The Pricing of Aggregate Volatility Risk in the Returns of the S&P 500

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in the Returns of the S&P 500

Master Thesis Finance

First semester (2014-‐2015)

Rijksuniversiteit Groningen

ABSTRACT

:

I research if aggregate volatility risk is priced in the returns of stocks in the S&P 500. The VIX index is used to proxy for changes in aggregate volatility. I do not find that stocks with a higher sensitivity to changes in aggregate volatility have lower average returns, which means that aggregate volatility is not a priced risk factor, where earlier research for other indices did find the opposite. This can be due to the small sample size and the fact that I only use stocks of the S&P 500. The S&P 500 only contains very large stocks and these can react differently to aggregate volatility than small stocks.

Studentnr.: s1792857

Name: W.A. van Guilik

Number of words: 7318

Supervisor: Prof. Dr. R.E. Wessels

Date: 25-‐02-‐2015

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1. Introduction

In this paper I investigate the influence of aggregate volatility in the pricing of stocks of the S&P 500. Aggregate volatility risk is the risk that all assets bear. Volatility of stock returns varies over time, and stocks react differently to this changing volatility. I investigate how aggregate volatility is priced in the returns of stocks of the S&P 500. Investors want to minimize risks and want to form portfolios that do this. If aggregate volatility is a priced risk factor, this is a variable that investor have to take into account when forming portfolios and calculating expected returns.

I use the VIX index as a proxy for aggregate volatility. The VIX index measures the expected volatility of the S&P 500 for the upcoming 30 days. Giot (2003) and Corrado and Miller (2005) have shown that the VIX is a good proxy for aggregate volatility.

Ang, Hodrick, Xing and Zhang (2006) investigate all stocks on AMEX, NASDAQ and the NYSE and find that stocks with a higher sensitivity to changes in aggregate volatility earn lower returns than stocks with a low sensitivity to changes in aggregate volatility. If expected volatility rises, the expected risk in the market has increased. This means that prices can potentially change by a greater extent. This risk works both ways, the stock prices can go up or down. But most investors do not prefer additional risk and uncertainty and will request a risk premium. This will lower stock prices. The stocks that have a high sensitivity for changes in aggregate volatility are the most risky stocks and should therefore earn a higher risk premium and this means that their stock price should fall by the larger extent.

I form five quintile portfolios with different sensitivities to changes in aggregate risk from the 500 stocks of the S&P. If these portfolios do have significant different sensitivities for changes in aggregate volatility, I perform an ordinary least squares regression, to calculate monthly returns for the five quintile portfolios.

My hypothesis is that the portfolios with different sensitivities to aggregate volatility will have significant different returns. This would mean that aggregate volatility is a priced risk factor in determining the expected returns of the S&P 500.

Accordingly I have the following research question:

Is aggregate volatility a priced risk factor in the cross-‐section of the

returns of the S&P 500?

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This research will give insight in the pricing of aggregate volatility risk in the returns of the most important barometer of the state of the American economy, the S&P 500. During the research I encounter one important difficulty. I use the VIX as proxy for aggregate volatility and compare this with the returns of the S&P 500. The VIX index is calculated from the implied volatilities of eight S&P 500 options. The price of these options is among others determined by the price of the underlying stocks of the S&P 500. The VIX index and the S&P 500 can thus have the same drivers. This problem is called endogeneity. This means that any correlation between these two variables can be caused by an unknown variable which is not included in my research. The results must thus be interpreted with caution.

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2. Literature review

In this paper I research if aggregate volatility is a priced risk factor in the cross-‐section of the returns of the stocks of the S&P 500. Volatility is a measure for the variation in the price of a financial instrument. Variation is measure for the risk. A financial instrument that has a high variation and thus a high volatility is said to be riskier. There a two sorts of risk: Aggregate volatility means the total risk of the market. This is the risk that all the assets bear. Idiosyncratic risk is the risk that is specific for only one asset or a small group of assets. In this paper I will only focus on aggregate volatility risk.

2.1 Volatility and returns

The relationship between volatility and returns can be explained in several ways. First, there is a positive relationship between the expected risk premium on stocks and the level of aggregate volatility. If expected risk premiums are positively related to predictable volatility, then a positive unexpected change in volatility (and an upward revision in predicted volatility) increases expected risk premiums (French, Schwert, & Stambaugh, 1987). Secondly, an asset’s expected return depends on risk from the market return, changes in the forecasts of future market returns, and changes in forecasts of future market volatilities. This means that if forecasts expect volatility to go up, that the risk premiums should increase (Chen, 2002). And thirdly, if aggregate volatility increases, this causes the risks to increase in the next period of Campbell (1993) his intertemporal-‐CAPM model. All these leads to lower stock prices.

