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Implied Volatility as a Risk Factor for Stock Returns

of Characteristic-based Portfolios

Student: Jorrit Kuilman

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Abstract

Implied market volatility has been proven to be a risk factor in the cross section of stock returns. The risk contained by implied market volatility is asymmetrically priced into stock returns. This means that the risk is only priced when the volatility, as implied by index options, when the implied volatility increases. The risk is not priced into stock returns when the implied volatility diminished. This paper assesses the asymmetric relation between stock returns and implied market volatility. Instead of focusing on the changes in the implied market volatility a new measure is calculated by subtracting the realized market volatility from the implied market volatility. This is a more appropriate technique to assess the expected change in short-term market volatility because it recognizes the current market volatility and the actual expected change in the volatility of the market. The findings show that the implied market volatility is priced into stock returns when it is below the current volatility of the market, the opposite, when the implied market volatility exceeds the current market volatility, is not priced into stock returns. The pricing of implied volatility is priced separate from the risk proxies measured by the Fama French and Carhart factors. However, small firms do have a larger exposure than big firms. This leads to the conclusion that a volatility risk discount is a priced upside risk factor into stock returns.

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Table of contents

1. Literature Review ... 8

1.1 Stock Return Models ... 8

1.2 Idiosyncratic Implied Volatility ... 11

1.3 Implied Market Volatility ... 13

1.4 Measurement Implied Volatility ... 16

1.5 Stock Market Anomalies ... 18

2 Data ... 23

2.1 Period Selection ... 23

2.2 Sample Selection ... 23

2.3 Dependent Variables ... 24

2.4 Volatility Premiums and Discounts ... 24

2.5 Control Variables ... 25 2.6 Descriptive Statistics ... 26 3 Methodology ... 28 3.1 Research Approach ... 28 3.2 Regression Models ... 29 3.3 Portfolio Formation ... 30 3.4 Portfolio Performance ... 31 4 Findings ... 34

4.1 Results Regression for full sample’s returns ... 34

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4.3 Regression Results of size based portfolios’ returns... 38

4.4 Regression Results of value based portfolios’ returns ... 39

4.5 Regression results of size/value portfolios’ returns ... 41

4.6 Regression Results of past return based portfolios’ returns ... 42

4 Conclusion ... 44

Appendices ... 51

Appendix A; Sample Firms ... 52

Appendix B; Multiple Regression Analysis Requirements ... 54

Appendix C; Portfolio Descriptive Statistics ... 56

Appendix D; Size/Value Portfolios ... 60

Appendix E; Portfolio Coefficients ... 62

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Introduction

‘When the Chicago Board of Options Exchange Volatility Index (VIX) touched 80 percent last autumn (September, 2008) , financial markets were in for a rough ride. In its entire history (since January 1993) this indicator of volatility had never been higher. Could volatility rise again? It can and almost inevitably will rise again, at least temporarily’ (Engle, 2009). This excerpt from the Financial Times illustrates the predictive power of the VIX. The following 2008 financial crisis dramatically reduced market capitalizations and devastated stock returns.

This paper is concerned with the relation between expected change in short term market volatility and whether this is priced into stock returns. A large amount of literature exists on the behaviour of stock returns. The capital asset pricing model (CAPM) has been extended to incorporate other possible risk factors. Although this has lead to a large body of literature, few factors have proved to be proxies for systematic risk. Much research has examined the contemporaneous relation between risk and return, thereby disregarding that investors are interested in the behaviour of future stock return, not how the past returns can be explained. The current accepted pricing models do not incorporate a forecasting factor, therefore they are limited in their ability to explain future stock returns. This paper will take an intertemporal approach on examining stock returns and examine the influence of a forward looking factor on the future stock returns.

A commonly used forecasting factor for market volatility is attained from the implied volatilities of stock options. Stock options are priced based on the estimations of the future volatility of the underlying stock. An aggregate of this concerning a market can then be calculated, this is commonly referred to as the market’s view of the short-term future volatility of the market or more simply; the implied market volatility. This forecast is the market’s best forecast of the future volatility, assuming a reasonable level of efficiency from the stock options market.

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6 | P a g e impact on stock returns. The expected volatility of the market can be measured using the implied volatilities of stock options. Literature has only recently started to examine this relation between stock returns and implied volatilities. Findings show that the implied market volatility is priced into stock returns when the level of implied market volatility increases, no effect is found when the level is decreasing. Investors require a higher compensation for holding stock when the level of implied volatility increases. Regardless of the validity of the findings of these papers, they focus merely on the actual change in the level of the measured implied market volatility, the realized market volatility is not taken into account. The implied market volatility is then used as an investors’ ‘fear gauge’, and is therefore more a measure of investor sentiment than an actual expectation of future market volatility.

This paper assesses the effect expected change in the short-term market volatility by incorporating the realized market volatility in the measurement of expected volatility. The realized market volatility is subtracted from the implied market volatility, the residual is then regarded as the expected change in market volatility. This produces a factor which measures the expected change in the short term market volatility. Any positive product is then regarded as an expected increase in market volatility and a negative product as a decline in market volatility. Since literature suggest that the effect on stock returns is not the same for an increase and decrease in implied market volatility, two factors are created. One factor captures expected increases and one factor captures expected decreases in market volatility. Consequently this paper does not utilize the implied volatility as a measure of investor sentiment, but rather as an actual forecast for changes in market volatility. Such a measure allows for a more accurate understanding of the actual expected change in the market volatility. The relation between a firm’s stock volatility and realized market volatility is priced into the stock, the question then remains whether such a relation also exists when it concerns expected volatility.

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characteristics and the exposure to implied market volatility. Although they have been used as control factors, this paper extends this by creating characteristic-based portfolios.

This paper contributes to the body of literature of stock returns and the relation between implied market volatility. Up to this point literature has focussed on innovations of the implied volatility based on its change between two observations. No existing literature conditions a risk factor based on the difference between the realized and implied market volatility, as done in this paper. This paper studies the relation between the expected change in market volatility and whether the stock returns exhibit a relation to this. It therefore assesses the validity of a priced risk factor based on the expected change in volatility and not the relation between returns and the level of implied volatility. Furthermore this paper extend the literature concerning stock market ‘anomalies’ with an evaluation of expected changes in market volatility and the returns of characteristic based portfolios. Even though recent literature has used firm characteristic based risk proxies as control factors in their models, the possibility of causality between firm characteristics and the relation to the implied market volatility has not been addressed.

