Hedging performance of volatility index futures: a partial cointegration approach

Abstract

Using the Clegg–Krauss framework, this paper first examines a partial cointegration relationship between stock index futures and $VIX$ futures prices and then constructs a hedging strategy based upon this relationship. This paper argues that the stock index futures and the $VIX$ futures are both affected by unobservable investor sentiment and thus the price series should be modelled by the partial cointegration relationship. Our empirical results validate a partial cointegration relationship between stock index and $VIX$ futures prices. Based upon the partial cointegration relationship between stock index futures and $VIX$ futures prices, we demonstrate that the proposed strategy outperforms conventional strategies, e.g., $OLS$, $VAR$, and $VECM$, in terms of tail risk reduction and expected utility, especially when the length of the hedge horizon increases. In addition, the partial cointegration-based strategy becomes more dominant when the stock index futures price is near its historical high. Overall, our empirical evidence of the hedging effectiveness for different hedge horizons and market timing with $VIX$ futures provides valuable information for practitioners in risk management.

1 Introduction

Previous studies have examined how volatility derivatives on the volatility index $VIX$ can be used as trading and risk management tools for investors and traders. Chen et al. (2011) investigate whether investors can improve their investment opportunity sets through the addition of volatility-related assets in various grou**s of benchmark portfolios. Using the mean–variance spanning approach, it is found that adding $VIX$-related assets does lead to a significant enlargement of investment opportunity sets for investors. Jung (2016) explores whether $VIX$ futures is an investment asset with downside protection. The evidence shows that the portfolio insurance (PI) strategy with $VIX$ futures serves as an effective hedging tool for portfolio insurance funds on the stock index (e.g., S&P 500 index). Bordonado et al. (2017) examine the most traded $VIX$ exchange traded products (ETPs) regarding their performance, price discovery, hedging ability, and trading strategy. It is shown that, despite being negatively correlated with the S&P 500, ETPs perform poorly as a hedging tool. Specifically, the inclusion of ETPs in a S&P 500 portfolio decreases the risk-adjusted performance of the portfolio. Overall, prior studies have not offered a universally consistent conclusion. This paper attempts to evaluate the hedging performance of $VIX$ futures from the perspectives of the downside risk protection and expected utility approaches, using a different approach—the partial cointegration of Clegg and Krauss (2018). Specifically, this paper first examines whether the stock index futures prices and the $VIX$ futures prices are partially cointegrated, rather than being cointegrated. Upon confirmation, we explore hedging effectiveness based on partial cointegration.

While cointegration is well known and widely adopted, partial cointegration is less referred to in the literature. Cointegration prevails when a combination of two unit-root time-series variables becomes stationary. Suppose that the two time-series variables are not cointegrated. According to Clegg and Krauss (2018), they can be partially cointegrated if there is a combination which can be written as the sum of a unit-root time-series and a stationary time-series. More specifically, two variables are partially cointegrated if they do not share a common long run and mean reverting equilibrium; however, a particular linear combination of them, while nonstationary, shares a common long run and mean reverting equilibrium with a third variable. Clegg and Krauss (2018) use a state space model approach to identify linear combinations. As an example, below we show that partial cointegration may arise from a missing variable framework. Suppose that $x_{1}$ is affected by two stochastic trends, $w_{1} {\text{and}} w_{2}$, such that $x_{1} = w_{1} + w_{2} + \varepsilon_{1}$, where $\varepsilon_{1}$ is stationary. Let $x_{2} = aw_{1} + bw_{2} + \varepsilon_{2}$, where $\varepsilon_{2}$ is stationary, $a \ne b$ and at least one of them is nonzero. In this case, $x_{1}$ and $x_{2}$ are not cointegrated since we cannot find a linear combination to remove both stochastic trends. Now, suppose that $x_{3} = cw_{1} + dw_{2} + \varepsilon_{3}$, where $\varepsilon_{3}$ is stationary. A linear combination $x_{1} + \theta_{2} x_{2} + \theta_{3} x_{3}$ is stationary if $1 + a\theta_{2} + c\theta_{3} = 0$ and $1 + b\theta_{2} + d\theta_{3} = 0$; that is, $\theta_{2} = \frac{c - d}{{ad - bc}}$ and $\theta_{3} = \frac{b - a}{{ad - bc}}$ assuming $ad \ne bc$. In this case, $x_{1}$ and $x_{2}$ are partially cointegrated such that $x_{1} + \theta_{2} x_{2}$ can be written as the sum of a unit root variable $\left( { - \theta_{3} x_{3} } \right)$ and a stationary variable $\left( {\varepsilon_{1} + \theta_{2} \varepsilon_{2} - \theta_{3} \varepsilon_{3} } \right)$. Accordingly, a cointegration relationship among three I (1) variables becomes a partial cointegration relationship when one variable (${\text{either}} x_{1}$,$x_{2} , {\text{or}} x_{3}$) is omitted or unobservable. In this case, instead of the partial cointegration analysis, one may attempt to find and incorporate the omitted I (1) variable into the system resulting in a cointegration framework. However, the task would be daunting in most cases since it would be difficult to find a proxy for this variable.

Pertaining to our framework, previous studies have shown that market-wide return is correlated with investor fear (e.g., Kaplanski and Levy 2010; Da et al. 2015). In particular, Da et al. (2015) show that the FEARS index predicts short-term return reversals and temporary increases in market volatility index $VIX$, which is well known by practitioners as an “investor fear gauge” (Whaley 2000, 2009). The $VIX$ futures serves as an instrument to hedge volatility risks. Both Frijns et al. (2016) and Kao et al. (2018) indicate that the $VIX$ futures contains useful information in predicting the change in $VIX$, implying that the $VIX$ futures leads the $VIX$ in information spillover. Frijns et al. (2016) argue that $VIX$ futures has replaced $VIX$ in its role as the investor fear gauge. As a consequence, the stock index futures prices and $VIX$ futures prices are both affected by investor fear. Since investor fear is not observable, there may exist a partial cointegration relationship between the stock index futures and $VIX$ futures.¹ Suppose that the stock index futures price (e.g., E-mini S&P 500 index futures $SPF$) is a linear function of investor fear: $SPF_{t} = \alpha Fear_{t} + e_{s,t}$, where $SPF_{t}$ and $Fear_{t}$ are both I(1) and $e_{s,t}$ is I(0). Also, suppose that $VIX$ futures (VF) is a quadratic function of the investor fear:$VF_{t} = \delta_{1} Fear_{t} + \delta_{2} (Fear_{t} )^{2} + e_{v,t}$, where $VF_{t}$ is I (1) and $e_{v,t}$ is I (0). The partial cointegration approach recently introduced by Clegg and Krauss (2018) allows for not only transient but also persistent shocks to the long-run equilibrium relationship between two (or more variables). Specifically, Clegg and Krauss (2018) define two non-cointegrated I (1) variables as partially cointegrated if there is a linear combination which can be decomposed as the sum of an I (0) variable and an I (1) variable. Based on our setting, we can derive $\delta_{1} SPF_{t} - \alpha VF_{t} = \delta_{1} e_{s,t} - \alpha e_{v,t} - \alpha \delta_{2} (Fear_{t} )^{2}$. Since $\delta_{1} SPF_{t} - \alpha VF_{t}$ can be decomposed into I (0) variable ($\delta_{1} e_{s,t} - \alpha e_{v,t}$) and I (1) variable ($\alpha \delta_{2} (Fear_{t} )^{2}$), it suggests that $SPF_{t}$ and $VF_{t}$ are partially cointegrated as defined in Clegg and Krauss (2018).² Chen (2011) finds that the Dow Jones Industrial Average (DJIA) is affected by investor fear. As a result, in addition to using $SPF$ for analysis, this paper also employs the E-mini Dow Jones Industrial Average index futures ($DJF$) as robustness checks.

