Testing the risk-return relationship in the stock market

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Master Thesis

Testing the risk-return relationship in the stock market

MSc Finance

Faculty of Economics and Business

University of Groningen

Author: S.C. Gorter Student number: 1627902

Email: simon_gorter@hotmail.com Phone: 0648117422

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Abstract

This thesis examines the risk-return relationship in the stock market. This is done by looking into the relationship between expected volatility and the risk premium. Expected volatility of the market portfolio is considered to be a risk measure for investing in stocks. Volatility expectations are measured using conditional volatility based on a GARCH(1,1) model and the implied volatility obtained from the CBOE VIX. The MSCI World index is used as a

representation of the market portfolio. No relationship is found between expected volatility and realized excess returns. When returns are adjusted for risk, this paper shows that the returns in times of a below average VIX are significantly higher than in times of an above average VIX. This indicates the possibility for a trading strategy that increases the return to risk ratio which can be achieved by investing more heavily in stocks when the VIX is low.

JEL classification: C22, G12, G14, G15

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1. Introduction

The relation between risk and returns in the stock market is one of the most discussed topics in financial economics. Assets traded on stock markets yield higher returns in the long term than assets being close to riskless, for example Treasury Bills. This is caused by the risk aversion of investors. In order to have people investing in risky assets, an expected excess return is required. Although the diversifiable risk of individual stocks can be eliminated, the systematic risk that relates to the risk of the overall market is priced. This price of risk is referred to as the risk premium. Equity risk premiums are a central component of every risk and return model in finance and can therefore be considered as a fundamental concept in finance.

A vast amount of research has been conducted on this topic, but there are still many contradictory findings regarding the properties of the equity risk premium. In many studies the equity risk premium is assumed to be fixed in time1. This is a restrictive assumption since market risk changes over time. If market risk changes over time, the premium should move along as well, since investors demand a higher premium when the risk is higher.

This thesis investigates whether the expected volatility of the stock market has an effect on the returns. This will test the theoretical risk-return relationship in which expected volatility represents the riskiness of the stock market2. The expected volatility is measured by daily conditional volatility using a GARCH(1,1)3 model and the monthly implied volatility from the CBOE VIX4. The MSCI World Index is used to represent the world’s equity market. The effect of expected volatility on stock returns and risk-adjusted stock returns is tested with an OLS regression. In addition, (risk-adjusted) stock returns in different levels of expected volatility are compared. This will give more insight into the risk-return relationship in the stock market. Knowledge about the risk premium’s behaviour in different market conditions could be helpful for investors to time the stock market in case inconsistencies are found.

Most studies thus far generate expected volatility with forecasting models that only use historical data. There has been much less research focusing on the relationship between

1_{See, e.g., Fernandez (2010); Damodaran (2009); Welch (2000). These authors summarize findings of}

several studies that try to estimate the equity risk premium using historical averages.

2_{The use of expected volatility as a measure of risk is discussed in section 2.} 3

The GARCH(1,1) (Generalized Autoregressive Conditional Heteroskedastic) model as introduced by Bollerslev (1986) is considered in some detail in section 4.

4_{The CBOE Volatility Index (VIX) is a measure of near-term volatility derived from S&P 500 index}

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the implied volatility and equity returns. Important extensions to prior work are made by using the implied volatility to measure expected monthly volatility and adjusting stock returns for risk based on volatility expectations.

The main results can be summarized as follows. Neither the daily conditional volatility, nor the monthly implied volatility show evidence of a relationship with returns. When returns are adjusted for risk, the returns in times of a below average implied volatility are significantly higher than in times of an above average implied volatility. This indicates the possibility for a trading strategy that increases the return to risk ratio which can be achieved by investing more heavily in stocks when the VIX is low.

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2. Review of the literature

2.1 Fundamentals of the risk premium

Since this thesis focuses on the risk-return relationship, I begin with the theoretical reasons for the existence of an equity risk premium and explain the factors that determine its size.

2.1.1 Risk

The risk associated with the world’s equity market may seem like a simple concept that does not need further clarification. But since risk is a phenomenon that is not directly observable it is surprisingly difficult to quantify. In finance, risk does not have one single definition. In this thesis, the view advocated by Damodaran (2009) is followed. He defines risk as “The likelihood that we will receive a return on an investment that is different from the return we expected to make”. Thus, risk does not only include bad outcomes, but good

outcomes as well. Damodaran (2009) refers to this as downside risk and upside risk. Risk in investments can therefore be defined as the probability distribution of future returns.

