## Verifying the low-volatility anomaly in the long-term and

## testing it in the short-term in times of crisis

Bachelor’s thesis in Finance and Organization University of Amsterdam

Faculty of Economics and Business Author: Michael van Amstel Student nr: 10437223

Date: February 2, 2015 Field: Finance

Supervisor: R.C. Sperna Weiland MSc Coordinator: dr. P.J.P.M. Versijp

2

**Table of contents **

**1 Introduction **

**3 **

**2 Theoretical background **

**4 **

**3 Data **

**9 **

**4 Methodology **

**10 **

**Equal-weighted strategy**

**11**

**Heuristic strategy**

**11**

**Minimum-variance strategy**

**12**

**Rebalancing vs. buy-and-hold**

**14**

**Performance evaluation**

**15 **

**5 Results and empirical findings **

**18 **

** General results **

**18**

** Results on significance **

**20**

**6 Shortcomings and future research **

**22 **

**7 Conclusion **

**23 **

**8 References **

**24 **

**9 Appendix A **

**26 **

**10 Appendix B **

**27 **

Hierbij verklaar ik, Michael van Amstel, dat ik deze scriptie zelf geschreven heb en dat ik de volledige verantwoordelijkheid op me neem voor de inhoud ervan.

Ik bevestig dat de tekst en het werk dat in deze scriptie gepresenteerd wordt origineel is en dat ik geen gebruik heb gemaakt van andere bronnen dan die welke in de tekst en in de referenties worden genoemd.

De Faculteit Economie en Bedrijfskunde is alleen verantwoordelijk voor de begeleiding tot het inleveren van de scriptie, niet voor de inhoud.

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**Introduction **

One of the basic principles in finance theory is the trade-off between risk and expected return; if an investor is willing to take on risk, there is the reward of higher expected returns (Bodie, Kane & Marcus, 2011). However, empirical studies show the long-term success of low-volatility and low-beta stock portfolios (Baker et al., 2011). Blitz and van Vliet (2007), for example, find that even relatively simple low-volatility investment strategies generate significantly higher returns than the market portfolio. This phenomenon of low-volatility stocks that out-perform the market is called the low-volatility anomaly (Baker et al., 2011).

The papers of Ang et al. (2009), Blitz and van Vliet (2007), Baker et al. (2011), and Dutt and Humphery-Jenner (2013) all test whether the low-volatility anomaly exists. Ang et al. (2009) do not only focus on the stock market of the United States but also on international markets. Their main finding is the strongly statistically significant global phenomenon where low-volatility stocks have higher returns relative to high-low-volatility stocks. The main result of Blitz and van Vliet (2007) is the superiority of risk-adjusted returns of stocks with low historical volatility, both in terms of Sharpe ratios and in terms of CAPM alphas. Dutt and Humphery-Jenner (2013) link the higher stock returns of volatility stocks to the fact that low-volatility firms have higher operating performance. The article of Baker et al. (2011) focuses on the deeper understanding of the behavioral explanation for this anomaly. Besides the behavioral finance diagnosis given in this article, Baker et al. (2011) also implies a practical prescription for investors who want to maximize returns subject to total risk.

In this report, we will test if we can verify the existence of the low-volatility anomaly for both long-term and short-term horizons. We use a data set that includes the full period of a crisis and its after effects so that a significant long period of crisis is included. The data set we will use is the Standard & Poor’s 500 for the period 1990 till 2014. Blitz and van Vliet (2007) state that efficient markets theory has been challenged by the finding that relatively simple portfolio construction strategies generate statistically significantly higher returns relative to the market portfolio. This market efficiency is, according to Blitz and van Vliet (2007), also challenged if some simple investment strategy generates a return that is similar to that of the market, but at a systematically lower level of risk. This statement is tested using three kinds of portfolios; the equal weighted portfolio, the heuristic way of constructing a portfolio and the minimum-variance portfolio. The latter one makes a distinction between a portfolio with 5%

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position limit and without the position limit. These different strategies will give us more than just one result to compare with the market portfolio to finally draw a conclusion about the low-volatility anomaly.

* *

The remainder of the paper is organized as follows. In section 2, we will discuss the low-volatility anomaly in more detail. In section 3, we will briefly explain the data that we used for our empirical part and we will describe the methodology in section 4. The empirical findings and results are discussed in section 5. Section 6 describes shortcomings of this research and opportunities for future research. Section 7 concludes.

**Theoretical background **

With the low-volatility anomaly, the thing that is different from what is usual is that firms with a low stock return volatility out-perform firms with a high stock return volatility. The part that the anomaly is not in agreement with is for example the leading theory of the Capital Asset Pricing Model (CAPM). The low-volatility anomaly can best be illustrated by considering the predictions of the CAPM. This model is widely used, although it does not fully withstand empirical tests, because of the insight it offers and because its accuracy is deemed acceptable for important implications. It is the best method available because of the logic of the distinction between systematic and firm-specific risk. The second reason is the efficiency of the market portfolio that may not be all that far from being valid. The CAPM is an accepted norm in the U.S. and many other developed countries, despite its empirical shortcomings like the low-volatility anomaly (Bodie, Kane & Marcus, 2011).

The low-volatility anomaly is a puzzle because one of the most basic principles of capital market theory is that equilibrium market prices are rational in that they rule out arbitrage opportunities. Such an arbitrage opportunity arises when an investor can earn riskless profits without making a net investment. There should be no arbitrage opportunity if the principle of capital market theory would hold (Bodie, Kane & Marcus, 2011). This paradox is thus the puzzle of the low-volatility anomaly.

The main implication of the CAPM is that it gives a precise prediction of the relationship that should be observed between the risk of an asset and its expected return, which is still shared with all other asset pricing models. The CAPM was developed in articles by William Sharpe, John Lintner, and Jan Mossin after Harry Markowitz laid down the foundation of modern

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portfolio management 12 years earlier in 1952. The most familiar expression of the CAPM to
*practitioners is the following (Bodie, Kane & Marcus, 2011): *

𝐸(𝑟_{𝑖}) = 𝑟_{𝑓}+ 𝛽_{𝑖}[𝐸(𝑟_{𝑀}) − 𝑟_{𝑓}]
Where:

𝐸(𝑟𝑖) = the expected return of stock i 𝑟𝑓= the risk-free interest rate

𝛽_{𝑖} = the beta of stock i

𝐸(𝑟𝑀) = the expected return of the market

The beta is the expected percent change in the excess return of a security for a 1% change in the excess return of the market (or other benchmark) portfolio. Nonsystematic risk, sometimes called specific risk or idiosyncratic risk, can be diversified away by constructing a well-diversified portfolio. This is a portfolio that is well-diversified over a large number of securities, with each weight small enough that for practical purposes the nonsystematic variance is negligible. Specific risk across firms cancels out in well-diversified portfolios. The theory expects that investors would not be rewarded for bearing risk that can be eliminated through diversification. For this reason, only systematic risk, often called undiversifiable risk, should cover a risk premium in market equilibrium (Berk & DeMarzo, 2011). A stock with a high beta is referred to as a risky stock. This follows from the equation where the excess return is multiplied by a higher beta. The risk in the CAPM is defined as the volatility of the returns of a stock. The low-volatility anomaly is all about stocks with a low volatility, thus stocks with low risk, stocks with a low beta. Following the theory of the CAPM, investing in a risky stock should give a higher return. A statistical failure of the CAPM equation is that the average of the risk-adjusted performance of high-beta securities is lower than that of the low-beta securities (Bodie, Kane & Marcus, 2011).