Based on above discussion, this paper follows the approach of Clegg and Krauss (2018) and validates a partial cointegration relationship between stock index and $VIX$ futures prices. A hedging strategy is developed from the estimated partial cointegration relationship. The hedging effectiveness is measured by the tail risk reduction and expected utility gain. For both effectiveness measures, we find the partial cointegration-based strategy outperforms the cointegration-based and other strategies (such as OLS and VAR approaches). The outperformance increases with the length of the hedging period. Finally, the dominance of the partial cointegration-based strategy becomes more significant when the stock index futures price is near its historical high.

This paper contributes to the literature in two ways. First, this is the first study to examine the existence of a partial cointegration relationship (e.g., partial equilibrium) and hedging performance between the stock index futures and the $VIX$ futures. Specifically, we show that the residuals for the stock index futures and $VIX$ futures equation estimated by Engle and Granger (1987) are not purely mean-reverting. The series contain a stochastic trend as well, suggesting that the partial cointegration model of Clegg and Krauss (2018) should be adopted. Accordingly, this paper constructs a hedging strategy based upon the partial cointegration relationship. Previous literature has investigated whether the derivatives on the index $VIX$ can be used as trading and risk management tools for investors and traders (see Chen et al. 2011; Jung 2016; Kim et al. 2023; among others). However, the partial cointegration between the stock index futures and the $VIX$ futures has not been taken into consideration. This paper argues that the stock index futures and the $VIX$ futures are both affected by unobservable investor sentiment and thus the price series should be modelled by the partial cointegration relationship, instead of either ignoring the long-term equilibrium (e.g., vector autoregression, $VAR$) or employing the commonly-adopted cointegration relationship (e.g., vector error correction model, $VECM$). Furthermore, Alexander and Barbosa (2007) find no evidence that complex econometric models, such as time-varying conditional covariance (e.g., generalized autoregressive conditional heteroscedasticity, $GARCH$), can improve upon the OLS (ordinary least squares) hedge ratio. Fabozzi and Fabozzi (2022) point out that employing advanced econometric models to estimate the slope coefficient offers little improvement in hedging effectiveness and even if there is some improvement, the modeling cost may not justify the extra effort. Kim et al. (2023) also find that, to hedge S&P 500 exposure with $VIX$ futures, the OLS approach shows better results than the DCC-GARCH approach. As such, using the OLS, $VAR$, and $VECM$ as baseline models, this paper focuses on the comparison of hedging performance across the OLS, $VAR$, and $VECM$, and partial-cointegration approach.³ The empirical evidence demonstrates that the partial-cointegration between the stock index futures and the $VIX$ futures is not only statistically significant but also economically significant in the hedging decision.

Second, the hedging strategy based on the patrial cointegration is compared to other popular alternative strategies in terms of hedging effectiveness over different hedge horizons and hedge timing. Sultan et al. (2019) reveal that hedgers make hedging decisions based on their preferences for a certain hedge horizon and risk tolerance. Moreover, Li and Yu (2012) and Yuan (2015) show the historical high is negatively related to its subsequent returns. Lee and Piqueira (2017) point out that short selling is positively associated with the nearness to the historical high. The timing of historical high, therefore, may affect the hedging effectiveness as well. We extend the analysis of hedging effectiveness for different hedge horizons and market timing with the partial cointegration approach. This paper sheds lights on the $VIX$ hedging literature for different hedge periods (e.g., 1-week and 12-week) and the timing of hedging strategies. The empirical results show that the partial cointegration approach improves the hedging effectiveness over other strategies as the length of the hedge period increases.⁴ This dominant effect is particularly strong when the current index futures price is close to its historical high ratio. Our empirical evidence provides useful information for $VIX$ futures hedgers with a longer time-horizon.

The rest of this paper is organized as follows: Sect. 2 presents the data. Section 3 introduces the partial cointegration modelling and hedging effectiveness measures. Section 4 reports the empirical results. We offer concluding remarks in Sect. 5.

2 Data

This paper uses the E-mini S&P 500 index futures ($SPF$), E-mini Dow Jones Industrial Average index futures ($DJF$) and $VIX$ futures $\left( {VF} \right)$ for our analysis. Because the CBOE (Chicago Board Options Exchange) introduced the $VIX$ futures in March 2004, this paper collects the daily data for the $SPF$, $DJF$, and $VF$ during the period from March 26, 2004 to January 19, 2022. The continuous price series data are collected from the Datastream International database.⁵ Following Chen et al. (2011), the weekly data are used to examine the hedging effectiveness between the stock index futures and the $VF$.⁶ This study considers data on Wednesday (Wednesday close to Wednesday close) each week for analysis (see Griffin et al. 2007; Chuang et al. 2014).

3 Methodology

3.1 Partial cointegration

Partial cointegration (Clegg and Krauss 2018) is a weaker form of cointegration and splits up the residual series into permanent and transient components. Following Clegg and Krauss (2018), a partially cointegrated system for the $SPF$ (or $DJF$) and the $VF$ is specified as follows:

$$LP_{i,w} = \beta_{i} \times LP_{VF,w} + W_{i,w} ,{ }i = SPF {\text{or }}DJF;$$

(1a)

$$W_{i,w} = M_{i,w} + R_{i,w} ;$$

(1b)

$$M_{i,w} = \rho_{i} \times M_{i,w - 1} + \epsilon_{{M_{i} ,w}} , \epsilon_{{M_{i} ,w}} \sim N\left( {0,\sigma_{{M_{i} }}^{2} } \right);$$

(1c)

$$R_{i,w} = R_{i,w - 1} + \epsilon_{{R_{i} ,w, }} \epsilon_{{R_{i} ,w }} \sim N\left( {0,\sigma_{{R_{i} }}^{2} } \right).$$

