Is There a Risk and Return Relation?

Traditional finance theory posits that the relationship between the risk and return of stocks is positive. Furthermore, investment practice is often based on the central contention of the Capital Asset Pricing Model (CAPM) that high (low) beta stocks earn higher (lower) returns. However, this fundamental return and risk relationship is questioned by a several researchers who assert that the relationship is, in fact, negative. Consequently, a growing body of research examines the nature of the stock return-risk relationship using both market- and firm-level data. The results of this research are mixed. The purpose of this paper is to shed further light on this relationship by (i) examining both market- and firm-level price data; (ii) employing a battery of tests, including individual market, panel and quantile regressions; and (iii) analysing the nature of the relationship during periods of high and low volatility and in bull and bear markets. The results indicate that there is no single robust relationship between risk and return. Of note, our results suggest a positive relationship when returns are high and during bear markets. Furthermore, the finding of a positive relationship is stronger (i) at the market-level than the firm-level; and (ii) over long time periods. However, the analysis indicates that a negative relationship exists at low return levels, during bull markets and, more so, at the individual firm level. Overall, the results suggest that the risk-return relationship is switching in nature and is primarily driven by changing risk preferences. Notably, a positive relationship exists when macroeconomic risk plays a larger role.

At the heart of finance theory is the notion that there is a positive relationship between risk and return. Indeed, this notion underpins one of the most important theories in finance -the Capital Asset Pricing Model (CAPM), which was pioneered by Sharpe (1964), Lintner (1965) and Mossin (1966). The CAPM asserts that the required rate of return on a stock is determined by the sensitivity of that stock to the market portfolio, where beta measures sensitivity. The theory contends that investors are risk averse and demand a premium for bearing risk. However, this fundamental belief in a positive trade-off between risk and return has been repeatedly questioned over time. For example, research dating back to Black (1976), Christie (1982), French et al. (1987) and Bekaert and Wu (2000) questions the existence of a positive risk-return relationship. Indeed, this stream of research argues that returns and volatility are negatively related and, moreover, that the nature of the relationship may be asymmetric across market states. Since this early research, the nature of the risk-return relationship has attracted extensive scrutiny in the academic literature. 1 Nevertheless, the risk-return trade-off remains one of the most hotly-debated puzzles in finance.
The purpose of this paper is to shed further light on the risk-return trade-off puzzle. The paper is innovative in a number of ways. First, the paper bridges the gap in the substantive literature by examining the nature of the risk-return relationship using both market-and firmlevel data. Second, the paper examines the relationship for a broad cross-section of developed and emerging stock markets. As emerging markets typically exhibit high expected returns and high volatility (Bekaert et al., 1998), it is reasonable to expect that the risk-return relationship in these markets is different from that in more mature markets. In addition, the inclusion of a 1 Most of this recent research effort has investigated the time series variation in the risk-return tradeoff (Salvador et al., 2014;Wu and Lee, 2015;Aslanidis et al., 2016;Badshah et al., 2016;Frazier and Liu, 2016;Ghysels et al., 2016;Hedegaard and Hodrick, 2016). Recent studies that have focused on the cross-sectional risk-return relationship include Ang et al. (2006Ang et al. ( , 2009, Bali and Cakici, (2008), Huang et al. (2010Huang et al. ( , 2012, and Wang et al., (2017).
wide selection of markets in the analysis may help resolve the current debate surrounding the risk-return trade-off. Third, the paper investigates whether the nature of the risk-return relationship is linked to different market states, including periods of high and low volatility and bull and bear markets. Finally, a number of approaches are employed to model the relationship.
Specifically, the paper utilises standard linear and quantile regression analysis, as well as different measures of risk and return. Thus, the paper seeks to provide firm evidence on the nature of the risk-return relationship.
The remainder of the paper is organised as follows. To establish a background for the analysis, a brief review of the literature examining the risk-return relationship is presented in section 2. Section 3 introduces the dataset, while section 4 discusses the research methods employed in the paper. The empirical results are detailed in sections 5 and 6. The final section offers a number of concluding observations and outlines the implications of the results for asset pricing.

Review of the Literature.