In the stock market, risk can be reduced by diversification. Theories about diversifiable risk date back to the work of Markowitz (1952). In Markowitz’s theory the fundamental concept is that the assets in an investment portfolio should not be selected on its individual risk and return characteristics. Rather, investors should consider how the price of each asset is behaving relative to price changes in the portfolio as a whole. The Capital Asset Pricing Model (CAPM), introduced by Sharpe (1964) and Lintner (1965), is a general

equilibrium application of Markowitz’s theory. It states that the market portfolio, a portfolio of assets consisting of a weighted sum of every asset in the market, is the optimal portfolio in terms of the risk-return ratio. There is criticism on the CAPM due to its many assumptions, such as the use of Gaussian distributions to model the return properties and the fact that this model neglects taxes and transaction fees. It is however still widely used because of its simplicity which makes it suitable for empirical research.

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higher return. When the market shifts to the other direction there is a capital gain. Because investors anticipate on the gains and losses caused by changing volatility states a lower rate of return in the high volatility state is accepted and a higher rate of return is demanded in the low volatility state.

The lack of liquidity is another risk that is considered to exist in addition to market risk. When stocks lack liquidity, it can force investors to sell at large discounts or pay high transaction costs to liquidate their positions. Several studies, such as those by Gibson and Mougeot (2004) and Bekaert, Harvey and Lundblad (2006) concluded that liquidity accounts for a significant part of differences in risk premiums between countries.

The literature often considers these additional risks to exist alongside systematic risk. However, it can be argued that these risks are part of the systematic risk. If there is a chance that a volatility state change occurs or a chance that stocks become illiquid which results in a capital gain or loss, it can be included in the probability distribution of future stock prices.

Although its significance is questionable, it is worth mentioning that not all risk factors relate to changes in the stock price. Some risks only manifest in the investor’s net value. Changes in taxes and transaction costs are examples of those risks. Imagine an asset that has a guaranteed pay-out. If there is a probability that the government will change the tax rate on this pay-out, then in terms of the net value, this asset is susceptible to risk.

2.1.2 Risk aversion

Why is there a premium for risk? The explanation for the existence of the risk

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losing is possible (Laughhunn, Payne and Crum, 1980; Schoemaker, 1990; Myagkov and Plott, 1997; Heath, Huddart and Lang, 1999). As with all theoretical models, the utility models are not without limitations. The models that measure utility are approximate,

incomplete, and subject to many assumptions. One reason is that utility cannot be measured or observed directly, so instead economists devised a way to infer utilities from observed choice. Choice is a constructive process which makes choices highly dependent on its context. The discussion about the validity of these studies and the assumptions that are made are outside the scope of this thesis. Although there is no solid proof, based on the above literature, it is safe to assume that most investors are risk averse. If most investors are risk averse, the higher the risk, the higher the risk premium should in theory be. In effect, the risk premium is the gain in expected value that investors demand for taking risk. Although risk aversion may vary across investors, it is the aggregate level of risk aversion that determines the market’s equity risk premium. The aggregate level of risk aversion is not fixed and can change over time. Bakshi and Chen (1994), for instance, found that the risk premium increased in the United States due to investors’ aging.

2.1.3 Behavioural factors

The above paragraphs point out that in theory the degree of risk aversion and the amount of risk are factors that determine the size of the risk premium. However, beside the rational risks, there are irrational and behavioural components which can affect the risk premium. Behavioural ﬁnance argues that some ﬁnancial phenomena can be understood using models in which some agents are not fully rational. If investors do not behave in accordance with the theoretical framework, the risk premium can become inconsistent with the size that is suggested by financial theory.

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found when investors evaluate returns, looking at nominal values rather than in risk-adjusted terms.

Another example of irrationality is the leverage aversion. If an investor can choose between a portfolio having a particular expected return and volatility without leverage and another portfolio that has the same expected return and volatility with the use of leverage, investors prefer the portfolio without leverage. The mean-variance utility function does however not distinguish between these two portfolios (Jacobs and Levy, 2013).

