Alternative measures of physical variance in variance risk premium and expected S&P 500 returns

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University of Amsterdam, Amsterdam Business School

MSc Finance, Asset Management

Master Thesis

Alternative Measures of Physical Variance in

Variance Risk Premium and Expected S&P 500

Returns

Student: Deksnys Simonas

Student Number: 10630287

Thesis supervisor: dr. Eiling Esther

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Statement of Originality

This document is written by Student Simonas Deksnys who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This thesis expands the current literature on the variance risk premium by employing the out-of-sample methodology. The variance risk premium is defined as the difference between risk-neutral variance as measured by the squared VIX index and the physical variance. The paper also proposes three alternative measures for the physical variance as opposed to the realized variance measured by the sum of squared intra-day returns. The alternative measures of the physical variance are: the sum of squared daily returns, and two conditional variance forecasts based on EGARCH(1,1) model using daily and monthly returns individually. It is found that the first and the third alternative measures of the physical variance improve the out-of-sample predictive power of the variance risk premium. Finally, it is determined that when combining the different measures of the variance risk premium with the price-earnings and price-dividend ratio only the third alternative measure remains economically useful, when defining the variance risk premium, for the mean-variance investor.

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1 Introduction

The asset pricing is a key component of finance. Being able to predict future market returns allows investors to exploit profitable investment opportunities. This in turn leads to a better performance of corporate and private portfolios, which is beneficial for their clients and their future investment plans, as they have more money to invest. Also sound investment decisions by the pension funds matter for the general public as their pension contributions depend on them. Finally, academics care about predicting the asset returns, which appear unpredictable at a first glance, because the market returns predictability poses several puzzles that are still unanswered. Welch and Goyal (2008) show that the traditional factors, such as the price-to-earnings ratio, the dividend yield, the dividend-to-price ratio, which in theory explain asset returns, perform poorly empirically both in-sample and out-of-sample, when predicting returns. On the other hand, Campbell and Thompson (2008) criticize their paper and show that small restrictions on the out-of-sample regressions lead to better investment strategies using the same standard variables as in Welch and Goyal (2008) when compared to the historical average. Although their proposed out-of-sample R2_{estimates are}

small, they claim that the results they find are useful for the mean-variance investor and thus carries economic significance. Furthermore, more recent studies such as Eiling, Kan and Sharifkhani (2016), Rapach, Ringgenberg and Zhou (2016) and Bollerslev et al. (2009) show high predictive power of the new alternative predictors: cross-sectional volatility of the industry portfolios, short interest and variance risk premium respectively. This thesis focuses on the third one.

The variance risk premium is defined as the difference between the risk-neutral (Q) variance and the physical (P) variance (Drechsler and Yaron, 2011). The risk-neutral (Q) variance is a measure of variance used in the derivatives pricing, which is derived from the derivatives prices rather than the historical returns data of the underlying. It assigns risk-neutral probabilities to the market movements in order to eliminate the arbitrage opportunities in the derivatives market. The physical variance is a measure of the asset returns variation, which is used in the stock markets in order to determine the riskiness of the asset and in theory it is calculated using the actual probabilities assigned to the market movements. That is in order to determine it in practice, the historical data is used by assuming

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that it carries the information about the future variation in asset returns and thus the actual probabilities of the asset price movements. The variance risk premium itself is interpreted as an insurance against the market downturns. It can be seen as a negative mean return investment, which offers high positive returns during the recessions and high market uncertainty. Therefore, investors are willing to pay a premium in order to insure their portfolios against the unexpected shocks (Drechsler and Yaron, 2011). Authors such as Carr and Wu (2009), Drechsler and Yaron (2011), Bollerslev, Marrone, Xu and Zhou (2014) explain the variance risk premium pricing in terms of investors’ attitude towards the market uncertainty. Intuitively the variance risk premium measures the expected market uncertainty, which is unfavourable to the risk-averse investor. The higher the variance risk premium, the higher the market uncertainty. Bollerslev et al. (2009) also derive a theoretical model, which links the variance risk premium directly to the market risk premium. According to the current literature, the variance risk premium isolates the volatility-of-volatility factor which measures the temporal shocks in the consumption growth variance. In other words, it measures the shocks in the market uncertainty. Therefore, the variance risk premium should be able to predict future market returns. Most of the papers mentioned above use the squared VIX index as a proxy for the implied (Q) variance, derived from the S&P 500 option prices. The VIX index measures one-month ahead expected volatility of the S&P 500 market and its data is available publicly on the CBOE website. VIX index is considered as the best proxy for implied variance by all of the papers that are discussed in this thesis. This is because it calculates the implied variance directly from the derivatives data rather than using Black-Scholes model. As a consequence, it is also called model-free measure of the risk-neutral variance (Bollerslev, Tauchen, and Zhou, 2009). The physical variance, on the other hand, has numerous different ways of estimation. Currently, it is considered that the sum of intra-day squared returns acts as the best estimator of the asset variance. For the realized variance (P) in the variance risk premium, the researchers use similar but slightly different measures. However, all of them find economically meaningful results for the variance risk premium, both in terms of time-series predictability and cross-sectional factor models (Zhou and Zhu, 2009), (Carr and Wu, 2009). Carr and Wu (2009) also show that the variance risk premium is not explained by the traditional Fama and French SMB and HML factors, and its slope still remains positive in the multiple cross-sectional regressions, which means that it can explain the variation in the market returns that is unexplained by the other classical predictors.

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Given the significant explanatory results of the variance risk premium, it is considered important to further investigate its usefulness for the mean-variance investor. This thesis proposes three alternative measures of the P variance, by keeping squared VIX index as a proxy for the implied risk-neutral variance. The main reason for choosing to investigate alternative measures of the physical variance as opposed to the Q variance is that the VIX index is considered as a good proxy for the risk-neutral variance in the theoretical model. In theory the squared VIX index measures one month ahead expected risk-neutral variance, therefore the P variance estimate should also measure the one month ahead expectation. As in Drechsler and Yaron (2011) one of the realized variance measures proposed here is the sum of squared daily S&P 500 continuously compounded returns over one month. Unlike the authors, this paper uses the current month’s daily returns to estimate its P variance. This is done in order to investigate whether the realized variance in Bollerslev et al. (2009), estimated from high-frequency intra-day returns gives better performing variance risk premium than using daily frequency returns. This allows to assess whether the less accurate realized variance measure, but which requires less data, can be effectively used in the variance risk premium as an alternative for the more accurate measure. If that is the case, the variance risk premium could be used by smaller investors who do not have an access to the private databases. Authors assume that the expectation of the next month’s variance is equal to the current month’s variance and claim that this assumption does not reduce the empirical performance of the model.

Moreover, the next two conditional P variance measures are derived using the Nelson’s (1991) Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model, which is one of the GARCH family models. For this model monthly and daily S&P 500 returns are used. The EGARCH forecasted variance is used as this model predicts variance well and thus allows to match the expected implied variance with expected physical variance, thus providing a theoretically consistent measure of the market’s physical variance. The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model was first introduced by Engle (1982) who shows that it is useful for predicting variance. Since then several other GARCH models were introduced. Nelson (1991) criticize the GARCH models and distinguishes three major drawbacks. He introduces EGARCH model which performs better than the other GARCH family models in terms of forecasting market variance, as it accounts

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for all of the three major drawbacks. The GARCH models alone do not show significant results when predicting out-of-sample returns. For instance, Guo, Kassa and Ferguson (2014) show that EGARCH models performs poorly out-of-sample. Glosten, Jagannathan and Runkle (1993) find that only by adjusting EGARCH-M model for season patterns in volatility it shows some significant predictive power of market returns. Their findings match the current findings of the negative variance-return relation.