Two approaches have been adopted in the substantive literature to examine the risk-return relationship. The first approach tends to focus on index-level data and employs relatively sophisticated econometric techniques to examine the behaviour of, and interaction between, the conditional mean and conditional variance. By contrast, the second strand analyses firmlevel stock price data and typically examines the difference between returns of portfolios with different levels of volatility. In terms of the first approach, a recent study by Badshah et al. (2016) employs a quantile regression approach to examine the intra-day return-volatility relationship at return horizons of 1, 5, 10, 15, 60 minutes and one day using data for the S&P 500 over the period September 2003 to December 2011. They find evidence of a strong negative relationship between risk and return. Moreover, they find evidence of an asymmetric relationship, whereby the effects of positive and negative returns on volatility are different and more pronounced for negative returns and in the tails of the conditional distribution of volatility changes. Although, they note that this asymmetry tends to disappear at the daily return horizon.
The finding of a negative relationship between risk and return is also supported by Aslanidis et al. (2016) who employed a Markov-switching approach to study 13 European stock markets over the period 1986 to 2012. They find evidence of a negative risk-return trade-off that is strongest at the lowest quantile. The authors also document time variation in the trade-off that is linked to the state of the economy. 2 However, several studies argue that the relationship between risk and return is positive.
For example, Frazier and Liu (2016) use a copula approach and find evidence of a positive risk-return trade-off for four international stock market indices that is driven by market timing and skewness. Similarly, Bali et al. (2009) find evidence of a significant and positive relationship between downside risk and return for a portfolio of US equities, while Breckenfelder and Tedongap (2012) and Sevi (2013) validate this finding using intra-day highfrequency stock returns. In a more recent analysis, Chang (2016) documents a positive relationship between risk and return, the strength of which varies according to different stock market conditions; specifically, the relationship is stronger during bear markets as compared to bull market periods. By employing a common information set to measure expected excess return and conditional variance, Jiang and Lee (2014) detect a positive relationship that is robust across different time intervals. Hedegaard and Hodrick (2016) confirm this finding using an overlapping data inference approach to study the relationship using data for the US market.
Similarly, Ghysels et al. (2016) confirm the existence of the traditional risk-return relationship using a MIDAS approach. However, they find evidence of fundamental changes in the relationship during periods of financial crisis. Notably, the strength of the relationship varies 2 They further found that the state of the economy depends on more than the business cycle alone.
with the level of volatility, and is also documented by Salvador et al. (2014) for a sample of 11 European markets and by Wu and Lee (2015) for the US market. 3 Of course, some authors report mixed results. For example, in a comprehensive study of 37 stock markets, Bali and Cakici (2010) find that the risk-return trade-off varies across countries. Galagedera et al. (2008) find evidence of a positive risk-return relationship only (i) when risk is measured by downside co-skewness; and (ii) for longer timescales. Some authors attempt to provide clarity on the mixed nature of results. For example, Feunou et al. (2013) and Cheng and Jahan-Parver (2016) suggest that the failure to find a positive relationship may result from a need to model skewness behaviour in stocks as well as means and variances. In a different vein, Wang and Yang (2013) argue that the offsetting nature of a volatility feedback mechanism may lead to confounding effects on the positive risk-return relationship. 4 An alternative approach to investigating the risk-return relationship is to examine individual stock level data. This body of work typically involves constructing portfolios of stocks according to their degree of volatility and examining whether portfolios that are characterised by high volatility are associated with higher returns. This research dates back to the foundations of the CAPM and an examination of whether the security market line slopes upwards (Sharpe, 1964;Jensen, 1969). A series of papers have identified a negative risk-return relationship for US stock markets. For example, Haugen and Baker (1991) find that low risk portfolios earn higher returns relative to a market capitalisation-weighted benchmark. The issue has attracted renewed interest in recent years and the finding of a negative relationship between risk and return has been confirmed for the US stock market (Jagannathan and Ma 2003; Blitz 3 In a forecasting context, Christoffersen and Diebold (2006) and Anatolyev and Gospodinov (2010) argue that volatility and return sign dependence are linked. 4 The volatility feedback hypothesis was proposed by French et al. (1987) and Campbell and Hentschel (1992). Under this hypothesis, higher volatility in the current period induces expectations of higher volatility in the future, a higher expected return and a higher discount rate. The higher discount rate reduces the present value of future cashflows and, thus, causes the current stock price to fall. As a result, price falls tend to be contemporaneously related to high volatility. For studies that have attempted to decompose the risk-return relationship into a risk premium and a volatility feedback mechanism, the reader is referred to Yang (2011) and Wang and Yang (2013). and van Vliet, 2007), as well as other developed markets (Ang et al., 2006;Blitz et al., 2012;de Carvalho et al., 2012). More recently, Baker and Haugen (2012) document comprehensive evidence of a negative risk-return relationship for a sample of 33 developed and emerging markets that has been evident since 1990. This literature also ties in with the recent discussion surrounding a low volatility anomaly, which states that low volatility stocks earn higher returns than high volatility stocks (Ang et al., 2009;Baker et al., 2011). Further insight into this anomaly has recently been provided by Wang et al. (2017) who find evidence of heterogeneity in the risk-return trade-off. Specifically, they note a positive risk-return relation for firms in which investors face capital gains and a significant inverted risk-return relation among firms for which investors face a capital loss.