A final example of a behavioural flaw in investing is narrow framing. In CAPM theory, the risk of an investment is based on additional risk which the investment adds to the portfolio. Investors however tend to evaluate the risk of an investment in isolation of their portfolio. This leads investors to overestimate the risks of investments which occurs at both the buy-side and sell-side of the market. Barberis and Huang (2008) state that narrow framing could cause the risk premium to be higher than it should rationally be.

2.2 The realized volatility as a measure of risk

In paragraph 2.1 it is explained that the size of the risk premium is determined by the degree of risk aversion, the amount of risk, and behavioural components. This paragraph will look at whether or not the realized volatility qualifies for a risk proxy. Also, the literature about the relationship between realized volatility and returns is described.

When risks that are not related to stock price changes are neglected, risk can be defined as the probability distribution of future stock returns. The question is how to measure this distribution and convert it into a single-number risk measure. At first sight, realized volatility seems to be a perfect candidate for measuring historical risk, since it is a result of stock price movement. Bollerslev, Litvinova and Taugen (2006) found a negative relationship between realized volatility and realized returns. This might seem illogical, because theory suggests a higher return is demanded for risk-taking. However, a thorough look shows that this finding is not surprising at all. Realized volatility is not appropriate for measuring risk because risk is the probability distribution of future price changes, not the actual price change that occurs. Risk is by definition ex ante.

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a negative shock than by a positive shock. Poterba and Summers (1986) and Campbell and Hentschel (1992) argue that the volatility feedback effect causes the negative relationship between equity returns and volatility. An increase in volatility increases the discount rate and reduces the present value of future profits. This in turn will result in a decline in stock prices. This effect is similar to the effect found by Han (2011) referred to as volatility risk in section 2.1.2. The leverage effect, found by Black (1976), amplifies this volatility feedback effect. For a company with fixed debt, the debt to equity ratio increases when the stock price declines. In other words, the leverage increases. As a result the stock becomes riskier which reduces the present value of future profits.

2.3 The expected volatility as a measure of risk

In paragraph 2.2 it is explained that the realized volatility is not eligible for measuring risk. The expected volatility is better suited to represent risk. Expected volatility is basically the probability distribution of future stock prices with the assumption that the probabilities are normally distributed. Expected volatility is not equal to risk, mainly because of this restrictive assumption. In reality, the probabilities are not necessarily normally distributed. In fact, they can never be normally distributed, because the loss on stocks is limited at -100%.

Nevertheless, it is the best proxy for market risk that is currently known. Whaley (2000) states: "If expected market volatility increases, investors demand higher rates of return on stocks". Ghysels, Santa-Clara and Valkanov (2005) confirm this statement by arguing that the “first fundamental law of finance” is the positive relationship between the expected return and volatility of the market portfolio, because of the risk-return trade-off. There are two

commonly used methods for measuring expected volatility; conditional volatility and implied volatility. A fair amount of research is aimed at conditional volatility and its relation to returns. The literature aimed at implied volatility and its relation to returns is less extensive.

2.3.1 Conditional volatility

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models. Tse (1991) and Tse and Tung (1992) questioned the GARCH(1,1) for volatility forecasting and concluded that the exponential weighted moving average (EWMA) is more accurate in predicting future volatility in Japanese and Singaporean markets. In an

examination of the UK market, Dimson and March (1990) found that more simplistic models provide more accurate estimations of future volatility. They recommend using exponential smoothing and simple regression for volatility forecasting. Figlewski (1997) also argues that simple averages predict future volatility better than the more sophisticated volatility models, such as the GARCH(1,1) specification. More recent studies such as those from Gokcan (2000) and Srinivasan (2011) found that GARCH-based models are superior in forecasting volatility. Models using historical volatility values as input have a significant advantage in terms of applicability. The main disadvantage of these types of empirical models is that their theoretical underpinning is rather weak.

Empirical evidence has been mixed for the relationship between returns and conditional variance. A positive relationship is found by French, Schwert and Stambaugh (1987); Baillie and DeGennaro (1990); Guo and Whitelaw (2006); Lundblad (2007); Müller, Durand and Maller (2011), whereas a negative relationship is documented by Nelson (1991); Glosten, Jagannathan and Runkle (1993); Whitelaw (2000). All the above studies used ARCH or GARCH models for modelling conditional variance.