Pindyck (1984) has another explanation. He shows that the relative riskiness of investors’ net real returns from holding stocks has increased because the variance of the firm’s real gross marginal return has increased significantly since 1965. Thus returns from the stock market became more uncertain, which lowered stock prices.

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volatility. This all leads to lower stock prices for all stocks. Campbell and Hentschel (1992) find an asymmetry in the movement of volatility. They find that volatility is typically higher after the stock market falls than after it rises by the same extent, so stock returns are negatively correlated with future volatility. Volatility feedback, the finding that an increase in volatility requires higher stock returns and thus lowers stock prices, can explain the asymmetry in the movement of volatility. If there is a large piece of good news, this is mostly followed by other good news because volatility is persistent, so this news increases future expected volatility. This persistency is confirmed by the finding that a volatility shock has an implied half-‐life of about 7 months with post-‐war monthly data. Future expected volatility increases the required rate of return on stocks and lowers the stock price, thereby offsetting part of the positive returns of the good news. On the other hand if there is a large piece of bad news, the stock price falls, and again volatility rises, but now this increase in volatility increases the already negative effect of the bad news. If there is small news or no news, volatility will be lowered, because of the mean reversion process.

Stocks that do badly when volatility increases tend to have negatively skewed returns over intermediate horizons, while stocks that do well when volatility rises tend to have positively skewed returns. Campbell (1993) finds that large negative stock returns are more common than large positive ones, so stock returns are in general negatively skewed.

Chen (2002) and Campbell (1993) both find that stocks with the highest sensitivity for unexpected changes in expected aggregate volatility should earn a risk premium. These stocks are the most risky. This means that the prices of these stocks should fall more than the stocks with a lower sensitivity for unexpected changes in expected aggregate volatility.

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aggregate volatility carry a statistically significant negative price of risk of approximately -‐1% per year, when controlling for several other variables like the Fama-‐ French (1993) 3-‐factors (market, size and book-‐to-‐market), momentum factor and liquidity factor. This can be explained: investors want to hedge against changes in market volatility, because increasing volatility represents a deterioration in investment opportunities. The stocks with the highest sensitivities for aggregate volatility, are the most risky ones, and their stock prices should decline more than the stocks with lower sensitivities for aggregate volatility. When only controlling for the market they find a negative price of aggregate volatility risk of 1.04% per month.

Investors see stocks of small firms and value firms (stocks of these firms seem to be undervalued) as riskier because their returns correlate strongly with market returns at times of high volatility. On the other hand, stocks of big firms and growth firms (have potential to achieve high earnings growth) are seen as hedges against innovations in aggregate market volatility. (Barinov, 2012)

Fama and French (1992) find that the size of a company and their book-‐to market value are very good in explaining the cross-‐section average returns on NYSE, Amex and NASDAQ stocks for the period of 1963-‐1990. The size of a company is measured by their market capitalization, the number of shares multiplied by the share price, and the book-‐ to-‐market value is the book value, which is calculated by looking at the firm’s historical cost, divided by the market value, which is equal to its market capitalization. Between the average returns and the size of companies they find a negative relationship, which means that smaller firms earn higher average returns. They find a positive relationship between the average returns and the book-‐to-‐market value. This means that value stocks, stocks with a high book-‐to-‐market value earn higher average returns. These are stocks that seem to be undervalued. The book-‐to-‐market value has a stronger influence on the average returns than the size effect. Also they find that the market beta does not help in explaining the average returns of the stocks. The size and book-‐to-‐market value are very commonly used in controlling for the effects of other variables in the cross-‐ section of average stock returns.

2.2 The VIX index

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changes in the VIX because the VIX is highly serially correlated, with an autocorrelation of almost one. This makes the change in the VIX a suitable proxy for the change in expected aggregate volatility. Barinov (2012) also investigates aggregate volatility risk. He is using changes in the VIX index to proxy for changes aggregate volatility risk.

The VIX index is calculated by the Chicago Board Options Exchange (CBOE). The VIX index represents the S&P 500’s expectation of 30-‐day volatility. It measures the implied volatility of S&P 500 index options. Implied volatility is a forecast of future volatility. The implied volatility will increase when the market expectations are negative. Historical volatility refers to the actual price changes that have been observed in the past over a specific period of time. An option is the right to buy or sell an asset (for example a stock) within a predetermined time for a predetermined price. The value of an option is based on the current price of the underlying asset, the strike price, the time to expiration, interest rates and the volatility of the underlying asset. There are two sorts of options namely call options, which gives you the right to buy an asset, and put options, which gives the right to sell an asset. For a call option the price of the option will rise if the price of the underlying asset will rise. For a put option the opposite will happen: the price of a put option will fall if the price of the underlying asset will rise. The VIX seems a good proxy for aggregate volatility but Jiang and Yisong (2007) find that the VIX may be not that accurate. The VIX may underestimate the true volatility by almost 1.98% or overestimate it by about 0.79%. If these errors are true, these are economically significant.