The anomaly is about volatility while stocks are ranked on their beta. These two terms are not the same. The volatility is the standard deviation of a return (Berk & DeMarzo, 2011). The beta of a stock is the sensitivity coefficient, its sensitivity to the index. It is the amount by which the stock’s return tends to increase or decrease for every 1% increase or decrease in the return on the index (Bodie, Kane & Marcus, 2011). As mentioned above, stocks will be ranked on their historical beta in this paper. Blitz and van Vliet report that ranking stocks on

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their historical asset pricing model betas is related to ranking stocks on their historical volatility. This relation follows theoretically from the fact that the beta of a stock is equal to its correlations with the market portfolio times its historical volatility divided by the volatility of the market portfolio. Blitz and van Vliet (2007) also observe empirically that portfolios consisting of stocks with low volatility exhibit a low beta as well. Chow et al. (2014) state that an equity portfolio’s volatility is driven by its market beta and therefore the out-performance of low-volatility portfolios directly relates to the anomaly that low-beta stocks deliver higher returns than high-beta stocks.

Dutt and Humphery-Jenner (2013) find evidence that the low-volatility effect exists across most markets outside the U.S., including the emerging markets. They also give several possible explanations. Some of the provided explanations are limits to arbitrage and operating performance. The main subject of their paper is the link between stock return volatility, operating performance, and stock returns. The stocks observed are classified on the location of the stock exchange. The four markets firms can belong to are: Emerging Asia1, Emerging EMEA2, Latin America3 and Ex-U.S./Canada Developed4. This research uses daily stock price data over the period 1990 – 2010 and also uses quintiles to distinguish low-volatility stocks. Both Baker et al. (2011) as Ang et al. (2009) sort firms into quintile portfolios. Blitz and van Vliet (2007) create decile portfolios with similar risk and return characteristics that are based on a straightforward ranking of stocks on their historical return volatility. In contrast with the articles of Dutt and Humphery-Jenner (2013) and Ang et al. (2009) the paper of Blitz and van Vliet (2007) uses monthly data. The index used is the FTSE World Developed index which consists of approximately 2,000 stocks on average.

The exploitation of security mispricing in such a way that risk-free profits can be earned is called arbitrage. One of the most basic principles of capital market theory is that equilibrium market prices are rational in that they rule out arbitrage opportunities (Bodie, Kane & Marcus, 2011). An assumption of the CAPM is that all investors are rational while behavioral finance starts with the assumption that they might not be. Baker et al. (2011) describe three biases that show the irrational preference for high volatility.

1_{ Emerging Asia comprises exchanges in India, Indonesia, Malaysia, Pakistan, the Philippines, South Korea, Taiwan, and Thailand. We }

exclude China from the sample on grounds that there are significant restrictions on foreign investors investing in Chinese-listed companies.

2_{ Emerging EMEA comprises the Czech Republic, Egypt, Hungary, Israel, Jordan, Morocco, Poland, Russia, South Africa, Turkey, and }

Argentina.

3_{ Latin America comprises Argentina, Brazil, Chile, Colombia, Mexico, Peru, and Venezuela. }

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The developed markets are Australia, Austria, Belgium, Bermuda, Britain, Cyprus, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Israel, Italy, Japan, Mauritius, Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, and Switzerland.

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One of these biases is the preference for lotteries. Baker et al. (2011) compare the preference for lotteries with a gamble with a 50% chance of losing $100 versus a 50% chance of winning $110. Most people would say no to this gamble. This behavior is called “loss aversion” which suggests that investors would shy away from volatility for fear of realizing a loss. Something remarkable happens as the probabilities shift. Most people take the gamble in the case where you are offered a gamble with a near-certain chance of losing $1 and a small (0,12 percent) chance of winning $5,000. As in the first example, this gamble has a positive expected pay-off of around $5. The preference for lotteries is thus shown by comparing it with buying a low-priced volatile stock. Another quote in their research states that volatile stocks are overvalued because of a lottery preference (Baker et al., 2011). Blitz and van Vliet related the preference for lotteries to the behavioral portfolio theory in Shefrin and Statman (2000). The latter two identified a low aspiration layer, which is designed to avoid poverty, and a high aspiration layer, which is designed for a shot at riches. This identification is in agreement with the earlier mentioned example of Baker et al. (2011).

An important part of the demand for volatile stocks is overconfidence (Cornell, 2009). This bias is called overconfidence; stocks with a wide range of opinions will have more optimists among their shareholders. These stocks will sell for higher prices, which leads to lower future returns (Baker et al., 2011).

The bias representativeness quoted in Baker et al. (2011) implies the following. A layman, someone who is not trained in or does not have a detailed knowledge of a particular subject (Online Cambridge Dictionary, 2015), is trying to think of great investments. He will ignore important information and buys speculative investments in new technologies. This results in a layman that overpays for volatile stocks. On the other hand, someone in the financial industry who uses mathematical methods (Online Cambridge Dictionary, 2015), a quant, analyzes the full sample of such speculative investments in new technologies. This quant will generate a more representative investment strategy compared to a layman.

A possible explanation mentioned by Blitz and van Vliet (2007) is the problem of leverage. Leverage is needed in order to take full advantage of the attractive absolute returns of low-risk stocks. Many investors are not allowed or unwilling to actually apply leverage. The result of this scarcity of leverage usage is that the opportunity presented by low-risk stocks is not easily arbitraged away.

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Some papers make statements about possible explanations for the low-volatility anomaly. A commonly heard explanation is the limit on arbitrage. This explanation is seen from the point of view of the institutional equity management. Contracts for institutional equity management state that the information ratio5 relative to a specific, fixed capitalization-weighted benchmark has to be maximized without using leverage. The conclusion of this explanation is that a benchmark makes institutional investment managers less likely to exploit the low-volatility anomaly (Baker et al, 2011). Asset managers have an incentive to tilt toward high-beta or high-volatility stocks, as this is a relatively simple way for asset managers to generate above-average returns (Blitz & van Vliet, 2007). One of the appealing features of this contract is that the skill of an investment manager, and the risks taken, is easier to understand by comparing returns with those of a well-known benchmark. This explanation is especially strong for U.S. markets because the US SEC requires funds to disclose a relevant benchmark. Such rules are not existent in all markets over the world so the explanation might be weaker for these markets (Dutt & Humphery-Jenner, 2013). But these advantages come at a cost. A benchmark makes institutional investment managers less likely to exploit the low-volatility anomaly.

