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The performance of beta sorted portfolios using implied

volatility market timing for large cap and small cap stocks

Marc Brouwer

University of Groningen, Groningen, The Netherlands

Abstract

This study shows that index returns can be enhanced by implementing market timing based on implied volatility levels for the large cap S&P 500 index and the small cap Russell 2000 index. Moreover, this study examines the performance of different beta sorted portfolios created from index constituents and shows for the S&P 500 that the highest and low beta portfolios do not outperform the index, but that medium to high beta portfolios do outperform the index when implementing market timing based on the implied volatility. None of the beta portfolios created from Russell 2000 constituents outperform the Russell 2000 index when using the implied volatility market timing strategy. The observed pattern in returns for the beta sorted portfolios for large cap and small cap stocks is similar and hence, beta sorted portfolios do not react differently for large cap stocks in comparison to small cap stocks in a low implied volatility environment.

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1. Introduction

Implied volatility (IV) is the volatility for a particular stock that is observed in the market by using the option pricing formula of Black and Scholes (1973). Hence, implied volatility is referred to as the markets opinion about the volatility of a particular stock and thus, is forward looking whereas historical volatility is backward looking (Hull, 2018). Besides calculating the IV for a particular stock it can also be calculated for a certain stock index using options on that index. A well-known example is the VIX that is calculated based on the S&P 500 index1. The Chicago Board

Options Exchange (CBOE) volatility index (VIX) is a measure of the expected volatility of return of the S&P 500 (SPX) stock index over the next 30 calendar days2. High levels of VIX indicate high degrees of market turmoil and hence, VIX is often referred to as the “Investor fear gauge” (Whaley, 2000).

The relationship between expected returns and volatility is a heavily studied subject in the world of finance. However, previous literature provides mixed results on the relation between expected risk premiums and volatility. French, Schewert, and Stambaugh (1987) is an often-cited study that documents empirical evidence in favor of a positive relationship between market return and volatility. On the other hand, Glosten, Jagannathan and Runkle (1993) and thereafter, Fleming, Ostdiek and Whaley (1995) provide evidence that the conditional volatility and the risk premium are negatively correlated. Moreover, Ang et al. (2006) documents a negative relationship between stock volatility innovations and returns for firms with high systematic risk exposure and firms with high idiosyncratic volatility in a cross-sectional analysis.

Forecasting future volatility using implied volatility gained interest in the industry after Christensen and Prabhala (1998) provided evidence that implied volatility outperformes the use of historical volatility to forecast future realized volatility. This result is supported by Poon and Granger (2003) that review 93 studies regarding the forecasting of volatility in financial markets. It concludes that most information about future volatility is contained by implied volatility because it is a market based volatility forecast.

Existing literature reviewing the relationship between stock index returns and implied volatility often only focus on the S&P and the VIX. Moreover, most studies have examined the relationship before 2004 when the old VIX (i.e. currently known as VXO) was calculated using stock index options on the S&P 100. Over the years it has been established that there is a negative and significant relationship between a stock index and the implied volatility index (see, e.g., Whaley,

1 The CBOE started calculating the VIX from S&P 500 (SPX) stock index options on September 22, 2003. The

levels of VIX before that day are calculated from S&P 100 (OEX) stock index options and is now referred to as “VXO”.

2 A full review of the VIX calculations can be found on the CBOE website

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2000). This means that positive (negative) stock index returns lead to decreased (increased) implied volatility levels. It can be assumed that this is a causal relationship based on Wang, Keswani and Taylor (2006) that provide evidence for causality between returns and realized volatility and on Christensen and Prabhala (1998) that show that implied volatility is an accurate measure of future realized volatility. Giot (2005) and Dennis, Mayhew and Strivers (2006) extend the result of a significant negative relationship by documenting that this relation is asymmetric for the S&P 100 and the VXO. This asymmetry means that the VXO is more sensitive to negative changes in the S&P 100 than it is for positive changes and it is probably caused by market-wide factors. Moreover, Giot (2005) documents that positive (negative) future returns are to be expected for extremely high (low) implied volatility levels and hence, concludes that the VXO has a predictive ability for future stock returns. This predictive ability is the strongest for a holding period 60-days. Banerjee, Doran and Peterson (2007) find that there is a statistical significant relation between future returns and both implied volatility levels and innovations for most of their 12 different portfolios sorted on beta, size, and book-to-market equity. This is even the case when the model is corrected for the Fama-French and Carhart factors which are introduces by Fama and French (1993) and Carhart (1997), respectively. It is stated that the results tend to be stronger for high beta portfolios.

Copeland and Copeland (1999) examine that portfolio return can be enhanced when implementing a market timing strategy based on the VXO where a portfolio is rotated according to size (i.e. large cap versus small cap) and style (i.e. value versus growth). After an increase (decrease) in expected future volatility, the portfolio is shifted into value (growth) stocks and large cap (small cap) stocks. The strategy behind the rotation between small cap and large cap is based on the believe that small cap stock, having a higher beta in general, perform worse when volatility increases based on the theoretical prediction that volatility and market return are negatively correlated.

Sarwar (2012) examines the intertemporal relationship between the VIX and the returns of the S&P 100, S&P 500, and S&P 600 over the timespan of 1992-2011, which is divided into three subperiods. The study concludes that the intertemporal relationship is the strongest for the subperiod 2004-2011 which is the subperiod experiencing the highest and most volatile VIX. Sarwar (2012) concludes that the strength of the VIX-return relation is dependent on the mean and volatility of the VIX.

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The aim of this research is to contribute to academic literature as well as portfolio management and trading practice by extending the research on the relationship between a stock index and its implied volatility index, creating a market timing tool based on levels of implied volatility, and examine the performance of beta sorted portfolios compared to index performance over the period of January 1st 2004 to July 1st 2018. Beta in this case is the measure of volatility of the stock in comparison to the stock index or in other words it represents the sensitivity of the stock to movement in the index. Moreover this will be examined for the large cap S&P 500 index and the VIX as well as for the small cap Russell 2000 index and its implied volatility index, the RVX, which has not yet been investigated. This study deliberately focusses on US stock indices based on the findings of Bekiros et al. (2017) that the strongest intertemporal relationship between implied volatility and index return is found in the US market. The main hypothesis of this research is that high beta stocks have a higher future upside potential in periods of low volatility because of a negative overreaction in high volatility periods.

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This risk premium is calculated as the excess return of the market times the beta, which indicates that a higher beta would result in a higher expected return. Lastly, several studies indicated that using the VXO (old VIX) the relationship between the S&P 100 and the VXO is asymmetric (see, e.g., Giot 2005; Dennis, Mayhew and Strivers 2006). A possible explanation for this asymmetry is that the market has a more aggressive reaction to negative returns. This could mean that more risky stock (i.e. high beta stocks) experience an overreaction and hence, the market excessively rotates away from those stock resulting in potentially higher future returns.

The structure of the paper is as follows. Section 2 describes the data that is used in the analysis. Section 3 explains the methodology. Section 4 describes the main results. Section 5 provides the conclusion.

2. Data

2.1. Index and IV data

Daily adjusted closing data of the SPX and RUT has been obtained from DataStream for the period of 1-1-2004 until 29-6-2018. The daily close values of the VIX and RVX are downloaded from the Chicago Board Options Exchange (CBOE) website3. The index returns at time t of the S&P 500 and the Russell 2000 are calculated using the following equation:

𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 = LN (

𝑣𝑖𝑛𝑑𝑒𝑥,𝑡 𝑣𝑖𝑛𝑑𝑒𝑥,𝑡−1

) (1)

Where 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is the return of the index at time t, 𝑣𝑖𝑛𝑑𝑒𝑥,𝑡 is the index value at time t, and 𝑣𝑖𝑛𝑑𝑒𝑥,𝑡−1 is the index value at time t-1.