2.3.2 Implied volatility

The implied volatility is the market equivalent of expected volatility derived from option pricing. Black and Scholes (1973) argue that an option price is based on a set of factors. Except for the expected volatility, all these factors are known. When the price of the option is given, the volatility expectations can be solved with the Black-Scholes formula. Because this implied volatility expectation is market driven it is expected to be superior in its forecasting power. Note that there is an opportunity to profitably trade options with volatility forecasting models that are less biased and more accurate than the implied volatility.

Christensen and Prabhala (1998) and Blair, Poon and Taylor (2001) confirm that the implied volatility performs well at forecasting volatility. Pong, Shackleton and Taylor (2004) found that implied volatilities are at least as accurate as the historical forecasts in one-month forecasts.

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3. Hypotheses

Although academic theory certainly suggests a link between the excess returns5 and the volatility expectations, research has discovered weak and mixed results showing both positive and negative relationships. The behavioural flaws in investing and lagged effects of the leverage effect and the volatility feedback effect possibly weaken the positive risk-return relationship that should exist in theory. Because the majority of studies have found a positive relationship I expect to confirm their findings in this thesis.

Hypothesis 1a: There isa positive linear relationship between excess returns and expected volatility.

In addition to the linear relationship, I will also compare returns in times of low versus high expected volatility. This is done because the effect of expected volatility on returns is not necessarily linear. Possibly, differences in returns are only found in times of very high or very low volatility expectations, due to for example behavioural flaws. Since a positive linear relationship is expected, it is also expected that excess returns in times of high expected volatility are higher than returns in times of low expected volatility.

Hypothesis 1b: Excess returns in times of high expected volatility are higher than excess returns in times of low expected volatility.

Thus far, no study has adjusted excess returns for the corresponding market risk measured by expected volatility. In theory, the risk-adjusted equity returns should be unaffected by

volatility expectation levels. However behavioural flaws that weaken the risk-return relationship, such as leverage aversion, will cause the relationship between risk-adjusted excess returns and expected volatility to be negative.

Hypothesis 2a: There is a negative relationship between the risk-adjusted excess returns and expected volatility.

5

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As with hypothesis 1, I will also compare excess returns in times of low versus high expected volatility. Since a negative linear relationship is expected in hypothesis 2a, it is also expected that adjusted excess returns in times of high expected volatility are lower than risk-adjusted excess returns in times of low expected volatility.

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4. Model

To test these hypotheses, excess returns of the market portfolio need to be measured. Because the market portfolio is unobservable, a world stock index is assumed to be a fully diversified equity portfolio. To measure the excess returns, the ex-ante risk-free rate is

subtracted from the nominal equity returns. The riskiness of the equity market is estimated by a volatility forecast using both a VIX based forecast and a GARCH(1,1)-modelled forecast. The VIX based forecast is considered superior to the GARCH(1,1) forecast due to it being market driven and because the VIX-based expected volatility can be capitalized. However, the GARCH(1,1) forecast has benefits in data availability because forecasts can be made using only historical stock data. Moreover, frequencies other than monthly forecasts are possible with the GARCH(1,1) forecasting.

Risk-adjustments are done under the assumption that investing a portion of the

portfolio in a risk-free asset is a feasible way of lowering the investor’s risk. An investment in stocks with a VIX of 20 is considered to be equally risky as an investment of 50% invested at a risk-free asset and 50% invested in stocks with a VIX of 40. Therefore the following

equation is used to convert excess returns into risk-adjusted excess returns.

_̅ (1)

Where is the risk-adjusted excess return, is the expected volatility of month t, ̅ is the average volatility of the sample period, and is the excess return of month .

To test the validity of the risk-adjustments, the standard deviations of the risk-adjusted returns in different levels of expected volatility are compared. If the standard deviations of the classes are not significantly different after risk-adjustments, it can be concluded that the procedure is appropriate.

An OLS regression is performed to test for a linear relationship between excess returns and expected volatility (equation 2) and risk-adjusted excess returns and expected volatility (equation 3).

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In addition, returns are compared in different levels of expected volatility. Expected volatility levels are ranked from low to high. Next, the dataset is classified into a high and a low expected volatility state. The corresponding returns and risk-adjusted returns of the high and low volatility state are compared and statistically tested. Different definitions of high and low volatility states are used. Firstly, the returns in times of above average expected volatility are compared to the returns in times of below average volatility. In addition to the 0-50% versus the 50-100% of the expected volatility ranked data, the returns of top 20% and top 10% of the ranked data are compared with the returns of the bottom 20% and 10% respectively.

