2.1. Minimum-variance hedge

The objective of the MV hedge is to minimize risk in terms of portfolio variance in Eq. (1):

(3)$$ \min_{b} \sigma^{2}= \min_{b} (\sigma^{2}_{{\rm S}} - 2 b \sigma_{{\rm SF}} + b^{2} \sigma^{2}_{{\rm F}}), $$

where $\sigma ^{2}_{{\rm S}}$, $\sigma ^{2}_{{\rm F}}$, and $\sigma _{{\rm SF}}$ are the variances of $r_{{\rm S}}$ and $r_{{\rm F}}$ as well as the covariance between $r_{{\rm S}}$ and $r_{{\rm F}}$. Alternatively, Ederington's [Reference Ederington15] regression is the most practical approach employed to test the MV hedge:

(4)$$ r_{{\rm S},t} = a + b r_{{\rm F},t} + \epsilon_{t}, \quad \epsilon_{t} \sim N(0, \sigma^{2}). $$

The ordinary least squares method estimates the MV hedge ratio, hedged variance, and hedged return:

(5)$$ \begin{aligned} \hat{b}_{L} & =\hat{b}_{{\rm M}}= \frac{\hat{\sigma}_{{\rm SF}}}{\hat{\sigma}^{2}_{{\rm F}}}, \\ \hat{a}_{L} & =\hat{\mu}_{{\rm M}}= \frac{\hat{\mu}_{{\rm S}}\hat{\sigma}^{2}_{{\rm F}}-\hat{\mu}_{{\rm F}}\hat{\sigma}_{{\rm SF}}}{\hat{\sigma}^{2}_{{\rm F}}},\\ \hat{\sigma}^{2}_{L} & = \frac{\sum^{n}_{t=1}[r_{{\rm S},t}-\hat{a}_{L} -\hat{b}_{L} r_{{\rm F},t}]^{2}}{n-2} = \frac{n}{n-2}\left (\frac{\hat{\sigma}^{2}_{{\rm S}}\hat{\sigma}^{2}_{{\rm F}}-\hat{\sigma}^{2}_{{\rm SF}}}{\hat{\sigma}^{2}_{{\rm F}}} \right )= \frac{n\hat{\sigma}^{2}_{{\rm M}}}{n-2}. \end{aligned} $$

Under the returns’ normality assumption, the simple regression approach also provides the sampling distributions of Eq. (5).

2.2. Mean-variance hedging

Previous studies also use intuitive expected utility maximization to determine the hedging policy based on either the quadratic utility function or the jointly normal return distribution. For example, Khoury and Martel [Reference Khoury and Martel17] maximize the expected exponential utility to determine the hedge ratio:

(6)$$ \max_{b} \left [\frac{1}{\gamma} (1- E(e^{-\gamma (r_{{\rm S}} - b r_{{\rm F}})}) ) \right]. $$

Assuming that the hedged return is normally distributed, this optimization is equivalent to maximizing MVH with a parameter $\gamma$ (see [Reference Hsin, Kuo and Lee16])Footnote ¹

(7)$$ \max_{b} Q= \max_{b} \left [\mu_{{\rm S}}-b\mu_{{\rm F}} - \frac{\gamma}{2} (\sigma^{2}_{{\rm S}}-2b \sigma_{{\rm SF}}+b^{2} \sigma^{2}_{{\rm F}} ) \right]. $$

In the MVH framework, investors consider the risk-aversion coefficient as the essential component while associating high levels of uncertainty with a greater probability of higher returns. The following results outline several well-known MVH properties for comparison with the VaR-based hedge.

Lemma 1. Assume that the MVH investor has a risk-aversion $\gamma$. The optimal MVH hedge ratio and the parametric system ($\sigma ^{2}_{U},\mu _{U}$) are given by:

(8)$$ b_{U}= b_{{\rm M}} - \frac{\mu_{{\rm F}}}{\gamma \sigma^{2}_{{\rm F}}}, \quad \sigma^{2}_{U}= \sigma^{2}_{{\rm M}} + \frac{\mu^{2}_{{\rm F}}}{\gamma^{2} \sigma^{2}_{{\rm F}}},\quad \mu_{U} = \mu_{{\rm M}}+ \frac{\mu^{2}_{{\rm F}}}{\gamma \sigma^{2}_{{\rm F}}}. $$

Moreover, the slope of the secant line reflects the steepness between the MVH and MV hedges

(9)$$ \frac{\mu_{{\rm M}}+ {\mu^{2}_{{\rm F}}}/{\gamma \sigma^{2}_{{\rm F}}}-\mu_{{\rm M}}}{\sigma^{2}_{{\rm M}} + {\mu^{2}_{{\rm F}}}/{\gamma^{2} \sigma^{2}_{{\rm F}}}-\sigma^{2}_{{\rm M}}}=\gamma. $$

As a result, the secant line between the MVH and MV hedges has an intercept in the mean-variance plane:

(10)$$ \tilde{C}= \mu_{{\rm M}} - \gamma \sigma^{2}_{{\rm M}}. $$

Equation (8) shows that the larger the $\gamma$, the closer the MVH hedge is to the MV hedge, and vice versa. The MVH investor chooses the MV hedge as $\gamma$ approaches infinity.