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1. Literature Review

1.1 Stock Return Models

The Capital Asset Pricing Model (CAPM) as developed by Sharpe (1964), Lintner (1965) and Mossin (1966) is the fundamental pricing model for explaining stock returns. The CAPM is a one factor model and therefore claims that all returns are explained by its single factor; the market beta. The market beta is the elasticity of the stock return compared to the market return. Equation (1a) depicts the standard CAPM model.

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Where is the expected return for common stock for firm i, the risk free rate at time t and the the return on the market at time t. The beta of the stock is the measure for the risk the stock has with relation to the market risk. Returns are based on the of market risk which in excess of the riskless return, the market premium, that is rewarded to the investor measured by the proportional risk of the stock to the market. In other words, the CAPM states that investors are proportionally compensated for bearing risks which are associated with the market of the stock.

The Intertemporal Capital Asset Pricing Model (ICAPM) as devised by Merton (1973) contradicts the CAPM based on the notion that the market is not static and can therefore not be captured by a single period model. An intertemporal approach to asset pricing entails taking into account the relationship between the current period returns and the returns that will be available in the future. The realized returns of a stock are based on increases in stock prices1. Stock prices are inversely related to the yield2 on that stock, e.g. a higher yield causes a decrease the stock price. ‘By holding a stock , the investor expects a higher return on the asset if, ex post, yield opportunities in the next period are lower than were expected (Merton,

1 Other factors also generate a return such as dividends but are excluded for the sake of the argument.

2

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1973:870). This intertemporal perspective is important because it recognises that investors have different time horizons and is a key mechanism on how stock prices react to changes in risk expectancies. When the risk for a stock rises so does the yield, the required future return for that stock. Investors who owned the stock prior to the yield increase have a lower realized return, yet a higher future expected return. Therefore these investors are only compensated for the ‘new’ risk from the point of the yield increase, they are not compensated for this risk prior to the yield increase. This is reasonable since the risk was not recognized by the market up to the point of the yield increase. This situation complicates when considering the fact that the market is capable of forecasting certain risks. It then becomes unclear when investors are compensated for the risk they take by buy stock. This issue is re-addressed when the hypotheses are set up. The intertemporal perspective therefore plays a large role when examining forecasts of risk as conducted in this paper. Regardless of the validity of the intertemporal perspective, the ICAPM did not add additional risk factors to the CAPM. This was done by two other models; the Fama/French three factor model and the Carhart four factor model3

.

Fama and French studied various factors that where argued to explain risk adjusted stock returns (Fama and French, 1992) . They concluded that only size measured by the market capitalization of firms and value measured by book-to-market effect have a significant effect on stock returns along with, but separate of, the CAPM beta. The resulting model argues that the stock return in excess of the risk-free rate can be explained by the market beta, a small minus big and higher minus low factors (Fama and French, 1993). They suggest that size and value are proxies for risks which are priced in stock returns in excess of the market beta. In a later paper they altered the BtM ratio for a price/earnings ratio as already done by Basu (1977) because it is a better measure of a value stock (Fama and French, 2005). They developed the following three factor model for estimating excess stock returns;

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3

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10 | P a g e Where is the common stock return in excess of the risk free rate, and is the result of a mimicking portfolio containing the value weighted proportion of all NYSE, AMEX, and NASDAQ stocks and represents the return of the overall market in excess of the risk free rate. SMB and HML are the factors for size and value respectively. In essence SMB represents the result of a portfolio that buys small sized stocks and sells big size stocks. Similarly, HML is the result of a portfolio that buys stocks with high BtM ratio’s and sells stock with low ones. It should be noted that the is the constant in the expression (2) but does not represent the risk-free rate as the constant in the CAPM model does. In fact, a well diversified portfolio should not have a constant if the model is accurate4.

In their paper of 1992, Fama and French did not include previous returns of a stock as a possible determinant of current return. Such a factor has been identified by Jegadeesh (1990) and Titman (Jegadeesh and Titman, 1993) who found that stock that did well in the previous six to twelve months, continue to do so in the coming months. Stocks that performed well maintained a one-year momentum and continued to do well in the short-run. Carhart (1995) incorporated the one-year momentum factor in the Fama French three factor model, creating a four factor model. The following equation (3) depicts the Fama French three factor model with a momentum factor added5

;

(3)

Where is the Winner Minus Loser factor which is the result of a portfolio buying high performers and selling low performs. High and low is defined as the upper and lower 30 percent returns.

Since the conception of the CAPM academic research has focused on either disproving or adding variables to the model. However, a more rigorous assessment of the volatility has been omitted. The assumption that the CAPM market beta incorporates the volatility of the market

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This also holds for the CAPM when based on returns in excess of the risk free rate, a constant (or intercept) would imply a return over the risk free rate which is not earned based on the risk relative to the market, thereby impeding the validity of the CAPM model.

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has lead researches to examine other sources of risk, without considering that the beta captures not all volatility information. The next section will discuss the information quality of implied market volatility based on stock options.

1.2 Idiosyncratic Implied Volatility

Implied volatility is implicitly contained in stock options prices. Stock options are priced on the four variables; the current stock price, intrinsic value, time to expiration, and volatility6. All factors are known except from the future volatility of the underlying stock. The volatility remains elusive and is an estimate based on the investors’ judgement7

. Once the stock option are priced it is possible to calculate the volatility used by the investors who traded the option (see Wilmott, 2001:205). Implied volatility for a given stock can be regarded as the market’s best estimate of the volatility of a given stock during the maturity of a stock option. The implied volatility is the best estimate of future volatility of the stock. If this would not be the case, ‘one could devise a trading strategy that could generate profits by identifying mispriced options’ (Jorion, 1995:507). As all the information regarding the stock is priced into the stock options the resulting price should be free of arbitrage opportunities. If an investor would somehow be able to calculate a more accurate volatility, the option would be mispriced. The investor would make use of this and buy underpriced or sell overpriced options8. The resulting price change would change the implied volatility resulting in a more accurate forecast.

Many academic papers have assessed the information content of idiosyncratic, firm level, volatilities regarding future volatility. Scholars find that implied volatility has information content and outperforms a historical model of forecasting (Christensen and Pabhala, 1998). Jorion (1995) is one of the first to test time series models against implied volatility and concludes that these models have no information content over the market’s view of volatility.

6

Cash dividends are omitted since such factors are not prime drivers and can be easily incorporated into a pricing formula.