Overall, a review of the extant literature indicates that a dichotomy exists between studies that examine the risk-return relationship at the market-level and those that analyse firmlevel data. Notably, the results from studies of market-level data are mixed; while several studies find evidence of a positive risk-return relationship, others document that the relationship is negative, and a few argue that the nature of the trade-off varies across market states. By contrast, analyses that employ firm-level data are more conclusive and assert that there is a negative relationship between risk and return. However, despite this evidence, there remains a belief that a positive relationship should exist. Indeed, Baker and Haugen (2012) argue that the positive risk-return paradigm should fall but remains rooted in our understanding of finance.
This paper provides a comprehensive analysis of the risk-return relationship by examining both market index and firm level data and utilising tools from both strands of the literature. Thus, the paper furthers our understanding of both the existence and the nature of the risk-return relationship and attempts to resolve one of the most fundamental puzzles in empirical finance. A clear understanding of the nature of this relationship is essential given the predominant belief of a positive relationship that is expounded by existing finance theory; as noted above, a positive risk-return relationship is embedded in current models of asset pricing and portfolio construction. This belief impacts investment appraisal and the required rate of return, the choice of stocks in portfolio and risk management and the investment of trillions of funds. Thus, the belief in a positive risk-return trade-off has significant implications for the real economy and an overturning of this belief would have profound implications for investment in real and financial assets.

Data.
Three broad datasets are employed to examine the stock market risk and return relationship. based on their inclusion in the main market indices of each market. Thus, the data consist of firms that are broadly equivalent across the sample markets. In addition, the data selection criterion excludes the potential for very small firms to be included in the analysis. Returns and risk for each sample firm are calculated in the manner outlined above, although here, a four- year rolling window is used to compute standard deviation.
The three datasets are included in the analysis in order to provide a comprehensive examination of the nature of the risk-return relationship. The two market-level datasets facilitate an examination of the risk-return relationship for a recent time period and for a wide range of both developed and emerging markets. These data are sampled at a frequency that is consistent with the time horizon that a portfolio manager would typically adopt, as well as a longer time horizon that encapsulates a greater number of market phases. The use of a longtime span is also in keeping with the argument of Lunblad (2007) that a lengthy time-series of data is needed to reliably estimate the risk-return relationship. The annual data also serve to reduce the influence of noise that is often apparent at higher frequencies. 6 In addition, the use 6 A mix of different data frequencies is employed in the existing literature to examine the risk-return relationship. While some studies use low frequency data, at quarterly and monthly intervals, to eliminate short-term noise (Frazier and Lu, 2014;Wu and Lee, 2015;Aslanidis et al., 2016), others utilise weekly (Guo and Neely, 2008), of daily, monthly and annual data allows for the fact that shocks at different levels of frequency may have a differential impact on the market. 7 The final dataset, which consists of individual firm-level data, captures different information. That is, investor behaviour is likely to be dominated by expected future economic performance and macroeconomic risk at the marketlevel while, at the firm level, information regarding cashflows, earnings, dividends and the future prospects of the firm is pertinent to investors (Jung and Shiller, 2005). Thus, marketand firm-level data capture different information and, consequently, provide a more complete picture regarding risk-return behaviour and the risk-return relationship.

Empirical Method.