Whaley (2008) explains that the VIX measures expected short-‐term market volatility. VIX is forward-‐looking, measuring volatility that investors expect to see:

VIX is like a bond’s yield to maturity. Yield to maturity is the discount rate that equates a bond’s price to the present value of its promised payments. As such, a bond’s yield is implied by its current price and represents the expected future return of the bond over its remaining life. In the same manner, VIX is implied by the current prices of S&P 500 index options and represents expected future market volatility over the next 30 days. (p. 1)

2.2.1 Calculation of the VIX index

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expiration. These implied volatilities are weighted in such a way that the VIX represents the implied volatility of a S&P 500 option with 30 days left to expiration.

The VIX is not based on calendar days, but on trading days. If the number of calendar days to expiration is 𝑁_!, the number of trading days, 𝑁_!, is computed as

𝑁! = 𝑁! − 2×𝑖𝑛𝑡

𝑁_! 7

The implied volatility rate is multiplied by the ratio of the square root of the number of calendar days to the square root of number of trading days, that is:

𝜎! = 𝜎!

𝑁_! 𝑁_!

Where 𝜎_! 𝜎_! is the trading-‐day (calendar-‐day) implied volatility rate.

To calculate the VIX index, we have to use the eight options that are the building blocks of the VIX index. The four options that are closest to expiration should have at least eight days to expiration. The second-‐closest are the options of the next succeeding contract month.

The exercise price just below the current price of the underlying asset, 𝑆, is denoted as 𝑋_!, the lower exercise price, and 𝑋_! is the upper exercise price, just above the current price. The eight implied volatilities are shown in Table 1.

Table 1: Implied volatilities of eight options

Closest to expiration Second-‐closest to expiration

Exercise Price Call Put Call Put

𝑋_! (< 𝑆) 𝜎_!,!!! _𝜎

!,!!! 𝜎!,!!! 𝜎!,!!!

𝑋_! (≥ 𝑆) 𝜎_!,!!! _𝜎

!,!!! 𝜎!,!!! 𝜎!,!!!

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and

𝜎_!!! ₌ 𝜎!,!

!!_{+ 𝜎}

!,!!!

2

The second step is to interpolate between the closest to expiration volatilities and the second-‐closest to expiration implied volatilities to get an at-‐the-‐money implied volatility for each maturity:

𝜎_! = 𝜎_!!! 𝑋! − 𝑆 𝑋_!− 𝑋_! + 𝜎!!! 𝑆 − 𝑋! 𝑋_!− 𝑋_! and 𝜎_! = 𝜎_!!! 𝑋!− 𝑆 𝑋_! − 𝑋_! + 𝜎!!! 𝑆 − 𝑋_! 𝑋_! − 𝑋_!

Two implied volatilities remain. The last step is to create a thirty-‐calendar day implied volatility out of the two implied volatilities. If 𝑁_!_! is the number of trading days to expiration of the closest to expiration contract, and 𝑁_!_! is the number of trading days of the second-‐closest contract, the VIX index is:

𝑉𝐼𝑋 = 𝜎_! 𝑁!! − 22

𝑁_!_!− 𝑁_!_! + 𝜎!

22 − 𝑁_!_! 𝑁_!_! − 𝑁_!_!

The VIX has high levels during periods of market turmoil. If expected market volatility increases, investors demand higher rates of return on stocks, so stock prices fall.

2.2.2 Relation of the VIX index with the S&P 500

A problem that arises is that the VIX index and the S&P 500 can have the same drivers. The VIX index is calculated from the implied volatilities of eight near the money S&P 500 options. If the price of the underlying asset changes, the implied volatilities will change and thus the VIX index will change. And the underlying asset is the S&P 500. This problem is called endogeneity. It can be that one force has influence on them both simultaneously. It than looks like there is a correlation between the changes of the VIX index and the returns of S&P 500, where in reality this is not true.