Despite the inability of the GARCH models to predict the future market returns, they are useful when predicting variance, and hence can be used for forecasting the physical variance in the variance risk premium. This thesis uses Nelson‘s (1991) EGARCH model for this purpose because it accounts for the asymmetric variance shocks due to the different signs of the market returns. The EGARCH model is also preferred because unlike in other papers it does not include the implied variance proxy as one of the explanatory variables for the realized variance, thus allowing to determine the P variance separately from the implied variance.

This paper extends the current methodology on the variance risk premium, by including the out-of-sample predictability measures. The current literature only focuses on the in-sample regressions. This is mainly due to the lack of data on the VIX index, which is available only from January 1990. This paper measure the explanatory power in terms of out-of-sample R2_{as defined by Campbell and Thompson (2008) and utility calculated from trading}

the strategy returns based on the predicted returns as in Rapach, Strauss and Zhou (2010). Finally, the differences between the utility of the proposed alternative variance risk premia measures and the utility of the original variance risk premium from Bollerslev et al. (2009) are tested using the t-statistic derived by Eiling et al. (2016). It is found that two of the alternative measures, the variance risk premium with daily squared returns and the one with the conditional variance forecasted from daily returns, perform better than the original variance risk premium when predicting the out-of-sample S&P 500 returns. These results hold only for longer horizons with the highest and statistically significant differences in utility for more than 12 month horizon. The better performance of the EGARCH forecasted P variance in the variance risk premium can be explained by the theoretical model, where both Q and P variances need to be the expectations of the one month ahead market variance, whereas there is no clear explanation why the sum of squared daily returns acts as a better variance

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measure in the variance risk premium. Also two of the alternative variance risk premia significantly outperform the historical benchmark strategy utility for horizons longer than 12 months, while the original variance risk premium never does. These findings are also supported by the positive out-of-sample R2_{s with values as high as 13.55% for 24 month}

horizon, when the P variance is predicted using the daily data. The robustness checks however, show that the classical predictors such as the earnings ratio and price-to-dividend ratio performed better during the out-of-sample period of January 2010 to December 2016 and that only one predictor is economically meaningful when combined with the standard predictors.

From this the research question follows: can EGARCH modelled physical variance improve the variance risk premium in terms of the future excess market returns predictability?

Apart from applying more elaborate time-series predictability methodology, this thesis contributes towards academia, by investigating the variance risk premium further, which appears to be a useful and economically meaningful predictor of the future excess market returns. Also, it introduces an alternative application of the EGARCH model, which alone does not perform well in explaining market returns as found in other papers. Finally, it also expands the in-sample variance risk premium analysis as it incorporates at least six more years of observations when compared to Bollerslev et al. (2009) and Bekaert and Hoerova (2014), for instance.

The thesis continues as follows. Section 2 reviews the current literature on the variance risk premium and the variance measures. Section 3 explains the theory behind the variance risk premium, the methodology to calculate the alternative measure of physical variance, proposed by the paper and the methodology to test the predictive power of the variance risk premium both in-sample and out-of-sample. Section 4 introduces the data used for the research and descriptive statistics of the variables used. Key results and their interpretation are presented in Section 5. Section 6 presents results of the robustness checks, where different sample period and classical predictors are tested together with the variance risk premia measures. Section 7 concludes.

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2. Related Literature

2.1 The Variance Risk Premium

The variance of the asset returns is an important measure for the derivatives markets that can be used to explain market uncertainty, investors’ attitude towards it and dynamics of the economy (Drechsler and Yaron, 2011). Drechsler and Yaron (2011) interpret the variance risk premium (VRP) as the difference between the price and the expected payoff of a trading strategy, where the price is higher than the payoff as in theory the VRP is positive. They explain its positive slope on the market excess returns in terms of the investors’ distaste for uncertainty. This means that people are willing to pay the risk premium for the assets that offer high payoff during the large variance shocks as an insurance. Therefore, when big market shocks are expected, the variance risk premium increases as people require higher premium for potential losses. Hence, the VRP slope is positive on the market excess returns. Zhou and Zhu (2009) support this explanation by showing that increase in the volatility is unfavourable for the investors. Bollerslev et al. (2014) test a global variance risk premium proxy, which is the value-weighted variance risk premium of eight countries that have dominant market indexes around the globe. They show that the global VRP proxy better predicts future market returns than the individual country-specific VRP. These results even hold for the individual countries, which means that the market returns in countries are driven by a global market uncertainty as well. This unexplained variation in returns due to the variance risk premium leads to another asset pricing puzzle. Carr and Wu (2009) question whether the positive slope of the VRP in cross-sectional regressions is a high market inefficiency in terms of the variance or other factor that is unknown, but which is priced heavily by the market participants and which is highly correlated with the VRP. They show that the VRP remains significant even in multiple regression model together with Fama and French benchmark SMB and HML factors. This thesis however does not answer this question, but rather tries to find better ways to exploit this risk premium.

Bollerslev et al. (2009) show that in theory the VRP isolates the volatility-of-volatility factor of the consumption growth model. This volatility-of-volatility measures the temporal variation in volatility of the consumption growth that is the measure of market uncertainty. From the price-to-dividend ratio equation they derive the equity premium equation, which is

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a function of consumption growth volatility and its volatility-of-volatility. Finally, they show that the difference between the implied expected Q variance and the expected P variance depends on the volatility-of-volatility. Their equations, therefore, show the direct link between the VRP and the equity premium. This allows to conclude that in theory the VRP should predict horizons of the market returns, for which, the disturbances in volatility-of-volatility factor has the highest effect (Bollerslev, Tauchen, and Zhou, 2009). This interpretation is consistent with the in-sample regression results, where Bollerslev et al. (2009), Bekaert and Hoereva (2014) and Drechsler and Yaron (2011) show that the predictive power of the VRP is higher for short-term horizons such as one to three months. Together with the theoretical model, their results imply that the volatility-of-volatility variation affects one to three months variation in consumption growth. And thus it is most influential on the market returns for short horizons. These results are however only based on the in-sample regressions and might not be the same for the out-of-sample tests as it is later shown by the empirical analysis of this paper.

Following the theory of the consumption growth and the price-to-dividend raio definition, together with an intuitive explanation of the investors’ attitude towards the market uncertainty, the VRP appears to be a strong predictor of the excess market returns. Another reason, why the VRP is an attractive predictor is that its R2_{s are easier to interpret}

for various return horizons than compared to the other more persistent predictors. According to Bollerslev et al. (2014) R2_{s increase for higher horizons even when no true predictability}

occurs if the predictor has high persistence. Compared to the other predictors, the VRP has lower persistence with AR(1)=0.27, while for others it is as high as 0.9.