7

Although this judgment could be based on mathematical models, the final input remains an estimate.

8Any deviation from the market’s volatility forecast would result in trading opportunities because the call and put can be

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12 | P a g e Nevertheless, the market’s view is biased and appears to be too variable against realized volatility (Jorion, 1995). Changes in the implied volatility are larger than changes in the actual realized volatility, meaning the markets tends to overreact and makes on average a too large adjustment to the implied volatility the needed. In a more recent study Mayhew and Stivers (2003), Yu et al. (2010), as well as Christensen and Pabhala (1998) conclude that implied volatility subsumes nearly all information about future firm volatility, including statistical models.

Since implied volatilities have a forecasting ability for the future volatility of a stock, literature has shifted towards examining its information content regarding future stock returns. Diavatopoulos et al. (2008) examine the effect of idiosyncratic volatility on the cross section of stock returns. They find that implied idiosyncratic risk positively predicts future stock returns while realized volatility does not. It should be noted that the research conducted in the field of idiosyncratic volatilities tends to use an approximation of implied idiosyncratic volatility. This is done by taking the standard deviation of the error term in a one factor market model (As done by Bali and Cakici, 2008). Ammann et al. (2008) claim to be the first to systematically analyze to information content of implied volatility on the cross section of returns using actual implied volatilities gained from option prices. They find highly significant positive relationships between returns and (lagged) implied volatilities. This relation is weaker for larger firms and independent of different value levels. Smaller firms seem to be effected more by volatility forecasts than larger firms, which is in line with Fama and French (1993). It also hints at the factor that small firms earn a premium because they are more sensitive to volatility, at least more than predicted with the CAPM.

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differently to certain risks. Using opposing firm relations to a particular risk, this risk is negated, it does not affect the overall portfolio. Because of this, studies examining risk and return use market level data to indentify risks and evaluate whether they constitute a basis for stock return. For example, Fama and French do not include a company’s own size in their regression model, this would make little sense since one then would try to estimate a firms’ sensitivity of stock returns on its own size. Instead market proxies (such as size) are actually market based portfolios mimicking a trading strategy based on the risk proxy, which generates a return on which a firm level return is measured.

1.3 Implied Market Volatility

Mayhew and Stivers (2003) included implied market volatility in their examination of stock returns. They argue that ‘for the majority of the firms, this index implied volatility contains reliable incremental volatility information beyond a model9 specification of firm’s returns’ (Mayhew and Stivers 2003:243). Market volatility as implied by (index) options is different from most of the variables used for predicting stock returns in at least two respects. ‘It is a real forward-looking variable measuring market participants’ expectations and it is a traded price, and therefore less likely to be affected by biases’ (Ammann et al. 2008:223). Ang et al. (2006a) conclude that there is a negative premium associated with the sensitivity to changes in the market volatility and that the implied market volatility is a priced risk factor after controlling for other factors. Ang et al. (2006a) focused on the change or innovations in implied market volatility index. Giot (2005) finds that the Volatility Index (VIX) of the CBOE is useful for predicting returns of the S&P 100. The VIX is the most commonly used as a proxy for measuring future market volatility expectations. Christensen and Nielsen (2005) find that the VIX is a dominant non-biased forecasting variable of future realized returns of the S&P 500. Banerjee et al. (2007) reach a similar conclusion and find that VIX levels and innovations predict the returns of characteristic based portfolios.

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14 | P a g e Ang et al. (2006a) focus on the changes in the level of the VIX and conduct a two factor regression analysis, the first beta being a ‘traditional’ market beta. They measure the sensitivity of a firm’s stock return to any change in the level of the VIX. After controlling for size, value, liquidity and momentum they estimate the effect of aggregate volatility to be -1 percent per annum (Ang et al. 2006). More general, an increase in VIX is followed by a (contemporaneous) decrease in stock prices. In general this proposition would hold due to the simple fact that in increase in VIX represents a downward sentiment in the market. The downward sentiment would encourage investors to demand a higher yield on the stock, decreasing the price. The premium demanded for holding the stock does increase, so one could expect to have higher future earnings. Examining this contemporaneous relation between the VIX level and stock returns does not prove that the VIX causes investors to change their demands for compensation for holding stock. It merely proves that the VIX is a useful indicator of general downward risk sentiment.

Whaley (2008) investigates whether the relationship between a VIX and stock returns is proportional. He finds that the relation is not proportional, the change in VIX rises in a higher absolute rate when the stock market falls than when it rises (Whaley, 2008). By regressing the change in the VIX innovations with two conditional beta’s, upward and downward market beta, he concludes that the innovation is twice as sensitive to an upward market movement. It should be taken into account that the formulated regression reports correlation rather than causality as the stock returns are not the dependent variable. However, the overall result suggests an asymmetric relationship between VIX innovations and the stock returns.

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risks, for example aggregate volatility innovations, have higher returns (see Ang et al., 2006b). They estimate the premium to be approximately six percent per annum which is not already captured by the market beta (Ang, et al., 2006b). They measure upside and downside risk by conditioning it on excess of market return over the mean market return. When return is greater than the mean return the beta is classified as an upside risk, when lower, a downside risk. They find that stocks returns reflect a premium for downside risk, even when controlling for size and value (Ang et al. 2006b).

Dennis et al. (2006) as well as Delisle et al. (2010) find a negative asymmetric effect between volatility and stock return. Delisle et al. (2010) find no relation between implied volatility declines and firm level returns. The asymmetry effect was found based on innovations of the VIX. ‘Investors require compensations for market volatility risk when it rises, but when market risk falls there is no differential return requirement’ (Delisle et al. 2010:20). They suggest it could be caused by investors overpaying for stocks that hedge increases in market volatility or investors requiring a premium for holdings stocks whose returns are negatively related to volatility increases (Delisle, et al. 2010).

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16 | P a g e the changes of the VIX, which could be expected of a indicator like the VIX. The next section will discuss the issue of the VIX measurement by these studies in more detail.