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Whaley (2008) finds that the S&P 500 option market has become dominated by hedgers who buy put options when they are afraid for a possible drop of the price of the S&P 500. They are more concerned by an unexpected drop in the stock price than an unexpected rise. They use the put options as a portfolio insurance for when they are afraid that the stock market will drop. They only use this for a drop, not for a rise: the demand for puts will thus be larger than the demand for calls. This is why Whaley (2008) found an asymmetry in the movements of the VIX caused by this portfolio insurance. If the S&P 500 rises by 100 basis points, the VIX falls by -‐2.99%. On the other hand if the S&P 500 falls by 100 basis points, VIX rises by 4.493%. Because of the demand for portfolio insurance, the VIX reacts stronger to a decline in the S&P 500, than to a rise in a S&P 500. Investors buy more puts when the S&P 500 falls, than that they buy calls when the S&P 500 rises by the same extent.

Giot (2003) shows that there is statistically significant negative relationship between the returns of the S&P100, which is very comparable with the S&P 500, and the VIX index. He also finds, like Whaley (2008), that for the S&P100 this is relationship is asymmetric, because negative stock returns produce bigger changes in the VIX than positive returns do. Dash and Moran (2005) test if a small allocation to the VIX index can be used for risk reduction. They find a negative correlation between hedge fund returns and changes in the VIX index, and this correlation is more negative in months when the hedge funds deliver negative returns.

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VIX have lower returns than stocks with a high sensitivity to changes in the VIX. Banerjee, Doran and Peterson (2007) test if the implied volatility, measured by changes in the VIX and the level of the VIX, has predictive power for future market returns. They show that this is true even when they control for the Fama and French (1993) factors. Changes in VIX and the level of VIX are significantly related to future returns, where the relationship is stronger for high beta portfolios. They conclude that their results suggest that aggregate volatility, as measured by the VIX, may be a priced risk factor.

The main prediction of Ang et al. (2006) was that stocks with different sensitivities on aggregate volatility risk have different average returns. This would mean that the aggregate volatility is a priced risk factor. I want to investigate this for stocks of the S&P 500, where the VIX index represents aggregate volatility risk.

Research question:

Is aggregate volatility a priced risk factor in the cross-‐section of the

returns of the S&P 500?

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3. Data and descriptive statistics:

I will first show the data that I have used and give descriptive statistics. I will also show if the data is properly fitted for the regression that I use.

Table 2: Summary of data selection

Country USA

Period January 2004 -‐ December 2013

Index Standard & Poor’s 500

Number of stocks 500

3.1 Description of the data

I will use the S&P 500 as my benchmark portfolio and I will use the closing values of the VIX index. These data are obtained from Thomson Financial Datatstream. The S&P 500 represents the 500 biggest American companies and gives the most reliable image of the state of the American economy. I have a sample period of 10 years, from January 2004 until December 2013. This is summarized in Table 2.

Table 3 shows the monthly data summary. In my sample period the S&P 500 has an average monthly change of 0.41% with a standard deviation of 4.30%. The standard deviation is ten times higher than the average change of the S&P 500. The VIX index has decreased on average a little with -‐0.16% and has a relatively high standard deviation of 18.28%. The VIX index is only used as an indicator of the expected (change in) aggregate volatility. The kurtosis measures the ‘peakedness’ of the distribution of the returns. A kurtosis of three means that the distribution is normal. A lower value means that the distribution very flat is, where a high value means that the distribution is peak shaped. The S&P 500 index has an kurtosis of 3.086, which means that the distribution is normal. The VIX index has a kurtosis of 0.665 and the distribution of the VIX is thus very flat and broad. A symmetric distribution has a skewness of 0. In table 3 can be seen that the S&P 500 has a skewness of -‐1.109, which means that the data has a long left tail and the mass of the distribution is concentrated to the right of the middle. This is confirmed by the fact that the median is higher than the average.

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Table 3: Monthly data summary (January 2004 – December 2013)

S&P 500 index VIX index

Average monthly change 0.41% -‐0.16%

Standard deviation 4.30% 18.28% Median 1.20% -‐2.44% Minimum -‐18.56% -‐38.51% Maximum 10.23% 64.58% Kurtosis 3.086 0.665 Skewness -‐1.109 0.641 Jarque-‐Bera 61.213 (P<0.001) 9.735 (P=0.008)

These are the simple returns (no excess returns). The risk free rate in the USA on 3-‐month T-‐Bills, in this period, was on average monthly 0.13%. Figure 1 shows the development of the S&P 500 index (distribution on the right axis) and the VIX index (left axis) during my sample period. According to CBOE, since 1990 the VIX has moved opposite of the S&P 500 88% of the times on the same day. In my sample, for monthly data, the VIX moved opposite of the S&P 500 in 96 months of the 120, that is 80%.

