## 1. Introduction

Asset pricing in financial literature is predicated on the importance of volatility in asset returns. Hence, financial market volatility is a key factor ranging from investment decisions to derivatives pricing and financial market regulation (

Poon and Granger 2003). Volatility, as a measure of market risk, is always an anxiety-generating factor for market participants not only in terms of its nature but also its level (

Giot et al. 2010). Admittedly, volatility is an unobservable variable and its effects on financial markets are hard to anticipate. This is one of the reasons that asset return volatilities (henceforth, volatility) are of utmost importance to empirical finance. In other words, it remains the ingredient in assessing asset or portfolio risk, playing an important role in asset pricing models which heavily depend on underlying asset return dynamics. Asset management and asset pricing models require the proper volatility modeling of financial assets.

Despite extensive studies on the relation between volatility and expected returns (

Campbell and Hentschel 1992;

Glosten et al. 1993;

Ang et al. 2006), the literature establishes empirical evidence of the existence of a link between asset returns and expected volatility, as volatility is associated with volatility risk premium.

1 In particular, volatility risk premium as a part of asset price can be linked to expected volatility either through risk-premium or leverage-effect (

Christie 1982). However, little has been done in examining jointly how realized volatility affects returns and vice versa.

Bollerslev and Zhou (

2006) provided theoretical framework for assessing the empirical linkages between returns and realized and implied volatility. We investigate the empirical linkages between returns and various realized volatility measures. In doing so, we extend previous works (e.g.,

Bollerslev and Zhou 2006) on this topic by considering recent data from the most common benchmarks for the broader US and UK equity markets, namely the Standard & Poor’s 500 Index (S&P500) and the Financial Times Stock Exchange 100 Index (FTSE100) indices.

According to

Bekaert and Wu (

2000), and

Black (

1976), two research questions arise respectively:

Q1: “If volatility is priced, does an anticipated increase in volatility raise the required return on equity, leading to an immediate stock price decline?”

Q2: “Does a drop in the value of the stock (negative return) increase financial leverage, so that it makes the stock riskier and increases its volatility?”.

To give full answers to the above questions, we empirically examine the magnitude and the sign of the linear relationship between market-based volatility and contemporaneous returns, the so-called volatility feedback effect both on S&P500 and FTSE100. Consequently, we study (1) the existence of positive relationship market-based volatility and contemporaneous returns and (2) the existence of negative relationship between the lagged returns and the volatility. According to

Carr and Wu (

2017), “this effect can show up as a negative correlation between the index return and its volatility, regardless of the market’s financial leverage level”. Furthermore, we test the relationship between the lagged returns and the current market-based volatility, the so-called leverage effect. We refer to the study implemented by

Zumbach (

2013) for a detailed analysis of the leverage effect. The literature often refers to a negative relation between equity return and return volatility as a

“leverage effect” (

Carr and Wu 2017).

In this paper, we provide empirical results from intraday data to derive a negative relation between returns and volatilities; to the best of our knowledge, this is the first study that considers several realized measures to explain these hypotheses. Asset management and asset pricing models require a proper volatility modeling of financial assets. High frequency data contains more information about the market, such as the intraday changes and the market microstructure, making realized measures more accurate than those constructed by employing lower frequencies (see

Hansen and Huang 2016). For modelling volatility with high-frequency data, see

Degiannakis and Floros (

2015). However, market participants face problems when modelling volatility using intraday data. They should choose the correct sampling frequency as well as whether to sample prices in calendar time (every

$n$ seconds) or tick time (every

$m$ trades). Nevertheless, when both quotation prices and transaction are available, the selection of which price to use arises too. What is more, some realized measures require selections about tuning parameters, such as kernel bandwidth. All realized measures are based on the minute-by-minute intraday data of S&P500 (US) and FTSE100 (UK) indices.

For the analysis, we examined a large number of realized measures across S&P500 and FTSE100 indices, we considered thirteen realized estimators from three classes, and we applied these to 17 years of data. Our objective was to compare a large number of available realized measures to examine the volatility and leverage effects, providing a comparison of realized measures in environments with varied price processes and market microstructures. The existing literature shows that realized measures estimated by high frequency data are the most appropriate tools to deal with good and bad volatility (see

Degiannakis and Floros 2015,

2016). The importance of the examination of volatility can also emerge from upside and downside volatility associated the positive and negative price increments (see

Bollerslev et al. 2019). In volatility modeling, it is necessary to decompose the volatility into two parts, the perceived as directional-persistent volatility (good volatility) and the jumpy-relatively hard to anticipate volatility (bad volatility; see

