small.
Table 2
Summery statistics of quintile portfolios sorted on past three-year monthly volatility
Panel A presents the ex-ante summery statistics of the volatility-sorted quintile portfolios. I construct all statistics by averaging 204 observations. The first row shows the mean average monthly volatility within the quintile portfolios. The second row shows the mean average market capitalization within the quintile portfolios. The third row shows the mean average book-to-market value within the quintile portfolios. I exclude negative market-to-book values since they heavily distort the average values. The last row shows the mean average 12-minus-1 month returns within the quintile portfolios.
Panel B presents the ex-post summery statistics of the monthly excess returns of the different types of portfolios.
Panel A: Ex ante average loadings of quintile portfolios sorted on past three-year monthly volatility
Low Q2 Q3 Q4 High
Average monthly volatility 5.4% 6.7% 7.9% 9.4% 12.9% Average size (million GBP) 17,903 14,762 11,101 8,954 8,209 Average value (book-to-market) 0.44 0.46 0.50 0.50 0.61 Average momentum 10.0% 11.6% 11.3% 11.9% 13.3%
Panel B: Summery statistics of dependent and independent variables (monthly observations)
Dependent variable Independent variable Name Low Q2 Q3 Q4 High High-low Market SMB HML UMD Symbol R1 R2 R3 R4 R5 R5-1 RMKT RSMB RHML RUMD Description Low volatility quintile (EW) High volatility quintile (EW) High-minus-low volatility portfolio (EW) Market factor (in excess of risk-free rate) Size factor Value factor Momentu m factor
Source Own calculation Own calculation
17 The mean average size, as reported in row three of Table 2 Panel A, reduces gradually across the volatility-sorted quintile portfolios. The low volatility-sorted portfolio consists of larger stocks than the high volatility-sorted portfolio. The high volatility-sorted portfolio has a significant SMB beta of 0.60 to 0.87, as reported in Table D1 Panel B in Appendix D. Thus, in this sample, the high volatility-sorted portfolio consists of more small firms which under-performed compared to large firms.
Value factor returns: The average monthly return of value (HML) factor in the sample period, as
reported in column ten of Table 2 Panel B, is 0.5%. Thus, value stocks have higher returns than growth stocks in the sample period as reported in Table C1 Panel B in Appendix C. Previous studies have also documented a positive relation between book-to-market ratios and average return (see e.g., Fama and French, 1993).
Table 2 Panel A shows that the high volatility-sorted portfolio consists, on average, of more value stocks. The average book-to-market values for the low and high volatility-sorted portfolios are 0.44 and 0.61 respectively. However, Table D1 in Appendix D shows that neither the low nor the high volatility-sorted portfolio has significant HML betas. Thus, there is no clear value effect apparent for volatility-sorted portfolios.
Momentum factor returns: The average monthly return of momentum (UMD) factor in the sample
period, as reported in column 11 of Table 2 Panel B, is 0.7%. The under-performance of losers stocks, as reported in Table C1 Panel C in Appendix C, causes this return difference.
The bottom rows of Table 2 Panel A shows no clear difference in ex-ante loading for the momentum factor. Although the mean average momentum is larger for high volatility stocks than for low-volatility stocks (13.3% and 10.0% respectively), this difference is small. Although not reported, the standard deviation of the average momentum of the high volatility-sorted portfolio is large. Therefore, the high volatility portfolio is likely to consist of both past winners and past losers. The last column of Table D1 in Appendix D shows that neither the low nor the high volatility-sorted portfolio has significant UMD betas. Thus, there is no clear momentum effect apparant for volatility-sorted portfolios.
4.3. Dependent variables
18
Fig. 1. Number of stocks over time. This figure plots the number of active constituents and the number of stocks included in
creating quintile portfolios based on past three-year monthly volatility for the January 31, 1996 to December 31, 2012 sample period.
Table 2 Panel B shows the summery statistics of the dependent variables. The mean monthly excess return varies between 0.2% and 0.6% for the different EW volatility-sorted quintile portfolios. For all quintile portfolios the return distributions are negatively skewed (-0.316 to -0.868) and the Jarque-Bera test rejects normality. Thus, all portfolios have a larger negative tail in their return distribution.