During the crisis period at the second half of 2008, the VIX rises to extremely high values and the S&P 500 drops to very low values. The two months in which the VIX

Figure 1: This figure shows the values of the S&P 500 index (right axis) and the values of the VIX

index (left axis) from January 2004 until December 2013.

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index had its highest changes, September and October 2008, where also the months in which the S&P 500 index had two of its three lowest returns. Due to abnormal returns in the period 2008-‐2009, these years are considered as outliers and will be discarded from the data.

Table 4: Monthly data summary (January 2004 – December 2013) without 2008 and 2009

S&P 500 index VIX index

Average monthly change 0.77% -‐0.14%

Standard deviation 3.30% 17.49% Median 1.21% -‐1.73% Minimum -‐8.55% -‐38.51% Maximum 10.23% 42.43% Kurtosis 3.671 2.829 Skewness -‐0.338 0.303 Jarque-‐Bera 3.595 (P=0.166) 1.567 (P=0.457)

Table 4 shows the monthly descriptive statistics without the years 2008 and 2009. The standard deviation of the S&P 500 is reduced by almost a quarter, and the Jarque-‐Bera of the S&P 500 and the VIX, show that both distributions are normal.

I have also considered to use daily data. The descriptive statistics of the daily data are shown in Appendix Table 1. The kurtosis of the S&P 500 is extremely high, which means that the distribution is peak-‐shaped. The variation is then caused by a few extreme values. The data has a very high standard deviation and this means that these extreme values that cause the variation are indeed very extreme. The Jarue-‐Bera statistic also shows that the data is far from being normally distributed. In comparison with the monthly data, the daily data is less better fitted for an OLS regression.

3.2 Robustness tests

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Table 5: Summary of statistics for checking if data is fitted to be tested with OLS, without the

years 2008 and 2009.

The Durbin-‐Watson is normal when it is close to 2. When it is above 3 or below 1 it is a warning sign that autocorrelation can be present. For the White test and the Jarque-‐Bera test, the P-‐values show if there is a sign of non-‐normality.

I have included a constant term in the regression, so this assumption is by definition never violated. The second one is the assumption of homoscedasticity. This means that the variance of the errors are constant. If the errors are not constant over time, they are said to be heteroscedastic. This can be tested with the White test. For all portfolios, I perform a White test. Column 3 and 4 in Table 5 show the results of the White test. The P-‐value indicates that all portfolios are heteroscedastic. This means that the ordinary least squares regression does not give the coefficients that are the best linear unbiased estimators. Heteroscedasticity does not cause the calculated betas to be biased but it can cause estimates of the standard errors of the betas to be biased. Thus, the regression will still provide an unbiased estimate for the relationship between the returns of the portfolios and the values of the S&P 500 and the VIX index. The betas that I calculate are not biased, but it can cause the estimates of the standard deviations of the betas of the VIX and the S&P 500 to be biased.

The third assumption is that the covariance between the error terms over time is zero. This means that the errors may not be autocorrelated. There may not be a repeating pattern, it must be random. This can be tested with the Durbin-‐Watson (DW) test. The DW is always between 0 and 4. When the DW is close to 2, there is little evidence of autocorrelation. There is cause for alarm when the DW is lower than 1 or higher than 3. As can be seen in column 5 of Table 5, the DW’s that I have calculated are all in the safe zone, so there is no reason to think that autocorrelation is present.

The fourth assumption is that all regressors, that are the explanatory variables, the S&P 500 and the VIX, are not correlated with the error term. This means that

Portfolio White F-‐

statistic P-‐value Durbin-‐Watson

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regressors should be exogenous. Endogeneity may not be present. Sources of endogeneity are: omitted variables, measurement error and simultaneity. A possible problem with this has already be mentioned. It could be that I find a correlation between the changes in the VIX and de returns of the S&P 500, where in reality they are both driven by the same variable which is not included in the model. The variance inflation factor (VIF) checks for multicollinearity. This means that two explanatory variables are highly correlated with each other, whereby one variable can be predicted with the model. The VIF is between 1.43 and 6.65, this means that there is no multicollinearity between the VIX and the S&P 500. This would be the case if VIF is above ten. I use the Ramsey RESET test, to test for omitted variables. The fifth column in Table 5 shows the P-‐values of the fitted terms, and these are all significant. This means that there is apparent non-‐linearity, and thus omitted variables. The linear model is not appropriate to explain the returns. I cannot remedy this problem, so the results should be interpreted with caution.