1.4 Measurement Implied Volatility

The VIX is most commonly used as a proxy for an aggregate expectation of future short-term market volatility. The index was introduced by Whaley (1993) and was originally based on the S&P 100 index option prices. Currently it is based on the S&P 500 and is ‘implied by the current prices of the S&P 500 index options and represents the expected future market volatility over the next 30 calendar days’ (Whaley, 2008:2). The VIX is often referred to as the investor’s fear gauge, an increase implies a greater amount of fear amongst investors. A more accurate understanding of the VIX would be in the vicinity of portfolio insurance. When investors have less confidence in the future market returns, and are not sure about the amount of future volatility, they start to buy portfolio insurances in the form of options. Investors will buy out-of-the-money puts when they fear a drop in the market, this increases the VIX level. Contrary, investors do not seem to buy out of the money calls when they expect a sudden rise, e.g. investors do not hedge against upwards risks. This is the most important argument in explaining the asymmetric effect as put forward by Delisle et al. (2010). The VIX is therefore not able to accurately represent the investor sentiment towards the market when this sentiment is positive, this is because a positive sentiment requires no insurances, therefore no options are traded on the basis of this sentiment and the VIX cannot change accordingly. The asymmetric is therefore caused by the actions of the investors by buying stock options in a downward sentiment and not buying stock options in an upward sentiment. Since stock options do respond to investors sentiment (Baker and Wurger, 2006) the VIX is unable to explain stock returns in a upwards market sentiment since it cannot capture this relation.

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should be priced into the stock. Assuming (semi-strong) market efficiency, the current volatility is priced into the stock, so the change in return should be caused by new information. The differential of both is the increase or decrease in market volatility of the following 30 days. The formula used by Ang et al. (2006) and Delisle (2010) is stated below (4). Delisle (2010) then separates the negative and positive changes in order to create two distinct variables that allow a measurement of both upside and downside risk, therefore taking into account the asymmetric phenomenon.

(4)

Where the is the VIX innovation at time t, based on the VIX for the coming period minus the VIX of the previous period.

This paper takes a similar approach as Delisle (2010) although the latter variable is replaced by the realized volatility in order to create a variable that can be interpreted as the expected change in market volatility.

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Where is the expected volatility change at time t, the is the implied market volatility expected for period t+1 and the volatility of the market for period t. The data section will go into more detail concerning the measurement of these variables.

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18 | P a g e expected change is labelled a volatility risk discount (VRD). The relation between the VRP and VRD are re-stated in hypotheses10 below.

H 1.a. Stock returns have a negative relation to a volatility risk premium. H 1.b Stock returns have a positive relation to the volatility risk discount.

Volatility is the principal driver of stock returns. It is a very good (the best so far) proxy for risk, which is why the CAPM is such a touchstone of financial theory. As discussed in the beginning of this review more proxies for risk have been developed and should therefore be considered in any research on stock returns. These proxies represent risks which will be discussed in the next section. Firms characteristics are not only included in the analysis as controlling variables, differences in the sensitivity to expected volatility could be explained by these characteristics.

1.5 Stock Market Anomalies

In the examination whether implied volatility is a risk factor in stock returns, the models presented in the beginning of this review will be used. Therefore, any observation relating to implied volatility is separate of the anomalies present in the two models. The fact that a implied volatility risk is separate from a risk proxy in the model (size, value, past return) does not indicate whether stock returns and the expected change in market volatility is not caused or related to one of these three anomalies. In this last section the three stock market anomalies will be reviewed and discussed starting with the size and value premiums included in the Fama French three factor model and the momentum effect based on past returns as included in the Carhart four factor model. The main goal is to examine how potential sources of risk can have an effect on the sensitivity of a firms’ return to both expected volatility factors.

The findings of Arisoy et al. (2006) suggest different relations to volatility for different classes of firms. For example, during high volatility periods, small firms and value firms are more prone to downside market risk, hence they are associated with negative volatility

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coefficients. Thus, ‘at times of high volatility, investors see value firms and small firms riskier than their growth and big counterparts and price this risk in their returns via an important factor, volatility risk’ (Arisoy et al., 2006:29) This could be a potential explanation to why growth firms underperform value firms and small firms outperform larger firms. Although this applies to volatility risk, such reasoning could also apply to the expected volatility risk. However, other potential reasons for premiums for firms classes should be discussed as these could be important sources of risk.

Banz (1981) is one of the first and well known contributors to what he specified as the ‘size effect’. The size effect entails that ‘the common stock of small firms had, on average, have a higher risk-adjusted returns than the common stock of large firms’(Banz, 1981:3) Although Banz can prove that small stocks have had higher returns in excess of the risk free rate, he cannot specify a theoretical model for why they do so. It could also be a proxy for an unknown additional factor which is correlated with the size of firms. One possible explanation is of the size effect is the availability of information. Klein and Bawa (1977) find that when information is insufficiently available for certain securities investors will avoid these. Therefore investors who are willing to bear the additional risk of insufficient information could earn a risk adjusted return. Another explanation of the size effect is given by Perez-Quiros and Timmermann (2000). They argue that the size serves as a proxy for tight credit market conditions. The conditions for small firms are tighter because they cannot use international and more favourable domestic bond markets. The willingness of investors to provide credit, e.g. buy stocks and bonds, is heavily affected by the market conditions. Therefore in unfavourable market conditions smaller firms are more sensitive to these market conditions. The higher return found for small firms is then a compensation for this additional risk.

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20 | P a g e results from Delisle et al. (2010) larger firms should be expected to have a lower sensitivity towards premiums and discounts. The reasoning behind the fact that smaller firms have larger sensitivities towards implied volatility is that they are more dependent on volatility itself. Whether size is a proxy for the restricted credit market opportunities or degree of information, a smaller firm is more sensitive towards general market conditions. The VRP and VRD factors predict changes in these market conditions, since small firms are more vulnerable to these market conditions, logic would dictate a higher relation to these factors as well. The following hypotheses are set up to test this relationship for both volatility risk discounts and premiums.

H 2.A The stock return of smaller firms is more sensitive to forecasted increases in short-term market volatility than the stock return of larger firms.

H 2.B The stock return of smaller firms is more sensitive to forecasted decreases in short-term market volatility than the stock return of larger firms.

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market which would lead to less tight credit market conditions, which is particular favourable for low value firms. Therefore the relation between the volatility factors can be different for different value stocks returns. The following hypotheses are set up to asses this theory.

H 3.a. The stock return of low value firms exhibit larger sensitivities towards forecasted increases in short-term market volatility (VRP) compared to the stock return of higher value firms.

H 3.b. The stock return of low value firms exhibit larger sensitivities towards forecasted decreases in short-term market volatility (VRD) compared to the stock return of higher value firms.

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22 | P a g e premium implies that markets will be getting more volatile and a discount a less volatile market, the proposition can be tested as stated by the following two hypotheses.