(1d)

where $LP_{i,w}$ and $LP_{VF,w}$ are the logarithms of prices for i and $VF$ at week $w$, respectively. $\beta_{i}$ is the partial cointegrating parameter. $W_{i,w}$ is the residual series which can be decomposed into two components: the permanent component, $R_{i,w}$, which is modelled as a random walk and the transient component, $M_{i,w}$, which is modelled as an autoregressive process with coefficient $\rho_{i}$. We assume $- 1 < \rho_{i} < 1$ to ensure stationarity. In addition, $\epsilon_{{M_{i} ,w}}$ and $\epsilon_{{R_{i} ,w}}$ follow mutually independent Gaussian white noise processes, with zero means and variances $\sigma_{{M_{i} }}^{2}$ and $\sigma_{{R_{i} }}^{2}$, respectively. To simplify the model estimation, this paper assumes $M_{i,0} = 0$ and $R_{i,0} = LP_{i,0} - \beta_{i} \times LP_{VF,0}$. Given the above residual decomposition, the proportion of the variance attributed to the mean-reversion can be calculated as:

$$R_{i,MR}^{2} = \frac{{Var\left( {\left[ {1 - {\text{ B}}} \right]M_{i,w} } \right)}}{{Var\left( {\left[ {1 - {\text{ B}}} \right]W_{i,w} } \right)}} = \frac{{2\sigma_{{M_{i} }}^{2} }}{{2\sigma_{{M_{i} }}^{2} + \left( {1 + \rho } \right)\sigma_{{R_{i} }}^{2} }},{ } 0 \le R_{i,MR}^{2} \le 1$$

(2)

where ${\text{ B}}$ denotes the backshift operator. To test whether the partial cointegration prevails between two nonstationary variables, the null hypothesis states there is no partial cointegration. Thus, the residual series $W_{i,w}$ is either a random walk ($H_{{0_{i} }}^{R}$) with $\sigma_{{M_{i} }}^{2} = 0$ and $R_{i,MR}^{2} = 0$; or a pure AR-process ($H_{{M_{i} }}^{R}$) with $\sigma_{{R_{i} }}^{2} = 0$ and $R_{i,MR}^{2} = 1$. The former case concludes the two variables are not cointegrated whereas the latter suggests they are cointegrated. Partial cointegartion is established only if both $H_{{0_{i} }}^{R}$ and $H_{{M_{i} }}^{R}$ are individually rejected. Clegg and Krauss (2018) suggest that the parameters of Eq. (1) can be estimated by reformulating the model in the state space framework, and, consequently, the parameter values are determined through maximum likelihood estimation along with Kalman filtering.

3.2 Hedging models

Traditionally, the optimal hedge ratio is defined as the ratio of the $VF$ position to the stock index futures position that minimizes the risk of a hedged portfolio. Conditional on the information set at week $w$, the hedged return at week $w + 1$ is calculated by using the risk-minimizing hedge ratio at week $w$. In the present study, four alternative methods are employed to calculate the optimal hedge ratios. First, the ordinary least squares ($OLS$) method is applied to derive the optimal hedge ratio over time (see Bordonado et al. 2017). The resulting $OLS$ hedge ratio corresponds to the coefficient of $R_{VF,w}$ in the following regression:

$$R_{i,w} = \varphi_{i,l} + h_{i,OLS,w}^{*} \times R_{VF,w} + \varepsilon_{i,w}$$

(3)

where $R_{i,w}$ is the stock index futures return, $i = SPF {\text{or }}DJF$, at week $w$. $R_{VF,w}$ is the $VF$ return at week $w$. $\varepsilon_{i,w}$ is the residual of the stock index futures $i$, $i = SPF {\text{or}} DJF$, at week $w$.

Second, bivariate vector autoregression ($VAR$) is used to estimate the optimal hedge ratio as follows (see Frijns et al. 2016):

$$R_{i,w} = \mu_{i,0} + \mathop \sum \limits_{j = 1}^{p} \delta_{ii,j} R_{i,w - j} + \mathop \sum \limits_{j = 1}^{p} \delta_{iVF,j} R_{VF,w - j} + \varepsilon_{i,w}$$

(4a)

$$R_{VF,w} = \mu_{VF,0} + \mathop \sum \limits_{j = 1}^{p} \delta_{VFi,j} R_{i,w - j} + \mathop \sum \limits_{j = 1}^{p} \delta_{VFVF,j} R_{VF,w - j} + \varepsilon_{VF,w }$$

(4b)

where $R_{i,w}$, $R_{VF,w}$, and $\varepsilon_{i,w}$ are analogously defined as in the $OLS$ method. $\varepsilon_{VF,w}$ is the residual of the $VF$ at week $w$. The optimal hedge ratio under the $VAR$ specification, $h_{i,VAR,w}^{*}$, is calculated as $h_{i,VAR,w}^{*} = \sigma_{{\varepsilon_{i,w} ,\varepsilon_{VF,w} }} /\sigma_{{\varepsilon_{VF,w} }}^{2}$, the covariance between $\varepsilon_{i,w}$ and $\varepsilon_{VF,w}$ divided by the variance of $\varepsilon_{VF,w}$.

Third, the bivariate vector error correction model ($VECM$) associated with $M_{i,w}$ from Eq. (1) is employed to obtain the optimal hedge ratio as (Vollmer et al. 2020)⁷:

$$R_{i,w} = \mu_{i,0} + \alpha_{i} M_{i,w - 1} + \mathop \sum \limits_{j = 1}^{p} \delta_{ii,j} R_{i,w - j} + \mathop \sum \limits_{j = 1}^{p} \delta_{iVF,j} R_{VF,w - j} + \varepsilon_{i,w}$$

(5a)

$$R_{VF,w} = \mu_{VF,0} + \alpha_{VF} M_{i,w - 1} + \mathop \sum \limits_{j = 1}^{p} \delta_{VFi,j} R_{i,w - j} + \mathop \sum \limits_{j = 1}^{p} \delta_{VFVF,j} R_{VF,w - j} + \varepsilon_{VF,w }$$

(5b)

where $R_{i,w}$, $R_{VF,w}$, $\varepsilon_{i,w}$, and $\varepsilon_{VF,w}$ are defined as in Eqs. (4a) and (4b). $M_{i,w}$ is estimated from the partially cointegrated model following Clegg and Krauss (2018) at week $w$, $i = SPF {\text{or }}DJF$. Accordingly, the optimal hedge ratio for the $VECM$ model, $h_{i,VECM,w}^{*}$, is calculated by using the residuals from the $VECM$ as $h_{i,VECM,w}^{*} = \sigma_{{\varepsilon_{i,w} ,\varepsilon_{VF,w} }} /\sigma_{{\varepsilon_{VF,w} }}^{2}$.

Finally, the optimal hedge ratio for the partial cointegration ($PCI$) system is estimated as $h_{i,PCI,w}^{*} = \beta_{i} \times \left( {LP_{VF,w} /LP_{i,w} } \right)$, $i = SPF {\text{or}} DJF$. $\beta_{i}$ is obtained from the partially cointegrated model (Clegg and Krauss 2018). $LP_{i,w}$ and $LP_{VF,w}$ are the logarithm of prices for futures i and $VF$ at week $w$. After the optimal hedge ratio is obtained at week $w$, the hedged return for week $w + 1$ is calculated as $HR_{i,l,w + 1} = R_{i,w + 1} - h_{i,l,w}^{*} \times R_{VF,w + 1}$, where $i = SPF or DJF$ and $l = OLS$, $VAR$, $VECM$, and $PCI.$

3.3 Metrics of hedging performance

The out-of-sample hedging performances of the four hedging strategies are examined with two approaches, namely downside risk reduction and utility-maximization. For the downside risk reduction, hedging performance is measured by the reduction in the downside risk of a hedged portfolio as compared with the downside risk of an unhedged portfolio. The utility-maximization approach compares the expected utility levels generated by various hedging strategies.

