# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

####
Need to know

- The authors propose LMS, a risk factor that connects between long and short risk horizons based on the VIX.
- LMS is able to predict future performance of equity portfolios.
- LMS contributes to traditional risk factors in terms of significance and Adj. R2.
- The results are robust for portfolios sorted by size, B/M and industry.

####
Abstract

The Chicago Board Options Exchange (CBOE) Volatility Index (VIX) and VIX-based risk measures, such as the variance risk premium, are predictors for future portfolio returns. In addition to the current level of the squared VIX and its changes over past periods (as previously identified by Banerjee and coworkers), the differential of expectations about future risk as indicated by the difference between long-horizon and short-horizon VIX indexes is another relevant pricing factor. We use the difference between the CBOE Standard & Poor’s 500 six-month and nine-day volatility indexes (the long-minus-short implied volatility measure) to proxy this risk differential. This measure captures how risk is expected to evolve and is thus directly related to the VIX term structure. Including it in a predictive portfolio regression model improves the fit by approximately two basis points in terms of the adjusted R2. The results hold for various portfolio sortings (industry, book-to-market, size) and over different sample periods. Overall, the results support the notion that volatility risk has multiple facets that are priced individually.

####
Introduction

## 1 Introduction

Volatility is a key factor in portfolio theory, options valuation and asset pricing models. In addition to the literature that relies on historical ex post volatility estimates based on realized volatility or generalized autoregressive conditional heteroscedasticity models (Ghysels et al 2005; Ludvigson and Ng 2007), there is a growing literature devoted to forward-looking, implied market volatility. While several proxies are available, the Chicago Board Options Exchange Volatility Index (CBOE VIX) is the main tool used to depict expectations about future volatility and market sentiment. It is frequently used in empirical work (see, for example, Bloom, 2009; Fleming, 1998; Huang et al, 2019a), particularly to describe both contemporaneous and predictive relationships between implied volatility (IV) and portfolio returns (Banerjee et al 2007; Bekaert and Hoerova 2014; Giot 2005; Rubbaniy et al 2014; Osterrieder et al 2019). Recently, the CBOE introduced volatility indexes that consider implied volatility between nine days and one year ahead (Chicago Board Options Exchange 2019). As these mirror the expected volatility for different look-ahead periods, they may jointly provide deeper insights into investors’ beliefs regarding future volatility.

In this paper, we construct a long-minus-short (LMS) implied volatility measure, which expresses the relationship between the IVs of long and short horizons. This measure is easily computed from publicly available data. By the use of the LMS measure, we may gauge the information provided by the VIX term structure with a single variable (which is easier to use than the option-based principal component proposed by Johnson (2017)) to improve the predictability of portfolio returns beyond the use of the VIX only. The economic rationale for this is provided by the model of Banerjee et al (2007), which links implied variance to stock prices. Based on Heston’s (1993) model of stock prices, Banerjee et al (2007) show that future excess returns can be explained by the levels of, and changes in, implied volatility today and in the future. When estimating their model, Banerjee et al (2007) had to set the relation to future IV to zero due to a lack of data. We propose to augment their model by using the LMS variable as a proxy for the expected change in IV. We follow their framework and test whether the LMS measure improves the prediction of excess returns of various portfolios over different horizons, controlling for market excess returns, the Fama and French (1993) risk factors (small-minus-big (SMB) and high-minus-low (HML)) and the Carhart (1997) momentum factor (MOM).

While the literature investigating the relationship between volatility and returns is large, it is usually confined to the contemporaneous link between realized volatility and the market risk premium. The equally important question of whether there is a predictive relationship between IV and future returns is less frequently studied. Banerjee et al (2007) state that the VIX possesses information content that is useful for predicting future market returns. Johnson (2017) finds that the second principal component, associated with the slope of the VIX term structure, summarizes nearly all the information about variance risk premiums. He shows that the slope of the VIX terms structure is an economically significant predictor of the variance of asset returns. By combining the two findings, our LMS measure contributes to the literature by offering an extension to the model proposed by Banerjee et al (2007) by including an important ex-ante factor that mirrors the relationship between short- and long-term VIX levels. We apply the slope idea described by Johnson as an informational tool for the prediction of future returns. Analogously to the yield curve, we suggest that the LMS measure, as a proxy for the slope of the VIX term structure, may reveal additional information regarding future returns compared with the spot VIX (either levels or past changes).

In addition, we contribute to the group of studies which focus on the predictive ability of the VIX yield curve obtained from futures on the VIX (see, for example, Lu and Zhu, 2010; Huskaj and Nossman, 2012; Mixon and Onur, 2019). However, the literature is still debating the relationship between spot and future prices. Liu (2014) argues that futures do not track the VIX perfectly due to the distinct characteristics of the futures market. For instance, it is argued that a shock in the spot market is usually muted in the futures market, and the VIX futures market is no exception. As a result, the beta of the VIX futures tracks only a portion of the VIX spot movement.^{1}^{1} 1 According to Liu (2014), the Standard & Poor’s 500 VIX Short-Term Futures Index, for example, has a beta of 48.78% with the VIX spot. In this spirit, Nossman and Wilhelmsson (2009), for example, test whether VIX futures have the capability to predict the VIX spot. They show that when the futures price is not adjusted with the risk premium the expectation hypothesis is rejected. On the other hand, Fassas and Siriopoulos (2011) conclude that VIX futures can be considered as unbiased and efficient estimators of the relevant spot VIX levels.

In contrast, we offer a spot-based (rather than futures-based) measure, which can easily be constructed from publicly available data. It thus mitigates potential idiosyncratic issues concerning the futures market and its relationship to the spot market. Still, our LMS measure achieves a connection between spot VIX prices from different horizons that reflects the differential of risk. Lastly, VIX futures are, as the name suggests, priced on the VIX (ie, related to the future expectations for a single horizon of 30 days), and do not consider different horizons. We thus argue that the proposed LMS measure, being a spot-based measure, avoids loss of information, as it captures the expectations today for future volatility in short and long periods.

In our empirical analysis, the LMS measure turns out to explain up to 2% of the market variation. The fit is best for long horizons. Regarding future portfolio returns, the LMS measure can serve as a factor in addition to the level and change in IV identified by Banerjee et al (2007), again particularly for long forecast horizons (in our case 90 days). This result holds for various portfolios (size, book-to-market, double-sorted and industry). In models that account for market excess returns and the Fama–French and Carhart risk factors, including the LMS measure can improve the adjusted ${R}^{2}$ by up to two percentage points (pp), but the effect is reduced as the traditional factors in our application are already able to explain the greatest part of future portfolio return variation. The effect also depends on the portfolios considered, the forecast horizon and the sample. Nevertheless, the documented range of ${R}^{2}$ improvement is similar to that of Carr and Wu (2016) or Wang (2019) for comparable horizons, while the size of the effect exceeds the results documented by Rubbaniy et al (2014). Overall, the explanatory power of the LMS measure is greater for longer look-ahead periods.

## 2 Literature review

The literature regards the VIX as an indicator of market and macroeconomic risk (Bloom 2009) or panic and fear (Whaley 2000) as well as market sentiment (Qadan and Aharon 2019b). Prior research has shown that the VIX provides nearly all of the relevant information and has the potential to reflect information that a model-based volatility forecast cannot convey (Fleming 1998; Blair et al 2001; Jiang and Tian 2005). Hence, the VIX has emerged as an important and useful tool for evaluating the degree of optimism or pessimism in the market and for predicting stocks’ and market performance.

Most studies focus on the VIX itself, which is based on the volatility for the next 30 days (VIX1M) in the context of explaining and predicting stock returns (Copeland and Copeland 1999; Giot 2005; Bekaert and Hoerova 2014; Smales 2017; Yun 2020), stock return volatility (Corrado and Miller 2005; Wang 2019), commodities (Jubinski and Lipton 2013; Qadan and Yagil 2012), exchange rates (Lu et al 2017; Daigler et al 2014) and bonds (Adrian et al 2019). Other volatility indexes that are calculated with different option maturities (for example, the recently introduced one-year volatility index (VIX1Y)) are still rarely used (Huang et al 2019b). In contrast, VIX-related measures such as the volatility of the VIX (VVIX) are readily employed. For example, Bu et al (2019) show that the VVIX may predict future stock returns in the US market. Huang et al (2019a) investigate how VIX and VVIX are related.

Another branch of studies decomposes volatility or the VIX and tests the predictive ability of the extracted variance risk premium (VRP). According to Bollerslev et al (2009), the VRP can be estimated as the difference between the ex ante risk-neutral expectation of future volatility over the time interval $[t,t+1]$ (proxied by the VIX) and the ex post realized volatility over the time interval $[t-1,t]$ (proxied by a realized variance measure). Using the VIX over the years 1990–2007, Bollerslev et al find that the VRP is a significant predictor for stock returns, especially on a quarterly basis. Using a similar decomposition, albeit with a different proxy for the expected volatility, Bekaert and Hoerova (2014) examine the predictive power of the VRP for stock market returns between 1990 and 2010. They use three different horizons for their predictions (monthly, quarterly and annually) and find that the variance premium proxy predicts stock market returns at all three horizons, with the predictive power being strongest at the quarterly horizon.

As mentioned in Section 1, the studies that are closest to ours include Copeland and Copeland (1999), Giot (2005), Banerjee et al (2007), Rubbaniy et al (2014) and Smales (2017). All of these studies consider the direct predictive ability of the IV on stock returns. The common feature in these studies is the employment of VIX as their main proxy for IV. Copeland and Copeland (1999) show that changes in the VIX between 1994 and 1997 are statistically significant leading indicators of daily market returns over the following periods ranging from 1 day to 20 days. They show that on days that follow increases (decreases) in the VIX, portfolios of large stocks outperform (underperform) portfolios of small stocks and that value-based portfolios outperform growth-based portfolios.