H 4.a. Firms with high past returns have a higher sensitivity to the forecasted increases in short-term market volatility than firms with low past returns.

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2 Data

2.1 Period Selection

Monthly observations will be used as is customary in stock return research. Using higher frequency data would increase the importance of the market efficiency assumptions. This research is not a test of market efficiency, but more of the general risk and return relation of stocks. Using less frequent data, quarterly, or even yearly, would dilute any finding of causality due to the long time span between observations.

Using the VIX as a proxy for the forecasted short term volatility restricts the sample period to after 1993, since the VIX was founded in that year. When examining the VIX and monthly realized volatility it becomes apparent that the first month where the implied volatility is higher than the realized volatility, is in 1998. Therefore to properly account for the asymmetric effect the sample period starts at January 1998. The end period of the sample is the end of 2007, which makes the total sample contain a 10 year period. Excluding the 2008 financial crisis from the sample period increases the reliability of the results since the ‘super’ volatility in that period would disrupt the outcomes severely. Since the VIX is a monthly forecast, using monthly observation accommodates in non-overlapping data as required in order to properly evaluate the effect of the VIX on stock returns.

2.2 Sample Selection

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24 | P a g e includes 500 firms of which 81 are financials and therefore rejected. Firms need to be listed the entire period in order to have monthly return data observations11, as well as have the total market capitalization and price/earnings ratio available. From the remaining 419 firms 108 failed to meet these criteria, reducing the final sample to 311 firms.

2.3 Dependent Variables

The stock returns are measured as the Return Index attained from Datastream. This index accounts for the stock’s return by creating an index with a base of 100 which incorporates the change of the stock price, the dividend payments and other factors representing a return for investors. The return is then calculated as the monthly rate of change of the index from one month to the next. The return is calculated as of the end of the month, so that the return for month t is the realized return in that month. In order to create a return based on the risk of the stock, the risk free rate is subtracted. The risk free rate is taken from the Kenneth French website12

. It represents the return from a close to maturity US Treasury bill which is considered as risk free.

2.4 Volatility Premiums and Discounts

The VIX is freely available at the CBOE website, the volatility needs to be calculated . The VIX looks forward 30 calendar days, not trading days. Therefore monthly periods are used, the data observations were taken at the first day of that month, which is also the last day of the previous month. Expression (6) depict the calculated volatility;

(6)

Where is the volatility for a month based on the standard deviation of the daily return for the market in that month13 . The SPX daily return is used as a proxy for the market return

11 Since the past return calculation is based on the year t-1 the monthly return for 1997 is also required.

12

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

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and only the trading days are included in the calculation. is the return of the market at the end of trading day i, is the mean of the market returns for the SPX index of month t.

The VIX is freely available14 for each month and measured at the beginning of month t, the premium or discount for any month in the sample period can be calculated.

; ( ) (7) ; ( ) (8) Where is the Volatility Risk Premium at time t and the Volatility Risk Discount at time t both calculated as the difference of the implied market volatility compared to the current market volatility at the beginning of month t. A VRP is conditioned on the fact that is higher than , the VRD on the fact that is lower than . When the condition is not satisfied, the factor takes the value of zero.. Both VIX and VOL are compared on a contemporaneous15 basis, however refer to the volatility in different periods. The VIX refers to the coming 30 day volatility (period t+1) , the VOL to the realized 30 day volatility (period t-1).

2.5 Control Variables

The control variables included in the regressions are the proxies for size value and momentum. The data for these control variables is freely available at the Kenneth French website. The measurements are returns for portfolios mimicking the size, value and momentum effects. The Fama/French factors are constructed using the six value-weight portfolios formed on size and book-to-market and past return. The portfolios essentially buy the upper thirty percent of the stock class and sell the lower 30 percent of stock class. The differential is then the return on a characteristic based portfolio and used in this paper as a control factor. The market premium (MKT) is the excess return on the market, which consists

14

The VIX is available for daily and monthly data at http://www.cboe.com/micro/vix/historical.aspx

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26 | P a g e of the value weighted return on all NYSE, AMEX, and NASDAQ stocks minus the one-month US Treasury bill rate (Fama and French, 1993).

2.6 Descriptive Statistics

The table (1) presents the descriptive statistics for the data used in the phase one regressions16. The descriptive concerns 120 observations over the ten year period for eight variables. The dependent variables, equal and value weighted return, are similar in range yet nonetheless have different means. This implies that on average the monthly returns of the overall portfolio are higher when all firms are equally included regardless of their size. By including the size of the firms in the measurement the mean is lower. This would imply that larger firms have a lower average monthly return.

The control variables, MKT, SMB, HML and WML are measures by the returns of mimicking portfolios. The mean returns of these portfolios lies below that of the market except for the WML portfolio, which is higher than value weighted returns nonetheless lower than equal weighted returns. Moreover, the WML portfolio is riskier as measured by its elevated standard deviation. In general the control variables have a similar or greater range than the market returns, the minima are lower and maxima higher.

The realized market volatility ( ) is based on 120 months and has a range 56,29. The range can be considered as large and is mainly due to the maximum observation of 62,31. The implied volatility for the coming period ( ) has a lower maximum (44,28) and therefore a lower range of 33,86. Both realized and implied market volatility have a mean of around 20. The measurement is in standard deviations *100 which is commonly used. The actual average standard deviation from both implied and realized market volatility is σ 0,20 percent. The variables or measuring the forecasted volatility, VRP and VRD are very dissimilar in their maximum observation. The VRP has 71 observations whereas the VRD has 49 non-zero observations. Therefore the in the 10 year period a premium is more common than a discount. The VRD has a maximum observation of twice as large a than the VRP, 36,64 and 18,44

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27 | P a g e

respectively. Their means are remarkably similar, and center around three. This represents that on average the monthly forecasted increase or decrease in market volatility is σ 0,03 percent.

Descriptive statistics for variables of the multivariate regression models for the period 01/01/1998 to 31/12/2007

N Range Minimum Maximum Mean Std.