Note that the application of the MVH requires knowledge of assigning the risk-aversion coefficient. However, it is difficult to state how the investor's risk attitude should be in practice. Previous studies usually attribute $\gamma$ between 1 (the lowest $\gamma$) and 20 (the highest $\gamma$) to an investor. For example, Pindyck [Reference Pindyck21] p. 183 highlights that an investor's reasonable estimate of $\gamma$ should range from 3 to 4. Aït-Sahalia and Brandt [Reference Aït-Sahalia and Brandt1] p. 1316 use $\gamma =2,5,10,20$ for empirical study. DeMiguel et al. [Reference DeMiguel, Garlappi and Uppal11] p. 1944 employ $\gamma =1,2,3,4,5,10$ to examine asset pricing robustness.

On the other hand, different studies could present divergent $\gamma$ estimates for the same portfolio. For example, an often-quoted estimate of S&P500 risk-aversion is Bliss and Panigirtzoglou's [Reference Bliss and Panigirtzoglou2] p. 429 option-implied $\gamma$. Their estimates are 4.36 for the 6-week horizon and 5.22 for the 5-week horizon. Das et al. [Reference Das, Markowitz, Scheid and Statman10] also suggest a VaR-constraint approach instead of using vague risk-aversion coefficients to combat the risk-aversion estimation in portfolio selection. They suggest that investors are better at expressing their threshold return for a portfolio and maximum collapse probability below the goal than stating their risk-aversion coefficient.

3.1. VaR-based futures hedge

Roy's [Reference Roy22] safety-first rule minimizes the collapse probability with a threshold return “$h$”. Equivalently, the VaR-constraint optimization maximizes the threshold return given a collapse probability “$\alpha$” in Telser [Reference Telser27]. The objective value function of the futures hedge is given as:

(11)$$ P(r_{{\rm S}}-br_{{\rm F}} \leq h) \leq \alpha. $$

The feasible set of Eq. (11) must satisfy the following inequality:

(12)$$ P \left (\frac{r_{{\rm S}}-br_{{\rm F}}-(\mu_{{\rm S}}-b\mu_{{\rm F}})}{\sqrt{\sigma^{2}_{{\rm S}}-2b \sigma_{{\rm SF}}+b^{2} \sigma^{2}_{{\rm F}}}} \leq \frac{h-(\mu_{{\rm S}}-b\mu_{{\rm F}})}{\sqrt{\sigma^{2}_{{\rm S}}-2b \sigma_{{\rm SF}}+b^{2} \sigma^{2}_{{\rm F}}}} \right ) \leq \alpha. $$

We apply the inverse of the cumulative normal function to Eq. (12) based on the return's normality assumption:

(13)$$ h \leq (\mu_{{\rm S}}-b\mu_{{\rm F}} ) + \Phi^{{-}1}(\alpha)\sqrt{\sigma^{2}_{{\rm S}}-2b \sigma_{{\rm SF}}+b^{2} \sigma^{2}_{{\rm F}}}, $$

where $\Phi (\cdot )$ denotes the cumulative standard normal distribution function. To obtain a feasible solution for Eq. (13), we formulate the objective function of the short hedge as follows:

(14)$$ \max_{b} h= \max_{b} (\mu_{{\rm S}}-b\mu_{{\rm F}} + \Phi^{{-}1}(\alpha)\sqrt{\sigma^{2}_{{\rm S}}-2b \sigma_{{\rm SF}}+b^{2} \sigma^{2}_{{\rm F}}} ). $$

The following result presents the main result of the VaR-based hedge.

Theorem 1. The necessary and sufficient conditions (NSC) for the VaR hedge are as follows:

(15)$$ \left\{\begin{array}{ll} \displaystyle \Phi^{{-}1}(\alpha) \lt{-}\dfrac{\mu_{{\rm F}}}{\sigma_{{\rm F}}}\lt0, & \text{if the futures' return is positive } (\hat{\mu}_{{\rm F}}\gt0), \\ \displaystyle \Phi^{{-}1}(\alpha) \lt \dfrac{\mu_{{\rm F}}}{\sigma_{{\rm F}}}\lt0, & \text{if the futures' return is negative } (\hat{\mu}_{{\rm F}}\lt0). \end{array} \right . $$

Under the NSC, the optimal VaR-based hedge ratio is given by:

(16)$$ b_{{\rm V}} = b_{{\rm M}}- \sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu_{{\rm F}}}{\sigma^{2}_{{\rm F}}}, $$

where the hedged return and variance are given by:

(17)$$ \left \{\begin{aligned} & \mu_{{\rm V}} = \mu_{{\rm M}}+ \sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu^{2}_{{\rm F}}}{\sigma^{2}_{{\rm F}}}, \\ & \sigma^{2}_{{\rm V}} = \frac{\sigma^{2}_{{\rm M}}[\Phi^{{-}1}(\alpha)]^{2}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}. \end{aligned}\right . $$

Proof. The first-order condition of the VaR-based optimization (14) is equivalent to solving the following identity:

$$-\mu_{{\rm F}} \sqrt{\sigma^{2}_{{\rm S}}-2b \sigma_{{\rm SF}}+b^{2} \sigma^{2}_{{\rm F}}} + \Phi^{{-}1}(\alpha) (-\sigma_{{\rm SF}}+ b \sigma^{2}_{{\rm F}} ) =0.$$

Denoting the variance relative to $b_{{\rm V}}$ as $\sigma ^{2}_{{\rm V}}=\sigma ^{2}_{{\rm S}}-2b_{{\rm V}} \sigma _{{\rm SF}}+b^{2}_{{\rm V}} \sigma ^{2}_{{\rm F}}$, this implies that the optimal hedge ratio is an implicit function of the hedged variance:

$$b_{{\rm V}} = \frac{\Phi^{{-}1}(\alpha) \sigma_{{\rm SF}}+\mu_{{\rm F}} \sigma_{{\rm V}}}{\Phi^{{-}1}(\alpha) \sigma^{2}_{{\rm F}}} = b_{{\rm M}}+\frac{\mu_{{\rm F}} \sigma_{{\rm V}}}{\Phi^{{-}1}(\alpha) \sigma^{2}_{{\rm F}}}.$$

To obtain the optimal hedge ratio, we first need to solve the hedged variance relative to $b_{{\rm V}}$:

$$\begin{align*} \sigma^{2}_{{\rm V}} & = \sigma^{2}_{{\rm S}}-2\sigma_{{\rm SF}} \left (\frac{\Phi^{{-}1}(\alpha) \sigma_{{\rm SF}}+\mu_{{\rm F}} \sigma_{{\rm V}}}{\Phi^{{-}1}(\alpha) \sigma^{2}_{{\rm F}}} \right ) + \left (\frac{\Phi^{{-}1}(\alpha) \sigma_{{\rm SF}}+\mu_{{\rm F}} \sigma_{{\rm V}}}{\Phi^{{-}1}(\alpha) \sigma^{2}_{{\rm F}}} \right )^{2} \sigma^{2}_{{\rm F}}\\ & =\frac{\sigma^{2}_{{\rm S}}\sigma^{2}_{{\rm F}}-\sigma^{2}_{{\rm SF}}}{\sigma^{2}_{{\rm F}}} + \frac{\mu^{2}_{{\rm F}} \sigma^{2}_{{\rm V}}}{[\Phi^{{-}1}(\alpha)]^{2} \sigma^{2}_{{\rm F}}} \\ & = \sigma^{2}_{{\rm M}} + \frac{\mu^{2}_{{\rm F}} \sigma^{2}_{{\rm V}}}{[\Phi^{{-}1}(\alpha)]^{2} \sigma^{2}_{{\rm F}}}. \end{align*}$$

Solving the above equation, we obtain:

$$\sigma^{2}_{{\rm V}} = \frac{\sigma^{2}_{{\rm M}}[\Phi^{{-}1}(\alpha)]^{2}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}.$$

Under the NSC, there are two possibilities of the optimal VaR-based hedge. For case (A) with $\Phi ^{-1}(\alpha ) \lt -\mu _{{\rm F}}/\sigma _{{\rm F}}\lt0$ or $\Phi ^{-1}(\alpha ) \lt \mu _{{\rm F}}/\sigma _{{\rm F}}\lt0$, the VaR-based hedge ratio and the hedged return are reduced to:

$$b_{{\rm V}} = b_{{\rm M}}- \sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu_{{\rm F}}}{\sigma^{2}_{{\rm F}}},$$

and

$$\mu_{{\rm V}} =\mu_{{\rm S}}-b_{{\rm V}} \mu_{{\rm F}}= \mu_{{\rm M}}+ \sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu^{2}_{{\rm F}}}{\sigma^{2}_{{\rm F}}}.$$

Additionally, the second-order condition confirms the optimality since:

$$\frac{\partial^{2} h}{\partial b^{2}} = \frac{\Phi^{{-}1}(\alpha)}{\sigma^{3}_{{\rm V}}} \left [\sigma^{2}_{{\rm V}} \sigma^{2}_{{\rm F}}-\left (\frac{\mu_{{\rm F}} \sigma_{{\rm V}}}{\Phi^{{-}1}(\alpha)} \right )^{2} \right]= \frac{\sigma^{2}_{{\rm F}}[\Phi^{{-}1}(\alpha)]^{2}-\mu^{2}_{{\rm F}}}{\Phi^{{-}1}(\alpha)\sigma_{{\rm V}}} \lt 0.$$