Two measures of expected volatility are used. Firstly, the implied volatility is used. The one-month implied volatility is readily available from the CBOE VIX. In addition, a GARCH(1,1) model, as introduced by Bollerslev (1986), is used to generate the expected volatility. The GARCH(1,1) model allows the conditional variance to change over time as a function of the previous error ( ) and the previous conditional variance ( ). The GARCH(1,1) model is described by the following equations:

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Where:

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And:

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5. Data

The data in this study are obtained from Thomson Reuters Datastream and the FRED (Federal Reserve Economic Data) database. The data codes for the data used can be found in the table below.

Table 1. Data overview

Name Source Code Data Frequency

3-Month Treasury Bill: Secondary Market Rate

Federal Reserve Economic Database

(FRED) TB3MS

01/01/2001-05/14/2013 Daily 3-Month Treasury Bill: Secondary

Market Rate

Federal Reserve Economic Database

(FRED) TB3MS

02/01/1990-02/01/2013 Monthly CBOE Volatility Index Thomson Reuters Datastream CBOEVIX(PI)

02/01/1990-02/01/2013 Monthly MSCI World Total Return Index Thomson Reuters Datastream MSWORLD$(MSRI)

01/01/2001-05/14/2013 Daily MSCI World Total Return Index Thomson Reuters Datastream MSWORLD$(MSRI)

02/01/1990-02/01/2013 Monthly

5.1 MSCI World Index

This study requires a measure of equity returns on a fully diversified portfolio, the so called market portfolio. Because the market portfolio cannot be observed, an observable portfolio has to be defined as the market portfolio. The MSCI World Index is used to describe the world’s equity market. The riskiness of an investment portfolio that tracks the MSCI World Index is considered to represent the systematic risk. The MSCI World Index is a representation of large and mid-cap companies across 24 developed markets. The index covers approximately 85% of the free-float adjusted market capitalization in each country. The MSCI World Index does not fully represent the world’s equity market, but it is the best representation available. Also, from the investor’s perspective, it may be practically

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5.2 US 3-Month Treasury Bill

For a measure of the risk-free rate the US 3-Month Treasury Bill rate is used. A clean risk-free rate does not exist in the real world, so a proxy needs to be found. US treasury Bills are commonly used as a risk-free rate proxy. The US 3-Month Treasury Bill is chosen because its short term nature allows an investor to adjust his portion invested in the risk-free rate on a frequent basis.

5.3 CBOE Volatility Index

The CBOE Volatility Index (VIX) is a measure of near-term volatility derived from S&P 500 index option prices. Since its introduction in 1993, the VIX has been considered to be the world’s leading measure of investor sentiment and market volatility. The VIX is quoted in percentage points and translates to the expected movement in the S&P 500 index over the next 30-day period, which is then annualized. The formula used in the VIX calculation is:

∑ [ ] (7)

Where:

σ

T Time to expiration

F The forward index level derived from index option prices The first strike below the forward index level, F

The first strike price of the out-of-the-money option; a call if and a put if ; both put and call if .

Δ Interval between strike prices – half the difference between the strike on either side of : Δ

R Risk-free interest rate to expiration

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5.4 Data adjustments

For both the MSCI World Index data and US 3-Month Treasury Bill data the logarithmic returns are calculated and converted into a yearly return. Non-trading days are excluded from the dataset. The return from Friday’s close till Monday’s close is therefore considered a one day return. The US 3-Month Treasury Bill logarithmic return is deducted from the MSCI World Index logarithmic return to represent the excess returns. The

descriptive statistics of these data can be found in table 2.

Table 2. Descriptive statistics excess returns

Excess returns Daily Monthly Mean 0,024 0,041 Maximum 23,739 1,668 Minimum -19,103 -2,492 Standard Deviation 2,884 0,542 Skewness -0,313 -0,733 Kurtosis 10,303 5,154 Jarque-bera 7221,941 147,387 Start date 1-1-2001 1-1-1970 End date 13-5-2013 1-5-2013 Number of observations 3226 521

There are several statistics that are worth mentioning. The mean excess return is higher in the monthly data. This is caused by the financial crises of 2008 that has a big impact on the daily data which is from 2001 to 2013. The maximum, minimum and standard

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6. Results

In this section, the forward-looking returns and risk-adjusted returns for different volatility expectations are given.

