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Low-Volatility Anomaly
Low-Volatility Anomaly
Author: Osama Abdeltawab
9490 Alvierweg 16 [email protected]
130212
Seminar paper
University of Liechtenstein
Graduate School
Programme: Master in Banking and Financial Management
Course: Advanced Research Methods in Finance
Module: Empirical Finance
Assessor: Dr. Georg Peter
Supervisor: Prof. Dr. Marco J. Menichetti
Working period: 27.09.2013 to 31.12.2013
Date of submission: 31.12.2013
Low-Volatility Anomaly
2
TABLE OF CONTENTS
1 INTRODUCTION 3
2 A LITERATURE REVIEW 4
3 DATA AND METHODOLOGY 5
4 EMPIRICAL RESULTS 6
5 CONCLUSION 8
REFERENCE LIST 9
Low-Volatility Anomaly
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1 Introduction and Motivation
Some theoretical models explain the relation between risk and return, such as the Capital Asset Pricing
Model (CAPM) which proves that the expected return of a security or a portfolio depends on the
risk-free rate and risk premium of the asset. Additionally, Modern Portfolio Theory (MPT) is the
theory that seeks to maximize the expected return of the portfolio for a given risk. Consequently, the
high-volatility is connected with higher future return and the low-volatility is connected with lower
future return.1
According to some analysts, investors choose the high volatility portfolio with high beta to achieve a
high expected return. However, some investors choose the low volatility portfolio with low beta to
achieve a higher risk–adjusted return. The main reason of this behavior is that the investor cannot
achieve a high expected return by investing in high volatility stocks, so the investor tends to
trade in low-volatility stocks. Generally risk-averse investors who have a long-term investment
horizon prefer to choose a low- risk strategy for acceptable performance levels.2
In this paper, we seek to understand the behavior for low-volatility anomaly by examining the
behavior of the investors that choose a high volatility portfolio with high beta and ignore the portfolio
with lower risk. An irrational preference for high volatility portfolio is the main reason for investors to
trade in high volatility stocks or risky portfolios due to the representativeness bias and over-confidence
bias.3 High volatility stocks are costly to trade in both position buy and short especially for stocks of
small firms, because the borrowing cost is very high and the number of shares that are available for
borrowing are not enough for all investors.4 The relation between the future return and risk is a
positive relationship but with the concept of the Low volatility anomaly, the relation between risk and
return is flat or negative which leads us to seek ways to resolve such conflict between financial
theories.5
In addition, an investor who has a long-term investment objective can outperform the market by
keeping mixes of financial assets. In other words, investors have probabilities to achieve high return
by exploiting anomalies strategies. Low volatility products expand the options available to equity
investors. Even better; some of them give investors cheap access to important investment anomalies.
Although low-volatility investments are likely to be used during expansionary market environments,
they are expected to achieve a high performance during declines and protect against large drawdown.
1 Yamada & Uesaki , 2009, p.2. 2 Yamada & Uesaki, 2009, p.11. 3 Baker, Bradley & Wurgler, 2010,p.5. 4 Baker, Bradley & Wurgler, 2010,p.8. 5 Vliet, Blitz & Grient, 2011,p.5.
Low-Volatility Anomaly
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2 Literature Review
Most of the previous studies in this domain tried to explain that both systematic risk and unsystematic
risk are equal to the total risk. Systematic risk refers to the market risk which we can’t mitigate with
diversification of financial products. For example the Capital Asset Pricing Model (CAPM) is used to
measure and calculate the systematic risk. The sensitivity of return to risk depends on some factors
that decide the performance. The most common of these factors are market, a momentum factor based
on the work of Carhart (1997), the size and the value based on the work of Fama and French
(1992, 1993).6 The Capital Asset Pricing Model (CAPM) and Modern Portfolio Theory (MPT) state
that the relation between risk and return is positive. However, this relation is not simple in reality.7
There are several arguments presenting different explanations of the Low-Volatility Anomaly. For
example, Black (1993) argues that investors choose low volatility products in case they face leverage
restriction. Blitz and Vliet (2007), Falkenstein (2010) and Baker, Bradley and Wurgler (2011) state
that investors with a relative return perspective do not support low volatility stocks even if they have
high Sharpe ratio.8 Also Siri and Tufano (1998) explained that fund managers seek to invest in high
volatility stocks and ignore investing in low-volatility stocks, which lead to the flattened relationship
between risk and return. Barberis and Huang (2008) state that it can be negative relation between risk
and return if we invest in stocks as gambling in lotteries, which has an effect on the stock price and
makes the stock overpriced. 9
Furthermore Ang, Hodrick, Xing, and Zhang (2006) find that stocks with a high-volatility achieve
lower future returns, as the main reason to invest in high-risk products is to hedge. 10 Baker, Bradley,
and Wurgler (2011) find that the behavioural explanation is the best way to describe the low-volatility
anomaly. Therefore investing in high-volatility products without objective, just for speculation or
gambling will increase the price of the stock as bubble and decrease the future return.11
6 Li-Lan & Feifei, 2013, p.3. 7 Yamada & Uesaki , 2009, p.2. 8 Vliet, Blitz & Grient, 2011,pp.13-14. 9 Barberis & Huang, 2008,p. 33. 10 Li-Lan & Feifei, 2013, p.3. 11 Baker, Bradley & Wurgler, 2010,p.5.
