I.         Introduction

 

One of the well-established features of financial data is the fact that firms with high book-to-market ratios, (commonly referred to as value firms) tend to deliver higher returns than firms with low book-to-market ratios (or growth firms). Robustness of this finding, known as the value premium, suggests that value investment strategies are likely to entail higher exposure to fundamental risks and, consequently, have to provide investors with higher compensation. This value premium was first identified by Fama and French (1992), and their conclusion is based on evidence that there is common variation in the earnings and returns of distressed firms that is not explained by market earnings and returns. Although empirically robust across time and across markets, this explanation fails to provide a satisfactory economic intuition.

 

On the other hand, economic theory suggests that if investors have mean-variance efficient preference structures, then they should care both about market returns and about aggregate market volatility. For example, Campbell (1993) and Chen (2002) argue that risk-averse investors want to hedge against market risk, but at the same time against innovations in market volatility as well. This is because most investors are unwilling to lose wealth at times of high volatility, which often represents a deterioration in investment opportunities, and which usually coincides with periods of low consumption and recessions. Thus, if investors are averse not only to market risk, but also to innovations in aggregate volatility, then stocks whose returns correlate positively with innovations in market volatility should produce lower expected returns. In other words, stocks that have positive covariance between their returns and variables that correctly relate to innovations in aggregate volatility can be viewed as hedges against volatility risk, and demanded by risk-averse investors, pushing their prices up, which in turn implies lower average returns.

 

In the paper, we try to explain how returns on value and growth stocks react to volatility risk, and whether volatility risk can account for the value premium observed in the Korean stock market. To test the above hypotheses, we need to decide on a proxy that takes into account investors’ true measure of volatility risk. To do so, we rely on a measure from options market, and employ returns on at-the money straddles written on the KOSPI 200 index. There are several reasons for using at-the-money straddle returns as a proxy for volatility risk. First of all, straddles are volatility trades, and straddle returns are sensitive to innovations in the volatility of the underlying assets. This makes straddle returns a good choice for examining the effect of volatility risk. Secondly, options give us an important insight about their underlying assets. Prices formed in option markets are forward looking, and thus reflect important information about investors’ expectations on the price dynamics of the underlying assets. Last but not the least, options are tradable assets. Therefore, using straddle returns as a proxy for volatility risk helps us avoid the problem of mimicking portfolios, and helps better represent a dynamically managed portfolio that corresponds to investors’ true investment opportunity set.

 

Our findings are as follows. First, both value and growth stocks have significant volatility betas, while growth stocks have much higher positive volatility betas. We know that there exists a positive relationship between straddle returns and volatility; at-the-money straddle returns therefore are relatively higher (lower) at times of high (low) market volatility. If we assume that investors are averse to innovations in aggregate volatility, investors would find growth stocks more attractive at times of high market volatility, since growth stocks have higher positive volatility betas than value counterparts. Investors then demand more growth stocks in order to protect themselves against innovations in volatility. This ‘‘flight-to-quality” phenomenon explains why growth stocks are on average priced higher and have lower returns, and seen as less risky compared to their value counterparts. This in turn implies that the sensitivity of value and growth stocks to volatility risk is an important determinant of the value premium observed in Korean stocks. Second, we document significant time variation in the volatility betas of value and growth stocks. When markets are classified into expected booms and recessions, volatility betas of value and growth stocks change considerably. Third, a conditional version of ICAPM of Merton (1973), which allows market beta and volatility beta depend linearly on observable conditioning variables, performs better than unconditional CAPM. Improvement in Jensen’s alphas for conditional ICAPM confirms the hypotheses that investors care not only about market risk but also about volatility risk, and volatility risk factor is time-varying. Overall, volatility risk is an important risk factor that drives the difference in the returns of value and growth stocks in the Korean stock market. 

 

These findings are related to recent studies that explain value premium within a rational expectations framework.  For instance, the business cycle explanation of Lettau and Ludvigson (2001), and Petkova and Zhang (2005) argue that it is the time variation in the conditional betas of value and growth stocks in bad and good times of the economy that drives the value premium. In conditional CAPM and CCAPM settings, respectively, the authors find that value stocks have lower market consumption betas during bad times relative to growth stocks, and therefore conclude that value stocks are riskier than growth stocks. This paper is similar in spirit, but differs from theirs by testing the observed value premium using a two factor model within the ICAPM framework of Merton (1973). Similar to Campbell (1993), and Chen (2002), we assume that investors are averse both to market risk and to aggregate volatility risk, and argue that it is the difference in the sensitivity of stocks to aggregate volatility, which drives the observed value premium. Conditional CAPM cares only about market risk and ignores investors ’ hedging demands due to intertemporal changes in the investment opportunity set. In contrast, a two-factor ICAPM is expected to be a more fruitful framework in helping us capture not only the market risk component, but also investors’ hedging need component against deteriorations in their wealth due to innovations in aggregate volatility.