The results of the paper of Dutt and Humphery-Jenner (2013) support the idea that operating performance might drive the relationship between volatility and returns. Low-volatility firms tend to have strong operating returns. Strong operating returns would increase expected stock returns. The explanation is that the stock price is bid up if the strong operating performance is unexpected. Shown is the fact that low-volatility stocks have stronger future operating performance. Dutt and Humphery-Jenner (2013) state that the explanation that higher operating performance might explain the low volatility effect can operate along-side the benchmarking explanation proposed by Baker et al. (2011).

Both the article of Baker et al. (2011) and Blitz and van Vliet (2007) give a practical prescription for the inefficient portfolios asset managers construct. Blitz and van Vliet (2007) propose the two-stage process by giving asset managers one single benchmark, such as fund-specific liabilities, plus a risk budget to deviate from that. Baker et al. (2007) state that investors who want to maximize returns subject to total risk must incentivize their managers to focus on the benchmark-free Sharpe ratio.

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**Data **

To verify if the low-volatility anomaly exists, we will first determine what the low-volatility stocks are by computing the betas for all S&P 500 companies. After this selection we will construct four different portfolios that are relatively easy to construct. Next, the Sharpe ratios of the portfolios are calculated to adjust the returns for risk. By doing this, we can test if simple low-volatility investment strategies out-perform the market.

A commonly used index for the United States stock market is the Standard & Poor's 500, the S&P 500. The CAPM uses proxies such as the S&P 500 index to stand in for the true market portfolio. This is a market-value-weighted index which means that the weight of each stock is proportionate to its market value (Bodie, Kane & Marcus, 2011). The S&P 500 is considered as one of the best representations of the United States stock market so this index will be seen as the market in this empirical research. At the time of writing this research, the S&P 500 consists of 502 stocks, issued by 500 large-cap companies which are traded on American stock exchanges. Two stocks (Google Class C and Navient) are removed from the data set because an error occurred for these stocks. This error is caused because these stocks completed their initial public offering in 2014 while the retrieved data is only until 2014. The final data set therefore consist of 500 stocks.

We obtain a data set with daily stock price data from Thomson Financial Datastream with values between January 1 1990 and December 31 2013. The data set consists of the S&P 500 index and all individual stocks of the S&P 500. The option ‘adjusted closing price’ is used as the daily stock price because it represents the official closing price. Datastream calculates the adjusted closing price by taking the ‘current’ prices at the close of the market that are stored each day. These stored prices are adjusted for subsequent capital actions, which results in the official closing price (Thomson Financial Datastream, 2015).

Datastream gives stock prices every single day of the year, except for weekends. Holidays are included in the data list, even if the market was closed. These holidays are omitted by hand to make sure that the calculation for the beta is accurate.

The total data set consist of 6,262 observations per stock, this is the period 1990-2013. The period 1990 till 1995, which has 1,305 observations in it, is used to determine the stock its beta. The remaining period that consists of 4,957 observations is divided in six different time frames. Five of them last three years, the last time frame consists of four years. This is to include the market crash of 2008 and all after effects of the crisis of 2008. The created portfolios are rebalanced at the beginning of every time frame to get the optimal weights.

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The progress of the S&P 500 between 1990 and 2013 is shown in figure 1. The figure shows the burst of the dot-com bubble in 2000. The S&P 500 index reached an all-time intraday high of 1,552.87 during this bubble (Reuters, 2013). The second visually apparent aspect is the financial crisis of 2007 – 2008. Figure 1 shows a large drop in the S&P 500 in both situations.

**Methodology **

**To verify the existence of the low-volatility anomaly, calculations have to be done. We focus **
on the exploitation of the low-volatility anomaly by using relatively simple low-volatility
investment strategies. These investment strategies are interesting because they can potentially
exploit the low-volatility anomaly in a significant way. Besides this, we test if the anomaly
exists in a short-term period in times of crisis. The first thing to do is determining which
stocks are classified as low-volatile. Following Ang et al. (2009), we will sort the stocks in the
S&P 500 into five groups according to their CAPM betas, which are a good indicator for their
volatility profile. Stocks with a high volatility end up in the fifth quintile while the lowest
volatile stocks end up in the first quintile. Ang et al. (2009) are not the only one using this
method, this approach is observed in other articles as well (Baker et al., 2011) (Dutt &
Humphery-Jenner, 2013). By calculating the betas for a certain period for all stocks in the
selected universe, a distinction can be made between low-volatile stocks and the remaining
stocks. The beta of investment i with portfolio M can be calculated by combining the
volatility and correlation terms as follows (Berk & DeMarzo, 2011):

𝛽_{𝑖}𝑀 _{=}𝑆𝐷(𝑅𝑖) × 𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅𝑀)
𝑆𝐷(𝑅_{𝑀})

Where:

𝛽_{𝑖}𝑀 measures the sensitivity of the investment i to the fluctuations of the portfolio M (the S&P
500 in this paper)
0
500
1000
1500
2000
19
90
19
92
19
94
19
96
19
98
20
00
20
02
20
04
20
06
20
08
20
10
20
12
**Va**
**lue S&P **
**5**
**0**
**0**
**Years **

Figure 1. The value of the S&P 500, January 1990 - December 2014.

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SD = the standard deviation

𝐶𝑜𝑟𝑟(𝑅𝑖, 𝑅𝑀) = the correlation between the return of the stock and the return of the market portfolio

First of all, the distinction between low-volatile stocks and the remaining stocks is made to determine what the low-volatility group of stocks is in this research. The beta of all stocks listed on the S&P 500 between January 1 1990 and December 30 1994 are calculated relative to the market, the S&P 500. From all 500 companies in the data set, 292 were listed over the complete period of January 1 1990 – December 31 2013. These 292 stocks are the only ones that are considered. By dividing these stocks into five groups, we get five quintiles. The lowest quintile is characterized by stocks with the lowest betas in the above mentioned period and contains 59 stocks. When creating the portfolios, only stocks from the lowest-quintile will be chosen. From this point, the remainder 441 stocks are not required anymore for the following calculations.

The second step is to actually create a portfolio. The optimal weights per stock for each portfolio are calculated within the group of 59 low-volatility stocks. There are various ways to construct a low-volatility portfolio (Chow et al., 2014), three of them will be used in this research. These three construction strategies are chosen because they are simple to construct and meanwhile effective to exploit the low-volatility anomaly. Blitz and van Vliet (2007) find that even relatively simple low-volatility investment strategies generate significantly higher returns than the market portfolio. By constructing such portfolios we verify if this statement holds and test if it holds in the short-term in times of crisis.