Moreover, the VRP better predicts short-term horizon returns as discussed above, when compared to the more traditional predictors, such as price-to-earnings ratio and dividend yield. Therefore, it is important in asset pricing as it can fill the gap of shorter horizon predictability. For robustness, Bollerslev et al. (2009), Bekaert and Hoerova (2014), Drechsler and Yaron (2011), also run multiple regression models, which include the VRP and other standard predictor variables. This way it is examined whether the VRP maintains its significance, or whether its predictive power is factored out by the other predictors. Bollerslev et al. (2009) find that including other predictors to the regression of the VRP, such as price-to-earnings ratio, price-to-dividend ratio and consumption-wealth ratio lead to even higher

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in-sample R2_{s. The adjusted R}2_{s in their results range between 1.89% to as much as 32.58%}

for multiple regressions, with highest predictability for one year returns horizon. Drechsler and Yaron (2011) test shorter horizons, but they find similar results. They show that including the price-to-earnings ratio to the VRP regression, the predictive power increases for one and three month horizons. R2_{increases from 1.46% to 8.30% for one month horizon, while the}

coefficient of the VRP still remains significant at 1% level. Finally, Bekaert and Hoerova (2014) also show that including the VRP in a multiple regression together with the conditional variance, three months T-bill rate, dividend yield, credit spread and term spread increases R2

from 2.9% to as high as 27.3% for one year horizon. However, coefficients on all other predictors except VRP and conditional variance are statistically insignificant, which might imply that including too many variables in a regression, artificially increases R2_without

carrying any true predictive power.

2.2 Variance Measures

It is now a common practice to measure the implied variance of the S&P 500 returns as the squared VIX index. Jiang and Tian (2005) find that the model-free variance measure includes the information content of both the Black-Scholes model and the past realized volatility data, therefore it is a better measure of the implied volatility than the volatility extracted from the model. According to the authors the S&P 500 option prices incorporate the information on historical volatility. The VIX index calculation includes options with all strike prices and different maturities and it is calculated in model-free fashion, meaning it is not bound to any option pricing model, such as the Black-Scholes model. On the other hand, the Black-Scholes implied volatility is measured only from close to at-the money option prices alone. Thus, the VIX index includes more data from option prices than the inverted volatility from the Black-Scholed model. According to Jiang and Tian (2005), one option price is not enough to capture the information content, which is incorporated in the option prices and thus the model-free calculation of the market variance is more accurate measure of the future expected variance. They finally show empirically that the model-free volatility estimate is a better predictor of future volatility than the Black-Scholes model implied volatility. Following these findings, this thesis sticks to the squared VIX index as a proxy for the implied risk-free market variance. This choice is also consistent with both the current literature and the

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variance risk premium theoretical framework as a predictor for the future returns, where the one month ahead expectation of the Q variance is required.

Unlike the commonly used squared VIX index as an implied variance measure, different authors use different measures for the physical variance in the variance risk premium. Bollerslev et al. (2009) measure it as a sum of the squared intra-day logarithmic returns over the months for which the P variance is estimated. Drechsler and Yaron (2011) also estimate the realized variance as the sum of squared intra-day returns but they run time-series regressions on the realized variance as a dependent variable and explain it by either its moving average or the VIX index and then subtract its one month ahead forecast based on the regressions. One of the P variance measures they employ is the sum of squared daily returns minus its moving average. Following their paper, one of the P variance measures proposed here is measured as continuously compounded squared daily returns. However, moving average is not subtracted in order to make sure that this measure can be directly compared to Bollerslev et al. (2009) realized variance measure. Showing the higher explanatory power of the proposed measure would imply that a more simplistic approach, which requires less and publicly available data could be used for the calculations of the VRP. Bekaert and Hoerova (2014) forecasts conditional variance using the squared VIX index, continuous and discontinuous jump components and negative returns of past day, week and month. That is returns for the model are either zero if their positive or negative. The dependent variable is the sum of squared intra-day returns. They test different combinations of the models by either including or excluding certain variables and test which model predicts the future variance most accurately. They then choose three models for the P variance forecasting and the VRP calculations. They find that two out of the three chosen models underperform the VRP of Bollerslev et al. (2009) with highest predictive power at three months horizon. In general, all of these papers incorporate the sum of intra-day squared returns as the measure for the realized variance. They either directly employ this measure in the variance risk premium or use it to forecast the one month ahead realized variance in order to match the timing of the implied variance, which is one month ahead expectation, as required by the theory. This choice of the realized variance measure is based on the several researches, such as Andersen and Bollerslev (1998) and Zhou (1996) that show the advantages of using high-frequency returns. Zhou (1996) explain that the variance is not

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constant over time because the information flow of the changes in the prices is not constant either, therefore the use of simple variance formula is not an accurate measure of the asset returns variation, since it is constant. The model-free calculation of the variance as the sum of the squared intra-day returns captures the information content embodied in the returns, and thus leads to an accurate measures of the realized variance. Barndorff‐Nielsen and Shephard (2002), however show that during the times of high market volatility, the realized variance measure has a large estimation error. Therefore, during the periods such as financial crisis of 2008, the realized variance measure, might not be the best way to calculate the P variance in the Variance Risk Premium.

The physical variance in this thesis is modelled as a conditional variance using the EGARCH(1,1) model, which estimates the variance using the historic returns of the S&P 500 index only. This model allows to measure the physical variance separately from the implied variance, while still matching the timing of the implied and P variances as required by the theoretical model derived in Bollerslev et al. (2009). Unlike the realized variance calculated from the intra-day squared returns, which uses only one month’s data, the EGARCH model uses all the available data on the S&P 500 returns to forecast the one month ahead P variance, thus the forecast carries more information content. Following this reasoning, it is believed that the EGARCH(1,1) model should lead to lower estimation error of the conditional variance and more accurately measure the variance during the market shocks, such as the 2008 financial crisis. According to Nelson (1991) the EGARCH model, unlike the other GARCH family models, captures the leverage effect in the market returns variation by assigning different magnitude of the variation for the negative and positive returns. This is consistent with the empirical findings, which show that during the market turndowns the variance in asset returns increases. Furthermore, Hansen and Lunde (2005) find that GARCH(1,1) model is not outperformed by 330 Autoregressive Conditional Heteroskedasticity (ARCH) models in terms of the ability to forecast the out-of-sample variance. However, they show that GARCH(1,1) model performs worse than the models that account for the leverage effects observed in the asset return variation. For this reason the EGARCH(1,1) model is used in order to forecast the one month ahead variance of the S&P 500 returns.

The main theoretical gap of the alternative P variance measures, proposed in this thesis, is the forecast of one month ahead variance with the EGARCH(1,1) model using daily

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returns. This approach is chosen due to the two reasons. The first one is that most of the researches use the EGARCH model for daily variance calculations, rather than monthly. The second reason is that daily returns are used because they incorporate more information about the variance shocks and thus can better forecast the future developments in the variance. Intuitively this approach is not that different from the realized variance calculation using the sum of squared intra-day returns. This is because summing the intra-day squared returns over the month is the same as summing the daily variance estimates. This method is also partially supported by Ang, Hodrick, Xing, and Zhang (2006) and Bali and Cakici (2008) who use daily data to estimate monthly variance. Therefore, the only major flaw of forecasting monthly variance with daily data using the EGARCH(1,1) model is the assumption of constant variance over the next month.

Following the current findings on the variance risk premium, this thesis expands the in-sample methodology and tests whether the VRP can actually outperform the benchmark historical average of the S&P 500 when applying the out-of-sample methodology. It also tests whether the alternative measures of the physical variance can improve the model and whether these newly defined variance risk premia can outperform the VRP as defined in Bollerslev et al. (2009). Based on the findings that the VRP has a positive coefficient on the excess market returns, it is expected that the positive sign will remain with ten additional years of monthly observations. Finally, following the theoretical model of the VRP, which

requires that the P variance at time t be measured as the expectation of the P variance for time t+1, it is hypothesized that EGARCH modelled conditional variance in the variance risk premium will improve the VRP model described in Bollerslev et al. (2009) and lead to a higher certainty equivalent return.