Banerjee et al (2007) investigate the relationship between future returns over 30 days and 60 days and current IV levels and innovations of the familiar VIX. They use portfolios sorted by book-to-market ratio, size and beta and show that returns are predictable even after controlling for the market and the SMB, HML and winners-minus-losers (UMD) factors. They show that, although the relationship with VIX levels and innovations is significant, this relationship is more pronounced for high beta portfolios and for 60-day time periods. Giot (2005) finds that for the Standard & Poor’s 100 (S&P 100) and Nasdaq 100 indexes there is a contemporaneous negative and significant relationship between the returns of the stock and IV indexes (the VIX and the CBOE Nasdaq Market Volatility Index VXN). He also tests a possible predictive (1-, 5-, 20- and 60-day) relationship in three subperiods representing different market states. According to Giot, the evidence is hardly conclusive, with weak evidence that positive (negative) forward-looking returns are to be expected for long positions triggered by extremely high (low) levels of the IV indexes.

Rubbaniy et al (2014) test the degree to which the VIX, VXN and VDAX predict their underlying index returns.^{2}^{2} 2 VDAX expresses the implied volatility of the Deutscher Aktienindex (DAX). Their findings suggest that VIX, VXN and VDAX are capable of predicting the following 20 days’ and 60 days’ returns but are found to be insignificant for shorter periods such as the next day’s and the next 5 days’ returns. Rubbaniy et al (2014) also extend their analysis to portfolios sorted by different sectors and find that the VIX is the best predictor for portfolios of different sectors. Similarly to Banerjee et al (2007), they also test whether the VIX, VXN and VDAX predict the returns of beta-, size- and book-to-market-sorted portfolios. They document a stronger relationship between the VIX and future returns on portfolios with higher beta stocks and a general predictive ability for the 20-day and 60-day time periods. They also find that the ability of IV indexes to forecast future returns weakens slightly when other risk factors are incorporated.

## 3 Methodology

### 3.1 A VIX-based measure for relative risk

The VIX indexes mirror the IV over a certain period of time. In general, a position that is open longer entails a greater exposure to risk. Hence, the CBOE S&P 500 six-month volatility index (VIX6M) should express a higher level of uncertainty than the CBOE S&P 500 nine-day volatility index (VIX9D) if all other market conditions are expected to be stable for (at least) six months. This pattern reflects a positive risk premium that compensates investors who are exposed to more risk over longer periods. Consequently, the shape of a curve constructed from volatility indexes for different time horizons will be upward sloping with time, a situation known as contango (ie, a reflection of increased uncertainty over longer time horizons). This rationale is rooted in the commodities theory of storage by Kaldor (1939), Working (1949) and Brennan (1958): longer-dated options are generally more expensive than shorter-term ones.

In times of high uncertainty and fear, such as the Covid-19 pandemic (ongoing at the time of writing) or the 2008 subprime crisis, expectations regarding increased volatility change sharply, and shorter options become more expensive than longer ones. This is due in part to the desire of investors to hedge their investments, but it might also reflect the increased panic and fear of investors. Thus, the shorter VIX indexes may react more intensively to bad news than their longer counterparts. In such situations, shorter IV indexes are higher and reflect a state known as backwardation. Thus, shifts from contango to backwardation in particular might be a signal for future equity performance. Therefore, the difference between far and near volatility expectations as captured by the LMS measure may reveal information about future returns in addition to the VIX levels or changes in VIX levels that consider exactly the expectation about volatility in the next 30 days.

Figure 1 shows the shape of the implied VIX term structure based on smoothed, daily estimates.^{3}^{3} 3 First, the different VIX indexes are regressed on linear, squared and cubic time variables on a daily basis and subsequently the term structure is predicted along the $x$-axis per day. Second, the daily estimates are smoothed using a generalized additive model with tensor product splines (Wood 2017). As can be seen, the plane increases in general with the time horizon of the VIX. Over time, we observe a high risk period after the 2009–10 euro crisis, followed by a low-risk period from 2014 to 2019; then the plane starts to increase again at the onset of the Covid-19 pandemic in early 2020.

To make the state of the market tractable in a single variable, we propose to use the difference between the short-term (nine-day) and the long-term (six-month) volatility indexes. Our LMS (ie, long minus short VIX horizon) measure is calculated as

$$\mathrm{LMS}=\mathrm{VIX6M}-\mathrm{VIX9D}.$$ | (3.1) |

We use the VIX6M instead of the VIX1Y, as the CBOE provides the VIX1Y only since 2018. The VIX6M thus provides more observations for our empirical analysis. As the lower bound we chose the VIX9D (instead of the VIX, which has a 30-day horizon) in order to be able to capture extreme states of contango and backwardation. The closer the maturities of the two VIX indexes are to each other, the less informative the LMS measure should be, as the differential should be smaller, by construction. Thus, VIX indexes at the very short and the very long end are desirable.

### 3.2 Modeling strategy

According to Banerjee et al (2007), the VIX is useful to predict future portfolio returns. They rely on the Heston (1993) model to show how IV and stock prices are related. Ultimately, they find that returns from time $t$ to time $t+\tau $ depend on the level of IV at time $t$, the change in IV at time $t$ and the level of IV at time $t+\tau $. This leads to their testable equation (Banerjee et al, 2007, Equation (15))

$${r}_{t,t+\tau}=\alpha +{\beta}_{1}{\sigma}_{\text{IV},t}^{2}+{\beta}_{2}\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}+{\beta}_{3}{\sigma}_{\text{IV},t+\tau}^{2}+{\nu}_{t},$$ | (3.2) |

where implied volatility ${\sigma}_{\text{IV},t}^{2}$ and changes in implied volatility $\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$ are proxied by the squared VIX and changes in the squared VIX, respectively. The residual ${\nu}_{t}$ is mean zero with time-varying variance. However, ${\sigma}_{\text{IV},t+\tau}^{2}$ is not observable, and Banerjee et al drop this term from the predictive equation (Banerjee et al, 2007, p. 3188).

To fill this gap, we argue that the LMS measure provides a market expectation of how the current ${\sigma}_{\text{IV},t}^{2}$ and future ${\sigma}_{\text{IV},t+\tau}^{2}$ implied volatilities are related. We augment (3.2) by our LMS measure to account for expectations about the future IV level. Whether the market is currently in contango or backwardation should provide an idea about the development of IV over the forecasting period, and therefore the LMS measure may reveal information not captured in the squared VIX or squared VIX changes available at time $t$.

To investigate whether the LMS measure provides additional forecasting power compared with the strategy in Banerjee et al (2007), we compare their basic models with an LMS-augmented model. First, we estimate two models that are directly related to (3.2):

${R}_{t,t+\tau}^{p}$ | $={\alpha}_{p}+{\beta}_{1,p}{\sigma}_{\text{IV},t}^{2}+{\beta}_{2,p}\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}+{\epsilon}_{1,t},$ | (3.3) | ||

${R}_{t,t+\tau}^{p}$ | $={\alpha}_{p}+{\beta}_{1,p}{\sigma}_{\text{IV},t}^{2}+{\beta}_{2,p}\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}+{\beta}_{3,p}{\mathrm{LMS}}_{t}+{\epsilon}_{2,t},$ | (3.4) |

where ${R}_{t,t+\tau}^{p}$ is the future portfolio return over the interval $[t,t+\tau ]$, $\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$ is the difference in the squared VIX from $t-\tau $ to $t$, and ${\mathrm{LMS}}_{t}$ and ${\sigma}_{\text{IV},t}^{2}$ are observed at $t$. Following Banerjee et al (2007), we use heteroscedasticity and autocorrelation consistent (HAC) standard errors (see Section 3.3 for details) and consider $\tau \in \{30,60,90\}$ days as forecasting horizons. To compare the models, we use the adjusted ${R}^{2}$, as we have the same dependent variable in (3.3) and (3.4). To test whether the model in (3.4) is better able to explain the variation in portfolio returns than the model in (3.3), we conduct an $F$-test. The implicit null hypothesis is therefore that the inclusion of the LMS measure does not lead to a better model fit.

Banerjee et al (2007) extend the model in (3.3) by the risk factors of Fama and French (1993) and the momentum factor of Carhart (1997). Thus, the extended specifications are

${R}_{t,t+\tau}^{p}$ | $={\alpha}_{p}+{\beta}_{1,p}{\sigma}_{\text{IV},t}^{2}+{\beta}_{2,p}\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}+{\beta}_{4,p}{\mathrm{MKT}}_{t,t+\tau}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{\beta}_{5,p}{\mathrm{HML}}_{t,t+\tau}+{\beta}_{6,p}{\mathrm{SMB}}_{t,t+\tau}+{\beta}_{7,p}{\mathrm{MOM}}_{t,t+\tau}+{\epsilon}_{3,t},$ | (3.5) | |||

${R}_{t,t+\tau}^{p}$ | $={\alpha}_{p}+{\beta}_{1,p}{\sigma}_{\text{IV},t}^{2}+{\beta}_{2,p}\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}+{\beta}_{3,p}{\mathrm{LMS}}_{t}+{\beta}_{4,p}{\mathrm{MKT}}_{t,t+\tau}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{\beta}_{5,p}{\mathrm{HML}}_{t,t+\tau}+{\beta}_{6,p}{\mathrm{SMB}}_{t,t+\tau}+{\beta}_{7,p}{\mathrm{MOM}}_{t,t+\tau}+{\epsilon}_{4,t},$ | (3.6) |

where the factors ${\mathrm{MKT}}_{t,t+\tau}$, ${\mathrm{HML}}_{t,t+\tau}$, ${\mathrm{SMB}}_{t,t+\tau}$ and ${\mathrm{MOM}}_{t,t+\tau}$ are calculated over the interval $[t,t+\tau ]$, which corresponds to the portfolio forecast horizon on the left-hand sides of the equations.