 

 

II.      Literature Review

 

Asset pricing literature has differing views on the empirically documented value premium. From a behavioral standpoint, Lakonishok et al. (1994), La Porta (1996), and La Porta et al. (1997) argue that the value premium in average returns arises because investors undervalue distressed stocks and overvalue growth stocks. When these pricing errors are corrected, value stocks have lower returns and growth stocks have higher returns. On the other hand, the proponents of rational asset pricing theory argue that the HML portfolio captures some macroeconomic distress or risk aversion factor. However, empirical studies thus far have not been able to point out what macroeconomic risk factors might be the source of this premium [1]. We argue that volatility risk is an important factor that investors take into account while pricing value and growth stocks in Korean stock market.

Campbell (1993) proposes a new way to generalize the insights of static asset pricing theory to a multi-period setting, by noting that intertemporal asset pricing theory has become unnecessarily tangled in complications caused by the nonlinearity of the intertemporal budget constraint. He uses a log-linear approximation to the budget constraint to substitute out consumption from a standard intertemporal asset pricing model. In a homoskedastic lognormal setting, the consumption-wealth ratio is shown to depend on the elasticity of intertemporal substitution in consumption, while asset risk premia are determined by the coefficient of relative risk aversion. Risk premia are then related to the covariances of asset returns with the market return and with news about the discounted value of all future market returns. An underlying mechanism here is that in the absence of frictions, the aggregate budget constraint restricts variations in market returns to affect aggregate consumption at some horizon. Hence, if a factor reflects the changes in the investment opportunity set, its risk premium should be linked to the amount of information that it conveys about the future. Furthermore, the risk premia across factors should be linked to each other through the willingness of investors to bear risk.

 

Chen (2002) notes that one possible explanation for the historically high returns found in the cross-section of stock returns is that these findings reflect compensation for exposures to adverse changes in the investment opportunity set, that is to say, an asset may earn a risk premium if it performs poorly when the prospects for the future turn sour. He then investigates this idea in terms of Campbell’s (1993) model. In so doing, Chen examines whether the historically high returns associated with the size effect, the book-to-market effect, and the momentum effect can be explained within an asset pricing framework suggested by Merton’s (1973) ICAPM, where the expected return and the volatility of the market are time-varying, and the pricing kernel depends upon the consumption growth rate and the aggregate market return. His model is then estimated using a multivariate VAR-GARCH model with non-Gaussian innovations, and yet the estimates suggest that the historical returns on the book-to-market effect and the momentum effect are too high to be explained as compensation for exposures to adverse changes in the investment opportunity set. However, he acknowledges that these results are only as valid as the assumptions of the model.

 

Lettau and Ludvigson (2001) employ an empirical test of the (C)CAPM in which the discount factor is approximated as a linear function of the model's fundamental factors. Instead of assuming that the parameters of this function are fixed over time, as in many previous studies, they model the parameters as time-varying by scaling them with a proxy for the log consumption-wealth ratio. In contrast to the simple static CAPM or unconditional consumption CAPM, they find that these scaled mul-tifactor versions of the CCAPM can explain a substantial fraction of the cross-sectional variation in average returns on stock portfolios sorted according to size and book-to-market equity ratios. This scaled consumption CAPM does a good job of explaining the celebrated value premium: portfolios with high book-to-market equity ratios also have returns that are more highly correlated with the scaled consumption factors they consider, and vice versa. Furthermore, the scaled consumption model eliminates residual size and book-to-market effects that remain in the CAPM. Taken together, these findings lend support to the view that the value premium is at least partially attributable to the greater non-diversifiable risk of high book-to-market portfolios, and not simply to elements bearing no relation to risk, such as firm characteristics or sample selection biases. Their results also help shed light on why the Fama-French three-factor model performs so well relative to the unsealed size: the data suggest that the Fama-French factors are mimicking portfolios for risk factors associated with time variation in risk premia. Once the (C)CAPM is modified to account for such time variation, it performs about as well as the Fama-French model in explaining the cross-sectional variation in average returns.

 

Petcova and Zhang (2005) study the relative risk of value and growth stocks. They find that time-varying risk goes in the right direction in explaining the value premium. Value betas tend to covary positively, and growth betas tend to covary negatively with the expected market risk premium.  As a result, value-minus-growth betas tend to covary positively with the expected market risk premium. Their inference differs from that of previous studies because they sort betas on the expected market risk premium, instead of on the realized market excess return. However, they find that the positive covariance between the value-minus-growth betas and the expected market risk premium is far too small to explain the observed magnitude of the value premium within the conditional CAPM. Nonetheless, they basically argue that that it is the time variation in the conditional betas of value and growth stocks in bad and good times of the economy that drives the value premium.

 

Barnov (2008) presents a simple real options model that explains why in cross-section high idiosyncratic volatility implies low future returns and why the value effect is stronger for high volatility firms. In the model, high idiosyncratic volatility makes growth options a hedge against aggregate volatility risk. Growth options become less sensitive to the underlying asset value as idiosyncratic volatility goes up. It cuts their betas and saves them from losses in volatile times that are usually recessions. Growth options value also positively depends on volatility. It makes them a natural hedge against volatility increases. In empirical tests, the aggregate volatility risk factor explains the idiosyncratic volatility discount and why it is stronger for growth firms. The aggregate volatility risk factor also partly explains the stronger value effect for high volatility firms. He also finds that high volatility and growth firms have much lower betas in recessions than in booms. He further shows that the aggregate volatility risk factor (the BVIX factor) explains the well-known underperformance of small growth firms.