3. Variance Risk Premium and Time-Series Predictability

3.1 Variance Risk Premium as a Measure of Temporal Market Uncertainty

Bollerslev et al. (2009) start explaining the VRP from the geometric consumption growth rate as defined in equation 1 below.

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𝜇_𝑔 is a constant mean growth rate, 𝜎_𝑔,𝑡 is a conditional variance, and 𝑧_𝑔,𝑡 is N(0,1) unpredictable innovation.

They further model variance of growth rate and its variance as in the equations 2 and 3 below.

𝜎_𝑔,𝑡+12 = 𝛼𝜎+ 𝜌𝜎𝜎𝑔,𝑡2 + √𝑞𝑡𝑧𝜎,𝑡+1 (2)

𝑞_𝑡+1 = 𝛼_𝑞+ 𝜌_𝜎𝑞_𝑡+ √𝑞𝑡𝑧𝑞,𝑡+1 (3)

𝑧𝜎,𝑡, 𝑧𝑞,𝑡 are independent N(0,1) innovations that are independent from 𝑧𝑔,𝑡. 𝜎𝑔,𝑡+12 measures

the time-varying market uncertainty and 𝑞𝑡+1 measures the temporal uncertainty in that

process.

With further calculations authors show that the equity premium depends on the volatility-of-volatility factor 𝑞𝑡.

𝜋𝑟,𝑡 = 𝛾𝜎𝑔,𝑡2 + (1 − 𝜃)𝜅12(𝐴2𝑞𝜑𝑞2+ 𝐴𝜎2)𝑞𝑡 (4)

Further derivations lead to the VRP equation, which show that the VRP is directly linked to the volatility-of-volatility factor.

𝐸_𝑡𝑄(𝜎_𝑟,𝑡+12 ) − 𝐸_𝑡𝑃_(𝜎

𝑟,𝑡+12 ) = (𝜃 − 1)𝜅1[𝐴𝜎 + 𝐴𝑞𝜅12(𝐴2𝜎+ 𝐴𝑞2𝜑𝑞2)𝜑𝑞2]𝑞𝑡 (5)

Where 𝐸_𝑡𝑄(𝜎_𝑟,𝑡+12 ) is a risk-neutral one-month ahead variance expectation derived from the derivatives market and measured as a squared VIX index and 𝐸𝑡𝑃(𝜎𝑟,𝑡+12 ) is the physical

variance.

Equations 4 and 5 present a direct link between the equity premium in the equation 4 and the variance risk premium, defined as the difference between the Q and P expected variances, in terms of the volatility-of volatility factor 𝑞𝑡, which measures the temporal

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variation in variation of the consumption growth rate. Therefore, as can be seen from the equation 5, higher VRP is associated with higher shocks in the temporal market uncertainty, and thus leads to a higher market risk premium in the equation 4.

The elaborate explanation of the model derivation is out of the scope of this thesis (see Bollerslev, Tauchen and Zhou, 2009 for full explanation and solution of the model).

3.2 Time-Series Predictability Methodology

The empirical analysis starts by calculating the implied variance as a proxy for risk-neutral variance. It is simply defined following, among others, Bollerslev et al. (2009) as a squared deannualized VIX index.

𝐼𝑉_𝑡 = 1 12×

𝑉𝐼𝑋2

10000

(6)

The squared index is divided by the factor 104_{in order to be comparable to the physical}

variance measures, that are derived from the returns measured in decimals. The first measure of the realized P variance is taken as given from Hao Zhou’s website, where it is defined as in Bollerslev et al. (2009). 𝑅𝑉_𝑡 = ∑ [𝑝 𝑡−1+_𝑛𝑗 − 𝑝𝑡−1+𝑗−1_𝑛 ] 2 𝑛 𝑗=1 (7)

It is a sum of squared intra-day five-minute interval returns, where 𝑝_𝑡−1+𝑗 𝑛

− 𝑝

𝑡−1+𝑗−1_𝑛 is the

logarithmic difference in S&P 500 levels 5 minutes apart. The VRP is then identified as the difference between equations 6 and 7.

𝑉𝑅𝑃_𝑡 ≡ 𝐼𝑉_𝑡− 𝑅𝑉_𝑡 (8)

The first alternative measure for the P variance, proposed in this paper is similar to the one in equation 7, but daily rather than intra-day squared returns sum is used.

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𝑅𝑉𝐷_𝑡= ∑[𝑟_𝑑]2 𝐷

𝑑=1

(9)

Where 𝑟𝑡 is continuously compounded S&P 500 return on day d. D is the number of days in

month t. The variance risk premium is then calculated accordingly:

𝑉𝑅𝑃𝐷𝑡≡ 𝐼𝑉𝑡− 𝑅𝑉𝐷𝑡 (10)

The next two P variance measures employ Nelson‘s (1991) EGARCH model, where conditional variance is modelled as a function of a constant κ, its lagged value, ARCH effect of the innovation in mean equation and the leverage effect, which accounts for different sign of the innovation as it is known that the negative returns lead to higher market volatility than compared to the positive returns.

Using monthly S&P 500 returns the conditional variance is modelled as follows:

𝜎_𝑡2 _{= 𝜅 + 𝛾 log 𝜎} 𝑡−12 + 𝛼 [ |𝜀𝑡−1| 𝜎_𝑡−1 − √ 2 𝜋] + 𝜉 ( 𝜀_𝑡−1 𝜎_𝑡−1) (11)

𝛾, 𝛼 and 𝜉 are the maximum likelihood parameter estimates. It is also assumed that the innovation process _𝜎𝜀𝑡−1

𝑡−1 follows a Gaussian distribution.

For every month t, the EGARCH model is estimated using all the available data through month t and then conditional physical variance for that month is measured as the expectation of variance in month t+1. This ensures that both IV and CV are forward looking.

𝐶𝑉_𝑡 = 𝐸_𝑡[𝜎_𝑡+12 _] ₍₁₂₎

Variance risk premium is then defined by equation 13.

𝑉𝑅𝑃𝐶𝑉𝑡 ≡ 𝐼𝑉𝑡− 𝐶𝑉𝑡 (13)

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15 𝜎_𝑑2 = 𝜅 + 𝛾 log 𝜎_𝑑−12 + 𝛼 [|𝜀𝑑−1| 𝜎𝑑−1 − √2 𝜋] + 𝜉 ( 𝜀_𝑑−1 𝜎𝑑−1) (14)

Where the definition of variables are the same as for equation 11, except in daily terms. To estimate monthly conditional variance from daily data, the EGARCH model is estimated on the last trading day of each month t and then it is used to forecast one day ahead variance. Here the assumption is imposed that the conditional variance remains constant over the next month and thus physical variance at month t is estimated as a forecast of variance at first day of month t+1 and it is multiplied by 30 as shown in the equation 15.

𝐶𝑉𝐷_𝑡 = 𝐸_𝑡,𝑑[𝜎_𝑡,𝑑+12 ] × 30 (15)

Where 𝐸𝑡,𝑑 is an expectation on the last trading day d of the month t. The factor of 30 is used

instead of 22, which would match the number of trading days in the month, because according to the Chicago Board Options Exchange (CBOE) white paper (2004), VIX is an expectation of 30 days ahead variance.