Equations (3.3) and (3.4) are the base models, which are intended to capture the roles that the level and innovations in VIX play in forecasting future returns, and the function of our LMS measure in particular. The main weakness of the base models is that they are restricted to the inclusion solely of the key variables of interest and may be biased as an outcome of an interaction or some degree of correlation to an omitted variable. Equations (3.5) and (3.6) instead contain the common risk factors in addition to the key variables in (3.3) and (3.4) in an attempt to minimize the potential bias of omitted variables. This will of course have an impact on the parameter estimates, and it is possible that the signs of the estimates change between models.

While the switching of signs between the base and augmented models is more likely to be an omitted-variable problem, the switching of the signs of the LMS measure estimate in the augmented models may be a signal for different information that the LMS measure transmits for different types of portfolios. In other words, the coefficients for small- versus large-cap stocks portfolios may not necessarily carry the same sign. In fact, the sign of the LMS parameter estimate may deliver different signals determined by the characteristic of the securities that a particular portfolio includes. In this spirit, Copeland and Copeland (1999, 2016) find evidence favoring small stocks during settled times, and large stocks under turbulent periods. In fact, both studies show that the well-known size effect holds when VIX declines, and when VIX increases the small size effect turns into a large stocks size effect. With this line of thinking it is plausible that the LMS coefficient will be different in its direction (sign) for small versus large stocks portfolios and value versus growth and will potentially be different for industries portfolios.

### 3.3 Details on estimation procedure

The model is estimated using ordinary least squares in R software (R Core Team 2021). As in Banerjee et al (2007), ${\sigma}_{\text{IV},t}^{2}$ is approximated by the squared VIX and $\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$ is the log-difference of the squared VIX. For all regressions, we report heteroscedasticity and autocorrelation robust standard errors. We rely on the estimator proposed by Newey and West (1987). The lag length is chosen automatically according to the procedure described in Newey and West (1994) and implemented in R by Zeileis (2006). Note that in general the lag length so determined is more conservative than the look-ahead period over which portfolio returns are calculated. We therefore set the lag length equal to $\tau $ as a robustness check but found no qualitative differences regarding parameter significance.

When it comes to estimating (3.3)–(3.6), Banerjee et al (2007) highlight a multicollinearity problem between squared VIX levels and changes. In our situation, these variables and the LMS measure are also highly correlated. Therefore, we follow Banerjee et al (2007) and separate the effects using two regressions:

${\sigma}_{\text{IV},t}^{2}$ | $=a+b\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}+c{\mathrm{LMS}}_{t}+{\zeta}_{t},$ | (3.7) | ||

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$ | $=a+b{\mathrm{LMS}}_{t}+{\xi}_{t}.$ | (3.8) |

Subsequently, we extract the residuals ${\widehat{\zeta}}_{t}$ as a representation of the squared VIX levels that are orthogonal to the squared VIX changes and the LMS measure. This new level variable replaces ${\sigma}_{\text{IV},t}^{2}$ when estimating (3.3)–(3.6). Similarly, we use the residuals ${\widehat{\xi}}_{t}$ as a representation of $\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$, which is now uncorrelated with the two other variables for estimation.

To obtain confidence intervals for the ${R}^{2}$ we rely on a bootstrap method. Our data are time series in nature, even though we do not estimate a time series model. We therefore use the maximum entropy bootstrap for time series proposed and refined by Vinod (2004, 2006). This allows us to construct 999 pseudo samples of our market and portfolio time series before all estimations and use the same samples for all necessary bootstraps later on. The maximum entropy bootstrap has favorable properties compared with a time series block bootstrap: it avoids the need to determine an optimal block length, and the bootstrapped time series have, by construction, the same length as the original time series. The reported confidence intervals are then the 2.5% and 97.5% quantiles of the bootstrap distribution of the adjusted ${R}^{2}$ values.

## 4 Data

For our analysis, we retrieve daily market and portfolio excess returns from Kenneth French’s website,^{4}^{4} 4 URL: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. along with the risk factors SMB, HML and MOM. The data, provided as percentages, are converted to basis points (bps). Daily VIX time series are obtained from the CBOE website^{5}^{5} 5 URL: http://www.cboe.com/. and rescaled to the interval $[0,1]$ by dividing by 100. The data set ranges from March 1, 2011 to January 31, 2020 and contains 2285 observations. The period is limited by the availability of VIX9D, which is provided from March 1, 2011 onward.

Figure 2 presents time series plots of the VIX indexes used to calculate the LMS measure (part (a)) and the LMS itself (part (b)). The normal, contango state is the most prominent. In effect, there are only 249 days (11% of the sample) when the LMS measure is negative. In general, such instances are related to an immediate surge in uncertainty in the short run, while investors seem to be relatively more optimistic about the longer-term future.

The four dates where the backwardation state is most prominent are August 8 and August 10, 2011, August 24, 2015 and February 5, 2018. The period from August 6, 2011 to August 10, 2011 was marked by the surprise news on August 6, 2011 (a Saturday) by the rating agency Standard & Poor’s that the United States would not be AAA rated anymore, which led to a strong stock market downturn the following Monday. In addition, the European sovereign debt crisis was at its peak and the European Central Bank had started to buy Italian and Spanish government bonds. August 24, 2015 was marked by a market crash, with the Dow Jones Industrial Average Index (DJIA) losing more than 6% after the opening of the US stock market, following bad economic news about China. It was the worst opening of the DJIA ever to have occurred at that time. Lastly, February 5, 2018 (another Monday), the DJIA again lost more than 6% as investors feared rising interest rates coupled with a still weak economic environment. Overall, these instances are examples where the immediate consequences were far less predictable then the long run, such that the immediate risk was perceived to be much greater than the long-run risk, leading to the backwardation state of the VIX term structure.

Table 1(a) presents selected summary statistics. The LMS measure is on average positive, in line with Figure 2. It is left skewed and exhibits heavy tails. Similarly, the risk-free market returns, MKT, and the momentum factor, MOM, are also left skewed. In contrast, the factors HML and SMB, as well as IV and changes in IV, are right skewed. All factors exhibit excess kurtosis. As all of our models require stationary data, we use augmented Dickey–Fuller tests to check this prerequisite. The null hypothesis that the data are generated by a unit root process is rejected at any significance level for all time series.

(a) Distributional characteristics | |||||||

Variable | Mean | Median | Max | Min | SD | Skew | Kurt |

LMS | 0.037 | 0.044 | 0.102 | $-$0.348 | 0.036 | $-$2.863 | 20.793 |

MKT | 0.001 | 0.001 | 0.051 | $-$0.070 | 0.009 | $-$0.476 | 7.801 |

HML | $-$0.000 | $-$0.000 | 0.031 | $-$0.019 | 0.005 | 0.434 | 5.114 |

SMB | $-$0.000 | $-$0.000 | 0.036 | $-$0.019 | 0.005 | 0.214 | 4.747 |

MOM | 0.000 | 0.001 | 0.036 | $-$0.038 | 0.007 | $-$0.342 | 4.991 |

${\sigma}_{\text{IV}}^{\text{2}}$ | 0.029 | 0.022 | 0.230 | 0.008 | 0.024 | 3.433 | 18.239 |

$\mathrm{\Delta}{\sigma}_{\text{IV}}^{\text{2}}$ | 0.000 | $-$0.010 | 1.536 | $-$0.628 | 0.156 | 1.125 | 9.908 |

(b) Correlations | |||||||
---|---|---|---|---|---|---|---|

LMS | MKT | HML | SMB | MOM | ${\bm{\sigma}}_{\text{\mathbf{I}\mathbf{V}}}^{\text{\U0001d7d0}}$ | $\mathbf{\Delta}{\bm{\sigma}}_{\text{\mathbf{I}\mathbf{V}}}^{\text{\U0001d7d0}}$ | |

LMS | 1.000 | ||||||

MKT | 0.302${}^{***}$ | 1.000 | |||||

HML | 0.032 | 0.019 | 1.000 | ||||

SMB | 0.055${}^{***}$ | 0.298${}^{***}$ | $-$0.087${}^{***}$ | 1.000 | |||

MOM | $-$0.030 | $-$0.074${}^{***}$ | $-$0.453${}^{***}$ | $-$0.119${}^{***}$ | 1.000 | ||

${\bm{\sigma}}_{\text{\mathbf{I}\mathbf{V}}}^{\text{\U0001d7d0}}$ | $-$0.643${}^{***}$ | $-$0.183${}^{***}$ | $-$0.021 | $-$0.023 | 0.008 | 1.000 | |

$\mathbf{\Delta}{\bm{\sigma}}_{\text{\mathbf{I}\mathbf{V}}}^{\text{\U0001d7d0}}$ | $-$0.305${}^{***}$ | $-$0.804${}^{***}$ | 0.008 | $-$0.186${}^{***}$ | $-$0.021 | 0.141${}^{***}$ | 1.000 |

Table 1(b) shows the correlations between the variables. It is interesting to note that LMS and the level VIX measure ${\sigma}_{\text{IV}}^{2}$ are strongly negatively correlated. Hence, a higher level of the VIX (ie, higher risk) is associated with a lower discrepancy in the risk measures over their respective time horizons. The correlation of LMS and the change in IV (over one day only) is also negative, but not as strong as for ${\sigma}_{\text{IV}}^{2}$. In contrast, the correlation of $\mathrm{\Delta}{\sigma}_{\text{IV}}^{2}$ with the market, MKT, is much stronger. The correlation between the LMS measure and the market return is stronger than between ${\sigma}_{\text{IV}}^{2}$ and the market, which gives rise to the assumption that the LMS measure can be a good predictor of market returns, as the VIX has been shown by Rubbaniy et al (2014) to fulfill this role. The correlation between the LMS measure and any of the other factors is weak and only statistically significant for SMB. While not perfectly linearly dependent, the correlation between our core variables (LMS, ${\sigma}_{\text{IV}}^{2}$, $\mathrm{\Delta}{\sigma}_{\text{IV}}^{2}$) is high. This result motivates the need to orthogonalize the variables, as outlined in Section 3.3.