Deviation Dependent Variables EWR 120 26,67 -11,82 14,84 1,0133 4,27 VWR 120 21,86 -8,85 13,01 0,6824 4,14 Volatility Variables VRP 71 18,44 0,00 18,44 3,0049 4,04 VRD 49 38,64 0,00 38,64 2,8596 5,82 Control Variables MKT 120 24,38 -16,20 8,18 0,3579 4,48 SMB 120 38,84 -16,85 21,99 0,2775 4,22 HML 120 26,24 -12,37 13,87 0,3810 3,86 WML 120 43,39 -25,04 18,35 0,8617 5,76 Measurement Variables 120 56,29 6,01 62,31 20,50 10,78 120 33,86 10,42 44,28 20,61 6,80

Table 1; Descriptive Statistic Regression Model data

Where EWR (Equal Weighted Return), the VWR (Value Weighted return) based on quarterly market capitalizations, both VWR and EWR are measured as monthly realized returns. The VRP (Volatility Risk Premium) and VRD (Volatility Risk Discount) are measured in standard deviations *100 as is used for the VIX, the ‘N’ column refers to the amount of non-zero observations. MKT is the monthly realized market return, the monthly realized return of a size based portfolio is SMB (small minus big). HML (high minus low) is the monthly realized return for a value based portfolio. WML (winner minus loser) and represents the monthly realized return for a one year return momentum based portfolio, all control

variables are in percentages. Both and are the realized market

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28 | P a g e

3 Methodology

3.1 Research Approach

The first phase of this research will concern hypothesis 1a and 1b and focuses the general relationship between stock returns and expected changes of the market volatility. For this phase time series regressions are run with three different models and two different dependent variables, and includes all 311 firms. The main goal of this phase is to identify an appropriate model for the next phase and evaluate the loadings found for the VRP and VRD. Three models are used, the CAPM, Fama French three factor and the Carhart four factor model. All models are run with the inclusion of the two volatility factors, VRP and VRD, as well as without these factors. This allows for a better evaluation of the return models and the appropriateness of forecasted volatility factors in the stock return models. As stated before, the regression models will be performed with two different dependent variables. The equal weighted and a value weighted return will be used. Although value weighted returns are more likely to achieve a higher model fit, since the market premium is value weighted as well, it could distort the models’ coefficients since a firm size is taken into account.

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the coefficient will be standardized17. Standardization allows for an understanding of the position of a coefficient with respect to the mean value as well as its position in the distribution. In order to argue that one coefficient significantly different than another coefficient, two criteria are set. The coefficient’s values should be on opposite sides of the mean as well as have a minimum distance of one standard deviation from each other. The emphasis will be on the upper and lower 10 percent of observations in any characteristic class (e.g. portfolios 1 and 10).

In addition to these portfolios a ´cross´ portfolio is created using the quintiles of both value and size. This is done in order to validate the results found in the separate portfolio regression. A portfolio based on a (binominal) combination of firm characteristics can isolate a combination of size and value of firms who are the most or least exposed to forecasted market volatility.

3.2 Regression Models

The effect of volatility risk premiums and discount on stock returns is examined in the three models presented in the literature review. The first model will incorporate the VRD and VRP as well as the market beta in order to capture the market volatility. The market beta as used in the CAPM is crucial since the two factors for implied volatilities should test the returns with their sensitivities towards expected changes to market volatility in the coming month. When the market beta would be omitted in the model, the sensitivities found would include the ‘normal’ CAPM beta’s and the it would merely test the accuracy of the volatility forecast . The following model (9) is similar to that of Whaley (2006) used for assessing the proportional relation of the VIX and used here to test premiums and discounts separately. This allows the model to deal with possible asymmetric relations between market volatility and stock returns.

(9)

17

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30 | P a g e Where is the excess return over the risk free rate of month t, the MKT market premium of month t.

The CAPM based model (9) does not include possible returns due to firm characteristics. The following two models are based on the Fama French (10) and Carhart (11) models with the inclusion of both the VRP and VRD factors. The models test for sensitivities of stock returns towards volatility forecast which are separate from returns which could be explained with firm characteristics.

(10)

(11)

Where is the excess return over the risk free rate of month t, the MKT factor the market premium of month t. SMB is the return for a size based portfolio, HML the return for a value based portfolio and WML the return for a portfolio based on passed returns.

A multivariate regression would assume a constant conditional variance. Since this assumption is often not satisfied in time series models, Bollerslev (1986) proposed an adjustment to allow for a conditional variance of the error. The CAPM based model (9) satisfies the homoscedasticiy requirement of regression analysis nevertheless the FAMA French (10) and Carhart (11) based models violate this assumption18. This is violation is resolved by adopting a Generalized Autoregressive Conditional Heteroscedasticity (GARCH)19 equation for the error term.

3.3 Portfolio Formation

Portfolios are created for size, value and past return , in addition a combination of both size and value allows for the creation of size/value portfolios. The initial portfolio formation is

18

Time series models require a test for ARCH errors in order to justify the adoption of GARCH, please refer to appendix B for the multivariate regression requirements.

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31 | P a g e

based on deciles of the measurements for size, value and past return for each year. For size the mean logarithm monthly market capitalization of the year t-1 is used. The monthly price/earnings ratio is used to proxy for firm value. Even though book-to-market ratio’s are common to use, Fama and French (2005) argue that book-to-market is a good proxy for growth stocks, not for value stocks. A complicating factor in using the P/E ratio on a monthly basis is that the variance is much larger that with market capitalizations. To mitigate the effect of outliers the median of year t-1 will be used instead of the mean20. Past return is measured based on monthly realized returns, the total return is then calculated as the geometric total

Stocks are subsequently sorted according to their relative size, value or return for every previous year (year t-1). Effectively a characteristic based portfolio contains 10 batches of stocks for each year, totalling 100 batches that forming the actual portfolios. As stated, the year t-1 is consistently used to form the portfolio’s for year t. This is a common approach in portfolio formation (see Fama and French, 1993). Since the regression period concerns 1998 to 2007 the measurement years for sorting the stock are from 1997 to 2006. Taking the previous year for the measurement as a basis for sorting the stocks is required in order to ensure that the factor can be regarded as independent and a forecasting variable for stock return. When contemporaneous data would have been used one could argue that the causality cannot be established since returns also have an impact on value, size and return measurements. When using the previous year’s observations (year t-1) those cannot influence the factors of the current year (year t). A similar approach is used for the 25 size/value portfolios. Formation is based on the quintiles of both size and value for previous year (t-1). Stocks are then sorted according to their relative size and relative value in that year.