This paper analyses the relationship between stock market returns and risk (typically defined as the standard deviation). Conventional finance theory posits that investors are risk-averse and that a positive relationship exists between risk and return (Sharpe, 1964;Lintner, 1965). The following regression model can be used to examine this contention: where rt denotes stock returns, vt is the measure of risk, or volatility, and εt refers to the random error term. Of key interest is the sign and significance of the parameter β, which determines the risk-return relation. Equation [1] is estimated for each of the sample markets and firms included in the analysis. In addition, a fixed effects panel model is estimated, such that Equation [1] extends to: where the subscript i refers to the individual markets or firms and γi is the cross-sectional fixed effects term.
daily (Darrat et al., 2011) or even intraday data (Sevi, 2013;Badshah et al., 2016). A small number of studies employ a MIDAS specification that allows for mixed data frequencies (Salvador et al., 2014;Ghysels et al., 2016). 7 Indeed, Galagedera et al. (2008) emphasised the importance of considering other time horizons besides the typical daily and monthly frequencies when investigating the risk-return relationship. to the level of volatility and whether the market was in a bull or bear phase.
A quantile regression models the quantiles (partitions or sub-sets) of the dependent variable given the set of potential explanatory variables (Koenker and Bassett, 1978;Koenker and Hallock, 2001). The quantile regression therefore extends the linear model in [1] by allowing a different coefficient for each specified quantile: where α (q) represents the constant term for each estimated quantile (q), β (q) is the slope coefficient that reveals the relationship between risk and return at each quantile, and εt is the error term.
As the risk-return relationship is designed to capture the behaviour of expected returns, as opposed to realised returns, the equations detailed above are also estimated using expected returns that are obtained from an asset pricing model. For the market index data, a GARCH(1,1)-in-mean model is estimated for the excess stock return and the fitted value obtained. This process is based on the view that the expected market index return is given by: where the risk-free rate (rf) is based on a short-term (three-month) Treasury bill and σt is the market standard deviation. Thus, the following GARCH(1,1)-M model is estimated: where the return process is defined as a function of the conditional mean, µ, the estimated conditional standard deviation, σ, and the disturbance term, t  . The conditional variance (σt 2 ) of the return series is given by the variance of the random error term (εt) conditional on the past information set 1 t   , such as: 2 = ( |Ω −1 ), with the GARCH model given by: where the non-negativity constraint must hold for all parameters in the model (  ,  , ) and the measure of persistence of shocks to volatility is given by α+β<1.
For the stock level data, expected returns are obtained from the standard CAPM, as well as the Fama-French three factor model (FF3). Thus, the fitted values from the following two equations are obtained: where rmt is the market return, and SMBt and HMLt are the small minus big and high minus low Fama-French market capitalisation and book-to-market factors, respectively.

5.
Risk-Return Results. Figure 1 shows a scatter plot of the average monthly return and standard deviation for the 43 international stock market indexes included in the study, along with the OLS regression line. 8

Evidence from 43 Stock Market Indexes
The graph supports a positive relationship between returns and risk; the regression coefficient is 0.116, with a White (heteroscedasticity) corrected t-statistic of 2.05. The correlation coefficient between returns and standard deviation is 0.47. Within this graph, we can observe that there is a small group of markets located in the North East of the plot which appear to exhibit different characteristics from the majority of the series, with noticeably higher returns.
Potentially, this grouping of markets may be driving the overall results. Therefore, Figure 2 presents results excluding the four high return markets from the analysis, 9 together with the market that presents a negative return. Within this figure we can see that although the scatter plot and regression line still indicate a positive relationship between return and risk, the regression coefficient is 0.039 with a White corrected t-statistic of only 1.59. In addition, the correlation between return and risk declines to 0.28. Overall, this preliminary analysis indicates that, at best, there is a weak positive relationship between risk and return and one that may be driven by the behaviour of only a few markets.
To consider this issue further and to introduce a time dimension into the analysis, the two-year rolling standard deviation for each market is calculated and the regression model in return and risk, even in the panel regression analysis. Indeed, for 40 of the markets, the coefficient on the standard deviation is not significant at any conventional level, while the slope coefficient is positive for twenty markets and negative for twenty-three markets. Furthermore, in only one market (China) is there a statistically significantly positive relationship between risk and return. Thus, the results suggest a minimal relationship between stock returns and their associated risk that, at best, is weakly positive.