## 5 Empirical results

### 5.1 LMS measure and the market

30 days | 60 days | 90 days | |||||||

(0) | (1) | (2) | (0) | (1) | (2) | (0) | (1) | (2) | |

LMS | $-$0.072 | $-$0.063 | $-$0.168${}^{*}$ | $-$0.132${}^{*}$ | $-$0.249${}^{**}$ | $-$0.244${}^{**}$ | |||

(0.079) | (0.067) | (0.092) | (0.074) | (0.108) | (0.095) | ||||

${\sigma}_{\text{IV},t}^{\text{2}}$ | 0.281 | 0.287 | 0.473${}^{**}$ | 0.489${}^{**}$ | 0.779${}^{***}$ | 0.795${}^{***}$ | |||

(0.184) | (0.185) | (0.211) | (0.206) | (0.231) | (0.202) | ||||

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.002 | $-$0.002 | $-$0.005 | $-$0.004 | $-$0.001 | $-$0.0001 | |||

(0.004) | (0.004) | (0.004) | (0.004) | (0.003) | (0.002) | ||||

Const. | 0.017${}^{**}$ | 0.015${}^{***}$ | 0.017${}^{**}$ | 0.036${}^{***}$ | 0.029${}^{***}$ | 0.034${}^{***}$ | 0.055${}^{***}$ | 0.046${}^{***}$ | 0.055${}^{***}$ |

(0.007) | (0.006) | (0.007) | (0.009) | (0.009) | (0.009) | (0.010) | (0.009) | (0.009) | |

Adj. ${R}^{\text{2}}$ | 0.003 | 0.020 | 0.022 | 0.012 | 0.057 | 0.064 | 0.022 | 0.061 | 0.082 |

[0.00; 0.02] | [0.00; 0.06] | [0.00; 0.07] | [0.00; 0.05] | [0.02; 0.12] | [0.02; 0.16] | [0.00; 0.07] | [0.01; 0.13] | [0.01; 0.20] | |

Wald | 0.871 | $p=\text{0.351}$ | 3.133 | $p=\text{0.077}$ | 6.605 | $p=\text{0.010}$ |

First, we consider the relationship between the LMS measure and the stock market. We implement the models in (3.3) and (3.4) using the S&P 500 returns as the dependent variable. In addition, we also consider a model that only contains the LMS measure to evaluate how much of an increase in adjusted ${R}^{2}$ we can expect when including the LMS measure in the competitor models.

Table 2 gives these results. The pure LMS model (labeled (0)) shows that the LMS measure is able to explain parts of the variability of the market on longer horizons. While, the measure is not statistically significant for 30 days ahead, it is statistically significant for 60 and 90 days ahead. In the latter case, the ${R}^{2}$ is 2.2%, so that we expect improvements of the competitor models of this order of magnitude. The sign of the parameter estimate of LMS is consistently negative in all models. As LMS decreases when present and future risk converge or move toward a backwardation state (which indicates a high risk environment), the sign is plausible, as investors require their risk to be compensated. As LMS decreases, the required rate of return must thus increase. Note that the sign here is not directly comparable with the sign of the correlation in Table 1 because we consider here the market return over the next $\tau $ days, while correlations are calculated on a contemporaneous basis.

Looking at the models labeled (1) in Table 2, we find that the level of implied volatility (captured by ${\sigma}_{\text{IV}}^{2}$) predicts future returns at forecasting horizons of 60 and 90 days, similar to Banerjee et al (2007), Giot (2005) and Rubbaniy et al (2014). In contrast, changes ($\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$) over the respective period in the past have no forecasting ability, again in line with the findings of Banerjee et al (2007).

Considering the fully specified models, our proposed LMS measure also turns out to be statistically significant. In addition, the sign of the estimate is again consistently negative. The model improvement is better for longer forecast horizons (compare all models labeled (2) in Table 2). The increase in the adjusted ${R}^{2}$ varies from 0.2bps in the 30 days forward regression to 2.1pp in the 90 days case. For 30 days, the Wald test indicates that the models with and without LMS perform equally well while the same null hypothesis is rejected for the other horizons on a 10% significance level.

Looking at the confidence bounds of the adjusted ${R}^{2}$, we observe that the inclusion of the LMS variable always leads to a higher upper bound. The lower bounds of the confidence intervals are, however, almost always (very close to) zero, so that for all models the bands overlap. We also observe that the bands get wider as the forecasting horizon increases. As the lower bound is nearly constant, this suggests that there is an opportunity for the model (in particular for the LMS measure included in the 90-day horizon) to perform really well, which motivates the use of the LMS measure in the subsequent portfolio regressions.

### 5.2 Portfolio regressions

Tables 3–5 present the estimation results for three portfolios where stocks are sorted according to market capitalization (MCAP). Specifically, part (a) of each table presents the findings of the basic model (3.3) and the augmented model (3.4), while part (b) reports the results of these models after adding the risk factors ((3.5) and (3.6)). Including LMS in the base setting results in a consistently negative parameter estimate, even though it is not statistically significant. This is in line with the above findings, indicating that LMS alone proxies risk expectations, and a higher risk (ie, a lower LMS) requires a higher return. In contrast, when accounting for the risk factors, LMS is positive in the low MCAP portfolio, while it is negative for the portfolios consisting of medium or high MCAP stocks.

More precisely, according to Table 3(a), the LMS measure does not play a significant role in explaining future returns for any of the three portfolios. The level and change in IV also do not turn out to be statistically significant, a finding that contradicts the results of Banerjee et al (2007). Regarding the signs, positive estimates for ${\sigma}_{\text{IV},t}^{2}$ and negative estimates of $\mathrm{\Delta}{\sigma}_{\text{IV},t}^{2}$ are consistent with their findings. As regards the model comparison, adding LMS improves the adjusted ${R}^{2}$ in the case of the Medium40 and High30 portfolios, but the inclusion does not result in a statistically significant better model fit (as indicated by the high $p$-values of the Wald test). Also, the confidence bounds of the adjusted ${R}^{2}$ indicate a generally poor model fit for all three portfolios.

(a) Basic and augmented models | ||||||
---|---|---|---|---|---|---|

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

${\sigma}_{\text{IV},t}^{\text{2}}$ | 0.3641 | 0.3660 | 0.3378 | 0.3473 | 0.2508 | 0.2571 |

(0.2709) | (0.2755) | (0.2532) | (0.2536) | (0.1772) | (0.1768) | |

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.0022 | $-$0.0020 | $-$0.0041 | $-$0.0033 | $-$0.0023 | $-$0.0018 |

(0.0056) | (0.0052) | (0.0055) | (0.0050) | (0.0041) | (0.0038) | |

LMS | $-$0.0201 | $-$0.0971 | $-$0.0645 | |||

(0.1298) | (0.1047) | (0.0640) | ||||

Const. | 0.0128 | 0.0136 | 0.0137${}^{*}$ | 0.0173${}^{*}$ | 0.0157${}^{***}$ | 0.0181${}^{***}$ |

(0.0095) | (0.0123) | (0.0078) | (0.0099) | (0.0052) | (0.0063) | |

Adj. ${R}^{\text{2}}$ | 0.0148 | 0.0145 | 0.0224 | 0.0261 | 0.0183 | 0.0211 |

[0.00; 0.05] | [0.00; 0.06] | [0.00; 0.07] | [0.00; 0.09] | [0.00; 0.06] | [0.00; 0.07] | |

Wald | 0.0240 | $p=\text{0.8769}$ | 0.8606 | $p=\text{0.3537}$ | 1.0146 | $p=\text{0.3139}$ |

(b) Basic and augmented models plus risk factors | ||||||

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

${\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.0612${}^{*}$ | $-$0.0661${}^{*}$ | $-$0.0462 | $-$0.0432 | $-$0.0095 | $-$0.0092 |

(0.0370) | (0.0371) | (0.0542) | (0.0562) | (0.0096) | (0.0098) | |

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | 0.0010 | 0.0006 | $-$0.0010 | $-$0.0008 | $-$0.0003${}^{*}$ | $-$0.0003 |

(0.0010) | (0.0010) | (0.0013) | (0.0013) | (0.0002) | (0.0002) | |

LMS | 0.0453${}^{***}$ | $-$0.0276${}^{*}$ | $-$0.0027 | |||

(0.0119) | (0.0159) | (0.0020) | ||||

MKT | 0.9842${}^{***}$ | 0.9864${}^{***}$ | 1.0532${}^{***}$ | 1.0519${}^{***}$ | 0.9925${}^{***}$ | 0.9924${}^{***}$ |

(0.0196) | (0.0192) | (0.0225) | (0.0198) | (0.0036) | (0.0036) | |

SMB | 1.1298${}^{***}$ | 1.1286${}^{***}$ | 0.5766${}^{***}$ | 0.5773${}^{***}$ | $-$0.1353${}^{***}$ | $-$0.1352${}^{***}$ |

(0.0320) | (0.0317) | (0.0334) | (0.0338) | (0.0093) | (0.0092) | |

HML | 0.2190${}^{***}$ | 0.2177${}^{***}$ | 0.0970${}^{**}$ | 0.0977${}^{**}$ | $-$0.0274${}^{***}$ | $-$0.0273${}^{***}$ |

(0.0306) | (0.0305) | (0.0443) | (0.0446) | (0.0078) | (0.0079) | |

MOM | 0.0259 | 0.0249 | $-$0.0439 | $-$0.0433${}^{*}$ | $-$0.0011 | $-$0.0011 |

(0.0275) | (0.0245) | (0.0269) | (0.0256) | (0.0054) | (0.0056) | |

Const. | 0.0014 | $-$0.0003 | $-$0.00004 | 0.0010 | 0.0006${}^{**}$ | 0.0007${}^{**}$ |

(0.0014) | (0.0012) | (0.0012) | (0.0015) | (0.0003) | (0.0003) | |

Adj. ${R}^{\text{2}}$ | 0.9833 | 0.9840 | 0.9763 | 0.9766 | 0.9985 | 0.9985 |

[0.96; 0.98] | [0.96; 0.98] | [0.94; 0.98] | [0.95; 0.98] | [0.97;1.00] | [0.97; 0.99] | |

Wald | 14.4713 | $$ | 2.9857 | $p=\text{0.084}$ | 1.7958 | $p=\text{0.180}$ |

Adding LMS to the estimation in addition to the risk factors (Table 3(b)) leads to a significant improvement in the adjusted ${R}^{2}$ for both the Low30 and Medium40 portfolios. For the High30 portfolio, LMS provides limited additional explanatory power (Wald test $p\text{-value}=0.180$). In the Medium40 and High30 portfolios, the effect of LMS is negative, as we would have expected from the previous results. Unlike in part (a), the sign switches in the Low30 portfolio. Similar to Banerjee et al (2007), we find the change in IV to now be statistically significant more often than in the base specification (part (a)) together with switching signs between the models.