Giot et al. 2010). As a consequence, many estimators of asset return volatility constructed using high-frequency price data have been suggested by

Andersen and Bollerslev (

1998),

Andersen et al. (

2001a),

Barndorff-Nielsen and Shephard (

2002,

2004),

Andersen et al. (

2007),

Barndorff-Nielsen et al. (

2008),

Hansen and Horel (

2009), and

Meddahi et al. (

2011), among others; these include realized variance, realized bi-power variation, and median realized variance. In this study, we also considered a simple realized variance estimator with a reasonable choice of sampling frequency, namely the 5-min realized variance or 10-min realized variance, and tested whether it was a “good” estimator, following the recent literature (see

Gkillas et al. 2019b). As both frequencies are “good” to construct realized estimators,

Liu et al. (

2015) found that the adequate balance between high and low sampling frequencies vary across different assets. However, the comparison of estimation accuracy shows that “it is difficult to significantly beat

${\mathrm{RV}}_{5}$ estimator” (

Liu et al. 2015); in this paper we also used the realized bi-power variation and median realized variance to deal with good volatility. On the other hand,

Low et al. (

2016) found that, by using copula-based models, several mean–variance-based rules exhibit statistically significant and superior performance improvements. For the superior performance of portfolio optimization, they follow

Low (

2018) by applying the Clayton canonical vine copula. For pairs trading strategy, see

Rad et al. (

2016).

The results for the S&P500 index show that eight estimators explain approximately 72% of the returns of the S&P500 index. As for the FTSE100 index, six estimators explain approximately 74% of the returns of the FTSE100 index. Also, in both indices, the returns negatively affect the realized volatility. Such findings are recommended to high-frequency financial analysts and volatility modelers. The findings are robust, using not only several realized measures estimated by high frequency data, but also using different sub-samples, providing evidence that realized measures have significantly impacted on return and vice versa.

This paper is organized as follows:

Section 2 details the data selection and introduces the realized measures used in this study.

Section 3 introduces the hypotheses and discusses the estimation results.

Section 4 discuss further practical implications of the results.

Section 5 concludes by summarizing the results of this study.

## 5. Summary and Conclusions

In this study, we investigated the impact of the realized measures on returns and vice versa, considering S&P500 and FTSE100 indices for the period spans from January 2000 to June 2017. Specifically, we empirically examined two research questions as follows: “If volatility is priced, does an anticipated increase in volatility raise the required return on equity on S&P500 and FTSE100 index?” and “Does a drop in the value of the stock (negative return) increase financial leverage, so that it makes the stock riskier and increases its volatility on S&P500 and FTSE100 index?”. We considered 10 realized measures from the Oxford-Man Institute of Quantitative Finance database to examine two hypotheses associated with the financial modelling and decision-making: (i) the volatility feedback effect as the relationship between the contemporaneous returns and the market-based volatility and (ii) the leverage effect as the relationship between lagged returns and the current market-based volatility.

As for the first research question, we found a positive relationship between the volatility and returns. The results were as follows for volatility feedback effect: for S&P500, most of the realized measures had a significant positive effect on daily returns

$BPV$,

$MedRV$,

$R{K}^{Barlet}$,

$RS{V}^{D}$,

$RS{V}^{D,SS}$,

$RV$,

$R{V}_{10}$,

$R{V}_{10}^{SS}$ and

$R{V}_{5}^{SS}$. For FTSE100, most of the realized measures had a significant positive effect on daily returns:

$BP{V}^{SS}$,

$MedRV$,

$RS{V}^{D}$,

$RS{V}^{D,SS}$,

$RV$,

$R{V}_{10}$ and

$R{V}_{5}^{SS}$. Both realized

$R{K}^{Barlet}$ and

$R{V}_{10}^{SS}$ had no effect on returns for FTSE100. An increase on “continuous-time” volatility raised the required return on both equity markets capturing the risk-return trade-off effect, which is consistent to

Bollerslev and Zhou (

2006). Furthermore, as for the second hypothesis, we found a negative relationship between the lag-returns and the volatility. With regard to the leverage effect hypothesis, both S&P500 and FTSE100 indices show that returns negatively affect realized volatility, which indicates that downside returns make the stocks riskier and increase their volatility. We confirmed the stylized fact of leverage effect, in which returns are negatively correlated with realized volatility (

Corsi et al. 2012). Such evidence for S&P500 and FTSE100 indices is consistent with the existing literature. Overall, we conclude that realized measures change and affect differently the daily returns of financial indices.

Future research should examine forecasting accuracy, that is, if implied volatilities provide unbiased and informationally efficient forecasts of the corresponding future realized volatilities.