Low-Volatility Anomaly
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3 Data and Methodology
Data
To assess the empirical validity of Low-Volatility Anomaly, we chose Standard & Poor's 500 because
it is one of the most commonly used benchmarks for the overall US stock market. We selected 30
stocks randomly from dataset of S&P 500 with a sample period of 20 years for the 1993 through 2013
period. Then we computed the returns of 30 stocks based on monthly returns and estimated the
standard deviation of monthly returns for each stock based on rolling 5 years window.
Methodology
In order to test the relationship between stocks returns and the previous month’s standard deviations,
we applied a two stage Fama-Macbeth regression to price how much return we expect to receive for a
Particular Beta. We assumed we have 30 stocks returns over 20 years with a particular stock’s excess
return in a particular time denoted . We would like to test whether the m factors , and
explain our stocks returns and the premia awarded to exposure to each factor. To do so, we must run a
two-stage regression. The first stage involves a set of regression equal in number to the number of our
sample. The second stage is a set of regressions equal in number to the number of time periods. The
first stage regressions are a set of time series regression of each stock on the factors.
We ran a cross-sectional regression between stocks returns and the previous month’s standard
deviations for each stock separately to calculate the parameters (Alpha ( ) and Beta ( ) of the model.
Now, we know to what extent each stock’s return is affected by each factor. Next, we will test whether
or not the time-series mean of coefficients is statistically significant different from zero. So we can
state our hypothesis, which is as follows:
The null hypothesis notes that our coefficients (Alpha ( ) and Beta ( ) are equal to zero,
states that there is no significant relationship between stocks returns and the previous
month’s standard deviations. In addition, the alternative hypothesis notes that our coefficients
(Alpha ( ) and Beta ( ) are not equal to zero, states that there is significant relationship between
Null Hypothesis
Alternative Hypothesis
Low-Volatility Anomaly
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stocks returns and the previous month’s standard deviations. If we don’t reject the null hypothesis ),
this does not necessarily mean that the null hypothesis is true, it only suggests that there is not
sufficient evidence against in favour of . Rejecting the null hypothesis then, suggests that the
alternative hypothesis may be true.
4 Empirical Results
Based on our sample, we couldn’t find positive relationship between stocks returns and the previous
month’s standard deviations which can support the Capital Asset Pricing Model (CAPM). And we
couldn’t find negative relationship between stocks returns and the previous month’s standard
deviations which can support the Low-Volatility Anomaly. We can support our result by statistical
tests, which is as follows:
Exhibit (1)
Coefficients Mean 95% confidence interval p-value t-statistic
Alpha (α) -0.1339467 -1.2195262 to 0.9516327 0.8079 -0.2435
Beta (β) 0.1040446 -0.04600003 to 0.25408933 0.1729 1.3683
Table (1) shows that p-value of Alpha (α) 0.8079 is larger than the actual significance level (5%) of
our test, so we can’t reject the null hypothesis ) and confirms that there is no relationship between
stocks returns and the previous month’s standard deviations. In additional, the p-value of
Beta (β) 0.1729 is larger than the actual significance level (5%) of our test, so we can’t reject the null
hypothesis ) and confirms that there is no relationship between stocks returns and the previous
month’s standard deviations.
We can support our result by t-statistic, Table (1) shows that t-statistic of Alpha (α) -0.2435 is lower
than the critical value (1.96) of our test, so we can’t reject the null hypothesis ) and confirms that
there is no relationship between stocks returns and the previous month’s standard deviations.
Furthermore, t-statistic of Beta (β) 1.3683 is lower than the critical value (1.96) of our test, so we can’t
reject the null hypothesis ) and confirms that there is no relationship between stocks returns and the
previous month’s standard deviations.