The implied volatility innovations are calculated in a similar way using:

𝑟𝐼𝑉,𝑡 = LN ( 𝑣𝐼𝑉,𝑡

𝑣𝐼𝑉,𝑡−1) (2)

Where 𝑟𝐼𝑉,𝑡 is the return of the implied volatility index at time t, 𝑣𝐼𝑉,𝑡 is the implied volatility index value at time t, and 𝑣𝐼𝑉,𝑡−1 is the implied volatility index value at time t-1.

Table 1 and table 2 display the descriptive statistics of the SPX and VIX and of the RUT and RVX, respectively. Figure 1 plots the SPX and VIX levels for the period of 1-1-2004 till 29-6-2018. Figure 2 plots this graph for the RUT and RVX levels.

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Page 6 of 34 Table 1

Descriptive statistics SPX and VIX Statistic

SPX Adj

Close SPX Return VIX Close

VIX Innovation Mean 1559.561 0.024% 18.430 -0.003% Standard Deviation 484.926 1.138% 8.872 7.086% Sample Variance 235153.524 0.013% 78.713 0.502% Kurtosis -0.369 12.830 9.431 7.550 Skewness 0.776 -0.384 2.635 1.020 Minimum 676.530 -9.470% 9.140 -35.059% Maximum 2872.870 10.957% 80.860 76.825% Sum 89.704% -12.432% Count 3781 3780 3781 3780

Correlation SPX Adj Close–VIX Close -0.463 Correlation SPX Return–VIX Return -0.738

Table 1 displays the descriptive statistics for the daily S&P 500 and VIX data over the period of 1-1-2004 until 29-6-2018.

Table 2

Descriptive statistics RUT and RVX Statistic

RUT Adj Close

RUT Return RVX Close RVX Innovation Mean 892.150 0.028% 23.948 -0.006% Standard Deviation 297.533 1.471% 9.605 5.478% Sample Variance 88526.106 0.022% 92.253 0.300% Kurtosis -0.531 6.492 7.652 5.905 Skewness 0.679 -0.381 2.427 0.759 Minimum 343.260 -12.614% 11.830 -36.428% Maximum 1706.990 8.861% 87.620 54.045% Sum 107.487% -22.030% Count 3781 3780 3781 3780

Correlation RUT Adj Close–RVX Close -0.549 Correlation RUT Return–RVX Return 0.035

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Page 7 of 34 Figure 1

S&P 500 adjusted close and VIX levels for the sample of 1-1-2004 till 29-6-2018

Figure 2

Russell 2000 adjusted close and RVX levels for the sample of 1-1-2004 till 29-6-2018

0 10 20 30 40 50 60 70 80 90 0 500 1000 1500 2000 2500 3000 3500 1-04 1-05 1-06 1-07 1-08 1-09 1-10 1-11 1-12 1-13 1-14 1-15 1-16 1-17 1-18 V IX L E V E L S & P 500 L E V E L

SP500 Adj Close VIX Close

0 10 20 30 40 50 60 70 80 90 100 0 200 400 600 800 1000 1200 1400 1600 1800 1-04 1-05 1-06 1-07 1-08 1-09 1-10 1-11 1-12 1-13 1-14 1-15 1-16 1-17 1-18 R V X L E V E L R U S S E L L 200 0 L E V E L

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2.2. Constituents data

In order to construct portfolios based on the stock betas data has to be obtained for all individual stocks that have been a constituent of the index over the period of 1-1-2000 until 29-6-2018. The required list of historical SPX constituents including the stock ticker and the time interval that the stock was part of the S&P 500 index is downloaded from the Wharton Research Data Services4 (WRDS) database. This results in a list of 849 unique constituents over the period of interest for which daily adjusted close prices are extracted from DataStream for the dates 1-1-2000 till 29-6-2018. Some constituents had to be excluded after the data extraction because either the stock ticker was invalid or DataStream had no data available for that particular company. This reduces the unique company count to 818. The stock beta is estimated using five years of historical data before the moment of occurring in the index and hence, the constituents for which the historical price data is less than five years before first appearance in the index or before the first day of the analysis has to be excluded from the sample. The final number of unique constituents is 710. Approximating the S&P 500 index for each day in the period of interest creates constituents lists ranging between 427 and 463 out of 500 companies. These are assumed to be close enough approximations of the index for performing the analyses.

The constituents data of the Russell 2000 comes with considerable data restriction. The fact that the Russell 2000 is a small cap index and has 2000 constituents makes that there is a high frequency of replacements in the constituents list making the data less easily accessible. The data used in this study is an annual constituents list observed at the end of July for the years 2009 till 2018 which is obtained from FTSE Russell5. This entails that the observable time period for the RUT-RVX analysis has been decreased to 1-8-2009 till 29-6-2018, probably not all constituents over the period of interest are included, and it has to be assumed that the constituents observed at year t stay in the index up to year t+1.

The annual constituents list from FTSE Russell displays a total of 4,336 unique stocks that have been included in the index between 1-8-2009 and 29-6-2018. The price data for these companies is downloaded from DataStream for the timespan of 1-1-2004 up to and including 29-6-2018. The companies for which DataStream could not provide data are excluded from the sample. Moreover, in order to analyze the data with the assumption of a constant constituents list for each year additional exclusions are made. Besides the earlier stated requirement that price data needs to be available for 5 years previous to the observation in the index (i.e. year t-5 of first observation) price data also has to be available over the whole year that the stock is assumed to be a constituent (i.e. year t+1 of last observation). The final unique company count is 2220 that create daily Russell

4 https://wrds-web.wharton.upenn.edu/wrds/

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2000 proxies with a number of constituents between 1134 and 1246 out of 2000 between 1-8-2009 and 29-6-2018.

3. Methodology

3.1. Regression analysis

The relationship between the stock index and IV index is examined using two different regression analyses. The first regressions aims at displaying the direct relation between the SPX and VIX and between the RUT and RVX.

𝑟𝐼𝑉,𝑡 = 𝛼 + 𝛽1𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 (3)

Where 𝑟𝐼𝑉,𝑡 is the implied volatility innovation at time t and 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is the index return at time t. The second analysis is performed to reveal if the sample is subject to any asymmetries in IV innovation with respect to index return, which has been documented in existing literature by Giot (200), Dennis, Mayhew and Strivers (2006), and Bekiros et al. (2017). In order to document the asymmetry this study implements the same regressions as used by Giot (2005).

𝑟𝐼𝑉,𝑡 = 𝛽0−𝐷𝑡−+ 𝛽0+𝐷𝑡++ 𝛽1−(𝑟𝑖𝑛𝑑𝑒𝑥,𝑡𝐷𝑡−) + 𝛽1+(𝑟𝑖𝑛𝑑𝑒𝑥,𝑡𝐷𝑡+) (4) Where 𝑟𝐼𝑉,𝑡 is the implied volatility innovation at time t, 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is the return of the index at time t, 𝐷𝑡− is a dummy variable that is equal to 1 when 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is negative and 0 when 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is

positive, 𝐷𝑡+ is a dummy variable that is equal to 1 when 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is positive and 0 when 𝑟𝑖𝑛𝑑𝑒𝑥,𝑡 is negative.

The output of regressions (3) and (4) for the SPX-VIX and RUT-RVX are presented in table 3. With regards to the SPX and VIX, all coefficients except the intercept from regression (3) are significantly different on a 1% percent level. Hence, the VIX innovations are significantly affected by the S&P 500 returns in general and are significantly affected by both positive and negative S&P 500 returns. These results are in line with findings in existing literature. The adjusted R2 of

regression (3) is 0.544 meaning that 54.4% of the variation in VIX innovation is explained by the S&P 500 return.