For case (B) with $\Phi ^{-1}(\alpha ) \gt \mu _{{\rm F}}/\sigma _{{\rm F}}\gt0$ or $\Phi ^{-1}(\alpha ) \gt -\mu _{{\rm F}}/\sigma _{{\rm F}}\gt0$, then the hedge ratio is:

$$b_{{\rm V}} = b_{{\rm M}}+\sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu_{{\rm F}}}{\sigma^{2}_{{\rm F}}}.$$

However, the second-order condition is not met since:

$$\frac{\partial^{2} h}{\partial b^{2}} = \frac{\Phi^{{-}1}(\alpha)}{\sigma^{3}_{{\rm V}}} \left [\sigma^{2}_{{\rm V}} \sigma^{2}_{{\rm F}}-\left (\frac{\mu_{{\rm F}} \sigma_{{\rm V}}}{\Phi^{{-}1}(\alpha)} \right )^{2} \right]= \frac{\sigma^{2}_{{\rm F}}[\Phi^{{-}1}(\alpha)]^{2}-\mu^{2}_{{\rm F}}}{\Phi^{{-}1}(\alpha)\sigma_{{\rm V}}} \gt 0.$$

This completes the proof.

Comparing the MVH variance in Eqs. (8) to (17), we observe the equivalence between the VaR-based and MVH hedges:

$$\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}= \frac{1}{\gamma^{2}}.$$

Solving this identity, we obtain the inverse relation between $\alpha$ and $\gamma$:

(18)$$ \gamma = \sqrt{\frac{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}{\sigma^{2}_{{\rm M}}}} . $$

More importantly, the threshold return $h$ relative to $b_{{\rm V}}$ is linked to the intercept of the line passing through the MV and VaR-based hedges.

(19)$$\begin{align} h & = (\mu_{{\rm S}}-b_{{\rm V}} \mu_{{\rm F}} + \Phi^{{-}1}(\alpha)\sqrt{\sigma^{2}_{{\rm S}}-2b_{{\rm V}} \sigma_{{\rm SF}}+b^{2}_{{\rm V}}\sigma^{2}_{{\rm F}}} ) \nonumber\\ & = \mu_{{\rm M}}+ \sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu^{2}_{{\rm F}}}{\sigma^{2}_{{\rm F}}} + \Phi^{{-}1}(\alpha) \sqrt{\frac{\sigma^{2}_{{\rm M}}[\Phi^{{-}1}(\alpha)]^{2}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}}\nonumber\\ & = \mu_{{\rm M}}+ \sqrt{\frac{\sigma^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times (\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}} -[\Phi^{{-}1}(\alpha)]^{2} ) \nonumber\\ & = \mu_{{\rm M}} - \sqrt{\frac{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}{\sigma^{2}_{{\rm M}}}} \times \sigma^{2}_{{\rm M}} \nonumber\\ & = \mu_{{\rm M}} - \gamma \sigma^{2}_{{\rm M}} \nonumber\\ & = \tilde{C} .\end{align}$$

To understand the VaR-based hedge more fully, we now graph the VaR-based hedge geometry. Substituting different values of $\alpha$ into a parametric system (17) is equivalent to incorporating $\gamma$ into a parametric system (8), and Figure 1 shows the VaR-based hedge boundary. Note that the curve's vertex is the MV hedge with infinite risk-aversion. There are several implications about the hedge boundary. First, there are two possibilities for the futures positions. For the case of positive (negative) futures returns, it is intuitive that the VaR-based investor will short fewer (more) futures hedges than the MV hedge. On the other hand, Eq. (17) indicates that a higher reward (in terms of $\mu ^{2}_{{\rm F}}/ \sigma ^{2}_{{\rm F}}$) leads the VaR-based hedge away from the MV hedge but also increases the portfolio's risk ($\mu ^{2}_{{\rm F}}/\gamma ^{2} \sigma ^{2}_{{\rm F}}$).

Figure 1. The VaR-based hedge boundary. This figure depicts the equivalence between the VaR-based hedge and the MVH. The MV hedge with infinite risk-aversion localizes at the curve vertex. The MVH hedge ($\gamma _{0}$) and the MVH hedge ($\gamma _{1}$) denote the hedges in which the investor establishes the futures hedges using $\gamma _{0}$ and $\gamma _{1}$, respectively. The slope of the line connecting the MV hedge and the MVH hedge ($\gamma _{0}$) equals the risk-aversion coefficient $\gamma _{0}$ with a threshold return $h_{0}$. Similarly, the slope of the line connecting the MV hedge and the MVH hedge ($\gamma _{1}$) is equal to the risk-aversion coefficient $\gamma _{1}$ with a threshold return $h_{1}$. It is evident that the smoother the line connecting the MV and MVH hedges is, the lower the risk-aversion coefficient associated with the investor, the higher the risk tolerance, and the farther away the MVH hedge from the MV hedge, and vice versa.