Afterwards, we created three portfolios with low, medium and high volatility stocks based on our
sample which is 30 stocks. Each portfolio will cover a maximum of 10 stocks. Then we computed a
time series of monthly returns for each portfolio of the three groups by calculating the weighted
average of next month’s returns. We can provide a descriptive statistic of the time-series of returns for
each of the three volatility portfolios which is as follows:
Low-Volatility Anomaly
7
Exhibit (2)
Low Portfolio Med Portfolio High Portfolio
nobs 180 180 180
Nas 0 0 0
Minimum -25.196012 -35.392793 -35.594055
Maximum 24.707609 24.655112 29.26647
1. Quartile -1.375935 -2.043676 -2.297949
3. Quartile 3.510044 4.866799 5.002555
Mean 0.56099 0.823101 1.155686
Median 1.242904 1.673981 1.816925
Sum 100.978246 148.158184 208.023484
SE Mean 0.424156 0.502859 0.568108
LCL Mean -0.275999 -0.169193 0.034636
UCL Mean 1.39798 1.815395 2.276736
Variance 32.383509 45.516019 58.094307
Stdev 5.690651 6.746556 7.621962
Skewness -0.765326 -0.962803 -0.528061
Kurtosis 6.131932 5.688747 3.695875
It can be seen that high volatility portfolio had the highest median (Median=1.816925), on the other
hand the low volatility portfolio had the lowest median (Median=1.242904). Additionally, the high
volatility portfolio had the highest standard deviation which indicates that the data points are spread
out over a large range of values and high variation (SD=7.621962), on the other hand the low volatility
portfolio had the lowest standard deviation which indicates that the data points tend to be very close to
the median and low dispersion (SD=5.690651).
Table (2) shows that, the skewness of the three volatility portfolios is lower than zero which is
negatively skewed, so there are more negative values which indicates that the the degree and the
direction of asymmetry are skewed to the left. We can conclude that a risk-averse investor does not
like negative skewness. It can be seen that the kurtosis of the three volatility portfolios is high than 3
which is positive excess kurtosis (leptokurtic distribution). In case of a positive skewness, it is possible
to have a high excess kurtosis and to have no future extreme negative returns. The extreme returns will
only be positive. This is only possible when the skewness is positive. As soon as the skewness is nega-
tive, the impact of a high excess kurtosis affects the extreme negative returns.
-60 -50 -40 -30 -20 -10
0 10 20 30 40 50 60
1
7
13
19
25
31
37
43
49
55
61
67
73
79
85
91
97
10
3
10
9
11
5
12
1
12
7
13
3
13
9
14
5
15
1
15
7
16
3
16
9
17
5
Exhibit (3) Analysis of Volatility Portfolios
low volatility portfolio Med volatility portfolio High volatility portfolio
Low-Volatility Anomaly
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5 Conclusion
In theory, the positive relationship between stocks returns and the previous month’s standards
deviations can be supported by traditional models, for example, the Capital Asset Pricing Model
(CAPM) and the Modern Portfolio Theory (MPT).The main reason for those theories is the fact that a
rational investor would only accept a higher level of risk if it is offset by a higher degree of return. In
contrast, the negative relationship between stocks returns and the previous month’s standards
deviations can be supported by the Low-volatility Anomaly. On the other hand, there are many
empirical evidences suggest that the relationship between risk and return is flat or negative.
In the final analysis, we conclude that there is no relationship between stocks returns and the previous
month’s standards deviations to support the traditional models or the Low-Volatility Anomaly based
on our sample of 30 stocks. If we tried to run a cross-sectional regression between stocks returns and
the previous month’s standard deviations for many times and calculate our parameters (Alpha ( ) and
Beta ( ) of the model again, we could find a positive or negative relationship between stocks returns
and the previous month’s standards deviations.
Low-Volatility Anomaly
9
Reference List
Baker, M., Bradley, B., &Wurgler, J. (2010). Benchmarks as Limits to Arbitrage:Understanding the
Low Volatility Anomaly. NYU Working Paper No. 2451/29593. Retrieved from SSRN:
http://ssrn.com/abstract=1585031
Barberis, N., & Huang, M. (2008). Stocks as Lotteries: The Implications of Probability Weighting for
Security prices. American Economic Review, 98(5), 2066-2100. doi: 10.1257/aer.98.5.2066
Hsu, L. C., Kudoh , H., & Yamada, T. (2013). When Sell-Side Analysts Meet High-Volatility Stocks:
An Alternative Explanation for the Low-Volatility Puzzle. Journal of Investment Management, 11(2),
28-46. doi:10.2139/ssrn.2061824
Li-Lan, K., & Feifei, L. (2013). An Investor’s Low Volatility Strategy. The Journal of Index Investing,
3(4), 8-22. doi:10.3905/jii.2013.3.4.008
Vliet,P., Blitz, D., & Grient, B. (2011). Is the Relation between Volatility and Expected Stock Returns
Positive, Flat or Negative?. Retrieved from SSRN: http://ssrn.com/abstract=1881503
Yamada, T., & Uesaki, I. (2009). Low Volatility Strategy in Global Equity Markets. Retrieved from
http://www.saa.or.jp/english/publications/yamade_uesaki.pdf