Dimson-Marsh-Staunton Data
In the second part of the analysis, the Dimson, Marsh and Staunton (2002, 2008 dataset is used to examine the risk and return trade-off. This dataset expands the time horizon of available data with which to examine the risk-return relation; specifically, the dataset consists of annual returns for 17 markets over the period 1900 to 2010. The longer time frame may provide more robust results as it captures a greater number of market cycles. Figure 3 shows a scatter plot of the mean return and standard deviation for all markets over the sample period. It is apparent from the figure that there is a positive relationship between return and risk; the regression coefficient of returns on risk is 0.186 with a White corrected t-statistic of 3.32, and the correlation between the average return and standard deviation is 0.64. This result is, therefore, more supportive of a positive risk-return relationship, which may arise from the longer time span considered. In addition, the use of annual rather than monthly data is likely to smooth out shorter-term fluctuations and, thus, may reveal in greater clarity the nature of the longer-term relationship. Table 2 presents the regression results using time-varying standard deviations that are obtained by constructing 10-year rolling values. In contrast to the evidence reported in Figure   3, the results are not overwhelmingly in favour of a significant positive relation; 14 of the 17 sample markets have a positive risk-return relation, although this is significant at the five (ten) per cent level for only two (four) market(s). 10 Thus, there is no evidence of a significant and positive risk-return relationship in 11 of the 17 sample markets. The results from the fixed effects panel model show a positive risk-return relationship; these results could be considered as more reliable due to the increase in the degrees of freedom or, alternatively, they could be driven by a subset of the markets considered. Hence, even with a longer time series of data, the view of a universal positive risk-return relation seems doubtful.
10 Specifically, the results are significant at the five per cent level for Japan and the UK, and at the ten per cent level for France, Germany, Italy and the Netherlands.

Quantile Regression Analysis
The above regression analyses utilise OLS and thus focus on the conditional mean point estimate in order to garner information about the risk-return relationship. However, this approach ignores the possibility that deviations in the risk-return relationship may occur in the tails of the distribution due to heterogeneity in investor beliefs. 11 Thus, in order to allow for such heterogenous investor beliefs and, therefore, differences in the trade-off across the distribution, quantile regression analysis is conducted. The results from conducting this exercise are reported in Table 3 for the monthly returns data across 43 markets and in Table 4 for the Dimson-Marsh-Staunton annual data for 17 markets. A visual inspection of Table 3 reveals an interesting pattern in the risk-return relation for different return values. In particular, the table shows that there is an exclusively negative relation at the lowest return quantile (Q1), which is statistically significant at the five per cent level for 28 of the 43 markets considered, and for a further six markets at the ten per cent level. 12 As a mirror image, there is an (almost) exclusively positive risk-return relationship at the highest return quantile (Q9). 13 This positive relation is statistically significant at the five per cent level for 34 markets, and significant at the ten per cent level for an additional three markets. 14 By contrast, the pattern of results for the middle return quantile (Q5) is very mixed. Of the 43 markets, 24 exhibit a negative relationship, 11 The importance of heterogeneity in the beliefs of investors to asset pricing is cogently argued by Shefrin (2008).
Here, expected stock returns in the US are bi-modal and fat-tailed due to the extreme beliefs of optimistic and pessimistic investors. 12 Specifically, there is a negative risk-return relationship that is statistically significant at the five per cent level for Austria, China, Cyprus, the Czech Republic, Finland, Germany, Greece, India, Indonesia, Ireland, Italy, Japan, Luxembourg, Malaysia, Mexico, the Netherlands, New Zealand, the Philippines, Poland, South Africa, South Korea, Sweden, Taiwan, Thailand, Turkey, the UK, the US and Venezuela. The relationship is significant at the ten per cent level for Australia, Belgium, Brazil, Norway, Pakistan and Singapore. 13 The Czech Republic is the only market that has a negative risk-return relationship at the highest return quantile, although it is not statistically significant. 14 The 34 markets that show a positive risk-return relationship that is significant at the five per cent level include Australia, Austria, Belgium, Brazil, China, Denmark, Finland, France, Germany, Greece, Hong Kong, Indonesia, Italy, Japan, Malaysia, Mexico, the Netherlands, Norway, New Zealand, Pakistan, the Philippines, Poland, Russia, South Africa, South Korea, Singapore, Sri Lanka, Sweden, Taiwan, Turkey, the UK, the US and Venezuela. The relationship is positive and significant at the ten per cent level in Canada, Cyprus and Spain.
while 19 show a positive relation. Furthermore, the relationship is significant at the five per cent level for only two markets (Cyprus and Greece); in both cases, the risk-return trade-off is negative. This result for Q5 is consistent with the conditional mean results reported in Table 1, which showed scant evidence of a significant risk-return relationship.