3.4 Portfolio Performance

The monthly realized returns for the three portfolio´s based on firm characteristics is based on the 120 realized return observations in the ten year period. The range of the size returns is around 30 percent, from around -15 to15 percent per month. The mean of monthly returns

20

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32 | P a g e varies greatly amongst the portfolios. The ‘small size’ portfolios have a larger average return that the ‘larger firm’ portfolios. The difference between the smallest firm and highest firm portfolio is 1,94 percent, 2,34 and 0,40 percent respectively. This implies that smaller firms earn larger returns as is suggested by the literature. The return can also be partly explained by the larger amount of risk associated with this portfolios. As the standard deviation of portfolio ´size 1´ is the highest (5,69) a higher return could be expected. The return of the size portfolio’s are consistent with the literature claiming that small size firms outperform large size firms.

The value based portfolios have a similar range, minimum and maximum as the size based portfolios. The mean returns are lower, the highest being 1,24 percent per month. On average a value portfolio has a realized monthly return of about one percent. No large differences in mean returns exist between portfolios, this implies that higher or lower value firms have similar returns in the measurement period. The low value firm portfolio has the highest standard deviation (5,09) , meaning the highest risk is associated with this portfolio. However, investors are not compensated for this since the mean return is not higher than in other portfolios.

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33 | P a g e

Portfolio monthly realized returns descriptive statistics for the 31/01/1998 – 31/12/2007 period

Portfolio Range Minimum Maximum Mean Std.

Deviation Size 1 (Small) 33,47 -15,87 17,60 2,34 5,69 Size 2 31,42 -12,85 18,56 1,31 4,61 Size 9 29,42 -16,61 12,82 0,37 4,62 Size 10 (Big) 26,90 -12,09 14,82 0,40 4,69 Value 1 (Low) 30,23 -15,79 14,43 1,00 5,09 Value 2 23,29 -12,12 11,16 1,19 4,50 Value 9 27,49 -11,92 15,57 0,90 4,24 Value 10 (High) 29,42 -13,47 15,95 0,99 4,82 Return 1 (Loser) 61,36 -21,33 40,03 1,55 8,05 Return 2 35,33 -14,52 20,81 0,94 5,24 Return 9 33,25 -12,46 20,79 0,82 4,90 Return 10 (Winner) 42,63 -15,26 27,36 1,46 6,97 Table 2; Portfolio Descriptive Statistics

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4 Findings

4.1 Results Regression for full sample’s returns

The findings of phase one concern the regression models based on the value – and equal weighted returns of all 311 firms in the sample. The table (3) provides an overview of the findings that are discussed below.

The model fit of the regressions can be considered as good, on average around 75 percent ( ) of the variance in the stock returns is explained by the model. The Carhart model has a better fit than both the CAPM and Fama French models, indicated by the higher adjusted . Incorporating the implied volatility factors into the model does not change the superiority of the Carhart, in fact the Carhart based model including the VRD and VRP has the highest fit of all equal weighted models, being 0,799 and adjusted being 0,787. The value weighted return models have a slightly higher fit ( ). This is as expected since the market factor is a value weighted index, making the dependent variable value weighted should increase the fit of the model. Interestingly the fit of the six factor model is only slightly higher value weighted than equal weighted, 0,803 and 0,797 respectively. As with the equal weighted models, the inclusion of the VRD and VRP improves the overall stock return model. The WML factor does not improve the model since it is not significant ant the adjusted is slightly lower than the Fama French based model.

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35 | P a g e

The fact that the six factor model has no significant alpha’s does not signify that both VRP and VRD factors contain an adding value to the overall model. Any variable should be significantly different from zero to have any explanation adding-value over the dependent variable. None of the VRP factors have a significant coefficient. Therefore the VRP is found to have no influence on stock returns. Even though the factor coefficients are not significant, the sign is unanticipated. The positive sign would indicate that the VRP cause an increase in the stock return. One would expect a decrease arguing the risk return trade-off principle.

In all three models with both value - and equal weighted return the VRD is significant to a less than one percent level, except for the CAPM with equal weighted returns. In this model the VRD is significant to a less than ten percent level. When examining the Fama French three factor and Carhart model types the factor coefficients are similar, 0,125 and 0,119 respectively. The coefficient are somewhat lower when using value weighted returns and are 0,101 and 0,105.

The control variables are based on the three models as presented in the literature review. The market premium (MKT) has been included in all models and is significant to a less than one percent level throughout all models. The market coefficient is lower in equal weighted return models than in value weighted return models. The highest value is nearly one (0,999) and the lowest 0,800. An important observation is that the inclusion of both expected volatility factors does not dramatically change the MKT coefficient. This means that the VRD does not include a risk already measured by the market beta, conversely to opposite can be observed. The VRD factor contains information not captured by the market beta.

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36 | P a g e Risk proxy coefficients based on the equal and value weighed return of 311 non-financial firms of

the S&P 500 in the period 31/01/1998 to 31/12/2007

Regression Coefficients based on Equal Weighted Return

VRP and VRD Included Models as in literature

CAPM Fama/French Carhart CAPM Fama/French Carhart

Α 0,152 -0,118 0,027 α 0,555* 0,408* 0,591* VRP 0,07 0,055 0,054 VRD 0,085*** 0,125* 0,119* MKT 0,918* 0,999* 0,927* MKT 0,908* 0,957* 0,892* HML 0,472* 0,424* HML 0,430** 0,397*** SMB 0,097** 0,127* SMB 0,087* 0,090* WML -0,130* WML -0,123* 0,635 0,759 0,797 0,616 0,740 0,779 0,625 0,748 0,787 0,613 0,733 0,772

Regression Coefficients based on Value Weighted Return

VRP and VRD Included Models as in literature

CAPM Fama/French Carhart CAPM Fama/French Carhart

Α -0,127 -0,074 -0,068 α 0,208 0,291** 0,298* VRP 0,001 0,042 0,053 VRD 0,132* 0,101* 0,105* MKT 0,814* 0,831* 0,817* MKT 0,800* 0,821* 0,817* HML -0,058 -0,071 HML -0,067* -0,069* SMB -0,124* -0,118** SMB -0,167 -0,166* WML -0,046 WML -0,005 0,787 0,803 0,803 0,764 0,794 0,795 0,781 0,795 0,792 0,762 0,789 0,788

Table 3; Overview Findings phase one. Significance level; *** < 0,10 ** <0,05 * <0,01

Where is the equal weighted return for the full sample. The MKT is the return of the market as used in the

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37 | P a g e

4.2 The effect of VRD and VRP on stock returns

This research uses a asymmetric approach as done by Delisle et al. (2010) and Dennis et al. (2006). The regressions on the full sample of the 311 firms show significant findings for the volatility risk discount yet not for the volatility risk premium. Investors are willing to accept a lower yield on the stock which in turn increases the stock price when a lower market volatility is expected. This effect is separate of the effects based on the three market anomalies. The VRD is therefore not a risk measure already included by these risk factors.