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In order to test for the presence of the asymmetric effect a Walt Test is performed to test if coefficient β1- is significantly different from β1+ in regression (4) for both the SPX-VIX and

RUT-RVX. The test results are presented on table 4.

Table 3

Results of regression analyses (3) and (4) for the S&P 500 – VIX and Russell 2000 – RUT S&P 500 - VIX Regression (3) 𝛼 𝛽1 𝑅̅2 𝑛 0.001 -4.595*** 0.544 3780 (0.001) (0.178) Regression (4) 𝛽0𝛽0+ 𝛽1𝛽1+ 𝑅̅2 𝑛 0.014*** -0.015*** -4.216*** -3.318*** 0.573 3780 (0.002) (0.002) (0.323) (0.248) Russell 2000 - RVX Regression (3) 𝛼 𝛽1 𝑅̅2 𝑛 0.000 0.130* 0.001 3780 (0.001) (0.178) Regression (4) 𝛽0𝛽0+ 𝛽1𝛽1+ 𝑅̅2 𝑛 0.004 0.001 0.398** -0.039 0.003 3780 (0.002) (0.001) (0.162) (0.112)

Table 3 presents the OLS output of the regression analyses as describes by equation (3) and equation (4) where (3) displays the direct relation between the index return on the IV innovation and (4) examines the possible asymmetry in effects for negative and positive index returns. In regression (4) β0- and β1

-represent the intercept and slope coefficient, respectively, when the index return is negative and β0+ and β1+ represent the intercept and slope coefficient, respectively, when the index return is positive. The 𝑅̅2 is the adjusted R-squared and n is the number of observations. White heteroskedastic robust standard errors are given in parentheses.

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Page 11 of 34 Table 4

Results of Wald Test β1- = β1+

Value S&P 500 - VIX -0.898** (0.407) Russell 2000 - RVX 0.436** (0.197)

The Wald Test results display that for both the SPX-VIX and RUT-RVX there is an asymmetric effect in reaction to index return. Hence, the asymmetry that negative returns in index have a larger effect on IV innovation that has earlier been documented for the SPX-VIX is confirmed in this sample on a 5% level. The positive value for the Wald Test output for the RUT-RVX data indicates that positive returns in RUT have a greater effect on RVX than negative returns. This asymmetry is significant on 5% level.

3.2.1. Determining high and low implied volatility

Figures 1 and 2 display that high and low levels of implied volatility are relative concepts. For example a VIX level of 30 would have been considered low in the beginning of 2009 and extremely high in 2017. Therefore, this study implemented a dynamic IV ranking model based on a system introduced by Giot (2005). Every observation of implied volatility is ranked based on the previous 500 observations. These observations are divided into 20 equally spaced quantiles and the hence, the IV observation at time t is assigned into a quantile based on the implied volatility levels of the previous 500 days. When the IV level at time t is very low compared to the 500 levels before it is assigned into the lower quantiles close to 1, a very high level will be assigned in a higher quantile near 20 and if the observed level at time t is higher than observed over the past 500 days it is assigned quantile 21. Table 5 displays the count of IV observations that has been assigned to each quantile and the corresponding proportion with respect to the total number observations for both the VIX and the RVX.

Table 4 presents the results of the Wald test in order to test the presence asymmetry in effects of positive and negative index returns on implied volatility innovations. The standard errors are reported in parentheses.

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Page 12 of 34 Table 5

IV count and proportion of total observation

VIX VIX RVX RVX

Quantile Count Proportion Count Proportion

1 547 16.67% 246 7.50% 2 494 15.06% 416 12.68% 3 530 16.15% 551 16.79% 4 364 11.09% 443 13.50% 5 239 7.28% 299 9.11% 6 160 4.88% 200 6.10% 7 128 3.90% 177 5.39% 8 108 3.29% 132 4.02% 9 112 3.41% 119 3.63% 10 113 3.44% 132 4.02% 11 99 3.02% 114 3.47% 12 77 2.35% 82 2.50% 13 66 2.01% 72 2.19% 14 48 1.46% 79 2.41% 15 52 1.58% 56 1.71% 16 37 1.13% 44 1.34% 17 35 1.07% 39 1.19% 18 20 0.61% 25 0.76% 19 11 0.34% 16 0.49% 20 14 0.43% 14 0.43% 21 27 0.82% 25 0.76%

Table 5 presents the count and proportion of total implied volatility observations that has been assigned to each of the 21 quantiles based on the previous 500 observations. Because of the use of 500 days as the reference period, the table displays this for the VIX and RVX over the period of 02-12-2005 till 29-6-2018.

3.2.2. Return per quantile

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and Peterson (2007)6. The presented S&P 500 results are not in line with the findings of Giot (2005) who found positive returns for high levels of VIX and negative returns for low levels of VIX. This distribution of positive and negative results might be dependent on the timeframe and the market developments in the period and hence, it is of importance to have a dynamic model.

Please note that in tables 6 and 7 the number of observations for each number of forward looking return is decreased with the corresponding number days with respect to the representations in table 5. This is because it is not possible to calculate for example the 60 days forward looking return for the last observation due to the fact that there are not enough future observations.

Table 6

Forward looking S&P 500 returns per VIX assigned quantile Quantile 1 Day Return 1 Day Std. Error 30 Days Return 30 Days Std. Error 60 Days Return 60 Days Std. Error 1 0.006% 0.023% 0.330% 0.151% 1.200% 0.211% 2 0.044% 0.030% 1.359% 0.154% 2.765% 0.197% 3 0.004% 0.034% 1.550% 0.157% 2.784% 0.225% 4 0.121% 0.044% 1.392% 0.250% 2.826% 0.326% 5 -0.025% 0.054% 1.386% 0.258% 2.387% 0.366% 6 0.035% 0.090% 0.795% 0.339% 2.387% 0.453% 7 0.147% 0.106% 1.069% 0.401% 2.921% 0.452% 8 -0.068% 0.118% 1.559% 0.555% 2.520% 0.690% 9 -0.191% 0.131% -1.259% 0.736% -0.557% 1.059% 10 0.125% 0.164% -1.327% 0.848% -1.869% 1.398% 11 -0.284% 0.174% -0.671% 0.877% -1.029% 1.412% 12 0.040% 0.192% 0.615% 0.917% -1.160% 1.535% 13 -0.195% 0.241% -1.352% 0.815% -5.817% 1.450% 14 0.255% 0.259% -0.867% 1.119% -4.165% 1.740% 15 -0.326% 0.289% 0.531% 1.026% -2.215% 1.118% 16 0.198% 0.371% 0.587% 0.970% -0.804% 1.023% 17 0.008% 0.449% -0.459% 1.578% -1.158% 2.009% 18 0.732% 0.427% -0.418% 2.212% -1.225% 2.610% 19 -1.246% 1.243% -2.586% 4.559% -4.236% 5.443% 20 0.095% 0.587% -8.580% 3.309% -9.272% 3.551% 21 1.257% 0.827% -0.649% 2.220% -0.054% 1.910%

Table 6 presents the 1, 30, and 60 days forward looking S&P 500 returns and corresponding standard errors for the 21 quantiles the observed VIX levels are assigned to in the time period January 1st 2004 – June 29th 2018 where the first 500 observations are used as a reference period.