The fifth and last assumption is that the error terms are normally distributed. The most commonly used test for normality is the Jarque-‐Bera test. The Jarque-‐Bera test uses the skewness and the excess kurtosis to calculate if they are jointly zero and the distribution is ergo normal. A normal kurtosis is three. The normal excess kurtosis is zero and is calculated by subtracting three from the normal kurtosis. The null hypothesis is of normality, and this will be rejected if the residuals from the model are either significantly skewed, have a high or low kurtosis or both. Column 6 and 7 show the results from the Jarque-‐Bera test. As can be seen, none of the P-‐values in column 7 are significant, so the errors are normally distributed.

The R-‐squared is very low, between 0.12 and 0.18 for all quintile portfolios. This means that the two variables, the S&P 500 and the VIX only explain about 15% of the returns of the quintile portfolios. This finding supports to control in the regression for other variables like the size and book-‐to-‐market value of Fama and French (1993), to investigate if these variables can help to explain the returns by a larger extent.

The problem of endogeneity is not solved. This means that the results of this research have to be interpreted with caution.

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4. Statistical model

I want to investigate whether stocks with different sensitivities to changes in the VIX index have significant different returns. In the first place I will not control for the Fama and French (1993) variables because adding other factors in constructing portfolios may add a lot of noise. First I only regress for the market and the aggregate volatility.

4.1 Regression with S&P 500 and the VIX

To do this I need to calculate the changes in the S&P 500 and the changes in the VIX index. I use all the monthly stock prices of all the stocks that are included in the S&P 500. I also need the monthly closing values of the VIX index. I calculate the monthly returns of the S&P 500 with a simple formula:

𝑅_!" = 𝑙𝑛 𝑃!" 𝑃_!"!!

Where 𝑃_!" is the share price adjusted for stock splits and dividend payments of company i on the end of month t. I use the monthly data from January 2004 until December 2013, excluding the years 2008 and 2009, and will form quintile portfolios at the end of each month.

The change in VIX is calculated in a similar way:

∆𝑉𝐼𝑋_! = 𝑙𝑛 𝑉𝐼𝑋!

𝑉𝐼𝑋_!!!

Where 𝑉𝐼𝑋_! is the value of the VIX index at the end of month t.

Ang et al. (2006) test whether stocks with different sensitivities to changes in the VIX have different average returns. They use the following regression:

𝑟_!! _{= 𝛽}

!+ 𝛽!"#! 𝑀𝐾𝑇!+ 𝛽∆!"#! ∆𝑉𝐼𝑋!+ 𝜀!! (1)

Where 𝑀𝐾𝑇 is the market excess return, ∆𝑉𝐼𝑋 is the change in the VIX index and is the instrument they use for changes in the aggregate volatility factor, and 𝛽_!"#! _{and
𝛽}

∆!"#!

are loadings on market risk and aggregate volatility risk, respectively. They sort firms on their 𝛽_∆!"#! _{coefficients
over
the
past
month
using
the
regression
(1)
with
daily
data.}

I will follow the next steps in forming the quintile portfolios, based on the paper of Ang et al. (2006):

-‐ I sort stocks on their 𝛽_∆!"#! _{over
the
past
month
using
the
regression
(1)
with}

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-‐ At the end of each month I sort stocks into equally weighted quintile portfolios based on their 𝛽_∆!"#! _{loadings.
Stocks
in
the
first
quintile
have
the
lowest
betas;}

stocks in the fifth quintile have the highest betas. I calculate the returns, standard deviations, 𝛽_!"#! _{,
𝛽}

∆!"#! and the standard deviation of 𝛽∆!"#! of each quintile

portfolio. The returns are calculated for the present month.

-‐ Every month I form new quintile portfolios based on the 𝛽_∆!"#! _{.
So
every
month
I}

create five portfolios.

-‐ The results of all quintile portfolios are averaged to form one result for the quintile portfolios from the data based on the full sample.

My goal is to test whether stocks with different sensitivities to changes in the VIX index (aggregate volatility) have different expected returns. This would mean that aggregate volatility risk is factored in the pricing of stocks.

First, the 𝛽_∆!"#! _{’s
of
the
five
quintile
portfolios
should
be
significant
different
from}

each other. If the 𝛽_∆!"#! _{’s
are
not
different
from
each
other,
I
cannot
state
that
the
stocks}

in the portfolios have different sensitivities for aggregate volatility risk, and consequential that aggregate volatility risk is not a priced risk factor.

These are the first hypotheses:

𝐻_!: 𝛽_∆!"#!" = 𝛽_∆!"#!" 𝐻_!: 𝛽_∆!"#!" ≠ 𝛽_∆!"#!"

Where 𝛽_∆!"#!",! is the beta of the change in the VIX index of respectively quintile portfolio number i,j where 𝑖 ≠ 𝑗. I perform a Students t-‐test to see if the differences are significant.