5. Results
This part presents the results of this paper. It is organized as follows: Section 5.1 discusses the full sample results in which I control for the classic size, value, and momentum effects. Section 5.2 compares the volatility effect with the beta effect. Section 5.3 reports the results for different formation and holding strategies. Section 5.4 reports the sub-sample results to determine how the volatility effect changes over time and how it behaves in different markets trends.
5.1. Full sample results
This section presents the full sample results of this paper. I present the main results by comparing the ex-post return characteristics of the different quintile portfolios sorted on past three-year monthly volatility. In these results, I control for size, value and momentum effects by a regression-based methodology and a double-sorted methodology.
Table 3 shows the out-of-sample performance of different quintile portfolios sorted on past three-year monthly volatility.3 The first two rows show the geometric and arithmetic mean excess returns for
3Appendix E shows similar results as Table 3 Panel A using the factor returns from the website of Kenneth French.
80 85 90 95 100 105 N u m b e r o f sto cks
Month of portfolio formation
19 the different quintile portfolios and the FTSE 100. The returns across the volatility-sorted portfolios show a kind of inverted U-shape where returns first increase and then decrease with volatility. The low volatility-sorted portfolio has a geometric and arithmetic mean excess return of 4.7% and 5.3% respectively. The high volatility-sorted portfolio has a geometric and arithmetic mean excess return of -1.8% and 2.4% respectively. A strategy that goes long in the high volatility-sorted portfolio and short in the low volatility-sorted portfolio has an annualized geometric and arithmetic mean excess return of -6.1% and -2.9%, respectively. These results indicate that the risk-return relation is not linear as predicted by CAPM. However, the empirical risk-return relation can take on any shape, as reported in Appendix F, based on a 95% confidence level of the mean returns of the quintile portfolios. Because the returns of the portfolios are very volatile, it is difficult to derive an unambiguous relation between risk and return.
In contrary to the mean returns, the ex-post standard deviations are increasing with ascending order of the quintiles. Thus, historic volatility is a good approximation of future volatility. The low sorted portfolio is about three-fourth as volatile as the market, while the high volatility-sorted portfolio is almost twice as volatile.
If we look at a risk-adjusted performance perspective, the results become more interesting. Low-risk stocks clearly outperform high-Low-risk stocks as measured by the Sharpe ratio. The returns across the volatility-sorted portfolios are flat or decreasing and the standard deviations are increasing, leading to decreasing Sharpe ratios across the portfolios. The Sharpe ratios for the low volatility-sorted portfolio and second to lowest volatility-sorted portfolio are significantly higher than the market, while the high volatility-sorted portfolio has a significant lower Sharpe ratio than the market (p<0.01). Fig. 2 Panel A shows this relation between volatility and return for the different portfolios graphically.
The signs of the alphas, as reported in Table 3 Panel A, indicate that low-risk stocks outperform and high-risk stocks under-perform compared to the CAPM. However, the evidence is rather weak because of the large standard errors in calculating the alphas. The market betas and CAPM alphas show an increasing beta and a decreasing alpha across the volatility-sorted portfolios. Fig. 2 Panel B displays the relation between beta and return for the different portfolios. Similar to the findings of Blitz and Van Vliet (2007), the theoretical (CAPM) security market appears to be violated. The low volatility-sorted portfolio has a low beta of 0.60 and a marginally significant positive alpha of 3.3% (p<0.10). At the other end, the high volatility-sorted portfolio has a high beta of 1.61 and a negative, but not statistically significant alpha of -2.8%. The high-minus-low volatility-sorted portfolio shows a statistically insignificant alpha of -6.1%.4 The hypothesis that all five alphas are jointly equal to zero cannot be rejected based on the one-factor model. Thus, although CAPM appears to be violated in Fig. 2 Panel B, we cannot reject the CAPM.
4 A sharp-eyed reader might notice that on average the alpha of the different quintile portfolios is larger than zero. The quintiles are equally
20
Table 3
Main results: Out-of-sample performance of past three-year volatility-sorted quintile portfolios.
At the end of each month for the January 31, 1996 to December 31, 2012 sample period, I sort all constituents into quintile portfolios. I sort these quintile portfolios on past three-year monthly return volatility on ascending order. I calculate the return of each equally weighted portfolio in excess of the risk-free rate in the month following portfolio formation, where I use the one-month GBP LIBOR as a proxy for the risk-free rate.