6 The analysis is performed for any number of forward looking return between 1 and 500. The pattern of increased

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Table 7

Forward looking Russell 2000 returns per RVX assigned quantile Quantile 1 Day Return 1 Day Std. Error 30 Days Return 30 Days Std. Error 60 Days Return 60 Days Std. Error 1 0.231% 0.046% 0.980% 0.323% 0.115% 0.418% 2 0.243% 0.039% 0.054% 0.242% 0.563% 0.346% 3 0.132% 0.044% 1.591% 0.239% 2.587% 0.325% 4 0.102% 0.053% 1.993% 0.242% 2.622% 0.391% 5 -0.052% 0.073% 1.737% 0.332% 3.380% 0.467% 6 0.059% 0.086% 1.725% 0.413% 3.887% 0.513% 7 0.020% 0.097% 2.481% 0.509% 5.933% 0.511% 8 -0.280% 0.124% 2.959% 0.528% 5.028% 0.600% 9 0.012% 0.137% 3.102% 0.700% 5.912% 1.152% 10 0.322% 0.206% 1.945% 0.905% 6.168% 1.539% 11 0.000% 0.182% 0.497% 0.854% 2.290% 1.342% 12 0.017% 0.244% 0.155% 1.184% 1.635% 1.874% 13 -0.022% 0.243% -1.584% 0.846% 0.093% 1.535% 14 0.015% 0.229% -1.758% 0.979% -6.413% 1.227% 15 -0.008% 0.332% -0.402% 1.007% -5.748% 1.354% 16 -0.049% 0.490% -1.774% 0.906% -5.088% 1.275% 17 -0.363% 0.418% 0.707% 1.854% -2.152% 2.058% 18 -0.786% 0.767% -0.355% 2.316% -1.492% 2.935% 19 -0.932% 1.185% -4.129% 4.405% -3.286% 3.436% 20 -0.430% 0.677% -8.647% 4.140% -8.478% 4.389% 21 -1.267% 0.697% -16.901% 3.126% -15.667% 3.710%

Table 7 presents the 1, 30, and 60 days forward looking Russell 2000 returns and corresponding standard errors for the 21 quantiles the observed RVX levels are assigned to in the time period January 1st 2004 – June 29th 2018 where the first 500 observations are used as a reference period.

3.3. Creating portfolios sorted on beta

In order to examine the hypothesis that high beta stocks experience a negative overreaction in returns during high volatile periods and thus, have a higher future upside potential in low volatile periods, this study estimates the beta at each day of the period for each company that is assumed to be a constituent of the index given the constituents data explained in section 3.27. The beta is explained as a measure of volatility of the stock compared to a benchmark and is estimated using the following formula:

7 This extensive every-day beta estimation is performed to examine the different beta portfolios throughout the

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𝛽 =𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝑟𝑖, 𝑟𝑚) 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝑟𝑚)

(5) Where 𝑟𝑖 is the daily return of the individual constituent and 𝑟𝑚 is the daily return of the index, which in this analysis will be the S&P 500 and the Russell 2000 for the SPX-VIX and RUT-RVX analysis, respectively8. The S&P 500 and the Russell 2000 are chosen as the benchmark because the model implements a trading strategy based on the SPX-VIX and RUT-RVX relation and hence, this study is interested in investigating portfolios that are more or less subject to that relation as a result of the difference in sensitivity to movement in the index. The stock betas for the S&P 500 are calculated starting at January 1st 2006, because of the use of a 500 day (i.e. almost two years)

reference period for the VIX. The calculations for the Russell start at August 1st 2009, given the data restrictions explained in section 3.2. For each day in the analysis the beta of the constituents will be estimated using five years of historical returns, ranked from high to low beta and split into five quantiles. The use of five years of historical data is to achieve a certain level of accuracy regarding the estimation. However, there still will be high individual betas that are over estimated in relation to their true beta and low individual betas that are under estimated in relation to their true beta according to Jensen, Black and Scholes (1972). These estimation errors are expected to average out by the use of five quantiles resulting in portfolios of around 88 stocks for the S&P 500 and around 239 stocks for the Russell 2000 stocks, given the exclusions as explained in section 3.2. The use of five quantiles is in line with the analysis of Fama and French (1996). The stocks in the beta portfolios are equally weighed in order to solely examine the effect of the beta without including capitalization effect. Table 8 displays descriptive statistics for each of the five beta portfolios for both the S&P 500 and the Russell 2000.

8 The calculation of the beta is also performed using the MSCI USA and MSCI World as benchmark. The use of the

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Page 16 of 34 Table 8

Descriptive statistics of the beta portfolios

High 4 3 2 Low

S&P 500 Number of Portfolios 3,260 3,260 3,260 3,260 3,260 Average Constituents per Date 89 88 88 88 89 Total Beta Observation 289,113 287,513 287,983 287,477 289,965 Maximum Beta 1.049 0.637 0.523 0.432 0.349 Average Beta 0.576 0.448 0.371 0.298 0.207 Minimum Beta 0.348 0.275 0.213 0.162 -0.005 Russell 2000 Number of Portfolios 2,326 2,326 2,326 2,326 2,326 Average Constituents per Date 239 239 238 239 239 Total Beta Observation 556,175 555,444 554,731 555,445 556,949 Maximum Beta 1.109 0.509 0.392 0.314 0.228 Average Beta 0.541 0.381 0.296 0.223 0.113 Minimum Beta 0.315 0.225 0.158 0.105 -0.069

Table 8 presents descriptive statistics regarding the five on beta sorted portfolios for the S&P 500 and the Russell 2000. The portfolios are created on each day of the period of interest using the method explained in section 3.3. The number of portfolios is equal to the number of days in the timespan, the average constituents per date displays how many stocks on average are included in each portfolio at each day of the period. The total beta observations reflects the sum of the number of stocks per date for each day in the period. For the S&P 500 the portfolios are formed over the period of January 1st 2006 to June 29th 2018 and for the Russell 2000 from August 1st 2009 to June 29th 2018 as explained in section 3.2.

3.4. Market timing and portfolio comparison

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Page 17 of 34

strategy, because the starting position is neutral and one would be able to borrow money and invest it when the strategy signals to invest.

The main function of the long only market timing strategy is that it provides one of three signals based on the assigned quantile of the observed implied volatility level. The signals are buy, sell, and do not act. Therefore, the basis of the strategy is to define the range of IV assigned quantiles in which the strategy will be invested (i.e. buy) and the range of quantiles where the strategy has no position (i.e. sell). These ranges will be referred to as the invest quantiles and divest quantiles, respectively. For example if the invest quantiles are defined by the range [1,10] it means that the strategy takes a long when the observed implied volatility at time t has been assigned a quantile between the quantiles 1 up to and including 10 based on the 500 previous IV observations as explained in section 3.2.1. and if the divest quantiles range from [11,21] it means that the strategy takes a neutral position, so either sells a current long position or stays uninvested, if it encounters an observed implied volatility at time t that is assigned a quantile ranging from 11 up to and including 21 based on the assigning explained in section 3.2.1. The invest quantile range and divest quantile range are not necessarily adjoint. The model for example also tests invest and divest quantiles of [1,7] and [9,21], respectively. In this case the same rule applies as before with the addition that if the observed implied volatility at time t is assigned a quantile that is neither in the invest quantiles, nor in the divest quantiles (i.e. in quantile 8) the model does not act. That means that if the strategy currently is invested it stays invested, if the strategy currently is not invested it stays not invested. In this situation the model has no opinion on the market movement. Moreover, based on the findings of Giot (2005), Banerjee, Doran and Peterson (2007) and the returns displayed in tables 6 and 7, once an investment is made it will be invested for a holding period of 60 days unless an IV level is observed that is in the divest quantiles and the model gives a sell signal, then the investment is sold prematurely. Again, if IV levels are encountered that are neither in the invest, nor in the divest quantiles the investment carries on without interruption until the 60 days are reached because the model does not act and stays invested if it already is invested. If an investment is hold for 60 days, the model will check again if an investment should be made based on the implied volatility level. This 60 days rule does not have much implication for investing in the index, but it has for investing in a beta sorted portfolio. If the model triggers an investment, that particular set of stocks, observed as being a certain beta portfolio at time t, will be the investment portfolio for the investment period up to the maximum of 60 holding days. This means that the portfolio will not be adjusted over those 60 days in order to avoid transaction costs due to every day adjustment and rebalancing of portfolios consisting of around 88 or 239 stocks.