Second, as mentioned earlier, VaR-based (MVH) investors may take a significant non-MV hedge with speculative positions. However, it is well known that transaction costs always impact hedging effectiveness. Because a particular VaR-based (MVH) hedge corresponds to one unique hedge ratio, for simplicity, we consider its location on the hedged boundary as the representative hedged cost. Following the usual analysis of the hedged boundary, we employ the comparative risk-return approach to analyze the hedged impact. In this vein, Eqs. (18) and (19) show the inverse relation between $\alpha$ and $\gamma$. Figure 1 shows that the more aggressive VaR-based hedge ($\alpha _{1}$) is equivalent to an MVH hedge ($\gamma _{1}$), in which the slope of the secant line between the MVH and MV hedges equals the risk-aversion coefficient ($\gamma _{1}$). The value of $\gamma$ also reflects the relative risk-return hedging performance: the steeper the secant line, the more efficient the VaR-based (MVH) hedge is. Likewise, compared to the aggressive VaR-based hedge ($\alpha _{1}$), the MVH hedge ($\gamma _{0}$) has a higher risk-aversion coefficient ($\gamma _{0} \gt \gamma _{1}$). In this situation, the conservative investor would build the VaR-based hedge ($\alpha _{0}$) with the lower risk tolerance parameter ($\alpha _{0}\lt \alpha _{1}$) and the lower threshold return ($h_{0}\lt h_{1}$) since:

(20)$$\begin{align} \alpha_{0}& = P(r_{{\rm S}}-br_{{\rm F}} \leq \mu_{{\rm M}} - \gamma_{0} \sigma^{2}_{{\rm M}})\nonumber\\ & = P(r_{{\rm S}}-br_{{\rm F}} \leq h_{0}) \nonumber\\ & \leq P(r_{{\rm S}}-br_{{\rm F}} \leq h_{1}) \nonumber\\ & = P(r_{{\rm S}}-br_{{\rm F}} \leq \mu_{{\rm M}} - \gamma_{1} \sigma^{2}_{{\rm M}})\nonumber\\ & = \alpha_{1} .\end{align}$$

Although the MV hedge is popular in reality from the perspective of risk reduction, futures also allow investors to take a riskier position for speculative purposes. Equation (20) indicates that the greater the risk-aversion coefficient, the more conservative the VaR-based (MVH) investor is, and the lower the collapse probability given a lower threshold return. In other words, both the return of the VaR-based hedge ($\gamma _{1}$) and its overall exposure to risk increase when the investor takes a speculative non-MV position with a generous amount of leverage. Nevertheless, its corresponding return would not offset the higher hedged risk such that the hedged boundary presents the concave-down property. Using the speculative hedge leads to a relatively inefficient risk-return performance, and the tradeoff depends on the investor's degree of risk aversion. This paper transfers the subjective $\gamma$'s determination into the estimated VaR-based judgment.

4.1. The hedge ratio test

The hedge ratio represents the hedged cost that also measures hedged effectiveness.Footnote ² Thus, we are first concerned about the equality test of hedge ratios over different collapse probabilities. Note that the VaR-based hedge ratio in Eq. (16) could be more or less than the MV hedge ratio. If futures returns are positive, we could possibly build a VaR-based hedge that uses a lower hedge ratio but is statistically equivalent to the MV hedge. In this case, the hedges other than the MV hedge always have smaller estimates of the hedge ratio. Thus, hypothesis (21) is given by:

(22)$$ H_{A0}:b=b_{0} \quad {\rm versus}\quad H_{A1}:b \gt b_{0}, \quad \text{if } \hat{\mu}_{{\rm F}}\gt0. $$

In another circumstance, the return is negative, and we build a VaR-based hedge that costs more but is statistically equivalent to the MV hedge. The hypothesis (21) is rewritten as:

(23)$$ H_{A0}:b=b_{0} \quad {\rm versus}\quad H_{A1}:b \lt b_{0}, \quad \text{if } \hat{\mu}_{{\rm F}}\lt0. $$

The following result summarizes the significance test based on the hedge ratio.

Theorem 2. To test the VaR-based hedge ratio in (22)–(23), the insignificant boundary of $\alpha$ is given as:

(24)$$ \alpha \leq \left \{\begin{array}{ll} \displaystyle \alpha^{*} \equiv \Phi \left (- \sqrt{\dfrac{n-2+t^{2}_{n-2;\delta/2}}{ t^{2}_{n-2;\delta/2}}} \times \dfrac{\hat{\mu}_{{\rm F}}}{\hat{\sigma}_{{\rm F}}} \right ), & \text{if } \hat{\mu}_{{\rm F}}\gt0,\\ \displaystyle \alpha^{*} \equiv \Phi \left (\sqrt{\dfrac{n-2+t^{2}_{n-2;\delta/2}}{ t^{2}_{n-2;\delta/2}}} \times \dfrac{\hat{\mu}_{{\rm F}}}{\hat{\sigma}_{{\rm F}}} \right ), & \text{if } \hat{\mu}_{{\rm F}}\lt0. \end{array} \right. $$