3.3.1 Downside risk comparison

This paper applies several downside risk measures to evaluate the hedging performance of the four hedging strategies $l$ ($l = LS, VAR, VECM, {\text{or}} PCI)$. Herein, the hedge performance is measured by the percentage of reduction in the downside risk level of a hedged portfolio ($HP$) from the downside risk level of an unhedged portfolio ($UHP$). The downside risk measures considered are semivariance (SV), lower partial moment of order 3 (LPM), value at risk (VaR), and conditional value at risk (CVaR).

The first performance metric is $HE_{i,l}^{SV}$, which is calculated as $1 - \left[ {Semivariance_{i,l}^{HP} /Semivariance_{i,l}^{UHP} } \right]$. $Semivariance_{i,l}^{HP}$ is calculated as $E\left\{ {\left( {max\left[ {0,\tau - HR_{i,l,w} } \right]} \right)^{2} } \right\}$, where $\tau$ is the target return and $HR_{i,l,w}$ is the hedged returns for stock index $i$ with the model $l$ at week $w$. $Semivariance_{i,l}^{UHP}$ is analogously calculated as $Semivariance_{i,l}^{HP}$, where the hedged return ($HR_{i,l,w}$) is replaced with the unhedged return ($RET_{i,w}$). $HE_{i,l}^{SV}$ measures the percentage of reduction in the $semivariance$ of the present hedge strategy as compared with a no-hedge position. For the measure of the down-side risk, this paper sets $\tau = 0$.

The second performance metric is $HE_{i,l}^{LPM}$, which is obtained as $1 - \left[ {LPM_{i,l}^{HP} /LPM_{i,l}^{UHP} } \right]$. $LPM$ (lower partial moment) is the generalization of semivariance where the power coefficient is generalized to any positive integer,$n$. Specifically, $LPM_{i,l}^{HP}$ is calculated as $E\left\{ {\left( {max\left[ {0,\tau - HR_{i,l,w} } \right]} \right)^{n} } \right\}$, where $\tau$ is the target return and $HR_{i,l,w}$ is the hedged returns for stock index $i$ with the model $l$ at week $w$. $LPM_{i,l}^{UHP}$ is analogously calculated as $LPM_{i,l}^{HP}$, where $RET_{i,w}$ substitutes for $HR_{i,l,w}$. This paper uses $n = 3$, which corresponds to a risk-averse investor. From a risk-management perspective the aim of a hedger is to avoid negative outcomes; therefore, a target return $\tau = 0$ is used (see Cotter and Hanly 2006). $HE_{i,l}^{LPM}$ measures the percentage of reduction in the LPM of the present hedge strategy as compared with a no-hedge position.

The third performance metric is $HE_{i,l}^{VaR}$, which is calculated as $1 - \left[ {VaR_{i,l,X}^{HP} /VaR_{i,l,X}^{UHP} } \right]$. $Value at risk$ ($VaR$) is the $\left( {100 - X} \right)$th percentile of the return distribution of the change in the asset or portfolio over the next $N$ periods. $VaR_{i,l,X}^{HP}$ is obtained as $F_{X}^{ - 1} \left( {LPM_{i,l,n = 0}^{HP} } \right) = F_{X}^{ - 1} \left( {E\left\{ {\left( {max\left[ {0,\tau - HR_{i,l,w} } \right]} \right)^{0} } \right\}} \right)$. The cumulative distribution function $F\left( {HR_{i,l,w} } \right)$ is the probability of the hedged portfolio return $HR_{i,l,w}$ being less than a given value ($\tau$) which is exogenous. Hence, $VaR_{i,l,X}^{HP}$ gives the hedged portfolio return, $HR_{i,l,w}$, that is exceeded with $\left( {100 - X} \right)\%$ probability. $VaR_{i,l,X}^{UHP}$ is calculated as $VaR_{i,l,X}^{HP}$, where $HR_{i,l,w}$ is replaced with $RET_{i,w}$. $HE_{i,l}^{VaR}$ measures the percentage of reduction in the $VaR$ of the present hedge strategy as compared with a no-hedge position. This paper uses $X = 1$.

The fourth performance metric is $HE_{i,l}^{CVaR}$, which is calculated as $1 - \left[ {CVaR_{i,l,X}^{HP} /CVaR_{i,l,X}^{UHP} } \right]$. $Conditional VaR$ ($CVaR$) measures the mean loss, conditional upon the fact that the $VaR$ has been exceeded. $CVaR_{i,l,X}^{HP}$ is calculated as a special case of $LPM_{i,l}^{HP}$ with $n = 1$ and the minimum target return $\tau$ set to be $VaR_{i,l,X}^{HP}$. $CVaR_{i,l,X}^{UHP}$ is calculated as $CVaR_{i,l,X}^{HP}$, where $LPM_{i,l}^{HP}$ and $VaR_{i,l,X}^{HP}$ are replaced with $LPM_{i,l}^{UHP}$ and $VaR_{i,l,X}^{UHP}$, respectively.

3.3.2 Utility-based comparison

Following Gagnon et al. (1998), the utility-maximization hedging performance for each stock index $i$ and model $l$ ($l = OLS$, $VAR$, $VECM$, and $PCI$) is estimated as: $Max\left[ {E\left( {HR_{i,l,w} |I_{w - 1} } \right) - 0.5 \times \emptyset \times Var\left( {HR_{i,l,w} |I_{w - 1} } \right)} \right]$, where $I_{w - 1}$ is the information set available at $w - 1$ and $\emptyset$ is the risk tolerance parameter. The utility-maximization is annualized.

A hedger makes hedging decisions stemming from the hedger’s preference for a certain hedge horizon and risk tolerance (Sultan et al. 2019). Conlon and Cotter (2012) show that hedgers with a longer time-horizon benefit from lower levels of downside risks, lower transaction costs, and higher utility. Sultan et al. (2019) indicate that hedging effectiveness with regards to the downside risk reduction metric and utility-based analysis tends to increase with the length of hedging horizon. Since the hedging performance varies with the length of the hedge period, we consider the hedging effectiveness for the 1-, 4-, 8-, and 12-week cases.⁸ Following Dong et al. (2022), this paper uses a 10-year moving window to estimate the hedge ratios and measures of the hedging effectiveness.⁹,¹⁰ The hedging performance metrics are measured from March 26, 2014 to January 19, 2022.