𝑉𝑅𝑃𝐶𝑉𝐷_𝑡 ≡ 𝐼𝑉_𝑡− 𝐶𝑉𝐷_𝑡 (16)

The four variance risk premia defined above are then used to run both in-sample and out-of-sample time-series predictive regressions as in equation 17.

𝑟_{𝑡+1:𝑡+𝑘} = 𝛼 + 𝛽𝑧_𝑡+ 𝜀_{𝑡+1:𝑡+𝑘} (17)

Where 𝑟𝑡+1:𝑡+𝑘 is k-month ahead S&P 500 value-weighted adjusted for dividend returns

excess of 30-day T-bill rate for period t to t+k, calculated using equation 18 below. 𝛼 and 𝛽 are constant term and the regression coefficient respectively. 𝑧𝑡 is the predictive variable, in

this case one of the four VRPs at month t and 𝜀𝑡+1:𝑡+𝑘 is the regression error term. The

coefficients are estimated using Ordinary Least Squares by minimizing the sum of squared residuals.

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16

For the in-sample regression all available data is used in the regression. The t-statistics for 𝛼 and 𝛽 are calculated following Newey and West (1987) to account for the serial correlation of returns and Hodrick (1992) to account for the overlapping returns for monthly horizons k>1. The standard errors are estimated using k-1 lags.

For the out-of-sample regressions all data through month t is used to estimate the next period‘s return. The k months ahead returns are then forecasted using the equation 19 below.

𝑟̂𝑡+1:𝑡+𝑘 = 𝛼̂ + 𝛽̂𝑧𝑡 (19)

Where 𝑟̂𝑡+1:𝑡+𝑘 is the predicted k-months ahead return 𝛼̂ and 𝛽̂ are estimated regression

coefficients from 17 and 𝑧𝑡 is predictor variable at month t. To investigate whether the

proposed predictor outperforms the historical market mean return out-of-sample R2_is

calculated following Campbell and Thompson (2008).

𝑅_𝑂𝑂𝑆2 = 1 − ∑ (𝑟𝑡+1:𝑡+𝑘 𝑚𝑘𝑡 _−𝑟̂ 𝑡+1:𝑡+𝑘𝑚𝑘𝑡 ) 2 𝑇−𝑘 𝑡=240+𝑘 ∑ (𝑟_{𝑡+1:𝑡+𝑘}𝑚𝑘𝑡 −((𝑟̅_1:𝑖𝑚𝑘𝑡+1)𝑘−1)) 2 𝑇−𝑘 𝑡=240+𝑘 (20)

The benchmark forecast is calculated as the mean S&P 500 return through period t and then compounded accordingly for k months as it is seen from the equation 20.

The out-of-sample R2_{is calculated using forecasted returns from the equation 19,}

making sure that the first forecast is made using no less than 240 months of data to run the regression. Therefore, the out-of-sample period is fixed to be from January 2010 to December 2016 for a total of 84 - k months of observations to estimate the out-of-sample R2_{s. This}

approach is chosen according to Rapach et al. (2016) who uses the same out-of-sample period for all monthly horizons, rather than fixed number of observations for the first predictive regression.

Note that the out-of-sample R2_{can be negative, which indicates that the proposed}

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17

3.3 Trading Strategies

To further investigate the economic significance of the variance risk premia, trading strategies are created following Rapach et al. (2010). At the end of each month t the weights assigned to the S&P 500 value-weighted portfolio are calculated as follows.

𝑤̂𝑡= 1 𝛾 𝑟̂_{𝑡+1:𝑡+𝑘} 𝜎̂_{𝑡+1:𝑡+𝑘}2 (21)

Where 𝛾 is a risk aversion level to which the value of three is assigned following their paper. 𝑟̂_{𝑡+1:𝑡+𝑘} is the forecasted k months ahead return using all the data up to time t, as defined in equation 19. 𝜎̂_{𝑡+1:𝑡+𝑘}2 is the forecasted variance of the excess S&P 500 returns. It is estimated as k times the rolling window of the past five year monthly returns variance as in Campbell and Thompson (2008). The weight 𝑤̂_𝑡 at the end of each month is then assign to the S&P 500 portfolio and the weight 1 − 𝑤̂_𝑡 is assigned to the risk-free asset. Finally, restrictions on weights assigned to the index are imposed following Rapach et al. (2016) so that the weights assigned to the S&P 500 index would be between -1.5 and 1.5. These restrictions make the trading strategies realistic in practical terms as the leverage of the trading strategy cannot be higher than 50%.

The strategy returns are then calculated by multiplying the estimated vector of the weights by the respective S&P 500 excess returns.

𝑟_{𝑡+1:𝑡+𝑘}𝑠 = 𝑤̂𝑡× 𝑟𝑡+1:𝑡+𝑘𝑆𝑁𝑃 (22)

Where subscripts s and SNP stand for the strategy and S&P 500 accordingly.

Then the utility for the mean-variance investor is calculated as a certainty equivalent using the equation below. The certainty equivalent measures the risk-free return associated to the trading strategy.

𝐶𝐸 = 𝑅̅_𝑠 −𝛾 2× 𝜎̂𝑠

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18

Where, 𝑅̅𝑠 is the annualized mean return of the strategy s, 𝜎̂𝑠2 is the annualized sample

variance of the strategy s and 𝛾 is the risk aversion level equal to three. The returns of the trading strategy are annualized as follows:

𝑅̅_𝑠 = (𝑅_𝑠,𝑘+ 1)

12

𝑘 _{− 1} (24)

Where 𝑅𝑠,𝑘 is the mean return of a trading strategy s, built by predicting k months ahead

returns. Therefore, it is a mean return of series of portfolios with overlapping returns, when k is higher than one.

Finally, in order to test the hypothesis, whether the alternative measures of the P variance improve the VRP, the one-sided test for differences in certainty equivalent are calculated as derived by Eiling et al. (2016)

𝑡 − 𝑠𝑡𝑎𝑡 = √𝑇𝐶𝐸1− 𝐶𝐸2 𝜎_𝑔

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Where 𝜎𝑔is the sample standard deviation of 𝑔̂, which is defined in 26.

𝑔̂ = 𝑟_1,𝑡−𝛾 2(𝑟1,𝑡− 𝜇̂1) 2 − 𝑟_2,𝑡+𝛾 2(𝑟2,𝑡− 𝜇̂2) 2 − (𝐶𝐸̂₁− 𝐶𝐸̂₂) (26)

3.4 Robustness Checks Methodology

Similar methodology is used for the models of multiple predictors used in robustness checks. Instead of running time-series predictive regression in equation 20, multiple regression model is used as in equation 27.

𝑟_{𝑡+1:𝑡+𝑘} = 𝛼 + ∑ 𝛽_𝑖𝑧_𝑖,𝑡

𝑖∈𝑆

+ 𝜀_{𝑡+1:𝑡+𝑘} (27)

Where 𝑟𝑡+1:𝑡+𝑘 is the simple arithmetic excess return on S&P 500 index from month t to

month t+k and S is a subset of predictive variables. The k months ahead returns are then forecasted using the equation below.

𝑟̂𝑡+1:𝑡+𝑘 = 𝛼̂ + ∑ 𝛽̂ 𝑧𝑖 𝑖,𝑡 𝑖∈𝑆

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19

Where 𝑟̂𝑡+1:𝑡+𝑘 is the forecast of k months ahead return, 𝛼̂ and 𝛽̂ are the estimated 𝑖

regression coefficients from the equation 28.