Proof. The standard regression theory (see [Reference Draper and Smith12]) provides the general sampling distribution for characterizing the regression coefficient's $t$-statistic with $n-2$ degrees of freedom:

$$t = \frac{\hat{b}_{L}-b_{0}}{\sqrt{\hat{\sigma}^{2}_{L}/\sum^{n}_{t=1}(r_{{\rm F},t} - \hat{\mu}_{{\rm F}})^{2}}} \sim t(n-2).$$

Thus, the endpoints of the $(1-\delta )\times 100 \%$ confidence interval for the hedge ratio are:

$$\hat{b}_{L} \pm t_{n-2;\delta/2}\sqrt{\frac{\hat{\sigma}^{2}_{L}}{\sum^{n}_{t=1}(r_{{\rm F},t} - \hat{\mu}_{{\rm F}})^{2}}} = \hat{b}_{L} \pm t_{n-2;\delta/2}\sqrt{\frac{\hat{\sigma}^{2}_{{\rm M}}}{(n-2)\hat{\sigma}^{2}_{{\rm F}}}} .$$

Setting the VaR-based hedge ratio to be less than the confidence endpoint:

$$\sqrt{\frac{\hat{\sigma}^{2}_{{\rm M}}(\mu^{2}_{{\rm F}}/\sigma^{4}_{{\rm F}})}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \leq t_{n-2;\delta/2}\sqrt{\frac{\hat{\sigma}^{2}_{{\rm M}}}{(n-2)\hat{\sigma}^{2}_{{\rm F}}}}.$$

Straightforward rearrangement of this inequality yields a range of $\alpha$ in Eq. (24) over which the VaR-based hedge ratio is significantly different from the MV hedge ratio.

4.2. The hedged variance test

We next focus on the equality test from the risk-reduction view. Because the hedges other than the MV hedge always have a larger hedged variance, we assess whether the VaR-based hedge significantly increases the amount of variance compared with the MV hedge. In this case, the null hypothesis (14) is reduced to:

(25)$$ H_{B0}:\sigma^{2}=\sigma^{2}_{0} \quad {\rm versus}\quad H_{B1}:\sigma^{2}\lt\sigma^{2}_{0}. $$

The following result summarizes the significance test based on the hedged variance.

Theorem 3. To test the VaR-based hedged variance (16), the insignificant boundary of $\alpha$ relative to the $(1-\delta )\times 100 \%$ confidence level is given as:

(26)$$ \alpha \leq \left \{ \begin{array}{ll} \displaystyle \alpha^{**} \equiv \Phi\left (- \dfrac{\hat{\mu}_{{\rm F}}}{\hat{\sigma}_{{\rm F}}}\sqrt{\dfrac{n}{n-\chi^{2}_{n-2; (1-\delta/2)}}} \right ), & \text{if } \hat{\mu}_{{\rm F}}\gt0,\\ \displaystyle \alpha^{**} \equiv \Phi\left (\dfrac{\hat{\mu}_{{\rm F}}}{\hat{\sigma}_{{\rm F}}}\sqrt{\dfrac{n}{n-\chi^{2}_{n-2; (1-\delta/2)}}} \right ), & \text{if } \hat{\mu}_{{\rm F}}\lt0. \end{array} \right. $$

Proof. Seber [Reference Seber25] p. 97 indicates that the sampling distribution of $\hat {\sigma }^{2}_{L}$ is of the form:

$$\frac{(n-2)\hat{\sigma}^{2}_{L}}{\sigma^{2}}= \frac{n\hat{\sigma}^{2}_{{\rm M}}}{\sigma^{2}} \sim \chi^{2}_{n-2}.$$

Thus, the $(1-\delta )\times 100 \%$ confidence interval for the hedged variance is:

$$\left [\frac{n\hat{\sigma}^{2}_{{\rm M}}}{\chi^{2}_{n-2; \delta/2}}, \frac{n\hat{\sigma}^{2}_{{\rm M}}}{\chi^{2}_{n-2; (1-\delta/2)}} \right] .$$

The VaR-based hedged variance varies according to the value of $\alpha$. We evaluate the statistical insignificance using the $\chi ^{2}$ statistic, which tests the equality of the MV and VaR-based hedge variances. Assume that the VaR-based hedge's variance $\hat {\sigma }^{2}_{{\rm V}}$ must be less than the upper bound of the confidence interval:

$$\frac{\hat{\sigma}^{2}_{{\rm M}}[\Phi^{{-}1}(\alpha)]^{2}}{[\Phi^{{-}1}(\alpha)]^{2} -(\hat{\mu}^{2}_{{\rm F}}/\hat{\sigma}^{2}_{{\rm F}})} \leq \frac{n\hat{\sigma}^{2}_{{\rm M}}}{\chi^{2}_{n-2; (1-\delta/2)}}.$$

Solving this inequality, we obtain a range of $\alpha$ in Eq. (26) over which the VaR-based hedged variance is significantly different from the MV hedged variance.