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Page 18 of 34

period) and the beta portfolio returns will be compared to the index return of the corresponding market timing strategy. For the index this means that implementing market timing creates a string of daily returns with similar values to the string of returns without market timing. However, the string of returns when implementing market timing will contain values of return equal to zero when the model is not invested. The string of returns for the different beta sorted portfolios are created by using the same timing as the index timing for that particular timing strategy. In other words, the market timing is established on the index level based on the implied volatility levels and the beta portfolios are invested or not invested at the same moment as the index when implementing a specific strategy. Moreover, for days that a beta portfolio is invested the daily return is calculated as the average return of the individual stocks that are part of the investment portfolio at that point in time. Taking the average of the stocks creates and equally weighted portfolio and therefore, solely examines the effect of difference in beta and excludes any effects due to market capitalization. This again creates a string of daily returns over the full time period of which some values might be zero based on the market timing strategy that is used.

4. Results

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portfolios is with respect to the index return of that particular strategy. In panel B the indicated significance of the index return displays if the value is different from zero and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. Although tables 9 and 11 cannot be directly compared because of the difference in time periods, the patterns in returns can be examined.

4.1. S&P 500 and VIX

Table 9

Results of the market timing trading strategy for S&P 500 Panel A

Invest -

Divest # Index High 4 3 2 Low

0. [1,21] - 77.83% 61.44%*** 76.63%*** 57.57%*** 34.94%*** -20.64%*** (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 1. [1,8] - [9,21] 94 83.93%*** 65.12%*** 89.77%*** 84.72%*** 70.50%*** 61.95%*** (0.013%) (0.011%) (0.012%) (0.014%) (0.016%) (0.019%) 2. [1,7] - [9,21] 68 102.05%*** 79.95%*** 108.67%*** 105.74%*** 92.82%*** 86.05%*** (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.019%) 3. [1,8] - [13,21] 59 87.40%*** 73.37%*** 94.63%*** 88.95%*** 65.97%*** 47.06%*** (0.014%) (0.012%) (0.013%) (0.015%) (0.017%) (0.020%) Panel B Avoided

return Index High 4 3 2 Low

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Page 20 of 34

First of all, one can see that the beta portfolio returns together do not match the index return. This could be explained by the weighting of the portfolios. The beta portfolios are equally weighted where the index is weighted based on market capitalization. For example with regards to the reference return where all beta portfolio returns are lower than the index return, if large firms have outperformed small firms over the timespan, this is reflected in the index, but not in the portfolios. Hence, the index will perform better than the portfolios due to market capitalization weighting. Another cause might be that the analysis misses valuable information because of the companies that have been excluded. However, this is not likely to have had a huge effect because under 15% of the constituents had to be excluded. Three different implied volatility market timing strategies are presented in table 99. Strategy 1 refers to the findings in table 6 and includes the positive returns

in the invest quantiles and the negative returns in the divest quantiles for 60 holding days and thus, the two sets of quantiles are adjoint. Strategy 2 is chosen to display because it maximizes the index return and for this strategy the invest and divest quantiles are not adjoint. The third strategy is tested based on the findings in table 6 and focusses on the lower eight and upper nine quantiles leaving the quantiles in between neutral that have 60 days forward looking return lower than 2%.

Moreover, table 9 shows that the index return can be significantly improved on a 1% level using implied volatility market timing with an optimized return of 102.05% for strategy 2 compared to the reference return of 73.83% for investing when the VIX level is assigned quantile 1 up to and including 7, not acting when it is assigned quantile 8 and divesting when the observed VIX is in quantile 9 or higher. When implementing implied volatility market timing the trend can be observed that beta portfolios 4 and 3 significantly outperform the index on a 1% level for the three different strategies. The other portfolios (i.e. high, 2, and low) significantly underperform the index on a 1%. For all three strategies the high beta portfolio return is significantly higher than the low beta portfolio return on a 1% level. The significance is tested by computing the confidence intervals at a 1% significance level for all returns based on the standard errors. In all cases the confidence intervals do not overlap and thus, the results are significantly different on a 1% level.

Therefore, the highest beta portfolio in the S&P 500 does not outperform the index in low volatility periods. However, portfolio 4 and 3 do outperform the index in low volatility periods. Panel B of table 9 also shows that there is no negative overreaction in high beta stocks. The avoided returns display that by using market timing most negative returns are avoided for the low beta portfolios. The avoided index returns are significantly different from zero and the portfolio avoided returns are significantly different from the avoided index return, all on a 1% level.

9 More combinations of invest quantiles and divest quantiles have been tested, but these are chosen to display

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The difference in returns for high beta stocks and low beta stocks is further examined in table 10 which presents the return results when implementing the index optimizing strategy 2 for three beta portfolios. The three beta portfolios in table 10 contain the same portfolios as the analysis in table 9, but the portfolios high and 4 have been merged into one high beta portfolio and portfolio s 2 and low have been merged into one low beta portfolio. It can be observed in table 10 that for the reference return the three beta portfolios have significantly lower returns than the index on a 1% level. With the implementation of strategy 2 beta portfolio 3 has significantly higher returns than the index on a 1% level, but the merged portfolios high and low underperform the index on a 1% level. Again the merged high portfolio is significantly higher than the low portfolio on a 1% level. Figure 3 shows the graphical representation of the cumulative returns of the index without implementing market timing and of the index, the merged high beta and merged low beta portfolios when implementing implied volatility market timing strategy 2. It can be observed that implementing the market timing strategy eliminates the negative returns during the 2007-2009 financial crisis.

Table 10

Results of market timing strategy 2 for three beta portfolios for S&P 500 Panel A

Invest - Divest # Index High 3 Low

0. [1,21] - 77.83% 69.09%*** 57.57%*** 7.02%***

(0.021%) (0.020%) (0.023%) (0.028%) 2. [1,7] - [9,21] 68 102.05%*** 94.30%*** 105.74%*** 89.40%***

(0.012%) (0.011%) (0.013%) (0.017%) Panel B

Avoided return Index High 3 Low

2. [1,7] - [9,21] -24.23%*** -25.20%*** -48.17%*** -82.39%*** (0.017%) (0.016%) (0.019%) (0.022%) Table 10 displays in Panel A the reference return for investing for every level of VIX and the return results for market timing strategy 2 from table 9 and as explained in section 3.4. for the S&P 500. The three beta portfolios are constructed by merging the highest two beta portfolios from table 9 (i.e. high and 4), merging the lowest two portfolios from table 9 (i.e. low and 2), and keeping the center portfolio 3 unchanged. The count in the second column explains how many times a long position is taken for that particular market timing strategy and hence, shows the number of times the beta portfolios are rebalanced. The reference return displays no count because it represents a long position for every day in the sample. Panel B gives the return that have been avoided because of the implemented market timing strategy and is calculated as the reference return minus the return obtained using market timing. In panel A the indicated significance of the index return is with respect to the index reference return and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. In panel B the indicated significance of the index return displays if the value is different from 0 and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. The standard errors are given in parentheses. The timespan is January 1st 2006 till June 29th 2018.

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Page 22 of 34 Figure 3

Cumulative returns when implementing strategy 2 for the S&P 500

Figure 3 presents cumulative returns of the index without implementing market timing and of the index, the merged high beta and merged low beta portfolios when implementing implied volatility market timing strategy 2 for the S&P 500over the period of 1-1-2006 to 29-6-2018.