Panel A reports the annualized geometric and arithmetic mean excess return, standard deviation, Sharpe ratio, market beta, CAPM alpha, FF alpha, four-factor alpha and Newey-West adjusted t-statistics. I take the FTSE 100 index as a proxy for the market index. The four-factor model regression is given below. The three-factor and one-factor model are obtained by excluding the UMD, and both the UMD, HML, SMB factors respectively.
.
Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***. The last column reports the Gibbons, Ross, and Shanken test for a joint test of all five alphas equal to zero.
Panel B reports additional risk characteristics of the monthly excess returns of the different portfolios. The first row shows the maximum monthly drawdown. The second and third rows show the average return of the portfolios for up months versus down months, where I define an up month as a month in which the FTSE 100 has a positive return in excess of the risk-free rate.
Low Q2 Q3 Q4 High High-low Market GRS-test
Panel A: Excess returns of volatility-sorted quintile portfolios (annualized)
Geometric mean 4.7% 6.1% 3.7% 3.8% -1.8% -6.1% 2.2% Arithmetic mean 5.3% 6.9% 4.9% 5.6% 2.4% -2.9% 3.2% Standard deviation 11.4% 13.8% 16.1% 19.3% 28.9% 25.7% 14.5% Sharpe 0.46 0.50 0.30 0.29 0.08 -0.11 0.22 (t-value) (4.74)*** (6.86)*** (2.37)** (1.98)** (-3.19)*** (-5.02)*** - Market beta 0.60 0.82 0.97 1.17 1.61 1.01 1.00 CAPM alpha 3.3% 4.2% 1.8% 1.8% -2.8% -6.1% - p = 0.24 (t-value) (1.86)* (2.24)** (0.82) (0.88) (-0.66) (-1.16) - FF alpha 3.0% 4.1% 1.6% 2.5% -0.7% -3.7% - p = 0.09 (t-value) (1.76)* (2.39)** (1.00) (1.41) (-0.18) (-0.75) - 4-factor alpha 2.8% 3.5% 1.9% 2.7% 0.2% -2.6% - p = 0.13 (t-value) (1.49) (2.06)** (1.13) (1.49) (0.04) (-0.50) -
Panel B: Risk analysis of volatility-sorted quintile portfolios (monthly returns)
Maximum drawdown -11.2% -13.5% -17.4% -20.0% -29.5% -29.3% -13.4% Return up market 2.1% 2.9% 3.2% 3.8% 5.1% 3.0% 3.2% Return down market -1.6% -2.2% -3.0% -3.6% -5.8% -4.2% -3.3%
21
Fig. 2. Empirical versus theoretical relation between risk and return. At the end of each month, for the January 31, 1996 to
December 31, 2012 sample period, I sort all constituents of the FTSE 100 into quintile portfolios. I sort these quintile portfolios on past three-year monthly return volatility on ascending order. I calculate the return of each equally weighted portfolio in excess of the risk-free rate in the month following portfolio formation, where I use the one-month GBP LIBOR as a proxy for the risk-free rate. Panel A shows the annualized arithmetic mean excess returns against the annualized standard deviation for the FTSE 100, and the five quintile portfolios. Panel B shows the annualized arithmetic mean excess returns against the CAPM beta for the FTSE 100, and the five quintile portfolios.
Table 3 Panel B shows additional return characteristics of the different quintile portfolios. The maximum monthly drawdown for the low and high volatility-sorted portfolio is -11.2% and -29.5% respectively. Thus, the low volatility-sorted portfolio is also less risky in this regard.
In line with their low beta, low volatility-sorted portfolios under-perform the market during up months and outperform the market during down months in the total sample period. During up months the low volatility-sorted portfolio under-perform the market by an average of -1.1% (= 2.1% - 3.2%) This is offset by an outperformance of 1.7% (= -1.6% + 3.3%) during down markets. The high volatility-sorted portfolio shows the opposite behavior and outperforms the market during up markets by 1.9% (= 5.1% - 3.2%), however this is clearly offset by an average under-performance during down Q1 Q2 Q3 Q4 Q5 Market Theoretical relation (Capital Market Line)
0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% E x ce ss r etu rn ( an n u alize d ) Volatility (annualized)
Low Sharpe ratio of high volatility stocks High Sharpe ratio of low
volatility stocks
A
Q1 Q2 Q3 Q4 Q5 Market Empirical relation Theoretical relation (Security Market Line)0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 E x ce ss r etu rn ( an n u alize d ) CAPM Beta
Positive alpha in low volatility stocks
Negative alpha in high volatility stocks
22
Table 4
Out-of-sample performance of double-sorted portfolios.