4 Empirical results

4.1 Summary statistics and cointegration test

Table 1 provides summary statistics of the weekly prices and returns for $SPF$, $DJF$, and $VF$. The mean (median) values of $S$ PF and $DJF$ prices are 1929.2931 and 16,926.0097 (1549.5000 and 14,101.0000), respectively. The minimum and maximum of the $SPF$ price are 705.2000 and 4784.5000, whereas the corresponding values for the $DJF$ price are 6785.0000 and 36,381.0000. The mean and median of the $VF$ price are 19.9579 and 17.7925, respectively. The minimum and maximum are 11.2400 and 70.4750, respectively. Moreover, the maximum of the $VF$ price occurs during the COVID-19 period, suggesting that uncertainty and investor fear associated with the wide stock market fluctuations may have created a strong demand for hedging against market risks. Using the Augmented Dickey-Fuller (ADF) test, we fail to reject the presence of a unit root at conventional significance levels for (the logarithms of) the $SPF$, $DJF$, and $VF$ prices. As a complement to the ADF test, following Taylor (2019), this paper also conducts the KPSS test to examine whether $SPF$, $DJF$, and $VF$ prices are stationery.¹¹ The null hypothesis for the KPSS test is that the data is level stationary, and the alternative is that the data is not level stationary. Similar to the ADF test, when applied to the original prices series, we reject the null hypothesis of stationarity. We further conclude that the prices for $SPF$, $DJF$, and $VF$ are $I\left( 1 \right)$ process. In particular, our empirical evidence is consistent with Shu and Zhang (2012), Taylor (2019), and Fernandez-Perez et al. (2019) in that the $VF$ price is in the presence of a unit root.

Table 1 Summary statistics for stock index futures and VIX futures

As shown in Table 1, the means of weekly returns for $SPF$, $DJF$, and $VF$ are 0.0015, 0.0013, and 0.0002, respectively. The corresponding standard deviations are 0.0227, 0.0222, and 0.0895, respectively. The standard deviation of the $VF$ return is four times larger than those of $SPF$ and $DJF$, suggesting that the $VF$ return is more volatile than the stock index futures return. This result is also reflected in the minimum and maximum values for $SPF$, $DJIA$, and $VF$ returns. Specifically, the minimum and maximum values are − 0.1748 and 0.1109 for $SPF$, − 0.1683 and 0.1141 for $DJF$, and − 0.3181 and 0.6027 for $VF$. Finally, the ADF (KPSS) tests reject (cannot reject) the null hypothesis of a unit root (stationarity) for the returns of $SPF$, $DJF$, and $VF$, suggesting that all three series are stationary.

Table 2 reports the cointegration tests for $SPF$ and $VF$ as well as for $DJF$ and $VF$. Panel A of Table 2 displays the $ADF$ test results for the first stage regression residuals following Engle and Granger (1987). Panel B reports the trace statistic proposed by Johansen (1988, 1991). The results show that the null hypothesis that the number of cointegrating vectors is at most zero cannot be rejected. To be specific, Panel A of Table 2 shows that the $ADF$ statistics for residuals from $LP_{SPF}$ and $LP_{VF}$ as well as $LP_{DJF}$ and $LP_{VF}$ are − 3.2116 and − 3.2114, respectively, which are insignificant at the conventional levels. Moreover, the $KPSS$ statistics for residuals from $LP_{SPF}$ and $LP_{VF}$ as well as $LP_{DJF}$ and $LP_{VF}$ are 0.8774 and 0.8828, respectively, which are significant at the 1% level. The evidence of the $ADF$ and $KPSS$ statistics suggests that the residuals from $LP_{SPF}$ and $LP_{VF}$ as well as $LP_{DJF}$ and $LP_{VF}$ are nonstationary. Panel B of Table 2 shows the trace statistics for $LP_{SPF}$ and $LP_{VF}$ as well as $LP_{DJF}$ and $LP_{VF}$, with the null hypothesis that the number of cointegrating vectors is at most zero ($H_{0} :r = 0$), are 14.29 and 13.88, respectively. Both are insignificant at the conventional levels. Figure 1 illustrates the regression residuals for $SPF$ and $VF$ (top plot, $Residual_{SPF,VF,w}$) as a pair as well as for $DJF$ and $VF$ (bottom plot, $Residual_{DJF,VF,w}$) as a pair. The residuals display a stochastic trend together with mean-reverting behavior, implying that the residuals might be subjected to the structural breaks.¹² Using multiple structural change models suggested by Bai and Perron (2003), this paper finds the existence of the structural breaks for the residuals. Specifically, as displayed in Panel A of Table 2, the $supF$ tests for $Residual_{SPF,VF,w}$ and $Residual_{DJF,VF,w}$ are 9.2048 and 8.8643, and the $supF$ statistics are all significant at the 5% level. The break dates are 2007-07-18, 2010-03-17, 2012-11-14, and 2019-05-22. In summary, the findings of Table 2 and Fig. 1 suggest that the long-term relationship between the stock index futures prices and $VF$ cannot be captured by a cointegration system.

Table 2 Cointegration tests

Fig. 1

Residual from the methodology by Engle and Granger (1987). Top plot: residuals of $SPF$ and $VF$, $Residual_{SPF,VF,w}$. Bottom plot: residuals of $DJF$ and $VF$, $Residual_{DJF,VF,w}$

4.2 Partial cointegration between the stock index futures and VIX futures

Table 3 shows whether the stock index futures prices and $VF$ prices are partially cointegrated. Panel A of Table 3 reports the parameter estimates of the partial cointegration for the logarithm of $SPF$ (or $DJF$) and $VF$ prices. For the pair of $SPF$ and $VF$, the regression coefficient $\beta_{SPF}$ is -0.2611, which is significant at the 1% level. The regression coefficient $\beta_{DJF}$ for the pair of $DJIA$ and $VF$ is significantly negative at -0.2270. The findings that both $\beta_{SP}$ and $\beta_{DJIA}$ are negative and significant at the 1% level suggest that investors can hedge downward movements in the stock market with $VF$. Moreover, Panel A of Table 3 also shows that the regression coefficients for the transient component of the residual series, $\rho_{i}$, are all significant and positive at the 1% level. Specifically, the estimated values of $\rho_{SP}$ and $\rho_{DJF}$ are 0.4242 and 0.5049, respectively. As revealed in Panel A of Table 3, the estimate for $R_{SPF,MR}^{2}$ ($R_{DJF,MR}^{2} )$ is 0.4149 (0.3427); that is, the proportion of the variance attributable to mean-reversion is 41.49% (34.27%).

Table 3 Parameter estimates and tests of partial cointegration

Panel B of Table 3 displays the test results of partial cointegration for the logarithm of $SPF$ (or $DJF$) and $VF$ prices. As indicated by Clegg and Krauss (2018), the null hypothesis of no partial cointegration consists of two conditions. First, the residual series follow a pure random walk (no cointegration), $H_{{0_{i} }}^{R}$. Second, the residual series follow a pure AR(1) process (time-invariant linear cointegration), $H_{{M_{i} }}^{R}$.