The extension of the relationship to a 60-day look-ahead period is reported in Table 4. In the basic models (part (a)), the estimate for LMS is again consistently negative and is now also statistically significant at a 10% significance level. Adding LMS also improves the adjusted ${R}^{2}$ values for all portfolios considered up to 1.4bps in the case of the Medium40 portfolio regression. In addition, we find positive estimates for the level and negative estimates for the change in volatility, more or less in line with the model results of Banerjee et al (2007, Table 4). While the confidence bounds of the adjusted ${R}^{2}$ overlap, the upper bound always increases (up to 6bps in case of the Medium40 portfolio) when adding LMS to the benchmark model. The inclusion significantly improves the models, as indicated by the Wald test $p$-value, which is less than 0.1 in all models.

(a) Basic and augmented models | ||||||
---|---|---|---|---|---|---|

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

${\sigma}_{\text{IV},t}^{\text{2}}$ | 0.7166${}^{**}$ | 0.7395${}^{***}$ | 0.6290${}^{**}$ | 0.6574${}^{***}$ | 0.4000${}^{*}$ | 0.4143${}^{**}$ |

(0.2950) | (0.2838) | (0.2515) | (0.2337) | (0.2066) | (0.2015) | |

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.0017 | $-$0.0007 | $-$0.0044 | $-$0.0032 | $-$0.0058 | $-$0.0052 |

(0.0056) | (0.0053) | (0.0055) | (0.0052) | (0.0039) | (0.0036) | |

LMS | $-$0.1960${}^{*}$ | $-$0.2431${}^{***}$ | $-$0.1224${}^{*}$ | |||

(0.1086) | (0.0900) | (0.0721) | ||||

Const. | 0.0253${}^{*}$ | 0.0325${}^{**}$ | 0.0267${}^{**}$ | 0.0356${}^{***}$ | 0.0310${}^{***}$ | 0.0355${}^{***}$ |

(0.0151) | (0.0158) | (0.0125) | (0.0127) | (0.0081) | (0.0083) | |

Adj. ${R}^{\text{2}}$ | 0.0305 | 0.0377 | 0.0448 | 0.0594 | 0.0613 | 0.0677 |

[0.00; 0.10] | [0.00; 0.13] | [0.01; 0.11] | [0.01; 0.17] | [0.02; 0.13] | [0.02; 0.17] | |

Wald | 3.2585 | $p=\text{0.071}$ | 7.2891 | $p=\text{0.007}$ | 2.8833 | $p=\text{0.089}$ |

(b) Basic and augmented models plus risk factors | ||||||

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

${\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.128${}^{**}$ | $-$0.137${}^{***}$ | $-$0.089${}^{*}$ | $-$0.083${}^{*}$ | $-$0.019 | $-$0.018 |

(0.061) | (0.048) | (0.052) | (0.046) | (0.017) | (0.016) | |

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.0003 | $-$0.001 | $-$0.001 | $-$0.001 | $-$0.0003 | $-$0.0002 |

(0.001) | (0.001) | (0.001) | (0.001) | (0.0003) | (0.0003) | |

LMS | 0.062${}^{***}$ | $-$0.034 | $-$0.008${}^{**}$ | |||

(0.012) | (0.031) | (0.004) | ||||

MKT | 1.016${}^{***}$ | 1.016${}^{***}$ | 1.053${}^{***}$ | 1.053${}^{***}$ | 0.991${}^{***}$ | 0.991${}^{***}$ |

(0.022) | (0.021) | (0.019) | (0.017) | (0.007) | (0.006) | |

SMB | 1.112${}^{***}$ | 1.119${}^{***}$ | 0.572${}^{***}$ | 0.568${}^{***}$ | $-$0.137${}^{***}$ | $-$0.138${}^{***}$ |

(0.045) | (0.039) | (0.037) | (0.039) | (0.022) | (0.021) | |

HML | 0.240${}^{***}$ | 0.236${}^{***}$ | 0.129${}^{***}$ | 0.131${}^{***}$ | $-$0.036${}^{***}$ | $-$0.036${}^{***}$ |

(0.030) | (0.027) | (0.041) | (0.039) | (0.008) | (0.008) | |

MOM | 0.050${}^{*}$ | 0.046${}^{*}$ | $-$0.036 | $-$0.034 | $-$0.003 | $-$0.002 |

(0.026) | (0.025) | (0.035) | (0.034) | (0.010) | (0.010) | |

Const. | 0.001 | $-$0.001 | $-$0.0004 | 0.001 | 0.001 | 0.002${}^{**}$ |

(0.002) | (0.002) | (0.002) | (0.002) | (0.001) | (0.001) | |

Adj. ${R}^{\text{2}}$ | 0.986 | 0.987 | 0.977 | 0.977 | 0.998 | 0.998 |

[0.95; 0.99] | [0.95; 0.99] | [0.92; 0.98] | [0.92; 0.98] | [0.95; 0.99] | [0.95; 0.99] | |

Wald | 27.369 | $p=\text{0.000}$ | 1.196 | $p=\text{0.274}$ | 4.191 | $p=\text{0.041}$ |

Adding LMS to the augmented models that include the risk factors (Table 4(b)) gives a similar picture to the that in Table 3. The parameter estimate of LMS is positive (and statistically significant) in the Low30 portfolio model and negative in the Medium40 and High30 portfolio models. The improvement in terms of adjusted ${R}^{2}$ is weak as the risk-free market return and the factors are able to explain nearly all variations in the portfolio returns. Thus, any small improvement in the adjusted ${R}^{2}$ is good. In addition, the Wald test indicates that the inclusion of the LMS measure significantly improves the model fit for the Low30 and High30 portfolios.

Finally, the 90-day look-ahead regression results reported in Table 5 are in line with the results presented in Tables 3 and 4. The LMS is consistently negative and statistically significant (at least at a 10% significance level) in the base specification reported in Table 5(a). The adjusted ${R}^{2}$ increases when including LMS (up to 2.4bps), and the upper bound of the confidence interval rises substantially (up to 8bps). The Wald test also indicates a better fit of the models including LMS. As soon as the other risk factors are added, the additional gains from considering LMS (or the level and change in IV for that matter) are small. The parameter estimate for LMS is again positive and statistically significant for the Low30 portfolio but negative in the other cases and significant only in the High30 portfolio. In line with these findings, the inclusion of LMS helps to increase the model fit only for the Low30 and High30 portfolios. However, the general results so far support the addition of LMS as a helpful pricing factor.

(a) Basic and augmented models | ||||||
---|---|---|---|---|---|---|

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

${\sigma}_{\text{IV},t}^{\text{2}}$ | 1.319${}^{***}$ | 1.335${}^{***}$ | 1.053${}^{***}$ | 1.074${}^{***}$ | 0.657${}^{***}$ | 0.673${}^{***}$ |

(0.397) | (0.363) | (0.324) | (0.265) | (0.218) | (0.174) | |

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | 0.007${}^{*}$ | 0.008${}^{**}$ | 0.003 | 0.003 | $-$0.002 | $-$0.001 |

(0.004) | (0.004) | (0.004) | (0.003) | (0.003) | (0.002) | |

LMS | $-$0.256${}^{*}$ | $-$0.340${}^{***}$ | $-$0.245${}^{***}$ | |||

(0.140) | (0.112) | (0.080) | ||||

Const. | 0.042${}^{**}$ | 0.051${}^{***}$ | 0.043${}^{***}$ | 0.055${}^{***}$ | 0.049${}^{***}$ | 0.058${}^{***}$ |

(0.017) | (0.017) | (0.014) | (0.014) | (0.009) | (0.008) | |

Adj. ${R}^{\text{2}}$ | 0.065 | 0.074 | 0.052 | 0.075 | 0.059 | 0.083 |

[0.02; 0.14] | [0.02; 0.17] | [0.01; 0.13] | [0.01; 0.19] | [0.01; 0.14] | [0.01; 0.22] | |

Wald | 3.341 | $p=\text{0.068}$ | 9.262 | $p=\text{0.002}$ | 9.286 | $p=\text{0.002}$ |

(b) Basic and augmented models plus risk factors | ||||||

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

${\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.141 | $-$0.145${}^{*}$ | $-$0.165${}^{**}$ | $-$0.162${}^{**}$ | $-$0.021 | $-$0.020 |

(0.086) | (0.079) | (0.075) | (0.071) | (0.025) | (0.023) | |

$\mathrm{\Delta}{\sigma}_{\text{IV},t}^{\text{2}}$ | $-$0.001 | $-$0.001 | $-$0.001 | $-$0.001 | $-$0.0002 | $-$0.0002 |