**Equal-weighted strategy **

An equal-weighted portfolio is characterized by positions in stocks that all have the same weight (Bodie, Kane & Marcus, 2011). With a low-volatility portfolio consisting of 59 stocks, the total value invested in every single stock would be 1/59 (one divided by fifty-nine). No distinction is made between stocks using this way of creating a portfolio.

**Heuristic strategy **

The second way to create a portfolio is the heuristic way. The heuristic approach uses the so called inverse betas to compute the weights. An inverse beta is computed by dividing 1 by the

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beta of a stock (Chow et al., 2014). In this way, the stocks with the lowest beta will get the highest weights because 1 dividing by a small number will give a higher weight. This should be an improvement in terms of volatility relative to the equal-weighted portfolio. For example, a stock with a beta of 0.33 in the previous period will have an inverse beta of 3. This calculation is done for all 59 stocks and these inverse betas are then added up. To get the percentage weights, the inverse beta of an individual stock is divided by the total of added up inverse betas. For example, suppose that the total weights added up is 150, the stock with beta 0.33 (inverse beta of 3) in the previous period would get 2% of the capital invested in the portfolio of the following period.

Stocks with a negative beta do not get a negative weight because we assume no short sales occur. These stock’s weights are set to zero, no money would be invested in this stock. The assumption for no short sales is done because investors have a general reluctance or inability to short stocks relative to buying them. The relative scarcity of short sales among individual investors and even institutional investors is empirically evident (Miller, 1977). The main drawback of going short in a stock is the unlimited downside potential thus bearing unlimited losses. Another reason is that we look at relatively simple constructed portfolios, whereby going short in a stock makes is more complicated.

**Minimum-variance strategy **

A third way to create a portfolio is by making a minimum-variance portfolio. This is a portfolio whereby it is constructed to provide the lowest possible variance. By pooling stocks together, a lower level of risk can be attained, given the anticipated return and volatility. A minimum-variance portfolio has a standard deviation smaller than that of all of the individual component assets (Bodie, Kane & Marcus, 2011). The essential thing to do with pooling, is computing the efficient weights for every individual stock in the portfolio. The covariance6 is computed by taking the product of the deviation of each return from its mean. The variance7 is calculated by taking the sum of the squared deviations from the mean (Berk & DeMarzo, 2011). Because portfolios of two stocks are relatively easy to analyze, consider such a portfolio. The variance of the two-stock portfolio is:

𝜎_{𝑝 }2 _{= 𝑤}
𝐴2𝜎𝐴2+ 𝑤𝐵2𝜎𝐵2+ 2𝑤𝐴𝑤𝐵𝐶𝑜𝑣(𝑟𝐴, 𝑟𝐵)
Where:
6 1
𝑇−1∑ (𝑅𝑡 𝑖,𝑡− 𝑅𝑖)(𝑅𝑗,𝑡− 𝑅𝑗)
7 1
𝑇−1∑ (𝑅𝑡− 𝑅)
2
𝑇
𝑡=1

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𝜎_{𝑝 }2_{ = the variance of the portfolio }
𝑤𝐴 = the weight invested in stock A
𝜎_{𝐴}2_{ = the variance of stock A }

𝐶𝑜𝑣(𝑟_{𝐴}, 𝑟_{𝐵}) = the covariance between the return of stock A and the return of stock B

There are two things that change by adding a stock to the portfolio. First, one more factor of the squared weight of the added stock times the variance of the added stock needs to be added. The second and final thing is to add the last factor of the equation but now for the covariance between the added stock and the individual stocks that are already in the portfolio.

Consequently, the equation for a portfolio with three stocks is as follows:
𝜎_{𝑝 }2 _{= 𝑤}

𝐴2𝜎𝐴2+ 𝑤𝐵2𝜎𝐵2+ 𝑤𝐶2𝜎𝐶2 + 2𝑤𝐴𝑤𝐵𝐶𝑜𝑣(𝑟𝐴, 𝑟𝐵) + 2𝑤𝐴𝑤𝐶𝐶𝑜𝑣(𝑟𝐴, 𝑟𝐶)
+ 2𝑤_{𝐵}𝑤_{𝐶}𝐶𝑜𝑣(𝑟_{𝐵}, 𝑟_{𝐶})

Now we have the equation for three stocks but one can expand it to N stocks in a portfolio. As more stocks are in a minimum-variance portfolio, the harder to calculate the variance of the portfolio. A more intuitive way to show such formulas is a matrix notation (Zivot, 2014):

𝜎𝑝,𝑥2 = (𝑥𝐴 𝑥𝐵 𝑥𝐶∙∙∙ 𝑥𝑁)
(
𝜎_{𝐴}2 _{𝜎}
𝐴𝐵 𝜎𝐴𝐶 ⋯ 𝜎𝐴𝑁
𝜎𝐴𝐵 𝜎𝐵2 𝜎𝐵𝐶 ⋯ 𝜎𝐵𝑁
𝜎_{𝐴𝐶} 𝜎_{𝐵𝐶} 𝜎_{𝐶}2 _{⋯ 𝜎}
𝐶𝑁
⋮ ⋮ ⋮ ⋱ ⋮
𝜎𝐴𝑁 𝜎𝐵𝑁 𝜎𝐶𝑁 ⋯ 𝜎𝑁2 )(
𝑥𝐴
𝑥𝐵
𝑥𝐶
⋮
𝑥_{𝑁}_{)}
Where:

𝑥 = the weight of an individual stock in the portfolio, the subscript indicates which stock 𝜎 = the standard deviation of an individual stock, or the covariance when there are two subscripts

As mentioned above, the minimum-variance portfolio is constructed to provide the lowest possible risk. To determine the weights of all stocks in the portfolio that result in the lowest risk, an Excel solver can be used. The solver uses the weights of the low-volatility stocks as the variables that need to be optimized. The portfolio weights and the covariance matrix are used as input for the calculation of the variance. This variance is the objective function that has to be minimized. A condition for the solver is that the total weights of the stocks need to

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add up exactly 1. The lowest variance possible will be the result presented by the solver. Assuming investors do not short stocks, Excel is used to select a set of non-negative stock weights. The explanation for this assumption is given above under ‘Heuristic strategy’. By diversifying, thus creating a portfolio, the exposure to firm-specific factors is spread out. However, there is always volatility risk, which is the risk incurred from unpredictable changes in volatility (Bodie, Kane & Marcus, 2011). A 5% position limit is often imposed to make sure that a portfolio is diversified and not too much focused on a few single stocks (Chow et al., 2014). This paper discusses both minimum-variance portfolios; with and without 5% position limit. An additional condition for the solver when computing the minimum-variance portfolio with 5% position limit has to be entered in Excel; a maximum of 0,05 weight for all stocks.