This table reports the results of a double sorting methodology. In the double-sorting methodology, I first sort stocks into five groups on ascending order of the size, value, or momentum scores. Then I sort each of these five groups into five sub-groups on ascending order of their past three-year monthly volatility. Finally, I merge the 25 sub-groups into five quintile portfolios by combining the five lowest volatility sub-groups, the five next lowest volatility sub-groups, etc. I calculate the return of each quintile portfolio in the month following portfolio formation. I report the annualized arithmetic mean excess return for all portfolios. For the equally weighted high-minus-low quintile portfolio I report the CAPM alpha, FF alpha, four-factor alpha and Newey-West adjusted t-statistics. I take the FTSE 100 index as a proxy for the market index. The four-factor model regression is given below. The three-factor and one-factor model are obtained by excluding the UMD, and both the UMD, HML, SMB factors respectively. For comparison, I report the results of the volatility sorted portfolios in the last row.
Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***.
Low Q2 Q3 Q4 High High-low
Sorting method Arithmetic mean return Mean return
CAPM
alpha (t-value) FF alpha (t-value) 4-factor alpha (t-value) Size/vol. 5.9% 5.5% 5.4% 6.2% 2.4% -3.5% -6.4% (-1.32) -4.6% (-0.97) -3.2% (-0.65) Value/vol. 4.9% 7.8% 3.7% 5.8% 2.3% -2.5% -5.3% (-1.33) -3.4% (-0.89) -2.0% (-0.49) Momentum/vol. 5.7% 5.0% 5.4% 5.2% 3.1% -2.5% -4.7% (-1.33) -3.0% (-0.90) -2.3% (-0.64) Volatility 5.3% 6.9% 4.9% 5.6% 2.4% -2.9% -6.1% (-1.16) -3.7% (-0.75) -2.6% (-0.50)
markets of -2.5% (= -3.3 + 5.8%). A strategy that goes long in the 20% highest volatility stocks and short in the 20% lowest volatility stocks, under-performs the market by -0.2% (= 3.0% - 3.2%) during up months and -0.9% (= -4.2% + 3.3%) during down months. In the sample 54.9% were up months, where I define an up month as a month with a positive return of the FTSE 100 in excess of the risk-free rate.
Double-sorted results: controlling for size, value, and momentum factors
I also use a double-sorting methodology to neutralize the volatility for the classic size, value, and momentum factors ex-ante. Table 4 shows the results of a double sort on the classic factors and volatility.
Table 4 shows that a double sort on the classic factors and volatility has only minor impact on returns of the different quintile portfolios. The mean return and alphas of the high-minus-low volatility-sorted portfolio remain negative for all sorting methods. Thus, although none of the alphas are statistically significant, none of the double sorting methodologies can completely eliminate the negative alpha for the high-minus-low volatility portfolio. Therefore, none of the size, value or momentum effects can completely explain the volatility effect by itself.
5.2. Comparison with the beta effect
23
Table 5
Comparison out-of-sample performance of volatility- and beta-sorted portfolios.
The first two rows show the results of the volatility-sorted and beta-sorted strategies. At the end of each month, for the January 31, 1996 to December 31, 2012 sample period, I sort all constituents into quintile portfolios. I sort these quintile portfolios on ascending order of their past three-year monthly return volatility (V 36/0/1) and three-year beta (B 36/0/1). I calculate the return of each equally weighted portfolio in excess of the risk-free rate in the month following portfolio formation, where I take the one-month GBP LIBOR as a proxy for the risk-free rate. The last row reports the results of a double sorting methodology. In the double-sorting methodology, I first sort stocks into five groups on ascending order of their local beta. Then I sort each of these five groups into five sub-groups on ascending order of their past three-year monthly volatility. Finally, I merge the 25 sub-groups into five quintile portfolios by combining the five lowest volatility sub-groups, the five next lowest volatility sub-groups, etc. I report the annualized arithmetic mean excess return for all portfolios. For the equally weighted high-minus-low quintile portfolio, I report the CAPM alpha, FF alpha, four-factor alpha and Newey-West adjusted t-statistics. I take the FTSE 100 index as a proxy for the market index and to calculate the past beta of each stock. The four-factor regression is given below. The three-factor and one-factor model are obtained by excluding the UMD, and both the UMD, HML, SMB factors respectively.
Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***.
Low Q2 Q3 Q4 High High-low
Sorting
method Arithmetic mean return
Mean return
CAPM
alpha (t-value) FF alpha (t-value) 4-factor
alpha (t-value) V 36/0/1 5.3% 6.9% 4.9% 5.6% 2.4% -2.9% -6.1% (-1.16) -3.7% (-0.75) -2.6% (-0.50) B 36/0/1 5.6% 6.2% 4.0% 5.0% 4.3% -1.3% -4.5% (-0.92) -4.2% (-0.91) -0.8% (-0.18) Double B/V 6.7% 6.1% 3.9% 3.9% 4.5% -2.1% -3.7% (-0.81) -1.0% (-0.24) -1.1% (-0.26)
Table 5 shows the return characteristics of the different strategies. I find weak evidence of a beta effect with relative flat returns across the beta-sorted portfolios. The high-minus-low beta-sorted portfolio shows a negative, but statistically insignificant CAPM alpha of -4.5%. The return characteristics of the volatility-sorted and beta-sorted portfolios are very similar. Only the high volatility-sorted portfolio has a 1.9% (= 4.3% - 2.4%) lower average excess return than the high beta-sorted portfolio. The annualized arithmetic return difference between the high and low volatility-beta-sorted portfolios is -2.9%, while the return difference between the high and low beta-sorted portfolios is -1.3%. All the signs of the return differences and alphas indicate that low-risk stocks outperform high-risk stocks. However, none of these measures is statistically significant.
The double-sorting methodology reduces the negative CAPM alpha for the high-minus-low quintile portfolio from -6.1% to -3.7% with a return of -2.1%. Thus, sorting stocks on volatility within groups of stocks with similar betas helps to capture abnormal returns relative to the CAPM. Although both risk-measures are related, the under-performance of high-risk stocks appears to be stronger for volatility-sorted stocks. These results indicate that both the systematic and the idiosyncratic part of total volatility are mispriced.
5.3. Different L/M/N strategies
24
Table 6
Out of sample results for different L/M/N strategies.
This table reports the return characteristics for different L/M/N strategies. At the end of each month, for the January 31, 1996 to December 31, 2012 sample period, I sort all constituents of the FTSE 100 on ascending order into equally weighted quintile portfolios using an L/M/N strategy as described in Section 3.3. At month t, I calculate the total monthly volatility over an L month period from months t – L – M to month t – M. I hold these portfolios for N months, following the methodology of Jegadeesh and Titman (1993). I calculate the return of each portfolio in excess of the risk-free rate, where I use the one-month GBP LIBOR as a proxy for the risk-free rate. I report the annualized arithmetic mean excess return for all portfolios. For the equally weighted high-minus-low quintile portfolio I report the CAPM alpha, FF alpha, four-factor alpha and Newey-West adjusted t-statistics. I take the FTSE 100 index as a proxy for the market index. The four-factor regression is given below. The three-factor and one-factor model are obtained by excluding the UMD, and both the UMD, HML, SMB factors respectively.
Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***.