A similar picture emerges from Table 4 Overall, these results constitute compelling evidence that the risk-return relationship is negative at low levels of return and positive at high levels of return. This pattern is consistent with the disposition effect that motivates prospect theory. The disposition effect contends that investors are likely to take profit when returns are high and thus act in a risk-averse fashion, generating a positive relationship. However, when returns are low, investors are more likely to maintain their position in the hope that subsequent returns will increase, leading to a profitable trading position. This risk-taking behaviour can induce a negative relationship. 17 15 The results are significant at the five per cent level for Australia, Belgium, Denmark, Germany, the Netherlands, South Africa, Spain, Sweden, Switzerland and the UK, while Canada and Ireland show a negative relationship that is statistically significant at the ten per cent level. 16 Only the UK shows a positive relationship that is not statistically significant, while South Africa is the only market with a negative risk-return relationship. This latter result is statistically significant at the five per cent level. 17 Propect Theory was pioneered by Kahneman and Tversky (1979) and later extended by Tversky and Kahneman (1992). The theory has been used to explain the disposition effect, as well as other phenomena in finance, such as momentum and the equity premium puzzle. For a review of Prospect Theory and its application to economics, the reader is referred to Barberis (2013).

Individual Firms
The results from conducting a similar analysis using the firm-level data are reported in Table   5. Column two of the table reports the results from the fixed effects panel regression  6. Further Tests.

Alternative Measures of Risk and Return
Given the absence of a robust relationship between risk and return detected so far across both individual market level and individual firm level data, we now extend the analysis to consider alternative measures of risk and return. As alternative measures of risk, we consider both VIX and VaR values, while we use an asset pricing model to obtain expected (as opposed to realised) returns.
First, we consider the volatility index based upon option price implied volatility as an alternative measure of risk for the markets for which suitable data is available. Often referred to as the VIX measure, the volatility index is colloquially regarded as the market fear index and represents risk derived from investors behaviour in trading on options relating to a particular index. We obtain VIX measures for five markets, including three measures based on different US indexes. The use of the standard deviation, which covers the full distribution of the data, in measuring risk can be criticised as it does not necessarily accord with our perception of risk.
Specifically, risk is more associated with negative outcomes rather than positive outcomes (March and Shapira, 1987;Tversky and Kahneman, 1992;Unser, 2000) 19 and, particularly, if the distribution is skewed, the use of the standard deviation may not be appropriate (Barberis and Huang, 2008) 20 . One solution would be to use the semi-standard deviation where we calculate the standard deviation statistics for observations below a given threshold value (such as a return of zero). However, even this value may include small (albeit negative) return values, whereas we can consider risk as being associated with must larger losses. Therefore, we 18 The results from using lagged values of the VIX index are available from the authors on request. 19 The use of standard deviation as a measure of risk in practice has been studied extensively. For example, the behavioural approach to risk focuses on the notion of loss aversion and documents that, in practice, individuals weight losses twice as much as gains (Tversky and Kahneman, 1992). Similarly, in an investor context, Unser (2000) and Veld and Veld-Merkoulouva (2008) find that investors prefer to use shortfall risk or semi-variance, rather than standard deviation as a measure of risk. Finally, several studies have focused on managerial attitudes to risk and report that, in practice, risk is associated with negative outcomes and uncertainty regarding positive outcomes is typically not perceived as 'risk' (March and Shapira, 1987;Helliar et al., 2002). 20 Barberis andHuang (2008, p. 2069) argue that some investors place more value on positively skewed securities as it makes the distribution of their overall wealth more "lottery-like". Thus, the authors argue that some investors may be willing to pay a high price for positively skewed stocks and to accept a negative average excess return.
calculate the 95% Value-at-Risk (VaR) for our return series and consider its relationship with returns. The 95% VaR is the value associated with cutting-off the 5% left hand side tail. In order to obtain a time-varying VaR, we utilise the GARCH model in equation [6] and calculate the VaR as: VaRα = μt(r) + (φ(α)√(σt 2 )), where μt(r) is the conditional mean of the return series, φ is the cumulative distribution function, with a significance level given by α and √(σt 2 ) is the standard deviation obtained from the GARCH variance model.