Initially, the solver was set up for all 292 stocks. No initial selection of low-volatility stocks have to be made for this method because the solver will give a zero weight to the stocks that are sub-optimal. The input for the solver is thus 292 variables. Doing this, the Excel solver generates an error. This error is caused by the fact that there are too much variable cells. The solution for this complication is to only include the initial 59 low-volatility stocks as variable. Now the Excel solver is able to calculate the optimal weights so this is the method that is used. Because of the relation between the low-volatility stocks and low-beta stocks, using the 59 stocks with the lowest volatility should not give negative implications relative to the initial plan.

**Rebalancing vs. buy-and-hold **

By constructing the portfolios, we calculate the initial weights. These initial weights for each stock will change over time because of differently changing stock prices. A stock that rises more than others will gain relatively more weight over time compared to other stocks. Besides this, betas are shifting all the time so a solution for this is rebalancing the portfolio to keep it up to date. Rebalancing is the process of realigning the weightings of a portfolio of assets (HSBC, 2014-2015).

The first portfolio weights are calculated on January 2 1995 using data from 1990 till 1995. The historical prices of the stocks in the previous time frame are analyzed at the beginning of a new time frame. Using the analyzed data of the previous time frame, the new weights will be calculated for the coming period. The rebalancing procedure is done after each period so

15

the rebalancing interval is three years. The effectiveness for portfolio returns of rebalancing every three years is confirmed by Smith and Desormeau (2006).

Rebalancing for the equal-weighted portfolio implies that all stocks get 1/59 weight again, based on the portfolio capitalization of the last day of the previous period.

For the heuristic portfolio, the weights of the following period are based on a calculation that uses the betas of the previous period to calculate the new inverse betas. The new weights are then calculated by using the new inverse betas.

The rebalancing for the minimum-variance portfolio is done by using the stock prices of the previous period to calculate a new covariance matrix. This new covariance matrix is the input for the Excel solver that generates the rebalanced weights.

Besides the portfolios that are rebalanced every time frame, a portfolio of every construction method is constructed in 1995 and not rebalanced over time. This is the so-called buy-and-hold strategy, where stocks are purchased and held for long periods (Yu & Lee, 2011). How rebalanced portfolios relate to buy-and-hold portfolios is observed by also constructing portfolios without rebalancing. Passive investors expend their resources neither on market research nor on frequent purchase and sale of stocks (Bodie, Kane & Marcus, 2011).

**Performance evaluation **

The weights invested in every individual stock are constant for all buy-and-hold portfolios over the whole period. This implies that the total return of the portfolios with the buy-and-hold strategy over all six periods is calculated by multiplying the return of an individual stock over 19 years by its weight invested in it. All returns are added up which results in the total return of a portfolio holding it 19 years.

The return of the portfolios with rebalancing over all six periods is calculated as follows. The return of an individual stock from period 1 is multiplied by the weight invested in it. All returns from period 1 are added up which results in the total return for period 1. The total returns from all the individual periods are then multiplied with each other. This final result is the total return of the portfolio over the six periods.

The return of the S&P 500 is the percentage change between January 2 1995 and December 31 2013.

16

Note that two assumptions are made in this analysis. One assumption made in this paper is that transaction costs are ignored. Another assumption is that returns are reinvested in the portfolio. For this reason, we can multiply returns to compute the total return of a portfolio.

The anomaly states that the risk-adjusted return of low-volatility stocks is higher than that of the market portfolio. The Sharpe ratio is used to adjust the returns of the portfolios for risk. The Sharpe ratio is the excess return of an asset divided by the volatility of the return of the asset and is calculated as follows (Berk & DeMarzo, 2011):

𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 = 𝑅𝑃− 𝑟𝑓
𝑆𝐷(𝑅_{𝑃})
Where:

𝑅𝑃 = the return of the portfolio
𝑟_{𝑓}= the risk-free interest rate

𝑆𝐷(𝑅_{𝑃}) = the standard deviation (volatility) of the return of the portfolio

This is the first calculation where a risk-free interest rate is necessary. Like the paper of Ang et al. (2009) the risk-free interest rate is the one-month U.S. T-bill return, retrieved from Datastream (2015). These Sharpe ratios are calculated for a long-term period of 19 years so for this reason the average monthly risk-free interest rate is calculated and converted to an interest rate for 19 years.

The return of the portfolio is the return over a period of 19 years because the portfolios are compared in the long-term. As well as the return of the portfolio, the standard deviation of the return of the portfolio is calculated over a period of 19 years and it is finally converted to a standard deviation for 19 years.

To test for the statistical significance whether the low-volatility anomaly holds, whether the low-volatility portfolio out-performed the market, a Jobson and Korkie (1981) test with the Memmel (2003) correction is applied. The number of observations in this test statistic is the number of observations that were used to compute the Sharpe ratios, which is 4,785 for the whole period and 1,762 for period of crisis. This test statistic follows a standard normal distribution and is calculated as follows (Blitz & van Vliet, 2007):

17

𝑧 = 𝑆𝑅1− 𝑆𝑅2

√1_{𝑇 [2(1 − 𝜌}_{1,2}) + ½ (𝑆𝑅_{1}2_{+ 𝑆𝑅}

22− 𝑆𝑅1𝑆𝑅2(1 + 𝜌1,22 ))]

Where:

𝑆𝑅𝑖 = the Sharpe ratio of portfolio i

𝜌_{1,2} = the correlation between portfolios i and j
𝑇 = the number of observations

To give an insight in how diversified a portfolio is, the Herfindahl-Hirschman Index is calculated. This index is interesting because it gives insight in the weights of the portfolio. One of the reasons portfolios are created in this paper is because it results in diversification. This the exact aspect the Herfindahl-Hirschman Index gives information about. The Herfindahl-Hirschman Index is the sum of the squared portfolio weights8. This index gives insight in the concentration of the stocks in the portfolio; the higher the concentration, the higher the index (McIntosh & Hellmer, 2012).

Finally, the variances of the portfolios are tested against each other. This is done by doing an F-test which is an analysis of variance for two samples:

𝑃(𝐹 > 𝐹𝛼,𝑛1−1,𝑛2−1) = 𝛼
𝐹1−𝛼,𝑛1−1,𝑛2−1=
1
𝐹_{𝛼,𝑛}_{2}_{−1,𝑛}_{1}_{−1}
Test statistic:
𝐹 =𝑠1
2
𝑠_{2}2~𝐹[𝑑𝑓𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 = 𝑛1− 1 & 𝑑𝑓𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 = 𝑛2− 1]
Where:

𝑛_{𝑖} = the number of observations of portfolio i
𝑠_{𝑖}2 _{= the variance of portfolio i }

𝛼 = the alpha

𝑑𝑓_{𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟} = the degrees of freedom for the numerator
𝑑𝑓_{𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟} = the degrees of freedom for the denominator

8_{ 𝐻 = ∑} _{𝑤}

𝑖2

𝑁 𝑖=1

18

The first part of the F-test showed above is the chance that F is higher than the critical value using the chosen alpha. This chance is called the alpha. The second part shows how to calculate the critical value for a left-tailed test; the right-tailed critical value is reciprocated. Finally, the test statistic divides the variances of both portfolios. A conclusion about its significance is drawn by looking at the table with critical values of F by using the degrees of freedom and the chosen alpha.