4.3. The hedged return test

Additionally, we intuitively test the equivalence based on the hedged returns. Because the MVH is located at the upper hedge boundary, we assess whether the MVH will significantly increase the return compared to the MV hedge. In this case, hypothesis (14) is reduced to:

(27)$$ H_{B0}:\mu=\mu_{0} \quad {\rm versus}\quad H_{B1}:\mu\lt\mu_{0}. $$

The following result summarizes the significance test based on the hedged return.

Theorem 4. To test the VaR-based hedged variance (27), the insignificant boundary of $\alpha$ is given as:

(28)$$ \alpha \leq \left \{ \begin{array}{ll} \displaystyle \alpha^{***} \equiv \Phi \left (- \sqrt{\dfrac{ t^{2}_{n-2;\delta/2} (1+{\mu^{2}_{{\rm F}}}/{\sigma^{2}_{{\rm F}}} )+(n-2)({\mu^{2}_{{\rm F}}}/{\sigma^{2}_{{\rm F}}} )}{t^{2}_{n-2;\delta/2}(1+{\mu^{2}_{{\rm F}}}/{\sigma^{2}_{{\rm F}}} )}} \times \dfrac{\hat{\mu}_{{\rm F}}}{\hat{\sigma}_{{\rm F}}} \right ), & \text{if } \hat{\mu}_{{\rm F}}\gt0,\\ \displaystyle \alpha^{***} \equiv \Phi \left (\sqrt{\dfrac{ t^{2}_{n-2;\delta/2} (1+{\mu^{2}_{{\rm F}}}/{\sigma^{2}_{{\rm F}}} )+(n-2)({\mu^{2}_{{\rm F}}}/{\sigma^{2}_{{\rm F}}} )}{t^{2}_{n-2;\delta/2}(1+{\mu^{2}_{{\rm F}}}/{\sigma^{2}_{{\rm F}}} )}} \times \dfrac{\hat{\mu}_{{\rm F}}}{\hat{\sigma}_{{\rm F}}} \right ), & \text{if } \hat{\mu}_{{\rm F}}\gt0. \end{array} \right. $$

Proof. The simple regression theory (see [Reference Draper and Smith12] p. 27) characterizes that the regression coefficient is t-distributed with $n-2$ degrees of freedom:

$$\frac{\hat{a}_{L}-a}{\sqrt{\hat{\sigma}^{2}_{L}[(1/n)+(\hat{\mu}^{2}_{{\rm F}}/\sum^{n}_{t=1}(r_{{\rm F},t} - \hat{\mu}_{{\rm F}})^{2}})]} \sim t(n-2).$$

Thus, the endpoints of the $(1-\delta )\times 100 \%$ confidence interval for the hedged return are given by:

$$\hat{a}_{L} \pm t_{n-2;\delta/2}\sqrt{\hat{\sigma}^{2}_{L}\left [\frac{1}{n}+\frac{\hat{\mu}^{2}_{{\rm F}}}{\sum^{n}_{t=1}(r_{{\rm F},t} - \hat{\mu}_{{\rm F}})^{2}} \right]} = \hat{\mu}_{{\rm M}} \pm t_{n-2;\delta/2}\sqrt{\frac{\hat{\sigma}^{2}_{{\rm M}}}{n-2}\left (1+\frac{\hat{\mu}^{2}_{{\rm F}}}{\hat{\sigma}^{2}_{{\rm F}}} \right )}.$$

Setting the VaR-based hedge's return to be less than the confidence endpoint:

$$\sqrt{\frac{\hat{\sigma}^{2}_{{\rm M}}}{[\Phi^{{-}1}(\alpha)]^{2} -(\mu^{2}_{{\rm F}}/\sigma^{2}_{{\rm F}})}} \times \frac{\mu^{2}_{{\rm F}}}{\sigma^{2}_{{\rm F}}} \leq t_{n-2;\delta/2}\sqrt{\frac{\hat{\sigma}^{2}_{{\rm M}}}{n-2}\left (1+\frac{\hat{\mu}^{2}_{{\rm F}}}{\hat{\sigma}^{2}_{{\rm F}}} \right )}$$

and solving, we obtain a range of $\alpha$ in Eq. (28).

5.1. Estimation

We begin by providing an overall view of the estimated parameters. The full estimation period 2000–2019 is decomposed into two subperiods. The subperiods 2000–2009 and 2010–2019 have a total of 522 weekly returns for each index and index futures. All of the historical returns are downloaded from the website, https://investing.com/indices/indices-futures. Table 1 reports the summary statistics. The results show that both stock index and index futures increase from negative weekly returns during the 2000–2009 subperiod to positive weekly returns during the 2010–2019 subperiod. In particular, the 2000–2009 subperiod has a relatively small squared IR $(-0.0055)^{2}$ compared to the 2010–2019 subperiod's squared IR $(0.1150^{2})$.

Table 1. Summary statistics.