If both hypotheses, $H_{{0_{i} }}^{R}$ and $H_{{M_{i} }}^{R}$, are individually rejected, the null hypothesis of no partial cointegration is rejected. For the pair of $SPF$ and $VF$, the likelihood ratio test statistics for $H_{{0_{SP} }}^{R}$ and $H_{{M_{SP} }}^{R}$ are -9.5971 and -9.5983, respectively; both are significant at the 1% level. The $ADF$ test using the residuals $M_{SPF,w}$ and $ADF_{{M_{SPF} }}$, is -8.7216, which is significant at the 1% level. The $KPSS$ test for $KPSS_{{M_{SPF} }}$ is 0.0681, which is insignificant at the conventional level. Panel B of Table 3 shows that the likelihood ratio test statistics for $H_{{0_{DJIA} }}^{R}$, $H_{{M_{DJIA} }}^{R}$, and $ADF_{{M_{DJIA} }}$ are − 5.1867, − 5.1875, and − 8.3955, respectively; all are significant at the 1% level. $KPSS_{{M_{DJIA} }}$ is 0.0495, which is insignificant at the conventional level. In Fig. 2, we plot the extracted mean-reverting component ($M_{i,w}$) of $SPF$ and $VF$ (top plot, $M_{SPF,w}$) as well as the $DJF$ and $VF$ (bottom plot, $M_{DJF,w}$). This paper also examines the existence of the structural breaks for $M_{SPF,w}$ and $M_{DJF,w}$ and the corresponding $supF_{{M_{SPF} }}$ and $supF_{{M_{DJF} }}$ are 0.6664 and 0.2808, suggesting that there are no structural breaks for $M_{SPF,w}$ and $M_{DJF,w}$. Overall, Table 3 and Fig. 2 show that the stock index futures ($SPF$ and $DJF$) and $VF$ are partially cointegrated.

Fig. 2

Mean-reverting component, $M_{i,w}$, from the methodology by Clegg and Krauss (2018). Top plot: Mean-reverting component of $SPF$ and $VF$, $M_{SPF,w}$. Bottom plot: Mean-reverting component of $DJF$ and $VF$, $M_{DJF,w}$

4.3 Hedging performance

This section reports the results of the hedging performance of stock index futures returns ($SPF$ and $DJF$) with $VF$. The optimal hedge ratio for the stock futures index $i$ ($i = SPF {\text{or}} DJF$) when using $VF$ as the hedging instrument is denoted by $h_{i,l,w}^{*}$ for model $l$ ($l = OLS$, $VAR$, $VECM$, and $PCI$). As shown in Panel A of Table 4, for $SPF$, the means of the optimal hedge ratios ($h_{SPF,l,w}^{*}$) estimated by $OLS, VAR, VECM$, and $PCI$ are − 0.1802, − 0.1864, − 0.1851, and − 0.1955, respectively. On average, more $VF$ contracts are used by investors to hedge the volatility risk when using the $PCI$ approach. Furthermore, based on the minimum and maximum of the optimal hedge ratios, the $PCI$ hedge ratio is more volatile than other hedge ratios. For $DJF,$ the means of the optimal hedge ratios ($h_{DJF,l,w}^{*}$) under $OLS, VAR, VECM$, and $PCI$ approaches are − 0.1662, − 0.1708, − 0.1702, and − 0.1365, respectively. Unlike the $SPF$ case, the $PCI$ hedge ratio is smaller than other hedge ratios. On the other hand, the $PCI$ hedge ratio is more volatile when judged by the minimum and maximum of the hedge ratios, which is parallel to the $SPF$ case.

Table 4 Summary statistics and correlation matrix for hedge ratios

Panel B of Table 4 reports the relationships between the optimal hedge ratio and the historical high ratio of stock index futures prices. For $SPF$, the optimal hedge ratios, $h_{SPF,l,w}^{*}$, derived from $OLS, VAR$, and $VECM$ models are negatively related to the historical high ratios of $SPF$ ($HH_{SPF,w}$). That is, the optimal hedge ratios estimated by $OLS, VAR$, and $VECM$ decrease when the prices of the $SPF$ are close to the corresponding historical high, implying that the optimal hedge ratio decreases with optimistic investor sentiment (e.g., an increase in $HH_{SPF,w}$) and increases with pessimistic investor sentiment (e.g., an decrease in $HH_{SPF,w}$). By contrast, the $PCI$ hedge ratio ($h_{SP,l,w}^{*}$,$l = PCI$) is positively correlated to the historical high ratios of $SPF$ ($HH_{SPF,w}$), suggesting that the optimal hedge ratio increases with optimistic investor sentiment (e.g., an increase in $HH_{SPF,w}$) and decreases with pessimistic investor sentiment (e.g., an decrease in $HH_{SPF,w}$).

Panel B of Table 4 shows a similar result for $DJF$. More specifically, the optimal hedge ratios estimated by $OLS$, $VAR$, and $VECM$ all decrease when the prices of $DJF$ are close to the corresponding historical high. Thus, the optimal hedge ratio decreases with optimistic investor sentiment (e.g., an increase in $HH_{DJF,w}$) and increases with pessimistic investor sentiment (e.g., a decrease in $HH_{DJF,w}$). In contrast, the $PCI$ hedge ratio ($h_{DJF,l,w}^{*}$,$l = PCI$) is positively correlated to the historical high ratios of the $DJF$ ($HH_{DJF,,w}$), suggesting that the $PCI$ hedge ratio increases along with optimistic investor sentiment (e.g., an increase in $HH_{DJF,w}$) and decreases with pessimistic investor sentiment (e.g., an decrease in $HH_{DJF,w}$). In sum, the behavior of hedge ratios for the $PCI$ approach is different from the other approaches, and this finding implies hedging performance for the $PCI$ approach varies compared with other approaches. Accordingly, the hedging effectiveness of stock index futures when using $VF$ is analyzed as follows.

Table 5 reports the hedging performance for the E-mini $SPF$ using $VF$ as the hedging instrument. Panels A to D of Table 5 display the hedging performance with the hedge periods of 1-, 4-, 8-, and 12-week, respectively. Hedging performance of the four hedging strategies are examined with two approaches, namely downside risk reduction and utility-maximization. Panel A of Table 5 shows that, for the 1-week hedge period, the $PCI$ model underperforms others in terms of downside risk reductions. That is, when concerned with tail risks, the hedging performance cannot be improved upon when taking partial cointegration into consideration. For example, the reductions in semivariance from $OLS$, $VAR$, $VECM$, and $PCI$ are 0.7440, 0.7454, 0.7461, and 0.5086, respectively, showing that $PCI$ has the worst performance compared to other approaches. For the hedging performance based upon utility maximization, the $PCI$ approach generates the lowest utility regardless of the value of the investor’s risk tolerance parameter, $\varnothing$. For example, when $\varnothing =2$, the values of the utility for $OLS$, $VAR$,$VECM$, and $PCI$ are 0.1118, 0.1119, 0.1120, and 0.1063, respectively. This result presents a lowest utility for $PCI$ (${U}_{SPF,PCI}^{\varnothing =2}$) with the value at 0.1063. However, the values of utility for the unhedged portfolios are lower than those of hedged portfolios regardless of the value of the investor’s risk tolerance parameter, $\varnothing$. This finding suggests that the inclusion of $VF$ in a portfolio based on S&P 500 will increase the risk-adjusted performance of the portfolio.