The results for the return and VaR regressions are reported in Table 7, in the column headed VaR. These results present a similar picture to those observed above, in which there is very little evidence of a significant relationship between returns and risk, regardless of the sign of that relation. We can observe a positive coefficient sign for 25 markets and a negative sign for 18 markets, however, only for one market is the (positive) coefficient statistically significant at the 5% level (one further market is significant at the 10% level). To further consider the behaviour of the return and risk relationship in the left tail of the distribution, the results under the column headed exceedances are based on the return and risk regression in which we only include observation from the left 5% tail of the distribution (i.e., we only include observations that exceed the 5% VaR). There results of the coefficient signs here are universally negative and statistically significant for 37 markets at the 5% level (with three markets significant at the 10% level). These results confirm those reported in Table 3 in which low return values exhibit a negative relationship with risk.
In line with most of the literature, the foregoing analysis is based on realised returns.
However, the risk-return relationship should be based on expected, rather than realised, returns.  The results from estimating expected returns for the individual stocks using both a CAPM and FF3 model are reported in Table 9. As with the results for the stock market indexes, the pattern that emerges is one that does not support a positive risk-return relationship. Only three (one) of the nine sample markets 22 exhibit a positive (negative) and significant risk-return relation, while no significant relationship is detected for five markets. Similarly, using FF3 derived expected returns, only three (four) markets exhibit a positive (negative) and significant relation, and two markets do not exhibit a significant relation. 23

High v's Low Volatility
We also examine the nature of the risk-return relationship across different volatility states using individual stock level data. To that end, the data is partitioned according to the level of volatility, as measured by standard deviation, for each stock in each year, and the risk-return 21 Only five markets exhibit a relationship between expected return and risk that is not statistically significant: Brazil, Canada, Finland, Pakistan and Sweden. 22 South Korea was excluded from this analysis due to the unavailability of data needed to estimate the Fama-French three factor model. 23 A significant positive risk-return relationship is documented for Australia, Canada and Hong Kong using the CAPM, and for Australia, Canada and the US using the FF3 model. By contrast, there is a significant negative relationship for the UK using the CAPM, and for Germany, Hong Kong, Italy and the UK using the FF3 model. regression model is estimated for firms in the lowest quartile of volatility, the highest quartile and the middle 50 per cent of volatility. Table 10  The analysis of risk-return behaviour across different levels of volatility is repeated using CAPM and FF3 expected returns rather than realised returns. The results from this analysis, which are reported in Table 11, emphasise the lack of clarity about, and inconsistency in, the risk-return relationship that has been reported so far. Although there is evidence of a positive relationship when expected returns are estimated according to the CAPM, especially at low and medium levels of volatility, the results when expected returns are calculated using the FF3 model are much less convincing. Specifically, using CAPM expected returns, five (six) markets exhibit a significant positive relationship at low and medium levels of volatility, but this number reduces to only two markets for the FF3 results. Furthermore, the relationship is 24 Specifically, only South Korea and the UK exhibit a positive risk-return relationship that is statistically significant at the five per cent level, while the relationship is statistically significantly negative in Italy and Japan. 25 The relationship is statistically significantly negative (positive) in Italy and Japan (South Korea). 26 Only Canada and the US exhibit a statistically significant relationship between risk and return; this relationship is positive in both cases.
significantly negative for four (five) markets at low (medium) levels of volatility when expected returns are calculated using the FF3 model.
The analysis is further expanded to consider the returns earned by portfolios of firms that are organised according to the level of volatility for each year; these results are reported in Table 12. This analysis is useful for providing evidence on the low volatility anomaly that is reported in the literature and which finds that returns are higher (lower) for low (high) volatility stocks (Ang et al., 2006(Ang et al., , 2009Blitz and van Vliet, 2007;Baker et al., 2011;Baker and Haughen, 2012). The results from the current analysis are inconclusive. Although they indicate that returns increase with volatility for four of the sample markets, there is evidence that returns decrease with volatility in three markets. Furthermore, returns for three markets are highest at medium levels of volatility, suggesting the presence of a hump shaped risk-return profile.