**Results and empirical findings **

**General results **

Figure 2 gives the graphical representation of the course of all constructed portfolios and the S&P 500 over the period 1995 – 2013. In agreement with Table 1, the heuristic way of constructing a portfolio without rebalancing has a return that is well above the rest. The lowest return an investor would get in December 2013 if he invested in January 1995 is when he invested in the market index, the S&P 500.

0
10
20
30
40
50
1
9
9
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1
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**Va**
**lue o**
**f **
**$**
**1**
** inv**
**este**
**d in 1**
**9**
**9**
**5**
**Years **
Equal-weighted
Heuristic
Minimum-Variance without
constraint

Minimum-Variance with constraint Equal-weighted no rebalance Heuristic no rebalance Minimum-Variance without constraint no rebalance

Minimum-Variance with constraint no rebalance

S&P 500 Figure 2. The value of $1 invested in 1995 of the different portfolios in the period 1995 – 2013.

19

Table 1 provides an overview of the main results for the different constructed portfolios between 1995 and 2013. The returns of the portfolios are compared with that of the S&P 500. The value of the S&P 500 increased 302,46% in six periods, from 1995 until 2014. The three exceptional returns over six periods are those of the weighted portfolio, the equal-weighted portfolio without rebalancing and above all the heuristic way of portfolio construction without rebalancing. Striking is the fact that all methods of constructing a low-volatility portfolio generate a higher return than the benchmark over the six periods.

The second row of results shows the return in the crisis. The period called crisis is the time frame 2007 to 2013. Note that not all returns are higher than the S&P 500 return of 30,32% this time. A return of 30,32% might sound high but this is a return over a period of seven years. The annualized return of the S&P 500 in times of crisis is 3,86% while the annualized return over all 6 periods 7,60% is. This is an almost double annualized return of the S&P 500 over all six periods relative to that in times of crisis. Even though the minimum-variance portfolio with 5% position limit was constructed to get the lowest possible risk, its return in the crisis is the lowest of all. Its return is less than half than that of the Standard & Poor’s 500. The heuristic method of portfolio construction without rebalancing every three years has the highest return in the period of crisis. The highest and lowest weights show the extreme weights in the portfolios.

The rows that indicate the maximum and minimum weight of each portfolio show that the heuristic way of constructing a portfolio without rebalancing is not very diversified. Over 91% of the dollars invested is in only one stock somewhere in the data set. Prominent is the fact that the maximum weight of both the minimum-variance portfolios is lower when there is no rebalancing. In contrast, the equal-weighted- and heuristic portfolio do follow the general expectation that the maximum weight invested in only one stock is lower when there is rebalancing every three years.

An overview of the standard deviations, thus the volatilities, of the portfolios is shown in Exhibit 1. All standard deviations of the portfolios are higher in times of crisis relative to the complete data set of six periods. Besides this result, it is obvious that the standard deviation of the heuristic way of constructing a portfolio without rebalancing is higher than the others.

20
Portfolio
construction:
Equal-weighted
portfolio
Heuristic way of
portfolio
construction
Minimum-variance
without 5%
constraint
Minimum-variance with
5% constraint
Equal-weighted
portfolio, no
rebalancing
Heuristic way of
portfolio
construction, no
rebalancing
Minimum-variance
without 5%
constraint, no
rebalancing
Minimum-variance with 5%
constraint, no
rebalancing
Standard &
Poor’s 500
Return over 6
periods: 1058,972653 % 779,2168285 %
708,0255572 % 511,3668106 % 1626,384793
% 4720,096585 % 543,4864983 % 541,9380866 %
302,4560716
%
Return in
crisis: 49,04779453 % 46,60215999 % 49,72262668 % 14,15468219 % 42,3892875% 52,80834121 % 28,01524021 % 23,02337677 %
30,32221674
%
Standard
deviation: 10,05174729 8,910890818 8,109737767 8,446188375 11,86443688 15,15978116 10,00550827 9,991594094 10,31204763
Standard
deviation
crisis:
10,18936643 9,097818373 8,320194343 8,472466894 12,36370621 15,70391698 10,85454629 10,84566842 9,560876275
Sharpe ratio: 0,986737611 0,799120874 0,790280318 0,525962538 1,314226596 3,069284175 0,476095309 0,4752086 0,228206014
Sharpe ratio
crisis: 0,042073502 0,044433288 0,052336596 0,009415346 0,029288731 0,029693725 0,020118463 0,015532298 0,025253601
Highest
weight: 0,346663351 0,228218627 0,467510765 0,302734966 0,68427999 0,914811352 0,277726076 0,270375934 -
Lowest
weight: 0,001591393 0,000 0,000 0,0000 0,000342982 0,000113516 0,000 0,0000 -
z-score of
Jobson &
Korkie test9_{: }
66,54352019*** 53,47622745*** 49,46614087*** 30,72874487*** 70,67571692*** 86,38447553*** 29,12265464*** 29,62245359*** -
z-score of
Jobson &
Korkie test
crisis10_{: }
2,865467367*** 3,073784743*** 2,766720006*** -2,026262791** 0,270467469 0,204702393 -0,662101366 -1,293411677* -
Sign test: 2,50***
Herfindahl-Hirschman
Index11_{: }
0,022271237 0,022833412 0,124444147 0,044322992 0,101058125 0,281816576 0,058010743 0,045803782 -
Herfindahl-Hirschman
Index Crisis:
0,018422987 0,020242485 0,214399492 0,045125865 0,217097341 0,620262168 0,059057407 0,052142174 -
**Results on significance **

The z-scores as result of the Jobson & Korkie test showed in Table 1 gives information about the significance of the risk-adjusted return. We can state that using an alpha of 0,01 all portfolios did out-perform the S&P 500 in the period 1995 – 2013. The heuristic way of portfolio construction without rebalancing has the highest z-score, thus is the most significant portfolio that is higher than the S&P 500.

The results change when looking at the z-scores of the Jobson & Korkie test in times of crisis. Not every relatively simple constructed low-volatility portfolio has a significantly higher Sharpe ratio than the S&P 500. In contrast, three of them have an even lower Sharpe ratio than the S&P 500, of which two significant.