-80% -60% -40% -20% 0% 20% 40% 60% 80% 100% 120% 1-06 1-07 1-08 1-09 1-10 1-11 1-12 1-13 1-14 1-15 1-16 1-17 1-18 Cu m ul at iv e re tur n Date

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4.2. Russell 2000 and RVX

Table 11

Results of the market timing trading strategy for Russell 2000 Panel A

Invest -

Divest # Index High 4 3 2 Low

0. [1,21] - 106.61%*** 93.14%*** 84.12%*** 73.94%*** 50.20%*** -35.18%*** (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 1. [1,13] - [14,21] 56 151.36%*** 120.95%*** 124.28%*** 122.79%*** 101.28%*** 24.63%*** (0.024%) (0.021%) (0.023%) (0.025%) (0.026%) (0.027%) 2. [1,11] - [14,21] 50 154.15%*** 126.07%*** 128.97%*** 123.73%*** 105.68%*** 26.64%*** (0.023%) (0.020%) (0.023%) (0.024%) (0.026%) (0.026%) 3. [3,11] - [14,21] 50 135.99%*** 108.49%*** 108.28%*** 107.16%*** 88.72%*** 11.57%*** (0.023%) (0.020%) (0.023%) (0.024%) (0.025%) (0.026%) Panel B

Avoided returns Index High 4 3 2 Low

1. [1,13] - [14,21] -44.75%*** -27.48%*** -40.90%*** -46.76%*** -53.00%*** -54.76%*** (0.010%) (0.009%) (0.010%) (0.011%) (0.012%) (0.012%) 2. [1,11] - [14,21] -47.54%*** -32.61%*** -45.58%*** -47.69%*** -57.40%*** -56.77%*** (0.011%) (0.009%) (0.011%) (0.011%) (0.012%) (0.013%) 3. [3,11] - [14,21] -29.38%*** -15.02%*** -24.89%*** -31.13%*** -40.45%*** -41.70%*** (0.012%) (0.010%) (0.011%) (0.012%) (0.013%) (0.014%) Table 11 presents in panel A the return results of the long-only market timing strategy as explained in section 3.4. for the index and the five beta sorted portfolios of the Russell 2000. The first row of panel A represents the reference return, which is explained as the return that would have been obtained when being invested for any level of IV. The other rows indicate the return when implementing implied volatility market timing given the invest and divest quantiles. The numbers in front of the quantiles are strategy identifiers. The count in the second column explains how many times a long position is taken for that particular market timing strategy and hence, shows the number of times the beta portfolios are rebalanced. The reference return displays no count because it represents a long position for every day in the sample. Panel B gives the return that have been avoided because of the implemented market timing strategy and is calculated as the reference return minus the return obtained using market timing. In panel A the indicated significance of the index return is with respect to the index reference return and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. In panel B the indicated significance of the index return displays if the value is different from 0 and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. The standard errors are given in parentheses. The timespan is August 1st 2009 till June 29th 2018.

*** Significant at 1%

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Page 24 of 34

results in a higher return for the index compared to the beta portfolios. However, because of the higher rate of exclusions and the assumption that the companies are a constituent for a full year as mentioned in section 2.2. it is more likely that the data restrictions influence the portfolio return for the RUT analysis. Furthermore, table 11 again shows three different market timing strategies10. Strategy 1 is based on table 7 where all quantiles displaying positive returns for 60 holding days are included in the invest quantiles and the quantiles displaying negative returns are included in the divest quantiles, this makes for adjoint invest and divest quantiles. Strategy 2 is chosen because it maximizes the index return and strategy 3 focusses for the invest quantiles on the quantiles displaying a return greater than 2% in table 7.

Moreover, table 11 shows that implementing implied volatility market timing significantly improves returns for the Russell 2000 on a 1% level. Although, all beta portfolios significantly underperform the market on a 1% level one can still observe a similar pattern for the Russell 2000 beta portfolios as is observed for the S&P 500 portfolios which is that portfolio 4 has (except for strategy 3) the highest returns and that the low beta portfolio has the lowest return. Similar to what is found for the S&P 500 the high beta portfolios significantly outperform the low beta portfolios on a 1% level for each strategy.

Panel B of table 11 shows that the avoided returns for the index are significantly different from zero on a 1% level and that all the avoided returns for the beta portfolios are statistically different from the index avoided return on a 1% level. Moreover, panel B displays that low beta stocks experience greater negative returns in high volatility periods where high beta stocks experience less negative returns compared to the index.

Table 12 shows the results for the reference return and the return for implementing strategy 2 with the merged higher two and lower two beta portfolios. When strategy 2 is implemented the beta portfolios perform significantly worse than the index on a 1% level and the high beta portfolio return is significantly higher than the low beta portfolio return on a 1% level. Regarding the avoided return, it is shown that that the avoided returns for the high beta portfolio are significantly lower than the avoided index return in high volatility periods and the low beta portfolio experiences significantly more negative returns in a high volatility environment, both on a 1% level. Figure 4 shows the graphical representation of the cumulative returns of the index without implementing market timing and of the index, the merged high beta portfolio and merged low beta portfolios when implementing implied volatility market timing strategy 2.

10 More combinations of invest quantiles and divest quantiles have been tested, but these are chosen to display

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Page 25 of 34 Table 12

Results of market timing strategy 2 for three beta portfolios for Russell 2000 Panel A

Invest - Divest # Index High 3 Low

0. [1,21] - 106.61% 88.64%*** 73.94%*** 7.44%***

(0.026%) (0.024%) (0.027%) (0.029%) 2. [1,11] - [14,21] 50 154.15%*** 127.52%*** 123.73%*** 66.09%***

(0.023%) (0.021%) (0.024%) (0.026%) Panel B

Avoided return Index High 3 Low

2. [1,11] - [14,21] -47.54%*** -38.88%*** -49.79%*** -58.65%*** (0.012%) (0.024%) (0.027%) (0.029%) Table 12 displays in Panel A the reference return for investing for every level of RVX and the return results for market timing strategy 2 from table 11 and as explained in section 3.4. for the Russell 2000. The three beta portfolios are constructed by merging the highest two beta portfolios from table 9 (i.e. high and 4), merging the lowest two portfolios from table 11 (i.e. low and 2), and keeping the center portfolio 3 unchanged. The count in the second column explains how many times a long position is taken for that particular market timing strategy and hence, shows the number of times the beta portfolios are rebalanced. The reference return displays no count because it represents a long position for every day in the sample. Panel B gives the return that have been avoided because of the implemented market timing strategy and is calculated as the reference return minus the return obtained using market timing. In panel A the indicated significance of the index return is with respect to the index reference return and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. In panel B the indicated significance of the index return displays if the value is different from 0 and the indicated significance of the beta portfolios is with respect to the index return of that particular strategy. The standard errors are given in parentheses. The timespan is August 1st 2009 till June 29th 2018.

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Page 26 of 34 Figure 4

Cumulative returns when implementing strategy 2 for the Russell 2000

Figure 4 presents cumulative returns of the index without implementing market timing and of the index, the merged high beta and merged low beta portfolios when implementing implied volatility market timing strategy 2 for the S&P 500over the period of 1-8-2009 to 29-6-2018.

4.3. Robustness

In order to check the robustness of this analysis this study examines the different outcomes when some of the main parameters of the analysis are changed. Interesting to review is the quantile assignment of the IV levels when the number of reference days is changed. The main analysis includes 500 IV observations as reference for determining what high and low levels of implied volatility are. Tables 13 and 14 display the count, the proportion of total, the 60 days forward looking returns, and the 60 days standard error for the 21 quantiles using IV reference observations of 100, 250, and 500 for the S&P 500 – VIX and Russell 2000 – RVX, respectively. It can be seen that decreasing the reference days, in general, increases the count and proportion of IV observations for the highest and lowest quantiles. This is because the smaller the reference period, the higher the probability that the observed IV level is higher than observed in the reference period. A smaller reference window also results in lower and less distinctive 60 day forward looking returns because of an overestimation and underestimation in IV levels. For example a certain level of IV is assigned quantile 21 because it is higher than occurs in the reference window, but the observation is not truly high. However, the reference window is too small to provide a accurate estimation.