The rest of the methodology for calculation of the out-of-sample R2_{, estimation of the}

trading strategy weights and calculation of the certainty equivalent returns is the same as for one predictive variable.

4. Data Description and Summary Statistics

The VIX index data is acquired from the Chicago Board Options Exchange website. The value-weighted and dividend adjusted S&P 500 returns are downloaded from The Center for Research in Security Prices (CRSP), which is accessed through Wharton Research Data Services (WRDS) database. The monthly returns on one-month T-bills are downloaded from Kenneth French’s website. The original VRP and RV monthly data is downloaded from Hao Zhou’s website.

Furthermore, five other predictive variables are used for robustness checks. The first two are the cyclically adjusted log earnings ratio (logPE) and the log price-to-dividends ratio (logDP) of S&P composite index. The data for both variables is downloaded from Robert Shiller’s website. The third and fourth variables are the default spread (DFSP) defined as the difference between returns on Moody’s Baa and Moody’s Aaa bond yield indices and Term Spread (TMSP) defined as the difference between the ten-year and three-months Treasury yields. The data for the two variables is downloaded from the Federal Reserve Bank of St. Louis website. Finally, the fifth variable used in robustness checks is the demeaned risk-free rate (RF), where the one-month T-bill rate is demeaned by its previous 12 months moving average. As mentioned above, the risk-free rate data is downloaded from Kenneth French’s website.

Table I summarizes all the variables used in this thesis. First of all, it is observed that both RVD and CVD are higher in mean than compared to the RV, with mean values of 26.49, 34.15 and 20.45 respectively. On the other hand, CV has a lower mean of 17.98. The data in Table I for the variance measures is expressed in squared percentages. It is also interesting to

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20

note that the VRP defined using the conditional variance forecasted with the EGARCH model from monthly data has the minimum observation of -2.96, whereas the other VRP measures have minimum values lower than -147 and as low as -267.21 for the VRPD. This shows that the EGARCH model from monthly returns underestimates the conditional variance.

As it has been noted by Bollerslev et al. (2014) the VRP is less persistent predictor than compared to the other popular predictors, such as logPE and logDP. The VRPs have substantially higher standard deviations ranging from 17.26 for VRPCVD to 24.74 for the VRPD, while for the other five predictors used in this paper the standard deviations vary between 0.07 and 1.13. Also standard deviation for the VRP measures are higher than their means, unlike for the other five predictors. The persistence of variable can also be assessed by autocorrelation AR(1) measures in Table I, where it ranges from 0.27 to 0.74 for the different VRPs, while for the other predictors it ranges from 0.85 to even 0.99. This is important because high persistence in a predictive variable tends to overestimate the out-of-sample R2_{measure for monthly horizons higher than one month. The AR(1) measure for the}

excess market returns is 0.04, which shows that the market returns are serially uncorrelated and thus the historic data of the market returns does not carry information about the future expected returns.

Table II is the correlations table. Both conditional variance measures have higher correlation with the implied variance, with correlation as high as 0.93 for CVD than compared to the realized variance measures. This is consistent with the fact that the IV and conditional variance measures are forward looking, whereas the realized variance measures the variation of returns in the current month. The small correlation between the VRPCV and the other VRP measures indicates that the VRPCV tells different story and is not a good measure of the variance risk premium. This is also later confirmed in empirical results. All of the VRP measures have low correlations with other predictors, ranging between -0.33 to 0.54, but most of the correlations are below 0.25. This tells that the Variance Risk Premium can explain the part of variation in market returns that is unexplained by other mainstream predictors. On the other hand, the correlation between logPE and logPD is 0.9, which means that both of them capture similar variations in the excess market returns. Finally, the correlations between VRPD and VRPCVD with excess market returns are 0.11 and 0.4 respectively, compared to the correlation of -0.01 between the original VRP and the excess market returns which tells that

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21 Table I. Summary Statistics

MktRf is the value-weighted and dividend adjusted return on the S&P 500 index excess of the one-month T-bill rate. IV is the implied variance, measured as the squared, deannualized VIX index. RV is the physical realized variance calculated as the sum of the high-frequency five minute interval squared returns. RVD is the realized variance calculated as the sum of daily S&P 500 continuously compounded squared returns. CV and CVD are one-month ahead conditional variance forecasts from the EGARCH(1,1) model using monthly and daily data respectively. The VRP, VRPD, VRPCV and VRPCVD are the Variance Risk Premium measures calculated as the difference between the IV and the physical variances, accordingly. The logPE and logPD are the log differences between the price and earnings and price and dividends respectively of the S&P composite index. The DFSP is the default spread defined as the difference between the Moody’s Baa and Aaa bond yield indices. The TMSP is the term-spread calculated as the difference between the ten-year and the three-month treasury yields. The RF is one-month T-bill rate demeaned by its previous 12 months moving average. MktRf, RF, DFSP and TMSP are presented in percentages. The variance and the variance risk premium measures are expressed in squared percentages that is multiplied by 104_{factor. The sample consists of 324 monthly}

observations between January 1990 and December 2016.

the alternative VRPs might act as better predictors of the market risk premium than the original VRP from Bollerslev et al. (2009).

Figures 1, 2 and 3 show the developments in the variance risk premium over time for the original VRP, VRPCV, which is calculated using conditional variance forecasted by the

Variable N Mean Median Std. Dev. Min Max AR(1)

MktRf 324 7.73 14.57 65.66 -89.66 263.31 0.04 IV 324 37.01 27.03 33.76 9.05 298.90 0.80 RV 324 20.45 11.11 37.34 1.73 517.47 0.64 RVD 324 26.49 14.43 46.33 1.58 566.11 0.71 CV 324 17.98 14.58 10.93 5.44 78.63 0.91 CVD 324 34.15 21.09 43.41 4.13 404.05 0.78 VRP 324 16.56 13.45 20.69 -218.56 115.85 0.27 VRPD 324 10.52 10.02 24.74 -267.21 84.02 0.35 VRPCV 324 19.03 12.36 24.62 -2.96 223.94 0.74 VRPCVD 324 2.86 4.12 17.26 -147.82 42.32 0.54 logPE 324 3.20 3.21 0.24 2.59 3.79 0.99 logPD 324 3.91 3.93 0.28 3.25 4.50 0.99 DFSP 324 0.96 0.89 0.40 0.55 3.38 0.96 TMSP 324 1.84 1.97 1.13 -0.70 3.69 0.98 RF 324 -1.33 -0.21 6.53 -23.17 16.08 0.85

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22 Table II. Correlations

MktRf is the value-weighted and dividend adjusted return on the S&P 500 index excess of the one-month T-bill rate. IV is the implied variance, measured as the squared, deannualized VIX index. RV is the physical realized variance calculated as the sum of high-frequency five minute interval squared returns. RVD is the realized variance calculated as the sum of daily S&P 500 continuously compounded squared returns. CV and CVD are one-month ahead conditional variance forecasts from the EGARCH(1,1) model using monthly and daily data respectively. The VRP, VRPD, VRPCV and VRPCVD are the Variance Risk Premium measures calculated as the difference between the IV and the physical variances, accordingly. The logPE and logPD are the log differences between the price and earnings and price and dividends respectively of the S&P composite index. The DFSP is the default spread defined as the difference between the Moody’s Baa and Aaa bond yield indices. The TMSP is the term-spread calculated as the difference between the ten-year and the three-month treasury yields. The RF is one-month T-bill rate demeaned by its previous 12 months moving average.