(0.001) | (0.001) | (0.001) | (0.001) | (0.0003) | (0.0003) | |

LMS | 0.060${}^{***}$ | $-$0.033 | $-$0.011${}^{*}$ | |||

(0.018) | (0.041) | (0.006) | ||||

MKT | 1.032${}^{***}$ | 1.037${}^{***}$ | 1.081${}^{***}$ | 1.078${}^{***}$ | 0.987${}^{***}$ | 0.986${}^{***}$ |

(0.034) | (0.032) | (0.027) | (0.026) | (0.011) | (0.009) | |

SMB | 1.141${}^{***}$ | 1.140${}^{***}$ | 0.553${}^{***}$ | 0.553${}^{***}$ | $-$0.135${}^{***}$ | $-$0.135${}^{***}$ |

(0.065) | (0.045) | (0.038) | (0.036) | (0.031) | (0.025) | |

HML | 0.259${}^{***}$ | 0.249${}^{***}$ | 0.129${}^{**}$ | 0.135${}^{**}$ | $-$0.045${}^{***}$ | $-$0.043${}^{***}$ |

(0.040) | (0.037) | (0.061) | (0.056) | (0.011) | (0.010) | |

MOM | 0.062${}^{**}$ | 0.055${}^{**}$ | $-$0.046 | $-$0.043 | $-$0.003 | $-$0.002 |

(0.027) | (0.025) | (0.050) | (0.048) | (0.013) | (0.013) | |

Const. | 0.002 | $-$0.001 | $-$0.002 | $-$0.001 | 0.002 | 0.003${}^{*}$ |

(0.003) | (0.003) | (0.003) | (0.004) | (0.002) | (0.001) | |

Adj. ${R}^{\text{2}}$ | 0.987 | 0.987 | 0.973 | 0.973 | 0.997 | 0.997 |

[0.93; 0.99] | [0.93; 0.99] | [0.90; 0.97] | [0.88; 0.97] | [0.93; 0.99] | [0.92; 0.99] | |

Wald | 10.778 | $$ | 0.674 | $p=\text{0.412}$ | 3.099 | $p=\text{0.078}$ |

The findings in Tables 3–5 show that the sign of the LMS estimate is sensitive to the inclusion of the risk factors in the small-cap stocks portfolio. According to our results future returns of small stocks are expected to be higher (lower) when LMS is higher (lower), whereas for large stocks future returns are expected to be higher (lower) when LMS is lower (higher). Comparing this finding across portfolios means that when LMS is lower, ie, when volatility increases, large-cap portfolios are expected to outperform small cap portfolios. In contrast, when LMS is higher, ie, when volatility decreases, small-cap portfolios are expected to outperform large-cap portfolios.

This finding of a changing sign of the LMS estimate is consistent with earlier studies (see, for example, Copeland and Copeland, 1999, 2016; Copeland et al, 2017), which show that changes in VIX have a different impact on small stocks and large stocks. In addition, Copeland et al (2018) argue that the sensitivity of portfolios to the same volatility shock also differs. They use the analogy of an earthquake exposing some regions to high risk while other regions may be a safe harbor. If we use this analogy, an LMS earthquake may be more harmful for small firms, while large firms may offer a flight to safety when uncertainty spikes. Small firms are, by nature, associated with greater information asymmetry, less coverage, etc. Hence, when LMS signals for increased future volatility (lower LMS), ie, when the market is overwhelmed with uncertainty, it makes the problem of information risk even more severe. Thus, in times of overall market uncertainty, investors will probably try to minimize their exposure to such risk by searching for safe ground and will probably shift their wealth to large, solid and high value firms rather than small, uncertain and high growth stocks. This shift is actually translated through a decline in future returns for small firms, whereas it will eventually increase the demand for and the price of big firms. The decline in future returns for small firms is necessary to balance overpriced small firms’ prices to their fair equilibrium values.

A further aspect that needs to be considered to explain the switching sign phenomenon is rooted in psychological factors and how investors value certain companies under various circumstances. For instance, according to Forgas (1995), better investor sentiment reduces risk aversion, making investors willing to tolerate more risk. Thus, they are more willing to take risky investments, since they judge risky situations more positively. In this spirit, Qadan and Aharon (2019a) found that the size premium is correlated to, and is predicted by, different investor sentiment measures, including the VIX. In other words, when investor sentiment, proxied by the VIX (among others), is improved, small-cap stocks are overvalued compared with large-cap stocks, and vice versa.

To check whether the LMS measure also plays the same role for different types of portfolio sortings, we extend the analysis and estimate the models for book-to-market portfolios (see Table A1 in the online appendix), double sorted size and book-to-market portfolios (Table A2 in the online appendix) and industry portfolios (Table A3 in the online appendix). It turns out that the LMS measure plays a significant and consistent role in explaining future returns for nearly all of the portfolios. Notably, the relationship is clearer over longer time periods. In the short run (30 day horizon), we find that stocks in the low book-to-market deciles are not sensitive to the LMS measure, while those in the high deciles are. Similarly, the performance across industry portfolios for 30 days is mixed, as these portfolios comprise high and low book-to-market stocks. Hence, LMS is indeed a reliable predictor in the long run. In the short run, it is a better predictor for high book-to-market stocks or high MCAP stocks.

### 5.3 Robustness checks

#### 5.3.1 Specification of the LMS measure

The LMS as defined in (3.1) uses the VIX indexes with six-month and nine-day look-ahead periods. The intention behind this choice is to capture the steepness of the VIX term structure, in particular to improve the long-run forecasts. Of course, there are more VIX indexes with different maturities available. To evaluate the robustness of the decision to implement the LMS as in (3.1), we vary the minuend (VIX, VIX3M, VIX6M) and subtrahend (VIX9D, VIX1M). Table 6 shows the correlation of the LMS measures thus obtained. As can be seen, the correlation is very high, particularly between those LMSs with the same subtrahend.

LMS | 1M–9D | 3M–9D | 6M–9D | 3M–1M | 6M–1M |
---|---|---|---|---|---|

$\text{VIX1M}-\text{VIX9D}$ | 1.000 | 0.943 | 0.896 | 0.716 | 0.682 |

$\text{VIX3M}-\text{VIX9D}$ | 0.943 | 1.000 | 0.984 | 0.908 | 0.873 |

$\text{VIX6M}-\text{VIX9D}$ | 0.896 | 0.984 | 1.000 | 0.933 | 0.935 |

$\text{VIX3M}-\text{VIX1M}$ | 0.716 | 0.908 | 0.933 | 1.000 | 0.969 |

$\text{VIX6M}-\text{VIX1M}$ | 0.682 | 0.873 | 0.935 | 0.969 | 1.000 |

Using such LMS measures in the market regressions (as in Section 5.1) leads to qualitatively similar results, as reported in Table 2. In particular, the sign of the coefficient estimate is negative, and inclusion of the LMS leads to an increase in the adjusted ${R}^{2}$ of a similar order of magnitude. As the orthogonalization makes a direct comparison difficult, we look at the $p$-value of the Wald test and find that this is smallest when using the LMS measure as proposed in (3.1). In the portfolio regression, the results are such that we usually find fewer instances where LMS is statistically significant when using shorter horizon LMS measures. Nevertheless, it is possible that a particular portfolio with a specific forecast horizon might benefit from a differently calculated LMS. Our idea is to provide a single measure, and it turns out that the LMS measure as calculated in (3.1) performs very well given this restriction.

#### 5.3.2 Time period sensitivity

The models above are implemented for a fixed time period from March 1, 2011 to January 31, 2020. In order to rule out the notion that the results are purely driven by the choice of the sample, we conduct a rolling analysis using 1250 observations (roughly five years of data) in each subsample. We then repeat the estimation as outlined above and compute the difference in the ${R}^{2}$ between the base specifications (either (3.3) or (3.5)) and the LMS-augmented specifications ((3.4) or (3.6)). The orthogonalization of the main variables is carried out on each subsample rather than on the full sample.

Overall, we find that the inclusion of the LMS measure helps to raise the adjusted ${R}^{2}$. For the market regression, we provide the full results graphically: Figure 3 presents kernel density estimates for the ${R}^{2}$ differences for the 30-, 60- and 90-day forecast horizons. Overall, we do not find a single subperiod in which the ${R}^{2}$ decreases. On the contrary, the improvement can be as large as 8pp. On average, the increase is comparable with the differences observed in Table 2, which leads us to conclude that the results are qualitatively robust with respect to the sample period considered. Also, regarding the importance of the LMS measure as measured by an increasing ${R}^{2}$ or rejections of the Wald test null hypothesis, the selected time period does not oversell the effect.

As regards the portfolio regressions, we also find that the inclusion of LMS in general leads to a rising adjusted ${R}^{2}$. In contrast to the market regression, however, we also observe instances where the adjusted ${R}^{2}$ goes down. Similarly to the results documented in Section 5.2, the gains are more pronounced in the baseline specifications. When adding the factors (MKT, HML, SMB, MOM), the gains are very small. In total, we observe an increase in the ${R}^{2}$ in 91% of all the estimated baseline models and in 88% of all fully specified models. Hence, we conclude that the LMS is a robust factor in pricing the observed portfolios and that the results reported in Section 5.2 are not driven purely by the choice of the sample period.

#### 5.3.3 Relation to the variance risk premium

An important pricing factor identified in the literature (Bollerslev et al 2009; Bekaert and Hoerova 2014) is the variance risk premium (VRP). We therefore investigate how our LMS measure and the VRP are related, and we compare their relative performance in the predictive regressions. We follow Bekaert and Hoerova (2014) and implement the VRP as

$${\mathrm{VRP}}_{t}={\mathrm{VIX}}_{t}^{2}-{E}_{t}[{\mathrm{RV}}_{t+1}^{2}].$$ | (5.1) |

To this end, we rely on realized volatility data provided by the Oxford-Man Institute’s realized library (Heber et al 2009). The conditional expectation on the right-hand side of (5.1) is replaced by a prediction of the realized volatility. Again, we follow Bekaert and Hoerova (2014) and implement the logarithmic model as described in their Equation (5), preparing all the necessary variables according to their description. The model is estimated on 1000 observations to predict the realized volatility. Therefore, the time frame to estimate the models is shorter than in Sections 5.1 and 5.2, starting only in January 2015.