-20% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 8-09 8-10 8-11 8-12 8-13 8-14 8-15 8-16 8-17 Cu m ul at iv e re tur n Date

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Page 27 of 34

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Page 28 of 34 Table 13

Quantile information for different reference days of VIX

100 250 500

Quantile Count Proportion 60 Day Return

60 Day

Std. Err Count Proportion

60 Day Return

60 Day

Std. Err Count Proportion

60 Day Return 60 Day Std. Err 1 478 14.57% 2.58% 0.002 443 13.50% 1.04% 0.002 547 16.67% 1.20% 0.002 2 321 9.78% 1.49% 0.003 422 12.86% 1.74% 0.002 494 15.06% 2.77% 0.002 3 369 11.25% 2.24% 0.003 415 12.65% 2.24% 0.003 530 16.15% 2.78% 0.002 4 293 8.93% 1.70% 0.003 324 9.88% 2.84% 0.003 364 11.09% 2.83% 0.003 5 244 7.44% 1.64% 0.004 300 9.14% 2.45% 0.004 239 7.28% 2.39% 0.004 6 236 7.19% 1.51% 0.005 222 6.77% 0.89% 0.006 160 4.88% 2.39% 0.005 7 216 6.58% 0.12% 0.006 192 5.85% 0.82% 0.007 128 3.90% 2.92% 0.005 8 166 5.06% -0.43% 0.007 162 4.94% 1.66% 0.006 108 3.29% 2.52% 0.007 9 152 4.63% 0.26% 0.006 137 4.18% 0.34% 0.008 112 3.41% -0.56% 0.011 10 129 3.93% 0.95% 0.007 123 3.75% 2.52% 0.008 113 3.44% -1.87% 0.014 11 105 3.20% 1.76% 0.007 104 3.17% 1.03% 0.010 99 3.02% -1.03% 0.014 12 109 3.32% 1.10% 0.008 80 2.44% -0.61% 0.012 77 2.35% -1.16% 0.015 13 83 2.53% 1.94% 0.008 70 2.13% -0.13% 0.009 66 2.01% -5.82% 0.015 14 76 2.32% 0.29% 0.009 64 1.95% -1.15% 0.009 48 1.46% -4.17% 0.017 15 61 1.86% 1.77% 0.009 45 1.37% 0.30% 0.009 52 1.58% -2.22% 0.011 16 53 1.62% -0.53% 0.015 41 1.25% -1.18% 0.016 37 1.13% -0.80% 0.01 17 37 1.13% 2.69% 0.014 33 1.01% 1.22% 0.017 35 1.07% -1.16% 0.02 18 24 0.73% 3.31% 0.016 18 0.55% -1.29% 0.028 20 0.61% -1.23% 0.026 19 22 0.67% 0.68% 0.026 12 0.37% -2.45% 0.044 11 0.34% -4.24% 0.054 20 21 0.64% 0.83% 0.020 14 0.43% -5.37% 0.032 14 0.43% -9.27% 0.036 21 69 2.10% 1.95% 0.009 43 1.31% 0.79% 0.012 27 0.82% -0.05% 0.019

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Page 29 of 34 Table 14

Quantile information for different reference days of RVX

100 250 500

Quantile Count Proportion 60 Day Return

60 Day

Std. Err Count Proportion

60 Day Return

60 Day

Std. Err Count Proportion

60 Day Return 60 Day Std. Err 1 381 11.61% 4.15% 0.004 260 7.92% 1.85% 0.003 246 7.50% 0.12% 0.005 2 281 8.56% 3.42% 0.004 306 9.33% 0.95% 0.004 416 12.68% 0.56% 0.004 3 238 7.25% 2.53% 0.005 355 10.82% 2.12% 0.004 551 16.79% 2.59% 0.004 4 232 7.07% 1.04% 0.005 330 10.06% 1.36% 0.005 443 13.50% 2.62% 0.005 5 273 8.32% 2.30% 0.005 331 10.09% 1.65% 0.006 299 9.11% 3.38% 0.006 6 246 7.50% 0.93% 0.005 249 7.59% 2.20% 0.006 200 6.10% 3.89% 0.005 7 237 7.22% -0.50% 0.007 224 6.83% 1.82% 0.007 177 5.39% 5.93% 0.005 8 222 6.77% 1.00% 0.006 193 5.88% 3.31% 0.006 132 4.02% 5.03% 0.006 9 181 5.52% 1.79% 0.007 186 5.67% 2.86% 0.009 119 3.63% 5.91% 0.008 10 176 5.36% 1.55% 0.006 174 5.30% 3.01% 0.007 132 4.02% 6.17% 0.008 11 133 4.05% 3.11% 0.008 133 4.05% 3.87% 0.009 114 3.47% 2.29% 0.007 12 111 3.38% 1.35% 0.010 96 2.93% -0.05% 0.012 82 2.50% 1.64% 0.007 13 100 3.05% 1.89% 0.009 91 2.77% 1.63% 0.008 72 2.19% 0.09% 0.006 14 106 3.23% 0.79% 0.010 76 2.32% 0.23% 0.011 79 2.41% -6.41% 0.013 15 57 1.74% 0.89% 0.012 62 1.89% 0.89% 0.010 56 1.71% -5.75% 0.015 16 60 1.83% 0.62% 0.012 57 1.74% -1.41% 0.014 44 1.34% -5.09% 0.017 17 63 1.92% 2.06% 0.007 39 1.19% 0.56% 0.009 39 1.19% -2.15% 0.01 18 38 1.16% 1.58% 0.017 30 0.91% 0.48% 0.022 25 0.76% -1.49% 0.015 19 33 1.01% -0.37% 0.019 22 0.67% -4.08% 0.025 16 0.49% -3.29% 0.03 20 26 0.79% -0.35% 0.025 14 0.43% -5.74% 0.042 14 0.43% -8.48% 0.037 21 70 2.13% -4.69% 0.018 36 1.10% -10.02% 0.028 25 0.76% -15.67% 0.037

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Page 30 of 34 Table 15

Robustness different reference days and holding days for the S&P 500 and VIX

VIX days

Holding

days Strategy # Index High 4 3 2 Low

100 30 0 110 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 149 39.28% 14.28% 37.61% 39.02% 36.40% 21.78% (0.012%) (0.010%) (0.013%) (0.014%) (0.016%) (0.020%) 60 0 56 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 118 38.33% 13.44% 37.07% 39.02% 35.13% 19.06% (0.012%) (0.010%) (0.013%) (0.014%) (0.016%) (0.020%) 100 0 34 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 112 38.33% 13.06% 38.83% 37.35% 35.23% 21.06% (0.012%) (0.010%) (0.013%) (0.014%) (0.016%) (0.020%) 250 30 0 110 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 125 82.82% 64.55% 89.45% 92.24% 78.31% 65.41% (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.018%) 60 0 56 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 88 82.45% 64.52% 89.20% 92.72% 77.28% 63.18% (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.018%) 100 0 34 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 75 82.45% 63.37% 91.01% 94.66% 75.00% 60.12% (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.018%) 500 30 0 110 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 109 102.05% 81.40% 107.52% 104.50% 93.67% 88.20% (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.019%) 60 0 56 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 68 102.05% 79.95% 108.67% 105.74% 92.82% 86.05% (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.019%) 100 0 34 77.83% 61.44% 76.63% 57.57% 34.94% -20.64% (0.021%) (0.018%) (0.021%) (0.023%) (0.026%) (0.031%) 2 53 102.05% 79.68% 108.81% 107.10% 93.23% 84.76% (0.012%) (0.010%) (0.012%) (0.013%) (0.015%) (0.019%)