Variable MktRf IV RV RVD CV CVD VRP VRPD VRPCV VRPCVD logPE logPD DFSP TMSP RF

MktRf 1.00 IV -0.40 1.00 RV -0.36 0.84 1.00 RVD -0.35 0.85 0.97 1.00 CV -0.29 0.88 0.71 0.73 1.00 CVD -0.47 0.93 0.90 0.94 0.82 1.00 VRP -0.01 0.12 -0.44 -0.36 0.16 -0.11 1.00 VRPD 0.11 -0.24 -0.68 -0.71 -0.15 -0.49 0.84 1.00 VRPCV -0.42 0.98 0.83 0.85 0.77 0.91 0.10 -0.26 1.00 VRPCVD 0.40 -0.38 -0.63 -0.69 -0.32 -0.70 0.51 0.77 -0.38 1.00 logPE 0.01 -0.11 -0.14 -0.14 -0.18 -0.10 0.08 0.12 -0.07 0.05 1.00 logPD 0.00 -0.01 -0.05 -0.06 -0.04 -0.01 0.08 0.10 0.01 0.01 0.90 1.00 DFSP -0.09 0.63 0.60 0.59 0.74 0.62 -0.05 -0.25 0.54 -0.33 -0.45 -0.28 1.00 TMSP 0.02 0.05 0.08 0.07 0.16 0.05 -0.07 -0.07 0.00 -0.03 -0.51 -0.35 0.25 1.00 RF 0.08 -0.33 -0.24 -0.24 -0.40 -0.28 -0.10 0.01 -0.27 0.08 0.21 0.17 -0.40 -0.34 1.00

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23 Figure I. Variance Risk Premium

Figure shows the variation in the original VRP over the time period from January 1990 to December 2016. VRP is expressed in squared percentages.

Figure II. Variance Risk Premium with Monthly EGARCH CV

Figure shows the variation in the VRP calculated using the EGARCH(1,1) forecasted conditional variance from monthly data (VRPCV) for the time period January 1990 to December 2016. VRPCV is expressed in squared percentages.

-250 -200 -150 -100 -50 0 50 100 150

1990 Jan 1992 Sep 1995 Jun 1998 Mar 2000 Dec 2003 Sep 2006 Jun 2009 Mar 2011 Nov 2014 Aug

VRP VRP -50 0 50 100 150 200 250

VRPCV

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24 Figure III. Variance Risk Premium with Daily EGARCH CV

Figure shows the variation in the VRP calculated using the EGARCH(1,1) forecasted conditional variance from daily data (VRPCVD) for the time period January 1990 to December 2016. VRPCVD is expressed in squared percentages.

EGARCH(1,1) model with monthly returns and VRPCVD, which is calculated using the conditional variance forecasted by EGARCH(1,1) model using daily returns respectively. From Figures 1 and 3 it can be seen that there is a high decrease in the variance risk premium during the financial crisis of 2008, whereas VRPCV in Figure 2, does not predict the turndown as it remains highly positive during that period. This again implies that the conditional variance from monthly data is not a good measure for physical variance, as the VRP is supposed to be negative during the recession. This is due to the fact that market uncertainty, in terms of the realized return variation, during the crisis was extremely high, leading to negative risk premium as the returns on variance swaps have increased substantially, leading to profitable payoffs for the investors.

5. Results and Discussion

This section empirically analyses both in-sample and out-of-sample predictive power of the four different variance risk premia, defined as the difference between the VIX index

-200 -150 -100 -50 0 50 100

VRPCVD

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25

implied S&P 500 variance and the four different measures of the physical variance. All four VRPs are defined by the equations 8, 10, 13 and 16 in Section 3.2. First, predictive power of the VRPs is assessed in terms of the in-sample and out-of-sample R2_{measures. Then the four}

predictors are used in order to create trading strategies on the S&P 500 index to analyse whether they help create higher utility for the mean-variance investors in terms of certainty equivalent returns when compared to the historical mean benchmark trading strategy. The period under investigation is January 1990 to December 2016. This is because the VIX index data is only available from 1990 and the RV measure from Bollerslev et al. (2009) is available up to December 2016. Therefore, this time frame allows to compare the original VRP predictor with three alternative ones.

5.1 Predictive Regressions

To begin with, simple linear regressions are run in order to assess the explanatory power of the VRPs on the excess market returns. The regression model is defined in the equation 17 of the Section 3.2. Then the returns are forecasted using the equation 19 and finally the out-of-sample R2_{s are calculated according to the equation 20. The results are}

presented in Table III. As in the previous studies, the original VRP shows the highest in-sample predictive power for three months horizon with R2_{of 10.65%. The Hodrick t-statistics are}

significant for all monthly horizons at 1% level, which tells that the VRP has economic meaning for monthly horizons up to 24 months. The VRPD is also consistent with the previous findings, and what is interesting that it outperforms the original VRP in terms of the in-sample R2_for

up to six months horizon, but underperforms for longer monthly horizons. The t-statistics for one, three and six month horizons are all statistically significant at 1% level. As discussed in the Section 4, the VRPCV does not act as a good future returns predictor and does not show high predictive power in terms of in-sample R2_{. Also most of the t-statistics show low or non}

significance at all. The reason why VRPCV does not perform well might be because EGARCH(1,1) model is more accurate with more observations, which makes daily data superior to the monthly one, as monthly data uses only 395 observations to forecast the first conditional variance due to the data availability. Therefore, this does not allows to capture the clustering of the variance as well, which leads to the understimation of the market variance during the shocks such as 2008 financial crisis. On the other hand, the VRPCVD

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26

predictor, which uses daily returns to forecast monthly variance shows significant in-sample results for one, three and six months horizons with in-sample R2_{as high as 10.84% for the}

three months horizon. Despite that, it does not outperform the original VRP when considering the rest of in-sample results.

Bollerslev et al. (2009) discuss that the VRP should have the highest predictive power for monthly horizons, for which the volatility-of-volatility factor isolated by the VRP and which measures the temporal disturbance in the market uncertainty, has the highest effect. This is because the VRP isolates the factor as shown by the direct link between the VRP and volatility-of-volatility factor in Section 3.1. Even though, the in-sample results show that the VRP best predicts shorter monthly horizons, the ones up to 6 months, the out-of-sample R2_{s tell a}

slightly different story. For the VRP and VRPD the highest out-of-sample (OOS) R2_{appears at}

one month horizon, where OOS R2_{is as high as 12.78% for the VRPD. The VRPD also}

outperforms the VRP for longer horizons such as 12, 18 and 24 months, with higher OOS R2_s

of 11.70% for the VRPD as compared to 7.55% for the VRP. It is interesting to note that the VRPCVD performs poorly for shorter horizons up to six months, when compared to the VRP and VRPD, but outperforms all the other models for horizons longer than one year with the highest OOS R2_{of 13.55% at the two years horizon. In general for all the VRPs, except VRPCV,}

the best OOS performance occurs for horizons longer than one year. These results contradict the in-sample findings and show that the volatility-of-volatility factor captured by the VRP has the highest effect on the market returns for longer rather than shorter horizons. It is also interesting to note that the VRPCVD shows the highest predictive OOS power for six months when compared to other models. However, this result does not have a clear explanation, as the predictor did not show high predictive power for other horizons and in terms of in-sample R2_{. The reason for this might be that the conditional variance in the VRPCVD calculation carries}

more information content as it is estimated using all the available daily data. All of the VRPs have positive slope against the excess market returns for all the horizons, which is consistent with the theoretical model of Bollerslev et al. (2009). Drechsler and Yaron (2011) further explains that the variance risk premium should have positive relation with the excess market returns as it measures the market uncertainty. The higher the market uncertainty, the higher the returns are required by the investors, in order to achieve a reasonable risk-return balance.