6.1 VIX-based monthly volatility expectations

The VIX does not seem to be a determinant of forward-looking excess returns. The OLS regression does not give any significant results. Neither is there any statistical evidence for excess returns found in the constant. In the table below, the main results are summarized.

Table 3. OLS regression on monthly excess returns based on VIX values.

Dependent variable R

N 276

Variable Coefficient Standard Error t-Statistic p-Value Constant 0.0441 0.0960 0.4595 0.6452 VIX 0.0006 0.0044 -0.1320 0.8951 R-squared 0.0001

F-statistic (p-value) 0.0174 (0.8950)

Also, no evidence of a linear relationship is found when using the risk-adjusted returns. Because the risk-adjustments are based on the VIX itself, there should be a

relationship between either the VIX and non-adjusted returns or between the VIX and risk-adjusted returns. A possible explanation for not finding a relationship in both tests is the large random noise and the small sample of monthly returns.

Table 4. OLS regression on risk-adjusted monthly excess returns based on VIX values.

Dependent variable RAR

N 276

Variable Coefficient Standard Error t-Statistic p-Value Constant 0.1442 0.0886 1.6619 0.0977 VIX 0.0048 0.0040 -1.2034 0.2299 R-squared 0.0052

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vice versa. The average returns of high VIX market states should be significantly higher to compensate for the higher risk investors bear. Although the differences are large (6,89% on low VIX versus -0,01% on high VIX), there is no significant difference in the average returns. When looking at risk-adjusted measures, in both groups (top and bottom 50% and top and bottom 20%) there are large differences in the average returns, with the low expected volatility state yielding on average higher adjusted returns. In the top and bottom 50% the difference is significant on a 5% confidence level. In the top and bottom 20% the difference is close to significant.

The large standard deviation for monthly returns shows that the returns have a large amount of random noise. Also, the market returns are divided into groups based on their expected volatility, which causes the amount of data per group to become very small.

Therefore, it is difficult to give accurate estimates of the equity risk premium. Unfortunately there is no possibility for a larger timespan or larger frequency. The VIX is a one-month estimate of volatility and there is no VIX data available prior to 1990.

Table 5. Monthly excess returns in different levels of VIX.

Excess returns Risk-adjusted excess returns Low Risk High Risk Low Risk High Risk Low Risk High Risk Low Risk High Risk VIX 0-50% 50-100% 0-20% 80-100% 0-50% 50-100% 0-20% 80-100% Average Return 6.89% -0.01% 11.06% 6.16% 10.22% -0.82% 18.66% 3.97% Average VIX 14.6991 26.1218 12.2238 32.4105 14.6991 26.1218 12.2238 31.6182 Standard Deviation 0.3577 0.7304 0.2870 0.9109 0.4945 0.5444 0.4860 0.5415 T-test (p-value) 0.2106 0.3524 0.0396 0.0674

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Figure 1: VIX ranked realized excess returns.

Returns (left) and risk-adjusted returns (right). The returns are ranked from low VIX to high VIX. The monthly returns are converted into a yearly %.

6.2 GARCH(1,1)-based daily volatility expectations

The GARCH(1,1) model allows us to produce daily forecasts of volatility without limitations on VIX data availability. The model requires only historical equity returns as input. Volatility forecasts are performed with the use of daily data of the MSCI Total Return index from 01-01-2001 to 14-05-2013. The first year’s data is used to estimate an initial model for establishing the parameters. This model then produces the volatility forecasts for the year 2002 on a daily basis. Every year a new model is estimated including the data from the previous year. A total of 12 models are estimated to forecast volatility between 2002 and 2013.

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Table 6. OLS regression on daily excess returns based on GARCH(1,1) conditional volatility.

Dependent variable R

N 2870

Variable Coefficient Standard Error t-Statistic p-Value Constant 0.1335 0.1160 1.1511 0.2498 σ -0.0387 0.0401 -0.9651 0.3346 R-squared 0.0003

The OLS regression on risk-adjusted returns is also not showing evidence of a

relationship. The same reasoning applies here as with the lack of evidence on the relationship between the VIX and excess returns.

Table 7. OLS regression on risk-adjusted daily excess returns based on GARCH(1,1) conditional volatility.