Bull vs Bear Markets
In this part of the analysis, the risk-return relationship is examined to consider if it varies with the financial cycle of bull and bear markets. A three-year moving average of the stock index is used to define bull and bear regimes for each market; if the change in this moving average is positive then the market was characterised as a bull market, while if the change in the three- year moving average is negative, the market is in a bear phase. This definition is consistent with Chauvet and Potter (2000) who defined bull (bear) markets as those that correspond to periods of increasing (decreasing) market prices. The results for the 43 international markets are presented in Table 13, while Table 14 reports the results obtained from the analysis of individual stock level data.
In terms of bull market conditions, Table 13 shows that there is a positive risk-return relation for 15 of the 43 markets and that this relation is significant at the 5% level for only one market (Sri Lanka). Hence, under bull market conditions, a majority of the markets (28) exhibit a negative risk-return relation, albeit with limited statistical significance (only three markets at the 5% level). The opposite pattern emerges under bear market conditions. Here, a majority of markets (32) exhibit a positive risk-return relation, although this relationship is statistically significant for only three markets at the five per cent level (Italy, South Korea and the US) and a further four markets at the ten per cent level (Australia, France, Spain and Sweden).
A similar picture emerges from the analysis of firm-level data. Table 14 shows that, in bull market conditions, the risk-return relationship is predominantly negative. Indeed, eight of the ten sample markets exhibit a negative risk-return trade-off, and this relationship is significant at the 5% level in six cases, and significant at the 10% level for one further market.
By contrast, with the exception of Germany, the risk-return relationship is positive in bear markets. Furthermore, this relationship is positive and significant for five and two markets at the 5% and 10% levels, respectively. Taken together, the results from the analysis of bull and bear markets suggest strongly that risk aversion is state dependent. Furthermore, the results lend some credence to the countercyclical risk premium hypothesis according to which investors demand a higher compensation from holding a risky asset in a bear market. A similar finding is reported for the US market by Chang (2016) who notes that the magnitude of compensation for enduring risk is stronger during periods of unfavourable financial conditions. 27 Overall, the results from a battery of tests on market-and firm-level data indicate that the relationship between risk and return is not uniform across countries and different market states. Evidence is detected to suggest a positive risk-return trade-off over long horizons, when returns are high, and during bear markets. By contrast, the relationship is negative at the individual firm level, low return levels and during bull markets. Thus, the paper finds evidence to support a risk-return relationship that switches according to the level of market aggregation, the level of returns and the overall market state. 28

Summary and Conclusion.
The core of finance theory retains the belief in a positive risk-return trade-off and, while empirical evidence against this view is mounting, there remains a view that the failure to report a positive relation is due to modelling issues rather than the absence of an intrinsically positive relationship (Wu and Lee, 2015). Consequently, this paper has sought to examine the riskreturn relationship across a range of international stock markets at both the market and individual stock level; existing work in the area typically focusses on only one type of data.
Moreover, the paper examines the nature of the relationship using a range of different modelling techniques, including standard linear and quantile regression analysis, models designed to capture expected returns behaviour, different definitions of risk and models that vary according to the level of volatility. In addition, the paper investigates whether the riskreturn trade-off varies across different market conditions.
Overall, the results suggest that there is no systematic relationship between risk and return. Notwithstanding this general conclusion, seven key points emerge from the analysis. In conclusion, the paper is supportive of the recent, and increasing, view that a carte blanche notion of a positive risk-return relation does not exist; while evidence of a positive relation exists under certain circumstances, at least equal evidence of a negative relation is also uncovered. Thus, the relationship can be more accurately described as switching in nature. One possible explanation for this result is the disposition effect and changing risk preferences of investors, whereby investors exhibit a tendency to sell stocks whose prices have increased since purchase rather than those that have declined in value. The presence of changing risk preferences on the part of investors undermines current asset pricing models, which are based on standard preferences, and imply the traditional positive risk-return relationship. The future development of asset pricing models must, therefore, seek to incorporate the changing risk appetite of investors.  Entries are the slope coefficient (beta) and accompanying Newey-West t-statistic from the regression of returns on 2-year rolling standard deviation: rt = α + β sdt + εt as given in equation (1). The panel regression is given by equation (2), again the values in () are autocorrelation and heteroscedasticity adjusted t-statistics.             -.008 -.004 .000 .004 .008 .012 .016 .020 .024 .028 .032 .04 .06 .08 .10 .12 .14 .16