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Page 31 of 34 Table 16

Robustness different reference days and holding days for the Russell 2000 and RVX

RVX days

Holding

days Strategy # Index High 4 3 2 Low

100 30 0 79 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 102 240.12% 191.56% 209.62% 216.77% 210.95% 137.51% (0.021%) (0.018%) (0.021%) (0.022%) (0.023%) (0.024%) 60 0 40 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 70 239.93% 190.86% 209.74% 216.48% 208.88% 135.23% (0.021%) (0.018%) (0.021%) (0.022%) (0.023%) (0.024%) 100 0 25 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 59 239.93% 190.96% 208.04% 218.06% 207.69% 135.30% (0.021%) (0.018%) (0.021%) (0.022%) (0.023%) (0.024%) 250 30 0 79 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 92 204.28% 168.15% 176.24% 174.35% 160.73% 79.79% (0.022%) (0.019%) (0.022%) (0.023%) (0.025%) (0.026%) 60 0 40 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 56 204.28% 167.71% 176.51% 173.73% 159.17% 77.23% (0.022%) (0.019%) (0.022%) (0.023%) (0.024%) (0.025%) 100 0 25 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 44 205.57% 167.76% 174.67% 176.31% 163.48% 79.52% (0.022%) (0.019%) (0.022%) (0.023%) (0.024%) (0.025%) 500 30 0 79 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 88 154.15% 126.61% 128.32% 124.59% 107.72% 27.75% (0.023%) (0.020%) (0.023%) (0.024%) (0.026%) (0.027%) 60 0 40 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 50 154.15% 126.07% 128.97% 123.73% 105.68% 26.64% (0.023%) (0.020%) (0.023%) (0.024%) (0.026%) (0.026%) 100 0 25 106.61% 93.14% 84.12% 73.94% 50.20% -35.18% (0.026%) (0.022%) (0.025%) (0.027%) (0.029%) (0.030%) 2 38 155.43% 127.20% 127.12% 127.90% 111.65% 28.45% (0.023%) (0.020%) (0.023%) (0.024%) (0.025%) (0.026%)

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Page 32 of 34

5. Conclusion

This research uses the prior documented relationship between implied volatility and index return to create an implied volatility market timing strategy and to investigate whether high beta and low beta portfolios created from the index constituents experience different returns based on the hypothesis that in a high volatility environment investors rotate away from high risk (i.e. high beta) stocks resulting in a negative overreaction and hence, have a higher future upside potential in comparison to low beta stocks and the index. This is being researched for the large cap S&P 500 index and corresponding implied volatility index VIX and the small cap Russell 2000 index and corresponding implied volatility index RVX in order to compare the differences in effects for large cap and small cap stocks. This builds on previous research in three ways. First, the study reviews some of the earlier findings regarding the index return and IV relationship for the time period starting in 2004 that is documented to have the strongest intertemporal relationship (Sawar, 2012) and that uses the VIX based on the S&P 500 instead of the S&P 100. Second, This study pioneers the research on the index return and implied volatility relationship for the small cap Russell 2000 index. Third, this study analyzes index returns and the differences in return for portfolios solely sorted on beta when implementing implied volatility market timing.

It is shown that the beta sorted portfolio returns do not completely match the index return probably because of the market capitalization weighting in the index and equally weighting in the beta portfolios and probably partly because of data restrictions. This may occur when larger capitalized stocks outperform smaller capitalized stocks. This difference is greater for the small cap Russell 2000 index than for the large cap S&P 500 index.

Furthermore, implied volatility market timing significantly enhanced index return. However, it is important to choose the correct length of the IV reference period to accurately estimate high and low levels of implied volatility. High beta portfolios created from the index constituents significantly outperform low beta portfolios for both the S&P 500 and the Russell 2000. The highest beta stocks do not outperform the index return, but middle to high stocks do outperform the index in a low implied volatility environment for the S&P 500 index. For the Russell 2000 none of the beta portfolios outperform the index in a low volatility environment. Based on the avoided return it is shown that high beta stocks do not experience a negative overreaction in stock price in high volatility periods, but have lower negative return than the index. Low beta portfolios have larger negative returns in a high volatility environment. The observed pattern in returns for the beta sorted portfolios for large cap and small cap stocks is similar and hence, beta sorted portfolios do not react differently for large cap stocks in comparison to small cap stocks in a low implied volatility environment.

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Page 33 of 34

combinations of factors from Fama and French (1996) to determine which combination of book-to-market equity and market capitalization stocks is able to outperform the index. Furthermore, it might be interesting to optimize the implied volatility market timing strategy as it shows to protect against the negative returns in the 2007-2009 financial crisis.

References

Ang, A., Hodrick, R. J., Xing, Y., Zhang, X., 2006. The cross‐section of volatility and expected returns. The Journal of Finance 61, 259-299.

Banerjee, P., Doran, J. and Peterson, D., 2007. Implied volatility and future portfolio returns. Journal of Banking Finance 31, pp.3183-3199.

Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of political economy 81, 637-654.

Bekiros, S., Jlassi, M., Naoui, K., Uddin, G. S., 2017. The asymmetric relationship between returns and implied volatility: Evidence from global stock markets. Journal of Financial Stability 30, 156-174.

Carhart, M. M., 1997. On persistence in mutual fund performance. The Journal of finance 52, 57-82.

Copeland, M. M., Copeland, T. E., 1999. Market timing: Style and size rotation using the VIX. Financial Analysts Journal, 73-81.

Christensen, B. J., Prabhala, N. R., 1998. The relation between implied and realized volatility1. Journal of financial economics 50, 125-150.

Dennis, P., Mayhew, S., Stivers, C., 2006. Stock returns, implied volatility innovations, and the asymmetric volatility phenomenon. Journal of Financial and Quantitative Analysis 41, 381-406.

Fama, E. F., French, K. R., 1993. Common risk factors in the returns on stocks and bonds. Journal of financial economics 33, 3-56.

Fama, E. F., French, K. R., 1996. Multifactor explanations of asset pricing anomalies. The journal of finance 51, 55-84.

Fleming, J., Ostdiek, B., Whaley, R. E., 1995. Predicting stock market volatility: A new measure. Journal of Futures Markets 15, 265-302.

French, K. R., Schwert, G. W., Stambaugh, R. F., 1987. Expected stock returns and volatility. Journal of financial Economics 19, 3-29.

Giot, P. 2005. Relationships between implied volatility indices and stock index returns. Journal of Portfolio Management 31, 92-100.

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Hull, J., 2018. Options, Futures, and Other Derivatives, Global Edition. 9th ed. Harlow: Pearson, p.363.

Jensen, M. C., Black, F., Scholes, M. S., 1972. The capital asset pricing model: Some empirical tests.

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Poon, S. H., Granger, C. W., 2003. Forecasting volatility in financial markets: A review. Journal of economic literature 41, 478-539.

Sarwar, G. 2012., Intertemporal relations between the market volatility index and stock index returns. Applied Financial Economics 22, 899-909.

Sharpe, W. F. 1964., Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance 19, 425-442.

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Mossin, J., 1966. Equilibrium in a capital asset market. Econometrica: Journal of the econometric society, 768-783.

Wang, Y. H., Keswani, A., Taylor, S. J., 2006. The relationships between sentiment, returns and volatility. International Journal of Forecasting 22, 109-123.

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