First, we find that the LMS measure and the VRP are strongly negatively correlated (Pearson correlation coefficient $-$0.86). This indicates that the LMS and the VRP capture similar information, albeit from a different angle. A high variance risk premium indicates that the uncertainty about the future variance is high. In a market state where uncertainty is high, the VIX indexes on all forecast horizons will go up and the relative difference between long-run and short-run uncertainty will be smaller. Thus, we obtain a lower LMS measure in such circumstances which then results in the negative correlation with the VRP.

Adding the VRP to the market regressions (similar to the ones reported in Table 2), we find that the model fit is best when both the LMS measure and the VRP are included. While the adjusted ${R}^{2}$ in a model with the LMS only is similar to the order of magnitude reported in Table 2, the VRP fit is better. Nevertheless, including both LMS and VRP nearly doubles the adjusted ${R}^{2}$ compared with the VRP only model (up to 0.085 for the 60-days-ahead market returns). However, the full model exhibits switching of the sign of the LMS, which is negative when included as a sole variable (as above), but changes to positive in the full model.

Regarding the portfolio regressions, we restrict the presentation to the 60-day forward-looking regression presented in Table 7. The other regressions are qualitatively similar. We find that the VRP is a significant predictor of future portfolio returns. In a model that only contains the VRP, the adjusted ${R}^{2}$ takes values up to 0.061 for the Medium40 portfolio. Adding the LMS measure always increases the fit and we reject the null hypothesis that LMS is an irrelevant variable. Looking at the fully specified models that also contain the other factors (Table 7(b)), the VRP is only statistically significant in the Low30 portfolio model. The LMS is significant at a 10% significance level in all portfolios. The sign of the LMS is negative. Both the VRP and the LMS switch signs when MKT enters the model. Again, adding LMS in addition to the VRP increases the model fit (as indicated by the adjusted ${R}^{2}$) only marginally. As in Tables 3–5, the established factors explain the variation in the portfolio returns almost completely. Neither the LMS measure nor the VRP can add significant explanatory power in this case. However, due to the small number of observations, these results have to be interpreted with care.

(a) Basic and augmented models | ||||||
---|---|---|---|---|---|---|

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

VRP | 0.972${}^{**}$ | 2.327${}^{**}$ | 1.075${}^{***}$ | 2.241${}^{***}$ | 0.662${}^{**}$ | 1.670${}^{***}$ |

(0.386) | (0.914) | (0.405) | (0.762) | (0.291) | (0.572) | |

LMS | 0.681 | 0.586${}^{*}$ | 0.507${}^{**}$ | |||

(0.414) | (0.310) | (0.225) | ||||

Constant | $-$0.005 | $-$0.063 | $-$0.005 | $-$0.055 | 0.012 | $-$0.031 |

(0.024) | (0.047) | (0.017) | (0.035) | (0.014) | (0.029) | |

Adj. ${R}^{\text{2}}$ | 0.038 | 0.066 | 0.061 | 0.088 | 0.039 | 0.073 |

[0.02; 0.06] | [0.04; 0.09] | [0.03; 0.11] | [0.05; 0.14] | [0.01; 0.11] | [0.03; 0.13] | |

Wald | 2.705 | $p=\text{0.100}$ | 3.620 | $p=\text{0.057}$ | 4.714 | $p=\text{0.030}$ |

(b) Basic and augmented models plus risk factors | ||||||

Low30 | Medium40 | High30 | ||||

(1) | (2) | (1) | (2) | (1) | (2) | |

VRP | $-$0.148${}^{***}$ | $-$0.273${}^{***}$ | 0.091 | $-$0.187 | 0.004 | $-$0.065 |

(0.057) | (0.101) | (0.188) | (0.216) | (0.027) | (0.060) | |

LMS | $-$0.059${}^{*}$ | $-$0.131${}^{**}$ | $-$0.032${}^{*}$ | |||

(0.035) | (0.056) | (0.018) | ||||

MKT | 1.033${}^{***}$ | 1.036${}^{***}$ | 1.056${}^{***}$ | 1.063${}^{***}$ | 0.993${}^{***}$ | 0.995${}^{***}$ |

(0.031) | (0.032) | (0.028) | (0.023) | (0.006) | (0.006) | |

SMB | 1.046${}^{***}$ | 1.044${}^{***}$ | 0.506${}^{***}$ | 0.503${}^{***}$ | $-$0.119${}^{***}$ | $-$0.120${}^{***}$ |

(0.037) | (0.037) | (0.055) | (0.047) | (0.016) | (0.013) | |

HML | 0.256${}^{***}$ | 0.256${}^{***}$ | 0.110${}^{*}$ | 0.110${}^{**}$ | $-$0.048${}^{***}$ | $-$0.048${}^{***}$ |

(0.025) | (0.025) | (0.058) | (0.045) | (0.007) | (0.006) | |

MOM | 0.050 | 0.044 | $-$0.016 | $-$0.029 | $-$0.006 | $-$0.009 |

(0.036) | (0.035) | (0.048) | (0.042) | (0.012) | (0.011) | |

Const. | 0.005${}^{*}$ | 0.010${}^{***}$ | $-$0.004 | 0.008 | 0.002${}^{***}$ | 0.005${}^{***}$ |

(0.003) | (0.004) | (0.004) | (0.007) | (0.001) | (0.002) | |

Adj. ${R}^{\text{2}}$ | 0.989 | 0.989 | 0.973 | 0.974 | 0.998 | 0.998 |

[0.98; 0.99] | [0.98; 0.99] | [0.94; 0.98] | [0.94; 0.98] | [0.96; 0.99] | [0.95; 0.99] | |

Wald | 2.994 | $p=\text{0.084}$ | 5.471 | $p=\text{0.020}$ | 3.473 | $p=\text{0.063}$ |

## 6 Conclusion

In this paper we measured the relative expectations of future uncertainty by using the difference of long (six-month) horizon and short (nine-day) horizon VIX indexes, building on previous findings in the literature which document that the VIX term structure contains valuable information for portfolio pricing. The measure obtained, denoted by LMS, is used to augment the model of Banerjee et al (2007), which relates future returns to the levels of IV in the present and the future as well as to past changes in IV. Including LMS in the regression models results in a better fit of the model, increasing the adjusted ${R}^{2}$ by up to 2bps. When accounting for the market and various risk factors, LMS still turns out to be statistically significant, even though the market returns almost completely explain the portfolio return variation. This finding holds over various subperiods. We conclude that, in addition to the well-known risk–return relationship, the differential of expectations about future risk relative to the present can be a relevant pricing factor for portfolios of stocks.

More precisely, our results indicate different signals of the LMS measure for small- versus large-cap stocks. This observation conforms to the theoretical, psychological perspective proposed by Forgas (1995), according to which people tend to judge risky situations more positively under times of improved sentiment. Copeland and Copeland (2016) and Qadan and Aharon (2019a) show that valuation is subjective to the state of market sentiment, particularly for small stocks. The positive LMS observed for small stocks in the fully specified model thus indicates that, for small stocks, investors are more willing to take risk compared with an investment in the market portfolio.

The LMS measure is based on publicly available data, namely the CBOE VIX indexes. Therefore, its calculation is straightforward, and the measure is easily used in practical applications. As our empirical example provides evidence for the LMS measure’s pricing capability, it could be fruitful to include it in further forecasting and asset pricing models. Further research may also check how the LMS measure performs on a less than daily frequency or compare it to other pricing factors, such as the price/earnings ratio discussed by Welch and Goyal (2007). Such a study is currently hindered by the small number of observations, but may of course be feasible in due time.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## References