9

*=significant at a level of 0,1; **=significant at a level of 0,05; ***=significant at a level of 0,01 10

Red color indicates that S&P 500 Sharpe ratio is higher

11_{ This will give the average HHI for the portfolios }

21

Table 2 in Appendix A shows the p-values of the F-test that is used to compare the standard deviations among each other. In this table, the asterisks indicate the same significance as in Table 1. The arrow behind each p-value and asterisks points to the portfolio standard deviation that is the highest of the two. For example, the first result between Equal and Equal – no rebalance indicates that the standard deviation of Equal – no rebalance is significantly higher with an alpha of 0,01. The p-values in red indicate that these standard deviations do not differ significantly.

The aspect that is in line with the theory of the low-volatility anomaly is that most relatively simple constructed low-volatility portfolios have a significant lower standard deviation, thus volatility, than the S&P 500. Six out of the eight portfolios are significant with an alpha of at most 0,1. For five out of the eight portfolios an alpha of maximum 0,05 shows the significance. The case is, however, that the equal-weighted portfolio without rebalance and the heuristic portfolio without rebalance have a significantly higher standard deviation relative to the S&P 500 using an alpha of 0,01.

The graphical representation between the standard deviations in both the whole period as the period of crisis is shown in Exhibit 1. The p-values of the standard deviations relative to each other in times of crisis are shown in Table 3 in Appendix A. The same explanation as for Table 2 applies for Table 3 as regards the significance, asterisks, arrows and red color.

Exhibit 1. Overview of the standard deviations of the different portfolios in the complete data set and in the crisis.

0
2
4
6
8
10
12
14
16
**S**
**ta**
**n**
**d**
**a**
**rd**
** d**
**ev**
**ia**
**ti**
**o**
**n**
**Portfolio **

Standard deviation all 6 periods: Standard deviation crisis:

22

A sign test is computed for the returns of the portfolios relative to the S&P 500. We test if the returns of the relatively simple constructed portfolios are higher than that of the crisis. For every return of a portfolio that we constructed that is higher than that of the S&P 500, a + is noted; - is noted when it is lower and 0 when it is equal. For example, the return of the equal-weighted portfolio over 6 periods is higher than the return of the S&P 500 over 6 periods. Observing this, we can note a + for the equal-weighted portfolio over 6 periods. Looking at all portfolios, we observe 16 nonzero differences relative to the S&P 500. A z-score is calculated using the following test statistic:

𝑧 = 𝑋−0,5𝑛_{0,5√𝑛} if 𝑛 ≥ 10
Where:

𝑛 = the total number of nonzero differences

𝑋 = the number of positive differences; observations where a + is noted

The conclusion of the sign test is that the return of the relatively simple constructed low-volatility portfolios significantly higher is than the return of the S&P 500.

**Shortcomings and future research **

One of the shortcomings is that the selection of low-volatility stocks is made in only one period; 1990 until 1995. The stocks that were low-volatile during this period are the only one considered in this research. The fact whether a stock is low-volatile or not can change during these 19 years that are observed in this paper. A solution for this problem is taken into consideration but this is explained in more detail in Appendix B.

Another shortcoming of this research is the fact that the average one-month Treasury bill rate is used for the whole data set. These risk free interest rates oscillate between 0,01% and over 5% per year. We might be too unpunctual by taking the average.

A solution for this shortcoming that can be used in future research is that the Sharpe ratios of the portfolios are calculated every month. By taking the average of the Sharpe ratios, this approach might overcome this shortcoming of this research.

23

Possible future research can focus on the two practical prescriptions of Baker et al. (2011) and Blitz and van Vliet (2007). These prescriptions can be alternatives for the widely used information ratio that does not result in efficient portfolios.

**Conclusion **

The phenomenon of volatility stocks that out-perform the market is called the low-volatility anomaly. Relatively simple low-low-volatility investment strategies are found to generate significantly higher returns than the market portfolio. We verify this statement in the long-term and test it in the short-term in times of crisis.

By using the S&P 500 as a market portfolio, using daily data and looking at the data set 1995 – 2013, we conclude the following. All constructed low-volatility portfolios in this research did significantly out-perform the market in the long-term. This is in line with the low-volatility anomaly. Whether the portfolio is rebalanced every three years or not does not matter for the fact that it significantly out-performs the S&P 500. A conclusion about the low-volatility anomaly can only be drawn by looking at the adjusted return. This risk-adjustment is done by calculating the Sharpe ratios for the portfolios and compare it with the Sharpe ratio of the S&P 500. The comparison of the Sharpe ratios is finally done by using the Jobson & Korkie test with the Memmel correction. And as well for the risk-adjusted return we can say that every relatively simple low-volatility portfolio construction strategy gives a higher return than the S&P 500. We can conclude that the low-volatility anomaly is verified in the long term.

When we look at the period 2007 – 2013, in which the 2008 crisis and its after effects occurred, all portfolios that are rebalanced every three years did out-perform the S&P 500 except the minimum-variance portfolio with 5% position limit. This out-performance is significant when looking at the risk-adjusted return. There is no single conclusion about the short-term validity of the low-volatility anomaly in times of crisis when using portfolios that are not rebalanced.

Possible explanations for the low-volatility anomaly are limits to arbitrage, a higher operating performance of low-volatility stocks, the problem of leverage, representativeness, preference for lotteries and overconfidence.

24

**References **

Ang, A., Hodrick, R.J., Xing, Y. & Zhang, X. (2009). High idiosyncratic volatility and low
*returns: International and further U.S. evidence, Journal of Financial Economics, Vol 91(1), *
1-23.

Baker, M., Bradley, B. & Wurgler, J. (2011). Benchmarks as Limits to Arbitrage:
*Understanding the Low-Volatility Anomaly, Financial Analysts Journal, Vol 67(1), 40-54. *

*Berk, J. & DeMarzo, P. (2011). Corporate Finance (Second Edition). Essex: Pearson *
Education Limited.

Blitz, D.C. & Vliet, P. van (2007). The Volatility Effect: Lower risk without lower return.
*Journal of Portfolio Management, Vol 34(1), 102-113. *

*Bodie, Z., Kane, A. & Marcus, A.J. (2011). Investments (Ninth Edition). New York: *
McGraw-Hill Irwin.

*Online Cambridge Dictionary (2015). dictionary.cambridge.org. *

Chow, T., Hsu, C., Kuo, L. & Li, F. (2014). A Study of Low-Volatility Portfolio Construction
*Methods. The Journal of Portfolio Management, Vol 40(4). *

*Cornell, B. (2009). The Pricing of Volatility and Skewness: A New Interpretation, Journal of *
*Investing, Vol 18(3), 27-30. *

Dutt, T. & Humphery-Jenner, M. (2013). Stock return volatility, operating performance and
*stock returns: International evidence on drivers of the ‘low volatility’ anomaly, Journal of *
*Banking & Finance, Vol 37(3), 999-1017. *

McIntosh, C. & Hellmer, S. (2012). Necessary and sufficient conditions in merger control: the
*use of HHI and threshold value, Applied Economics, Vol 44(7), 867-878. *

25

*Miller, E. M. (1977). Risk, Uncertainty, and Divergence of Opinion, Journal of Finance, Vol *
*32(4), 1151-1168. *

*Shefrin, H. & Statman, M. (2000). Behavioral Portfolio Theory, Journal of Financial and *
*Quantitative Analysis, Vol 35(2), 127-151. *

The Hongkong and Shanghai Banking Corporation. Managed portfolios (2014-2015),
*Tailored solutions for your investment goals, www.hsbc.bm. *

*Valetkevitch, C., Key dates and milestones in the S&P 500’s history (May 6 2013). Reuters, *
www.reuters.com.