If the null hypothesis will be rejected, then the different portfolios have different sensitivities for aggregate volatility risk and I can continue with the second hypothesis: do the portfolios containing stocks with different sensitivities for aggregate volatility have significant different returns:

𝐻!: 𝑟!! = 𝑟!"

𝐻_!: 𝑟!" ≠ 𝑟!"

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If the returns of the quintile portfolios do not differ significantly, the null hypothesis is not rejected and the sensitivity to changes in aggregate volatility, measured by changes in the VIX index, is not a priced risk factor. But if the returns between the quintile portfolios differ significantly from each other, this means that the sensitivity to aggregate volatility is a priced risk factor. The null hypothesis will be rejected. For clarification: I use the VIX only as an indicator of aggregate volatility risk. 4.2 Regression with S&P 500, the VIX, SMB and HML

I also control for the Fama and French (1992) variables, in addition with the market variable and the aggregate volatility variable. I use the following regression:

𝑟_!! _{= 𝛽}

!+ 𝛽!"#! 𝑀𝐾𝑇!+ 𝛽∆!"#! ∆𝑉𝐼𝑋!+ 𝛽!"#! 𝑆𝑀𝐵!+ 𝛽!"#! 𝐻𝑀𝐿!+ 𝜀!! (2)

Where, again, the 𝑀𝐾𝑇 is the market excess return, ∆𝑉𝐼𝑋 is the change in the VIX index and is the instrument I use for changes in the aggregate volatility factor. The SMB and HML are the Fama and French (1992) variables for size and book-‐to-‐market value, and 𝛽_!"#! _{,
𝛽}

∆!"#! , 𝛽!"#! and 𝛽!"#! are loadings on market risk, aggregate volatility risk, size

and book-‐to-‐market value, respectively.

The SMB, is the small-‐minus-‐big portfolio, the variable for the size effect and the HML, is the high-‐minus-‐low portfolio, the variable for the book-‐to-‐market value, of Fama and French (1992). They calculated the SMB and HML values by creating 6 portfolios. They sort all stocks from the NYSE, Amex and NASDAQ based on their market capitalization and their book-‐to-‐market value. First the stocks are sorted in two groups based on their size. The median size is used as break point between the group with small stocks, and the group with big stocks. In these two groups, the stocks are sorted on their book-‐to-‐market value. These are broken down in three groups, the bottom 30% (low), middle 40% (medium) and top 30% (high). In total there are now have 6 portfolios: small size firms with low book-‐to-‐market values (S/L), small size firms with medium book-‐to-‐market values (S/M), small size firms with high book-‐to-‐market values (S/H), and big size firms with with also low (B/L), medium (B/M) and high (B/H) book-‐to-‐ market values. (Fama & French, 1993)

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related to the book-‐to-‐market value is the high-‐minus-‐low (HML) portfolio, this the simple average of the S/H and B/H portfolios minus the simple average of the S/L and B/L portfolios. In this case, it should be largely free of influences of size factors. Fama and French (1992) expect that stocks that are smaller earn a higher a return and that stocks with a higher book-‐to-‐market value should earn higher returns. The historical SMB and HML values are found on the website of Kenneth French, and these are the ones I use in my regression. I will again sort the stocks in portfolios based on their sensitivity for aggregate volatility.

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5. Results

Table 6: Monthly data of regression (1) without the years 2008 and 2009

Portfolio Return Std.Dev

Return 𝛽!"#!" 𝛽∆!"#!" Std.Dev 𝛽_∆!"#!" Q1 0.73% 4.29% 0.04 -‐0.27 0.1436 Q2 1.00% 4.12% 0.61 -‐0.09 0.0280 Q3 0.88% 4.00% 0.97 -‐0.01 0.0222 Q4 0.76% 4.14% 1.38 0.08 0.0318 Q5 0.57% 4.27% 2.52 0.27 0.1414

The results from regression (1) are shown in Table 6. Column 5 and 6 show the beta of the aggregate volatility proxied by changes in the VIX, and the standard deviation of the beta of aggregate volatility. The beta of quintile portfolio 1 is by construction the lowest, and the beta of quintile portfolio 5 is the highest. Their standard deviations are fairly low. My first hypothesis is that the betas of aggregate volatility should differ significantly. To find out if the betas of the aggregate volatility differ significantly I perform the Students t-‐test. The resulting t-‐values with corresponding P-‐values are shown in Table 7.

Table 7: t-‐values of differences between the betas of aggregate volatility of the quintile

portfolios, with between brackets the corresponding P-‐values with α=0.05.