- Adrian, T., Crump, R. K., and Vogt, E. (2019). Nonlinearity and flight-to-safety in the risk–return trade-off for stocks and bonds. Journal of Finance 74(4), 1931–1973 (https://doi.org/10.1111/jofi.12776).
- Banerjee, P. S., Doran, J. S., and Peterson, D. R. (2007). Implied volatility and future portfolio returns. Journal of Banking and Finance 31(10), 3183–3199 (https://doi.org/10.1016/j.jbankfin.2006.12.007).
- Bekaert, G., and Hoerova, M. (2014). The VIX, the variance premium and stock market volatility. Journal of Econometrics 183(2), 181–192 (https://doi.org/10.1016/j.jeconom.2014.05.008).
- Blair, B. J., Poon, S. H., and Taylor, S. J. (2001). Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns. Journal of Econometrics 105(1), 5–26 (https://doi.org/10.1016/S0304-4076(01)00068-9).
- Bloom, N. (2009). The impact of uncertainty shocks. Econometrica 77(3), 623–685 (https://doi.org/10.3982/ECTA6248).
- Bollerslev, T., Tauchen, G., and Zhou, H. (2009). Expected stock returns and variance risk premia. Review of Financial Studies 22(11), 4463–4492 (https://doi.org/10.1093/rfs/hhp008).
- Brennan, M. J. (1958). The supply of storage. American Economic Review 48(1), 50–72 (https://doi.org/10.1002/j.1551-8833.1958.tb16218.x).
- Bu, R., Fu, X., and Jawadi, F. (2019). Does the volatility of volatility risk forecast future stock returns? Journal of International Financial Markets, Institutions and Money 61, 16–36 (https://doi.org/10.1016/j.intfin.2019.02.001).
- Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance 52(1), 57–82 (https://doi.org/10.1111/j.1540-6261.1997.tb03808.x).
- Carr, P., and Wu, L. (2016). Analyzing volatility risk and risk premium in option contracts: a new theory. Journal of Financial Economics 120(1), 1–20 (https://doi.org/10.1016/j.jfineco.2016.01.004).
- Chicago Board Options Exchange (2019). CBOE volatility index. White Paper, CBOE. URL: https://cdn.cboe.com/resources/futures/vixwhite.pdf.
- Copeland, M., and Copeland, T. (1999). Market timing: style and size rotation using the VIX. Financial Analysts Journal 55(2), 73–81 (https://doi.org/10.2469/faj.v55.n2.2262).
- Copeland, M., and Copeland, T. (2016). VIX versus size. Journal of Portfolio Management 42(3), 76–83 (https://doi.org/10.3905/jpm.2016.42.3.076).
- Copeland, M., Copeland, T., and Copeland, T. (2017). Revising equity valuation with tail risk. Journal of Portfolio Management 43(4), 100–111 (https://doi.org/10.3905/jpm.2017.43.4.100).
- Copeland, M., Copeland, M., and Copeland, T. (2018). Industry rotation and time-varying sensitivity by VIX. Journal of Portfolio Management 44(6), 89–97 (https://doi.org/10.3905/jpm.2018.44.6.089).
- Corrado, C., and Miller, T. W., Jr. (2005). The forecast quality of CBOE implied volatility indexes. Journal of Futures Markets 25(4), 339–373 (https://doi.org/10.1002/fut.20148).
- Daigler, R. T., Hibbert, A. M., and Pavlova, I. (2014). Examining the return–volatility relation for foreign exchange: evidence from the euro VIX. Journal of Futures Markets 34(1), 74–92 (https://doi.org/10.1002/fut.21582).
- Fama, E. F., and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33(1), 3–56 (https://doi.org/10.1016/0304-405X(93)90023-5).
- Fassas, A. P., and Siriopoulos, C. (2011). The efficiency of the VIX futures market: a panel data approach. Journal of Alternative Investments 14(3), 55–65 (https://doi.org/10.3905/jai.2012.14.3.055).
- Fleming, J. (1998). The quality of market volatility forecasts implied by S&P 100 index option prices. Journal of Empirical Finance 5(4), 317–345 (https://doi.org/10.1016/S0927-5398(98)00002-4).
- Forgas, J. P. (1995). Mood and judgment: the affect infusion model (AIM). Psychological Bulletin 117(1), 39–66 (https://doi.org/10.1037/0033-2909.117.1.39).
- Ghysels, E., Santa-Clara, P., and Valkanov, R. (2005). There is a risk–return trade-off after all. Journal of Financial Economics 76(3), 509–548 (https://doi.org/10.1016/j.jfineco.2004.03.008).
- Giot, P. (2005). Relationships between implied volatility indexes and stock index returns. Journal of Portfolio Management 31(3), 92–100 (https://doi.org/10.3905/jpm.2005.500363).
- Heber, G., Lunde, A., Shephard, N., and Sheppard, K. K. (2009). Oxford-Man Institute’s realized library, Version 0.3. Oxford-Man Institute, University of Oxford. URL: https://realized.oxford-man.ox.ac.uk/.
- Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2), 327–343 (https://doi.org/10.1093/rfs/6.2.327).
- Huang, D., Schlag, C., Shaliastovich, I., and Thimme, J. (2019a). Volatility-of-volatility risk. Journal of Financial and Quantitative Analysis 54(6), 2423–2452 (https://doi.org/10.1017/S0022109018001436).
- Huang, Z., Tong, C., and Wang, T. (2019b). VIX term structure and VIX futures pricing with realized volatility. Journal of Futures Markets 39(1), 72–93 (https://doi.org/10.1002/fut.21955).
- Huskaj, B., and Nossman, M. (2012). A term structure model for VIX futures. Journal of Futures Markets 33(5), 421–442 (https://doi.org/10.1002/fut.21550).
- Jiang, G. J., and Tian, Y. S. (2005). The model-free implied volatility and its information content. Review of Financial Studies 18(4), 1305–1342 (https://doi.org/10.1093/rfs/hhi027).
- Johnson, T. L. (2017). Risk premia and the VIX term structure. Journal of Financial and Quantitative Analysis 52(6), 2461–2490 (https://doi.org/10.1017/S0022109017000825).
- Jubinski, D., and Lipton, A. F. (2013). VIX, gold, silver, and oil: how do commodities react to financial market volatility? Journal of Accounting and Finance 13(1), 70–88. URL: https://bit.ly/3IonHwI.
- Kaldor, N. (1939). Speculation and economic stability. Review of Economic Studies 7(1), 1–27 (https://doi.org/10.2307/2967593).
- Liu, B. (2014). Identifying the differences between VIX spot and futures. Strategy Paper 201, S&P Dow Jones Indices Practice Essentials Series. McGraw-Hill Financial. URL: https://bit.ly/3NW2V8O.
- Lu, X., Sun, X., and Ge, J. (2017). Dynamic relationship between Japanese Yen exchange rates and market anxiety: a new perspective based on MF-DCCA. Physica A 474, 144–161 (https://doi.org/10.1016/j.physa.2017.01.058).
- Lu, Z., and Zhu, Y. (2010). Volatility components: the term structure dynamics of VIX futures. Journal of Futures Markets 30(3), 230–256 (https://doi.org/10.1002/fut.20415).
- Ludvigson, S. C., and Ng, S. (2007). The empirical risk–return relation: a factor analysis approach. Journal of Financial Economics 83(1), 171–222 (https://doi.org/10.1016/j.jfineco.2005.12.002).
- Mixon, S., and Onur, E. (2019). Derivatives pricing when supply and demand matter: evidence from the term structure of VIX futures. Journal of Futures Markets 39(9), 1035–1055 (https://doi.org/10.1002/fut.22035).
- Newey, W. K., and West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703–708 (https://doi.org/10.2307/1913610).
- Newey, W. K., and West, K. D. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61(4), 631–653 (https://doi.org/10.2307/2297912).
- Nossman, M., and Wilhelmsson, A. (2009). Is the VIX futures market able to predict the VIX index? A test of the expectation hypothesis. Journal of Alternative Investments 12(2), 54–67 (https://doi.org/10.3905/jai.2009.12.2.054).
- Osterrieder, D., Ventosa-Santaulària, D., and Vera-Valdés, J. E. (2019). The VIX, the variance premium, and expected returns. Journal of Financial Econometrics 17(4), 517–558 (https://doi.org/10.1093/jjfinec/nby008).
- Qadan, M., and Aharon, D. Y. (2019a). Can investor sentiment predict the size premium? International Review of Financial Analysis 63, 10–26 (https://doi.org/10.1016/j.irfa.2019.02.005).
- Qadan, M., and Aharon, D. Y. (2019b). How much happiness can we find in the US fear index? Finance Research Letters 30, 246–258 (https://doi.org/10.1016/j.frl.2018.10.001).
- Qadan, M., and Yagil, J. (2012). Fear sentiments and gold price: testing causality in-mean and in-variance. Applied Economics Letters 19(4), 363–366 (https://doi.org/10.1080/13504851.2011.579053).
- R Core Team (2021). R: a language and environment for statistical computing, Version 4.1.2. R Foundation for Statistical Computing, Vienna, Austria. URL: http://www.R-project.org/.
- Rubbaniy, G., Asmerom, R., Rizvi, S. K. A., and Naqvi, B. (2014). Do fear indices help predict stock returns? Quantitative Finance 14(5), 831–847 (https://doi.org/10.1080/14697688.2014.884722).
- Smales, L. A. (2017). The importance of fear: investor sentiment and stock market returns. Applied Economics 49(34), 3395–3421 (https://doi.org/10.1080/00036846.2016.1259754).
- Vinod, H. D. (2004). Ranking mutual funds using unconventional utility theory and stochastic dominance. Journal of Empirical Finance 11(3), 353–377 (https://doi.org/10.1016/j.jempfin.2003.06.002).
- Vinod, H. D. (2006). Maximum entropy ensembles for time series inference in economics. Journal of Asian Economics 17(6), 955–978 (https://doi.org/10.1016/j.asieco.2006.09.001).
- Wang, H. (2019). VIX and volatility forecasting: a new insight. Physica A 533(3), Paper 121951 (https://doi.org/10.1016/j.physa.2019.121951).
- Welch, I., and Goyal, A. (2007). A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21(4), 1455–1508 (https://doi.org/10.1093/rfs/hhm014).
- Whaley, R. E. (2000). The investor fear gauge. Journal of Portfolio Management 26(3), 12–17 (https://doi.org/10.3905/jpm.2000.319728).
- Wood, S. N. (2017). Generalized Additive Models: An Introduction with R, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL.
- Working, H. (1949). The theory of price of storage. American Economic Review 39(6), 1254–1262. URL: http://www.jstor.org/stable/1816601.
- Yun, J. (2020). A re-examination of the predictability of stock returns and cash flows via the decomposition of VIX. Economics Letters 186, Paper 108755 (https://doi.org/10.1016/j.econlet.2019.108755).
- Zeileis, A. (2006). Object-oriented computation of sandwich estimators. Journal of Statistical Software 16(9), 1–16 (https://doi.org/10.18637/jss.v016.i09).

Only users who have a paid subscription or are part of a corporate subscription are able to print or copy content.

To access these options, along with all other subscription benefits, please contact info@risk.net or view our subscription options here: http://subscriptions.risk.net/subscribe

You are currently unable to print this content. Please contact info@risk.net to find out more.

You are currently unable to copy this content. Please contact info@risk.net to find out more.

Copyright Infopro Digital Limited. All rights reserved.

As outlined in our terms and conditions, https://www.infopro-digital.com/terms-and-conditions/subscriptions/ (point 2.4), printing is limited to a single copy.

If you would like to purchase additional rights please email info@risk.net

Copyright Infopro Digital Limited. All rights reserved.

You may share this content using our article tools. As outlined in our terms and conditions, https://www.infopro-digital.com/terms-and-conditions/subscriptions/ (clause 2.4), an Authorised User may only make one copy of the materials for their own personal use. You must also comply with the restrictions in clause 2.5.

If you would like to purchase additional rights please email info@risk.net