*Yu, J-R. & Lee, W-Y. (2011), Portfolio rebalancing model using multiple criteria, European *
*Journal of Operational Research, Vol 209(2), 166-175. *

*Zivot, E. (August 7 2014). Portfolio Theory with Matrix Algebra, University of Washington, *
*www.washington.edu. *

26 T ab le 2 . T h e P -v alu es o f th e st an d ar d d ev iatio n s relati v e to ea ch o th er . S ta n d a rd d e v ia ti o n s: 1 0 ,0 5 1 7 4 7 2 9 1 1 ,8 6 4 4 3 6 8 8 8 ,9 1 0 8 9 0 8 1 8 1 5 ,1 5 9 7 8 1 1 6 8 ,1 0 9 7 3 7 7 6 7 1 0 ,0 0 5 5 0 8 2 7 8 ,4 4 6 1 8 8 3 7 5 9 ,9 9 1 5 9 4 0 9 4 1 0 ,3 1 2 0 4 7 6 3 P -v a lu e s : E q u a l E q u a l - n o r e b a la n c e H e u ri s ti c H e u ri s ti c n o r e b a la n c e M V w it h o u t M V w it h o u t - n o r e b a la n c e M V w it h M V w it h n o r e b a la n c e S & P 5 0 0 1 0 ,0 5 1 7 4 7 2 9 E q u a l x 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0, 75 ← 0 ,0 0 0 *** ← 0, 67 8← 0 ,0 7 7 *↑ 1 1 ,8 6 4 4 3 6 8 8 E q u a l - n o r e b a la n c e x 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 8 ,9 1 0 8 9 0 8 1 8 H e u ri s ti c x 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ↑ 1 5 ,1 5 9 7 8 1 1 6 H e u ri s ti c n o r e b a la n c e x 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 8 ,1 0 9 7 3 7 7 6 7 M V w it h o u t x 0 ,0 0 0 *** ↑ 0 ,0 0 5 *** ↑ 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ↑ 1 0 ,0 0 5 5 0 8 2 7 M V w it h o u t - n o r e b a la n c e x 0 ,0 0 0 *** ← 0, 92 3← 0 ,0 3 7 **↑ 8 ,4 4 6 1 8 8 3 7 5 M V w it h x 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ↑ 9 ,9 9 1 5 9 4 0 9 4 M V w it h n o r e b a la n c e x 0 ,0 2 9 **↑ 1 0 ,3 1 2 0 4 7 6 3 S & P 5 0 0 x 1 0 ,1 8 9 3 6 6 4 3 1 2 ,3 6 3 7 0 6 2 1 9 ,0 9 7 8 1 8 3 7 3 1 5 ,7 0 3 9 1 6 9 8 8 ,3 2 0 1 9 4 3 4 3 1 0 ,8 5 4 5 4 6 2 9 8 ,4 7 2 4 6 6 8 9 4 1 0 ,8 4 5 6 6 8 4 2 9 ,5 6 0 8 7 6 2 7 5 P -v a lu e s : E q u a l E q u a l - n o r e b a la n c e H e u ri s ti c H e u ri s ti c n o r e b a la n c e M V w it h o u t M V w it h o u t - n o r e b a la n c e M V w it h M V w it h n o r e b a la n c e S & P 5 0 0 1 0 ,1 8 9 3 6 6 4 3 E q u a l x 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 8 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 9 *** ↑ 0 ,0 0 8 *** ← 1 2 ,3 6 3 7 0 6 2 1 E q u a l - n o r e b a la n c e x 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 9 ,0 9 7 8 1 8 3 7 3 H e u ri s ti c x 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ← 0 ,0 0 0 *** ↑ 0 ,0 0 3 *** ← 0 ,0 0 0 *** ↑ 0 ,0 3 7 **↑ 1 5 ,7 0 3 9 1 6 9 8 H e u ri s ti c n o r e b a la n c e x 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 0 ,0 0 0 *** ← 8 ,3 2 0 1 9 4 3 4 3 M V w it h o u t x 0 ,0 0 0 *** ↑ 0, 44 7↑ 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ↑ 1 0 ,8 5 4 5 4 6 2 9 M V w it h o u t - n o r e b a la n c e x 0 ,0 0 0 *** ← 0, 97 3← 0 ,0 0 0 *** ← 8 ,4 7 2 4 6 6 8 9 4 M V w it h x 0 ,0 0 0 *** ↑ 0 ,0 0 0 *** ↑ 1 0 ,8 4 5 6 6 8 4 2 M V w it h n o r e b a la n c e x 0 ,0 0 0 *** ← 9 ,5 6 0 8 7 6 2 7 5 S & P 5 0 0 x S ta n d a rd d e v ia ti o n s in c ri si s:

**Appendix A **

T
ab
le
3
.
T
h
e
P
-v
alu
es o
f
th
e
st
an
d
ar
d
d
ev
iatio
n
s
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v
e
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**Appendix B **

In this research we determine which stocks are low-volatile by looking at the period 1990 – 1994 of the S&P 500. We only consider the stocks that are in the lowest quintile when looking at the betas. In which quintile each stock is in later years does not matter. This approach for determining the low-volatility quintile is used because the initial approach did not give the desired result. The initial approach was as follows.

Every three years we checked which stocks ended up in the lowest quintile. The period looked at was 1995 – 2013, divided in 7 periods. We gave each stock a rank on the basis of its beta relative to the other stocks in the same period. The twenty percent with the highest rank number, thus the lowest beta, of the total stocks in that period would be selected for the low-volatility quintile. The stocks that are in the low-low-volatility quintile all periods would get the label ‘low-volatility stock’. Because of the fact that only three stocks got this label, we tried something else.

One of the aspects that caused this low number of low-volatility stocks is the fact that 142 companies did not exist in our data set in the first period. This means that you lose 28,4% of the S&P 500 companies. Next, the solution was the following. What if we do not just check for all 7 periods in the low-volatility quintile but also 4, 5 or 6 periods? The requirement was that if we check a stock for four consecutive periods it was only included in the data set for four periods. For example, a stock that was present in the data set for five complete periods, but only four of them in the lowest quintile, would not qualify for the low-volatility quintile. The result was that still only three stocks qualified for the low-volatility quintile portfolio. It is not possible to create a diversified portfolio by using only three stocks. The conclusion is that we use the approach discussed in this paper.