 

Corradi et al. (2008) develop a no-arbitrage model in which stock market volatility is explicitly related to a number of macroeconomic and unobservable factors. They show that stock volatility is linked to these factors through no-arbitrage restrictions, and under fairly standard conditions on the dynamics of the factors and risk aversion corrections, their model is solved in closed-form, and is amenable to empirical work. They further use the model to quantitatively assess how volatility and volatility-related risk-premia change in response to business cycle conditions.

 

Arisoy (2010) documents that systematic volatility risk is an important factor that drives the value premium observed in the French stock market. He documents significant differences between volatility factor loadings of value and growth stocks.  This paper is similar to Arisoy (2010) in that this paper also claims that systematic volatility risk is an important factor that drives the value premium, by using returns on at-the-money straddles as a proxy for systematic volatility risk. While Arisoy (2010) focusing on French stock market, however, we investigate Korean stock market, and see if we could document statistically significant estimates of volatility factor loadings of value and growth stocks with Korean stock market data. Furthermore, we examine whether the proposed two-factor ICAPM performs better in explaining the value premium, and whether the volatility beta for the HML strategy is negative and statistically significant in terms of Korean stock market data. The rest of the paper is organized as follows. Section 3 shows data and methodology used to test the hypothesis of whether volatility risk can explain the differences in returns on value and growth stocks in the Korean stock market. Section 4reports the associated empirical findings, and details the tests for time variation in market and volatility factor loadings. The final section offers concluding remarks.

 

 

III.    Data and Methodology

 

1. Data

 

The data covers the period January 2004 to December 2011. The options and stocks data were obtained from KRX, and all the financial information of the listed firms from KIS-Value II. Portfolio returns for value and growth stocks (denoted by BMH and BML, respectively) were formed in the same manner as in Fama and French (1993). For the risk-free rate, we used the rate on one year government bond, which was converted on the monthly basis and announced by the bank of Korea. In calculating the expected market premium in the context of Petkova and Zhang (2005), we used the dividend yield of the KOSPI 200 index obtained from KRX, the 3-month CD rate, and the difference between  the yields of 10-year and 1-year Korean government bonds, both of which are obtained from the bank of Korea.

 

2. Methodology

 

Straddles are volatility trades, and at-the-money straddle returns are known very sensitive to levels of volatility. When there is large fluctuation in the prices of the constituents of the KOSPI 200 index, the returns on at-the-money straddles are supposed to large, and when the market experiences relatively low levels of volatility, the buyers of at-the-money straddles are supposed to have lower returns. Due to their sensitivity to the level of volatility, returns on at-the-money straddles seem to be a good choice for studying the effect of volatility risk on portfolio returns.

 

The method to compute daily at-the-money straddle returns is as follows. First, options that significantly violate arbitrage-pricing bounds are eliminated. Then, options that expire during the following calendar month are identified. The reason for choosing options that expire the next calendar month is that they are the most liquid data among various maturities. Options that expire within 7 days are also excluded from the sample because they show large deviations in trading volumes, which shows doubt on the reliability of their pricing associated with increased volatility. [2]– exercise price) between -1 and +1, as at-the-money options. [3] The return on an at-the-money straddle is then the equally weighted average of the return on an at-the-money call and at-the-money put computed as above. Finally, daily at-the-money straddle returns are cumulated to monthly returns, which form the basis of the empirical tests.  Next, each option is checked whether it is traded the next trading day or not. If no option is found in the nearest expiry contracts, then options in the second-nearest expiry contracts are used. To calculate the daily return of an option, raw net returns are used. Once daily call and put returns are calculated, we classify options with money-ness level (=underlying asset price

 

Table 1 presents the monthly average returns on three test portfolios (BMH, BML, and HML), the market portfolio, and at-the-money straddles. During the same period, KOSPI 200 index earned 1.123% per month on average. Looking at portfolio returns, one can see that value stock portfolios, denoted by BMH, outperform growth stock portfolios, denoted by BML, in terms of higher average return, 1.775%, and lower standard deviation, 4.964%. HML is the portfolio strategy where it is composed of a long position in the value stock portfolio (BMH) and a short position in the growth stock portfolio (BML). [4] Table 1 shows that HML has the monthly average return=0.863%, and monthly standard deviation=3.439%. On the other hand, at-the-money straddles written on the KOSPI 200 index didn’t lose on average, and instead made 45.595% per month on average for the sample period studied, while their volatility jumping up to 191.569%. [5]

 

<Table 1> Descriptive Statistics

This table shows descriptive statistics of the monthly average returns on three test portfolios (BMH, BML, and HML), the market portfolio, and at-the-money straddles for the period from January 2004 to December 2011. BMH, BML, and HML are the value stocks portfolio that has high book to market ratios, the growth stocks portfolio that has low book to