Dependent variable RAR

N 2870

Variable Coefficient Standard Error t-Statistic p-Value Constant 0.1329 0.1008 1.3180 0.1876 σ -0.0299 0.0349 -0.8580 0.3909 R-squared 0.0002

No statistical differences in mean excess returns are found when looking at different conditional volatility levels. It is remarkable to see average returns in the low risk volatility states to be higher in both the 0-20% versus 80-100% and the 0-10% versus 90-100% conditional volatility ranked data. The results are summarized in the table below.

Table 8. Excess returns for different levels of GARCH(1,1) based conditional volatility.

Excess returns Low Risk High Risk Low Risk High Risk Low Risk High Risk GARCH ranked 0-50% 50-100% 0-20% 80-100% 0-10% 90-100% Average Return 0.83% 6.12% 11.29% -1,27% 15.10% -16.19% Average GARCH 1.7120 3.3892 1.4041 4.6105 1.2956 5.7677 Standard Deviation 1.6838 3.7855 1.3919 5.0435 1.4348 6.2554 T-test (p-value) 0.3146 0.5589 0.4982

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yearly return of 21.09% excess return while the top 20% (high risk) only yields a 0.74% excess return. In the bottom and top decile differences on average returns are even larger with 29.55% versus -7.59% respectively. Yet no statistical differences can be found due to

extremely large daily fluctuations.

Table 9. Risk-adjusted excess returns for different levels of GARCH(1,1) based conditional volatility.

Risk adjusted excess returns Low Risk High Risk Low Risk High Risk Low Risk High Risk GARCH ranked 0-50% 50-100% 0-20% 80-100% 0-10% 90-100% Average Return 4.27% 7.04% 21.09% 0.74% 29.55% -7.59% Average GARCH 1.7120 3.3892 1.4042 4.6106 1.2957 5.7677 Standard Deviation 2.4956 2.5981 2.5694 2.6086 2.8443 2.6920 T-test (p-value) 0.3856 0.0897 0.1898

As shown in the figure below, realized return volatility is constant for all volatility expectations when using the adjusted returns. The White test confirms homoscedasticity. Therefore risk adjustments are considered valid.

Figure 2. Conditional volatility ranked realized returns.

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7. Conclusion

This thesis examines the risk-return relationship in the stock market. This is achieved by looking at the relationship between the excess returns and expected volatility. The

expected volatility is considered to be a measure of risk. An OLS regression is performed to check for a linear relationship between expected volatility and returns. In addition, the returns in different levels of expected volatility are compared. Finally, the returns are adjusted for risk and the effect of expected volatility on risk-adjusted returns is tested again with the same procedure. Expected volatility is measured with both a GARCH(1,1) volatility forecasting model (daily expectations) and the implied volatility obtained from the CBOE VIX (monthly expectations).

Neither the GARCH-based nor the VIX-based expected volatility show evidence of a relationship between expected volatility and returns. It is hypothesized that the risk premium is higher when volatility expectations are high. The results are in contradiction to this

expectation. Results show that average returns in times of high expected volatility are lower than those in times of low expected volatility. Differences in average returns are large, but not statistically significant. When looking at risk-adjusted measures, these differences are even larger. Risk-adjusted returns in times of low expected volatility outperform the returns in times of high expected volatility. Significant results are however only found when the returns during below average VIX are compared with the returns during above average VIX. No significant results appear from the OLS regressions with the expected volatility as a dependent variable for both non-adjusted and risk-adjusted returns. There should be a

relationship in the former, the latter, or both. Note that if there is no relationship between non-adjusted returns and expected volatility, a negative relationship should result when risk-adjusted measures are used. This is because the risk-adjustments are based on the expected volatility itself. However, because of the large random noise in equity returns no relationship can be proven. With only a few decades of data it is hard, if not impossible to draw strong conclusions on the influence of expected volatility on equity returns.

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the findings of Giot (2005) who found positive forward-looking returns for extremely high levels of VIX. Possibly, the differences in the time period and the different stock index that is investigated cause results to differ.

This thesis makes a contribution by suggesting a VIX based investment strategy that increases the return to risk ratio. Outperformance could for example be achieved by investing in stocks at below average VIX levels and investing in risk-free assets when the VIX is above average.

There are a number of limitations to this research. Although the expected standard deviation of equity returns is appealing for quantifying risk, it requires stocks to have a normal distribution. This is clearly not the case. In the data section it is mentioned that stock returns are not normally distributed, which is generally accepted in literature as well.

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Testing the risk-return relationship in the stock market

Master Thesis