Q2 Q3 Q4 Q5 Q1 12.05 (<0.001) 17.53 (<0.001) 23.32 (<0.001) 26.25 (<0.001) Q2 21.94 (<0.001) 39.31 (<0.001) 24.47 (<0.001) Q3 22.74 (<0.001) 19.17 (<0.001) Q4 12.84 (<0.001)

All the t-‐values are significant, with a significance level of α=0.05. This means that the portfolios do have different sensitivities for aggregate volatility risk. The next step is to find out if these portfolios with different sensitivities for aggregate volatility have different returns.

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highest return, and the subsequent portfolio has a lower return, until finally, quintile portfolio 5 has the lowest return. I do not find exactly the same pattern, although from quintile 2 to 5 the returns are decreasing. Only quintile portfolio 1 is an exception, which does have the second lowest return. The standard deviation is relatively high in comparison to the return. To find out if the returns differ significantly from each other, I again calculate t-‐values. The results are shown in Table 8.

Table 8: t-‐values of differences between the betas of the returns of the quintile portfolios, with

between brackets the corresponding P-‐values with α=0.05.

Q2 Q3 Q4 Q5 Q1 0.445 (0.329) 0.251 (0.401) 0.049 (0.481) 0.259 (0.398) Q2 0.205 (0.419) 0.402 (0.344) 0.710 (0.240) Q3 0.204 (0.419) 0.519 (0.302) Q4 0.313 (0.378)

None of the t-‐values in Table 8 is significant. This means that I can’t reject the null hypothesis and thus the portfolios with significant different sensitivities for aggregate volatility do not have significant different returns. This means that aggregate volatility is not a priced risk factor in my research.

Column 4 of Table 6 shows the market beta. The market beta is very low for quintile portfolio 1 with 0.04, and incurs with the subsequent portfolios, with quintile portfolio 5 having a market beta of 2.52. Quintile portfolio 1 moves almost independent form the market, where quintile portfolio 5, moves 2.5 times as strong as the market. The R-‐squared is on average 0.15, which is very low, which means that the regression has barely any explanatory power.

Appendix Table 2 shows the monthly data, comparable with Table 6, but then calculated for all the years from 2004 until 2013, thus with the years 2008 and 2009 included. The standard deviation is much higher when 2008 and 2009 are included in the calculation showing that these years where very volatile in comparison to the other years. The differences between betas of aggregate volatility are significant, but again the differences between the returns are not significant.

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the VIX, the variable for aggregate volatility could not do this. The results are shown in Table 9.

Table 9: Monthly data of regression (2) controlling for SMB and HML without the years 2008

and 2009 Portfolio Return 𝛽_!"#!" 𝛽_∆!"#!" 𝛽_!"#!" 𝛽_!"#!" Q1 0.86% 0.97 -‐0.06 -‐0.11 0.03 Q2 0.70% 1.03 -‐0.02 -‐0.06 0.04 Q3 0.77% 1.07 -‐0.01 0.14 0.13 Q4 0.74% 1.14 0.01 0.07 0.14 Q5 0.87% 1.16 0.03 0.15 0.30

The beta for aggregate volatility, as proxied by the changes in the VIX, is close to zero for all portfolios. The value of the beta is not even a quarter of the value of the beta calculated with regression (1). The explanatory power of the beta for aggregate volatility is much lower, when the two control variables of Fama and French are included. The returns of the different quintile portfolios are close to each other, much closer than the returns of regression (1) which are shown in Table 6. In Table 6 the returns were not significant different from each other, so this is certainly not the case in Table 9.

The market beta, is close to one for all quintile portfolios, where in the first regression the market betas were very different. The betas for the SMB and HML are not very high, but have more explanatory power than the changes in the VIX index. The R-‐ squared is now on average 0.67. This is a large improvement in comparison to the low R-‐squared of 0.15 of regression (1). About two thirds of the returns is explained by the variables in the regression.

The Pricing of Aggregate Volatility Risk in the Returns of the S&P 500

in the Returns of the S&P 500

Master Thesis Finance

First semester (2014-­‐2015)

Rijksuniversiteit Groningen

:

Table of contents

1. Introduction

Is aggregate volatility a priced risk factor in the cross-­‐section of the

returns of the S&P 500?

2. Literature review

Is aggregate volatility a priced risk factor in the cross-­‐section of the

returns of the S&P 500?

3. Data and descriptive statistics:

4. Statistical model

5. Results

First semester (2014-‐2015)

Is aggregate volatility a priced risk factor in the cross-‐section of the

Is aggregate volatility a priced risk factor in the cross-‐section of the