Table 5 Hedging performance for the E-mini $SPF$

For longer hedge periods of 4-, 8-, and 12-week, Panels B to D of Table 5 present results contradicting the case of the 1-week hedge period illustrated in Panel A of Table 5. Because the results of Panels B and C are qualitatively similar, we focus only on Panel D. First, $PCI$ dominates other models for all for different metrics in terms of the downside risk reduction. For example, the reductions of lower partial moments ($LPM$) from $OLS$, $VAR$, $VECM$, and $PCI$ are 0.9564, 0.9610, 0.9605, and 0.9691, respectively. This finding shows the largest reduction in downside risk for $PCI$ with the value at 0.9691. For hedging performance based upon utility maximization, the $PCI$ approach generates the highest utility regardless of the value of the investor’s risk tolerance parameter, $\emptyset$. For example, when $\emptyset = 2$, the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.1198, 0.1202, 0.1202, and 0.1252, respectively. This result presents the largest utility for ${ }PCI$ ($U_{SPF,PCI}^{\emptyset = 2}$) with the value at 0.1252. Similar to the findings observed in Panel A of Table 5, the values of utility for the unhedged portfolios are lower than those of hedged portfolios regardless of the value of the investor’s risk tolerance parameter, $\emptyset$.

Table 6 presents the hedging performance of the E-mini $DJF$ using the $VF$ as the hedging instrument. Panels A to D of Table 6 display the hedging performance with the hedge periods of 1-, 4-, 8-, and 12-week, respectively. Hedging performance of the four hedging strategies are examined with downside risk reduction and utility-maximization. Panel A of Table 6 shows that, for the 1-week hedge period, the $PCI$ model underperforms others in terms of $SV$, $LPM$, $VaR$, and $CVaR$. For example, the reductions in semivariance from $OLS$, $VAR$, $VECM$, and $PCI$ are 0.6947, 0.7009, 0.7015, and 0.6817, respectively. As to the hedging performance based upon utility maximization, the $PCI$ approach generates the lowest utility regardless of the value of the investor’s risk tolerance parameter, $\varnothing$. For example, when $\varnothing =3$, the values of the utility for $OLS$, $VAR$,$VECM$, and $PCI$ are 0.0871, 0.0871, 0.0874, and 0.0866, respectively. However, as the length of the hedge period increases, the $PCI$ approach produces better hedging effectiveness than other approaches. Taking the 12-week hedge period for instance, the $PCI$ approach dominates all other models for different metrics in terms of downside risk reduction. For instance, the reductions of lower partial moments ($LPM$) for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.9128, 0.9230, 0.9230, and 0.9632, respectively. In addition, the $PCI$ approach generates the highest utility regardless of the value of the investor’s risk tolerance parameter, $\varnothing$. For example, when $\emptyset = 3$, the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.0951, 0.0954, 0.0956, and 0.0997, respectively.

Table 6 Hedging performance for the E-mini $DJF$

In conclusion, the empirical results of Tables 5 and 6 show that the $PCI$ approach dominates other models and achieves the greatest improvement of hedging effectiveness as the length of the hedge period increases. Hence, hedgers with a longer hedging horizon can benefit from lower levels of downside risk and higher levels of utility by using the $PCI$ approach to estimate optimal hedge ratios.

4.4 Hedging performance and historical high

Prior studies have shown that the psychological anchor of the historical high affects subsequent stock index returns (e.g., Li and Yu 2012; Yuan 2015). Li and Yu (2012) show that the historical high ratio is negatively related to its subsequent returns. Yuan (2015) shows that when the closing index level of the Dow Jones Industrial Average reaches a new record, individual investors sell more shares, leading to a decrease in the stock returns following Dow record events. In a similar vein, Lee and Piqueira (2017) present that short selling is positively associated with nearness to the historical high, and short sellers exploit other investors’ behavioral biases. Overall, these studies present a decline in stock index returns when the current stock index price is close to its historical high.¹³ In addition, as shown in Panel B of Table 4, the optimal hedge ratios estimated by $OLS$, $VAR$, and $VECM$ all decrease when the prices of the stock index futures, $SPF$ and $DJF$ are close to the corresponding historical high. On the contrary, the $PCI$ hedge ratios are positively correlated to the historical high ratios of the stock index futures, $SPF$ and $DJF$. Accordingly, we examine whether the hedging performance of the $PCI$ approach is better than those of the $OLS$, $VAR$, and $VECM$ approaches when the length of hedge period increases, using $VF$ as the hedging instrument.

Table 7 reports the hedging performance of the E-mini $SPF$ using $VF$ as the hedging instrument when the historical high ratio of the stock index futures, $HH_{SPF,w}$, is higher than its third quartile. Panels A to D of Table 7 present the hedging performance with the hedge periods of 1-, 4-, 8-, and 12-week, respectively. Panel A of Table 7 shows that, for the 1-week hedge period, the $PCI$ model underperforms others in terms of tail risk reduction. For example, the reduction in $CVaR$ from $OLS$, $VAR$, $VECM$, and $PCI$ are 0.4520, 0.4784, 0.4759, and 0.3322., respectively, However, for the hedging performance based upon utility maximization, the $PCI$ approach generates the highest utility regardless of the value of the investor’s risk tolerance parameter, $\emptyset$. For example, when $\emptyset = 2$, the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.2636, 0.2711, 0.2700, and 0.2764, respectively.

Table 7 Hedging performance for the E-mini $SPF$ when its historical high ratio is higher than the value of the third quartile

As to the longer hedge periods for 4-, 8-, and 12-week displayed in Panels B to D of Table 7, the result is different from the finding from the 1-week hedge period illustrated in Panel A of Table 7. Because the results of Panels B and C are qualitatively similar, we focus only on Panel D. We observe that, for the reduction in tail risk, the $PCI$ model provides the best hedging performance among all four different metrics.

For example, using $OLS$, $VAR$, $VECM$, and $PCI$ hedging strategies, the $SV$ ($CVaR$) reductions are 0.9098, 0.9203, 0.9192, and 0.9871 (0.6101, 0.6223, 0.6222, and 0.7026), respectively. As the evidence shown in Panel D of Table 5, the $SV$ ($CVaR$) reductions for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.8715, 0.8797, 0.8789, and 0.8866 (0.5498,0.5792, 0.5805, and 0.5666), respectively. As such, as compared to the evidence of Panel D for Tables 5 and 7, the percentages of improvement for risk reduction¹⁴ regarding $SV$ ($CVaR$) for $OLS$, $VAR$, $VECM$, and $PCI$ are 4.3947%, 4.6152%, 4.5853%, 11.3354% (10.9676%, 7.4413%, 7.1835%, and 24.0028%), respectively, when the historical high ratio of the stock index futures, $HH_{SPF,w}$, is higher than its third quartile. This result shows that $PCI$ has a greatest improvement for the percentage of risk reduction when the historical high ratio of the stock index futures, $HH_{SPF,w}$, is higher than its third quartile under the long horizon hedge period. For the hedging performance of utility-maximization, the $PCI$ approach provides the highest utility regardless of the value of the investor’s risk tolerance parameter, $\emptyset$. For instance, when $\emptyset = 2$, Panel D of Table 7 shows that the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.1392, 0.1432, 0.1425, and 0.1943, respectively. As shown in Panel D of Table 5, the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.1198, 0.1202, 0.1202, and 0.1252 when $\emptyset = 2$. The percentages of utility improvement for $OLS$, $VAR$, $VECM$, and $PCI$ are 16.1937%, 19.1348%, 18.5524%, and 55.1917%, respectively, when the historical high ratio of the stock index futures, $HH_{SPF,w}$, is higher than its third quartile. The evidence shows that $PCI$ has a greatest improvement for the percentage of maximum utility when the historical high ratio of the stock index futures, $HH_{SPF,w}$, is higher than its third quartile with a long horizon hedge period.