Low Q2 Q3 Q4 High High-low
L/M/N Arithmetic mean return Mean return CAPM alpha (t-value) FF alpha (t-value) 4-factor alpha (t-value)
V 12/0/1 5.7% 6.7% 7.0% 5.4% -1.3% -7.0% -5.7% (-1.04) -7.5% (-1.53) -5.8% (-1.13) V 24/0/1 5.1% 5.7% 7.2% 5.1% 1.4% -3.7% -3.8% (-0.73) -4.1% (-0.84) -2.8% (-0.55) V 36/0/1 5.3% 6.9% 4.9% 5.6% 2.4% -2.9% -6.1% (-1.16) -3.7% (-0.75) -2.6% (-0.50) V 36/1/1 5.9% 6.2% 5.7% 4.6% 5.3% -0.7% -4.0% (-0.71) -1.0% (-0.19) 0.9% (0.15) V 36/2/1 7.2% 5.4% 5.0% 6.0% 5.9% -1.3% -4.5% (-0.85) -1.5% (-0.29) 0.2% (0.04) V 36/3/1 7.5% 4.5% 4.7% 5.0% 5.4% -2.1% -5.2% (-0.95) -2.1% (-0.40) -0.5% (-0.09) V 36/0/6 6.4% 6.1% 4.5% 5.5% 4.7% -1.7% -5.0% (-0.92) -1.8% (-0.36) -0.2% (-0.03) V 36/0/12 6.3% 5.7% 3.9% 5.0% 3.3% -3.0% -5.7% (-1.04) -2.6% (-0.50) -1.0% (-0.19) V 36/0/24 5.6% 4.6% 3.8% 4.6% 3.3% -2.3% -3.8% (-0.73) -1.8% (-0.34) -0.4% (-0.08) V 36/0/36 5.9% 4.3% 4.3% 4.9% 3.1% -2.8% -3.7% (-0.76) -2.4% (-0.49) -1.2% (-0.24)
The trend of the volatility effect, shown as the returns of the different quintile portfolios in Table 6, is present for all different formation, waiting and holding periods. The return of the high-minus-low volatility-sorted portfolio remains negative. More specifically, the risk-return relation is consistently negative or shows an inverted U-shape. Again, I stress that these return differences are not statistically significant. None of the one-factor, three-factor, or four-factor alphas is statistically significant. The CAPM alpha of the high-minus-low quintile portfolio ranges between-3.7% and -6.1%. After controlling for size, value, and momentum most alphas reduce close to zero, with the exception of the V 12/0/1 strategy. Thus, these classic factors appear to do well in explaining the volatility effect over the total sample period.
For all different formation periods, as reported in Table 6, the return spread between the high and low volatility-sorted portfolios is negative. The largest annualized return difference (-7.0%) is measured for a 12/0/1 strategy, that is, for the one-year monthly volatility-sorted portfolios in which the return is calculated in the month following portfolio formation. The strong under-performance of the high volatility-sorted portfolio causes this return difference. This under-performance of the high volatility-sorted portfolio is less pronounced for longer formation periods.
25 portfolio declines from -2.9% to -0.7%. This decline in return difference is caused by less under-performance of high volatility-sorted portfolios. However, the return spread increases again to -1.3% and -2.1% for a 36/2/1 and 36/3/1 strategy respectively. The CAPM alpha of the high-minus-low quintile portfolio remains negative (-4.0% to -5.2%). There is no steady reduction in the CAPM alpha when months are skipped between formation and return calculation period. Therefore, the return differences are not driven by possible short-term mean reversals or microstructure noises.
The volatility effect is not only present in the short run. The annualized return difference between the high and low volatility quintiles remains between -1.7% and -3.0% with no real trend. All CAPM alphas remain negative between -3.7% and -5.7%. For longer holding periods, the CAPM alpha reduces slowly. Thus, low-risk stocks continue to outperform high-risk stocks even for longer holding periods.
5.4. Sub-sample results
This section looks at the sub-sample results. I explore how the volatility effect changes over time, how it is present in bull versus bear markets and I compare the effects for months in which the market went up and months in which the market went down.
Fig. 3 displays the compounded excess return for the low volatility portfolio, the high volatility portfolio, and the FTSE 100 for the total sample period. The low volatility portfolio has more constant returns than the high volatility portfolio for the total sample period. The low volatility portfolio did not dominate the high volatility portfolio from the start of the sample period. In the first part of the sample, the high volatility-sorted portfolio has higher returns than the low-volatility-sorted portfolio. To analyze the volatility effect over time, I divide the total sample period into two equal periods of 8.5 years or 102 months.
Furthermore, Fig. 3 shows the different market trends and the behavior of the different portfolios for these market trends. The burst of the technology bubble and the latest credit crunch are clearly distinguishable in the figure. There are five periods with different market trends visible (bull, bear, bull, bear, bull). I divide the total sample period into five unequal sub-periods based on the trend of the market to analyze the volatility effect during bull and bear markets.
26 Fig. 3. Compounded excess returns. At the end of each month, for the January 31, 1996 to December 31, 2012 sample period, I sort all constituents of the FTSE 100 into quintile portfolios. I sort these quintile portfolios on past three-year monthly return volatility on ascending order. I calculate the return of each portfolio in excess of the risk-free rate in the month following portfolio formation, where I take the one-month GBP LIBOR as a proxy for the risk-free rate. This figure shows the geometric excess returns for the FTSE 100, the low and the high volatility-sorted quintile portfolios.