In general, findings are in line with those of Ang et al. (2006), who conclude that there is a relation between stock returns and changes implied market volatility. However, Ang et al. (2006) used a single factor to measure the implied market volatility forecast. Delisle et al. (2010), and Dennis et al. (2006) address the asymmetric issue and find a negative significant relationship between positive VIX innovations, and no significant relationship between negative innovations. They create their implied volatility factors based on the previous implied volatility, whereas this research uses the realized market volatility. Therefore this paper assesses the relation between stock return and expected changes in market volatility. Although the previous literature would suggest that an increasing VIX has an effect on stock return, the excess level of the VIX over the volatility does not cause changes in stock returns. A VIX below the realized market volatility does cause a decrease in required future return. It should be noted that an increasing or decreasing VIX is not the same as a VIX above or below the market volatility. The VIX can increase or decrease while being above or below the market volatility. These findings show that when the expected volatility, as measured by the VIX, is below the current market volatility investors adjust their yield demands. So far literature did not find a downward market sentiment as measured by an implied volatility index.

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38 | P a g e decreases in short-term market volatility. Indicating that investors do accept a lower future market return when the implied volatility is lower than the current market volatility.

The regressions included the full sample and demonstrated that the VRD factor and not VRP factor increases the demanded return on stocks. A more rigorous approach is used in the second part of the research. Thus far the findings prove that the implied volatility is a separate effect than captured by the measurements of the risk proxies, the firm ‘characteristics’ could have different exposures to both expected volatility factors. In this phase portfolios are created using a Fama French approach described earlier. The equal weighted returns are used in the Carhart type (equation 11) regression in order to examine the effect of firm characteristics on both the VRP and the VRD factors.

4.3 Regression Results of size based portfolios’ returns

The table (4) depicts the result from the regression of a equal weighted returns based on the size of a firm for the lower and upper 20 percent of firms based on market capitalization21. None of the constants are significant indicating well diversified portfolios and no residual return unexplained by the factors. The volatility risk premium factor is not significant in nine out of ten portfolio’s. Only portfolio seven has a significant positive VRP loading. The volatility risk discount factor is significant to the less than one percent level in seven portfolios. Portfolio eight has a VRD with a five percent significance level, portfolio’s seven and nine are not significant. The coefficients do not display a particular pattern (please refer to appendix E). The largest VRD coefficient is 0,238 on portfolio three, whereas the smallest is 0,120 on portfolio two. The mean coefficient is 0,149 and the standard deviation 0,046. The difference between the largest and smallest size portfolio is 0,067.

As for the control variables the MKT factor is consistently significant throughout all ten portfolios and shows larger coefficient for smaller size portfolios than for bigger size portfolios. The value risk proxy (HML) is significant to a one percent level in most portfolios, its coefficient is lower for portfolios with larger firms. The size risk proxy (SMB) shows a

21

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39 | P a g e

similar pattern as the HML factor. The largest size portfolios shows a negative coefficient for both HML and SMB factors. The momentum factor (WML) shows various significant coefficients for larger size portfolios. The coefficient have a negative sign, indicating a negative relation with stock returns.

Table 4; Size Portfolio Significance level *** < 0,10 ** <0,05 * <0,01 The findings concerning the VRP are consistent with the findings of the phase one, which also concluded that firms’ return is not affected by the VRP. Therefore the hypotheses 2a, stating that smaller firms are more affected than larger firms by the VRP, cannot be accepted. Firms are in general not affected by the volatility risk premium. The return of firms is effected by a volatility risk discount since many portfolios where found to have a significant positive relation to the VRP factor. In order to compare whether small firms have a lower then large firms the coefficients are standardized. The small size portfolio has a z-scores of 1,05 and big size portfolio has a z-score of -0,41. Large firms have a coefficient below the mean whereas small sized firms have a coefficient above the mean. The difference is then 1,46 standard deviations, indicating that smaller firms have a larger exposure. This allows the acceptance of hypothesis 2b, smaller firms have a larger exposure towards the VRD factor.

4.4 Regression Results of value based portfolios’ returns

In order the examine a possible distinction between high and low value firms and their stocks relation to implied market volatility, ten portfolios based on firm value are regressed using the same six factor model as for the size portfolios, and the results are presented in the table (5). As with the size portfolios the constant is in mostly insignificant, and the volatility risk premium has no significant coefficients. The volatility risk discount has several

VRD coefficients for portfolios based on the upper and lower two deciles of firm size

VRP VRD MKT HML SMB WML Alpha

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40 | P a g e significant coefficients, portfolio one to five and seven, nine and ten. The VRD coefficients are of similar amount as with the size portfolio’s, ranging between 0,131 and 0,173 and has a mean of 0,144 with a standard deviation of 0,019. The market factor is significant and not very different from the market beta for size portfolios. For the remaining control factors, HML, SMB and WML, different significance levels can be observed. Not all portfolio’s show a significant relation to the control factors.

Table 5; Value Portfolios Significance level *** < 0,10 ** <0,05 * <0,01 Low value firms derive their return premium from the fact that they are more dependent on unfavourable economic conditions. These type of firms where therefore hypothesised to be more sensitive to expected changes in market volatility than higher value firms. The findings indicate that this is not the true when an increase in market volatility is expected. High and low value firms in general do not seem to have a relation to the VRP factor. Hypothesis 3a cannot be accepted since low value firms exhibit no larger sensitivities towards forecasted increases in short-term market volatility compared to higher value firms. Significant relations between firm portfolios where found between the volatility risk discount and stock returns. This confirms that stock prices do react to this factor, also when value based portfolios are created. The difference between low and high value firms and the amount of influence of the VRD can be assessed using standardized coefficients. The z-scores for the lowest and highest value portfolio is -0,05 and -0,53 the difference being 0,48 standard deviations. Both are below the mean and although the lowest 10 percent of value firms is higher than the highest value firms the difference is too small in order to confirm hypothesis 3b. Low value firms exhibit no larger sensitivities towards forecasted decreases in short-term market volatility

VRD coefficients for portfolios based on the upper and lower two deciles of price/earnings ratio

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