Table 8 reports the hedging performance of the E-mini $DJF$ futures with using $VF$ as the hedging instrument when the historical high ratio of the stock index futures, $HH_{DJF,w}$, is higher than its third quartile. Panels A to D display the hedging performance with the hedge periods of 1-, 4-, 8-, and 12-week, respectively. Panel A of Table 8 shows that, for 1-week hedge period, the $PCI$ model outperforms others in terms of tail risk reduction. However, for the hedging performance based upon utility maximization, the $PCI$ approach generates the lowest utility regardless of the value of the investor’s risk tolerance parameter, $\emptyset$. As to the longer hedge periods for 4-, 8-, and 12-week displayed in Panels B to D of Table 8, the empirical results show that the hedging performance of the $PCI$ approach dominates other models for all for different metrics in terms of downside risk reduction and utility-maximization.

Table 8 Hedging performance for the E-mini $DJF$ when its historical high ratio is higher than the value of the third quartile

Similarly, Tables 6 and 8 show that, as the length of hedge period increases, the $PCI$ has the greatest improvement for the percentage of risk reduction and maximum utility when the historical high ratio of the stock index futures, $HH_{DJF,w}$, is higher than its third quartile. For example, as presented in Panel D of Table 8, the $SV$ reductions for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.8171, 0.8345, 0.8346, and 0.9429, respectively, when the historical high ratio of the stock index futures, $HH_{DJF,w}$, is higher than its third quartile. As illustrated in Panel D of Table 6, the $SV$ reductions for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.7887, 0.8022, 0.8027, and 0.8570, respectively. The percentages of improvement for risk reduction regarding $SV$ for $OLS$, $VAR$, $VECM$, and $PCI$ are 3.6009%, 4.0264%, 3.9741%, and 10.0233%, respectively. Again, the evidence reveals that $PCI$ has the greatest improvement for the percentage of the risk reduction. When $\emptyset = 2$, Panel D of Table 8 shows that the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.108, 0.1121, 0.1119, and 0.1278, respectively. As compared with the evidence shown in Panel D of Table 6, the values of the maximum utility for $OLS$, $VAR$, $VECM$, and $PCI$ are 0.0991, 0.0993, 0.0995, and 0.1033, respectively, when $\emptyset = 2$. The percentages of the utility improvement for $OLS$, $VAR$, $VECM$, and $PCI$ are 8.9808%, 12.8902%, 12.4623%, and 23.7173%, respectively, when the historical high ratio of the stock index futures, $HH_{DJF,w}$, is higher than its third quartile. The evidence shows that $PCI$ has the greatest improvement for the percentage of the maximum utility.

In sum, the findings of Tables 7 and 8 are consistent with the results displayed in Tables 5 and 6 that the $PCI$ approach dominates other models as the length of the hedge period increases. Moreover, this dominance effect is particularly strong when the stock index futures price is close to its historical high. That is, hedgers with a longer hedge horizon can use the $PCI$ approach to achieve greater improvement of hedging performance when the stock futures price is near the historical high. Our findings presented in Tables 7 and 8 might be attributed to the fact that the $PCI$ hedge ratios are positively correlated to the historical high ratios of the stock index futures.

For the robustness check, this paper further examines the hedging performance when the historical high ratio of the stock index futures, $HH_{i,w}$ ($i = SPF or DJF$), is higher than its top decile. The results are reported in Table 9. Panels A1 and A2 of Table 9 report the hedging performance with hedge periods for 1- and 12-week for $SPF$. Panels B1 and B2 in Table 9 show the hedging performance with the hedge periods for 1- and 12-week for $DJF$. The empirical results show that, for both $SPF$ and $DJF,$ the $PCI$ approach dominates other models for all metrics in terms of downside risk reduction and utility-maximization as the length of the hedge period increases. Taking $SPF$ as an example, the hedging performance of the $PCI$ approach reported in Panel A2 of Table 9 is better than those of other approaches, in terms of tail risk reduction and utility-maximization. Moreover, the empirical results also reveal that the $PCI$ approach has the greatest improvement in hedging effectiveness when the stock index futures price ($SPF$ and $DJF$) is close to its historical high under the case of a longer hedge period.

Table 9 Hedging performance for stock index futures when its historical high ratio is higher than the value of the top decile

5 Conclusions

This paper investigates whether a partial equilibrium exists between stock index futures and $VF$, and then explores the effect of partial equilibrium on hedging effectiveness. Prior literature has investigated whether the volatility index $VIX$ can be used as a trading and risk management tool for investors and traders (e.g., Chen et al. 2011; Jung 2016; Bordonado et al. 2017). However, the empirical results are mixed for the viewpoints of tail risk reduction and risk-adjusted performance. Hence, this paper adds to the literature by exploring hedging performance of $VIX$ futures from the perspectives of the downside risk protection and expected utility approaches.

Previous studies have shown that the market-wide return is correlated with investor fear (e.g., Kaplanski and Levy 2010; Da, et al. 2015). As a futures market, the $VF$ serves as an instrument to hedge VIX, volatility risks. In particular, Frijns et al. (2016) argue that $VF$ has replaced $VIX$ in its role as an investor fear gauge. Consequently, stock index futures prices and $VF$ prices are both affected by investor fear. However, investor fear is not observable and only its effects are visible, such as an increase in $VIX$ or $VF$ prices (see Khuu et al. 2016). Hence, there might exist a partial cointegration system between stock index futures and $VF$ when investor fear is unobservable. As such, this paper uses the partial cointegration model suggested by Clegg and Krauss (2018) to examine whether a partial cointegration system exists between the stock index futures and $VF$. If a partial equilibrium exists between the stock index futures and $VF$, this paper further examines the hedging effectiveness of partial cointegration ($PCI$) for stock index futures. The stock index futures prices used in this paper are the E-mini S&P 500 ($SPF$) as well as the E-mini Dow Jones Industrial Average ($DJF$) index futures.

Our empirical results show that the traditional cointegration tests proposed by Engle and Granger (1987) and Johansen (1988, 1991) fail to capture a complete cointegration system for the stock index futures prices and $VF$. The estimation results by the partial cointegration model show the presence of a partial cointegration system between the stock index futures ($SPF$ and $DJF$) and $VF$. Furthermore, our empirical results show that, as compared with alternative models, the hedging effectiveness, measured by downside risk protection and risk-adjusted performance, based on the partial cointegration model is substantially improved along with the increase in the length of hedge periods. Specifically, the partial cointegration approach dominates other approaches as the length of hedge periods increases and this dominant effect is particularly strong when the current index futures price is close to its historical high ratio. Overall, our evidence indicates that hedgers with a longer time-horizon can use the partial cointegration approach to achieve the greatest improvement of tail risk reduction and risk-adjusted performance when the stock index futures price is near its historical high.

6 Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.