Equal sub-samples
In Table 7, I explore the volatility effect over time by looking at the return characteristics of volatility-sorted portfolios in the first and second half of the total sample (i.e., January 1996 to June 2004 and July 2004 to December 2012). Simply splitting the sample period into two equal samples avoids data mining and ensures enough observations. Both sub-samples consist of 102 monthly observations. The first period contains the burst of the technology bubble (2000 to 2002), while the second period contains the credit crisis (from 2008 onwards).
In the first half of the sample, as reported in Table 7 Panel A, the risk return relation appears to be flat with the exception of considerable under-performance of the high volatility-sorted portfolio. The high volatility-sorted portfolio has large negative returns during the burst of the technology bubble. This exceptionally bad performance causes an annualized arithmetic return for the high-minus-low volatility-sorted portfolio of -7.9%. In the second half of the sample, the risk return relation also appears to be rather flat. However, the high volatility-sorted portfolio has a higher mean return than the low volatility-sorted portfolio. The annualized arithmetic mean return of the high-minus-low volatility-sorted portfolio for the second period, as reported in Table 7 Panel B, is therefore a positive 2.1%.
The CAPM alpha for the high-minus-low volatility-sorted portfolio is negative in both sub-samples (-8.6% in the first half and -3.2% in the second half). In both sub-sub-samples, the alphas of the different quintile portfolios do not consistently reduce to zero after controlling for size, value, and momentum factors. Therefore, in these sub-samples, the classic size, value, and momentum factors appear not to contribute in explaining the returns of the different quintile portfolios.
-100% -50% 0% 50% 100% 150% Co m p o u n d e d e xcc e ss r e tu rn
27 Based on the return and CAPM alpha of the different portfolios, the volatility effect appears to be stronger in the first part of the sample. Therefore, I conclude that the volatility effect has not strengthened over time in the sample period. These findings are not in line with the hypothesis that agency issues involved with delegated portfolio management contribute to the volatility effect, since institutional ownership has increased over time.
Table 7
Out-of-sample performance of past three-year volatility sorted quintile portfolios for equal sub-samples.
At the end of each month, for the January 31, 1996 to December 31, 2012 sample period, I sort all constituents into quintile portfolios. I sort these quintile portfolios on past three-year monthly return volatility on ascending order. I calculate the return of each equally weighted portfolio in excess of the risk-free rate in the month following portfolio formation, where I use the one-month GBP LIBOR as a proxy for the risk-free rate. Panel A shows the results of the January 1996 to June 2004 sample period. Panel B shows the results of the July 2004 to December 2012 sample period. I report the annualized geometric and arithmetic mean excess return, standard deviation, Sharpe ratio, market Beta, CAPM alpha, FF alpha, four-factor alpha and Newey-West adjusted t-statistics. I take the FTSE 100 index as a proxy for the market index. The four-factor model regression is given below. The three-factor and one-factor model are obtained by excluding the UMD, and both the UMD, HML, SMB factors respectively.
Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***. The last column reports the Gibbons, Ross, and Shanken test for a joint test of all five alphas equal to zero.
Low Q2 Q3 Q4 High High-low Market GRS-test
Panel A: Period 1 (January 1996 – June 2004)
Geometric mean 2.0% 3.9% 1.4% 2.8% -9.9% -11.6% -0.5% Arithmetic mean 2.7% 4.8% 2.7% 4.7% -5.1% -7.9% 0.6% Standard deviation 12.0% 14.0% 16.0% 19.4% 32.0% 29.4% 14.8% Sharpe 0.23 0.34 0.17 0.24 -0.16 -0.27 0.04 (t-value) (2.50)** (4.75)*** (2.45)** (4.06)*** (-3.09)*** (-3.24)*** - Beta 0.60 0.77 0.94 1.16 1.70 1.10 1.00 CAPM alpha 2.3% 4.3% 2.1% 4.0% -6.2% -8.6% - p = 0.36 (t-value) (0.82) (1.43) (0.62) (1.61) (-0.97) (-1.07) - FF alpha 0.3% 3.0% 1.4% 5.4% 0.2% -0.2% - p = 0.35 (t-value) (0.15) (1.21) (0.56) (2.44)** (0.03) (-0.03) - 4-factor alpha -0.1% 2.1% 2.0% 5.9% -0.3% -0.1% - p = 0.35 (t-value) (-0.06) (0.81) (0.72) (2.47)** (-0.07) (-0.02) -
Panel B: Period 2 (July 2004 – December 2012)