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27

The OOS R2_{is a better measure for the VRP predictive power, since the VRP is less}

persistent than compared to the other classical predictors, as discussed in Section 2.1. However, these results should be assessed with caution, because the out-of-sample period considers only seven year of data. Therefore, this might lead to the inaccurate results. Also this does not tell whether the positive OOS R2_{s would be persistent for longer samples. This}

limitation might explain why the t-statistics of the regression coefficients for longer horizons have lower or no significance, while OOS R2_{s show higher predictive power as opposed to the}

in-sample R2_s.

Table III. In-Sample and Out-Of-Sample Predictability

The table presents the results for the following regression model: 𝑟𝑡+1:𝑡+𝑘 = 𝛼 + 𝛽𝑧𝑡+ 𝜀𝑡+1:𝑡+𝑘

Where 𝑟𝑡+1:𝑡+𝑘 is the monthly S&P 500 value-weighted and dividend adjusted returns excess

of the risk-free rate from month t to month t+k, 𝑧𝑡 is one of the variance risk premia at month t

and 𝛼 and 𝛽 are the regression coefficients.

The VRP is the variance risk premium defined as the difference between the squared VIX index implied variance and realized variance calculated as the sum of intra-day five-minute interval squared S&P 500 returns. The VRPD is the variance risk premium defined as the difference between the squared VIX index implied variance and the realized variance calculated as the sum of daily squared S&P 500 returns. The VRPCV is the variance risk premium calculated using the conditional variance forecast from the EGARCH(1,1) model using monthly S&P 500 returns and VRPCVD is the variance premium calculated using conditional variance forecast from the EGARCH(1,1) model using daily S&P 500 returns. Both in-sample and out-of-sample R2_{s are expressed in percentages. The predictors and}

the excess returns were expressed in decimals rather than percentages when running the regressions. The time period for the in-sample regressions is January 1990 to December 2016. The time frame for out-of-sample regressions is from January 2010 to December 2016. The six columns show the results for different monthly horizons: k=1, 3, 6, 12, 18 and 24 months. The table reports both Newey-West (1987) and Hodrick (1992) t-statistics in parentheses with k-1 lags. *, **, *** indicate 1%, 5% and 10% significance levels for the two-sided test respectively.

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28 Table III - continued

5.2 Trading Strategies

In order to investigate whether the positive OOS R2_{s actually carry economic}

usefulness, the trading strategies based on the four VRPs are considered. Considering the trading strategies allows to compare the OOS performance across different horizons. Every month t weight 𝑤̂_𝑡 based on the equation 21 in Section 3.2 is assigned to the S&P 500 index and weight 1 − 𝑤̂_𝑡 is assigned to the risk-free asset. Then the certainty equivalent returns are calculated according to the equation 23 in Section 3.2. Certainty equivalent (CE) measures the returns associated with trading strategy for which the risk averse investor would be indifferent between investing in the strategy and risk-free asset. Therefore, the CE essentialy

Horizon 1 3 6 12 18 24 Beta 4.68 11.54 14.46 13.25 13.14 14.98 t-statistic (Newey-West) (3.94)*** (5.17)*** (3.75)*** (2.25)** (1.70)* (1.77)* t-statistic (Hodrick) (3.94)*** (5.53)*** (5.70)*** (3.28)*** (2.91)*** (2.75)*** In-Sample R2 (%) 5.49 10.65 7.77 2.97 1.74 1.45 Out-of-Sample R2 (%) 9.04 2.58 2.78 3.39 6.14 7.55 Beta 4.43 9.96 10.23 6.01 5.95 6.79 t-statistic (Newey-West) (3.56)*** (5.99)*** (4.05)*** (1.49) (1.17) (1.40) t-statistic (Hodrick) (3.56)*** (8.35)*** (4.36)*** (1.53) (1.88)* (4.26)*** In-Sample R2 (%) 7.45 11.71 5.74 0.90 0.54 0.44 Out-of-Sample R2 (%) 12.78 1.96 2.76 7.69 8.70 11.70 Beta 0.23 2.15 7.10 10.40 11.50 14.89 t-statistic (Newey-West) (0.12) (0.42) (1.27) (1.97)** (1.79)* (1.62) t-statistic (Hodrick) (0.12) (0.36) (1.69)* (1.85)* (1.83)* (1.74)* In-Sample R2 (%) 0.01 0.49 2.53 2.47 1.76 1.87 Out-of-Sample R2 (%) -0.50 2.52 6.55 4.41 6.74 -0.38 Beta 5.72 14.62 16.81 13.92 19.36 20.31 t-statistic (Newey-West) (2.48)** (6.78)*** (3.50)*** (1.47) (1.42) (1.33) t-statistic (Hodrick) (2.48)** (6.27)*** (5.22)*** (1.68)* (1.82)* (1.70)* In-Sample R2 (%) 5.21 10.84 6.42 1.86 2.14 1.47 Out-of-Sample R2 (%) 4.64 -0.54 2.97 8.38 9.98 13.55 VRPCVD VRPCV VRPD VRP

(33)

29

measures the utility achieved by the investor who follows the proposed trading strategy. The weights assigned to the index are restricted to range between -0.5 and 1.5 as in Rapach et al. (2016) to make the strategies feasible in practice. The differences between CEs are also calculated. This difference show the extra risk-free return achieved by one strategy when compared to another. The Table IV below shows the results associated with the trading strategies based on the different VRP measures. The Panel A shows the mean, standard deviation and certainty equivalent returns for the strategies based on the original VRP and VRPCVD, which performs best in terms of OOS R2_{for horizons higher than one year, and}

historical mean benchmark, where returns are predicted using the average monthly return up to month t. The Panel B shows the differences between certainty equivalent of the three alternative VRPs proposed by this thesis and the original VRP. It also present the difference between all four VRP strategies and the benchmark strategy. Finally, it reports the respective t-statistics of the difference between the CEs as derived by Eiling et al. (2016).

As can be seen from Panel A in Table IV the original VRP based strategy has higher standard deviation than the benchmark strategy for all horizons and lower returns for most of the horizons except from one, 12 and 24 months. On the other hand, the VRPCVD based strategy achieves higher returns than the VRP and benchmark strategies, but with a bit higher standard deviations than the benchmark strategy and higher standard deviations for one, three and six months than compared to the VRP strategy. The results are consistent with the out-of-sample R2_{s in Table III, as the highest returns and CEs are achieved at one month}

horizon by both VRP and VRPCVD strategies with annualized mean returns of 14.45% and 15.05% respectively.

Panel B shows that the original VRP does not significantly outperform the benchmark strategy and underperforms it at 10% significance level for six month horizon. The VRPCV based strategy also does not outperform the benchmark nor the original VRP, which is consistent with the OOS R2_{results in Table III. Both VRPD and VRPCVD outperform}

benchmark and the VRP strategies in terms of certainty equivalent. The differences are statistically significant at 1% level for horizons higher than 12 months. Except the difference between the VRPD strategy and the benchmark strategy for 12 months horizon, which is significant at 5% level. The VRPD offers higher-risk free returns than the VRP strategy of 0.92, 0.53 and 0.74% annually for horizons of 12, 18 and 24 horizons respectively. It also offers