This table reports the key statistics for the S&P500 stock index and the corresponding index futures. It decomposes the estimation period into two subsamples. Specifically, both stock index and index futures increase from negative weekly returns during the 2000–2009 subperiod to positive weekly returns during the 2010–2019 subperiod. All the sample data are downloaded from the website, https://investing.com/indices/indices-futures.

To investigate the impact of futures’ IR, Table 2 illustrates the safety-first hedge ratio, return, variance, and risk-aversion coefficient at $5\%$, $10\%$, $20\%$, and $30\%$ collapse probability. Panel A shows that the safety-first hedge ratio is an increasing function of $\alpha$ when the 2000–2009 subperiod has negative futures returns. In this case, the safety-first hedge has more spot positions since the futures return brings an inverse effect to the hedged portfolio. On the other hand, Panel B indicates that the safety-first hedge ratio is a decreasing function of $\alpha$ when the 2010–2019 subperiod has positive futures returns. Panel C has a similar result as Panel B. In addition, Panel B shows that the $\gamma$ at the $5\%$, $10\%$ collapse probability is close to Bliss and Panigirtzoglou's estimates of 4.36 and 5.22.

Table 2. Estimated safety-first hedges at different collapse probabilities.

This table illustrates the S&P500 safety-first hedge for distinct collapse probabilities. Panel A shows that the safety-first hedge ratio is an increasing (decreasing) function of $\alpha$ ($\gamma$) when the futures return is negative. Panel B indicates that the safety-first hedge ratio is a decreasing (increasing) function of $\alpha$ ($\gamma$) when the futures return is positive. Panel C reports the full sample results that are similar to Panel B.

5.2. Significant test between the MV and safety-first hedges

To compute the bound of $\alpha$ in which the safety-first hedges are statistically equivalent to the MV hedge, this section performs significance tests. Table 3 reports the estimated collapse probability bounds at the $5\%$ significance level. The corresponding threshold returns ($h$) and risk-aversion coefficients ($\gamma$) are reported in parentheses and brackets, respectively. For the case of the 2000–2009 sample, column 2 reports the results using the hedge ratio $P(r_{p} \leq -0.0314)=47.464\%$ and $\gamma =0.1259$; column 3 reports the results using the hedged variance $P(r_{p} \leq -0.0069)=49.375\%$ and $\gamma =0.0292$; column 4 reports the results based on the hedged return $P(r_{p} \leq -0.0003)=49.782\%$ and $\gamma =0.0007$. In this subsample, there are wide regions in which the $\alpha$ bounds (and $\gamma$ bounds) of the safety-first hedge could be insignificantly different from the MV hedge.

Table 3. Upper bounds of insignificant collapse probability.

This table estimates the parameters given the $95\%$ confidence level ($\delta =5\%$). The upper bound of the collapse probability using the hedge ratio, hedged variance, and hedged return tests are computed according to Eqs. (24), (26), and (28). The corresponding threshold returns and risk-aversion coefficients are reported in parentheses and brackets, respectively. Taking the 2000–2009 hedge ratio test as an example, column 2 reports that $P(r_{p} \leq -0.0314)=47.464\%$ with a risk-aversion coefficient $\gamma =0.1259$. In all cases, the hedge ratio test yields the narrowest region in which the safety-first is statistically equivalent to the MV hedge. Then, the hedged variance test gives a wider region, and the hedged return test produces the widest insignificant region.

For the case of the 2010–2019 subsample, column 2 reports the results using the hedge ratio $P(r_{p} \leq -0.4205)=9.022\%$ and $\gamma =4.1945$; column 3 reports the results using the hedged variance $P(r_{p} \leq -0.0945)=37.064\%$ and $\gamma =0.9731$; column 4 reports the results based on the hedged return $P(r_{p} \leq -0.0456)=42.331\%$ and $\gamma =0.4891$. In this subsample, there are large differences in the three insignificant $\alpha$ bounds (and $\gamma$ bounds). Note that the results also depend on the sample size, and the results of the full period 2000–2019 are between two subsamples.

To deliver the results visually in the return-variance space, Figure 2 plots the sample hedged boundary and compares the insignificant hedged boundary from the hedge ratio test with boundaries from the hedged variance as well as return tests. We also add the conventional $5\%$ significance level lines in Figure 2, in which the lower parts of the sample hedged boundary below the $5\%$ significance line represent the statistical equivalence between MV and safety-first hedges corresponding to different tests. Figure 2(A) plots the hedged return in the $10^{-3}$ scale that indicates the small differences of the $5\%$ significance lines between tests. Thus, there are wide regions such that the $\alpha$ bounds (ranging from $47.464\%$ to $49.782\%$) of the safety-first hedge could be statistically equivalent to the MV hedge. In Figure 2(B), the return magnitude has been transformed to $10^{-2}$. This straightens out the graph and yields $\alpha$ bounds ranging between $9.022\%$ and $42.331\%$. There are large differences in the three upper bounds of $\alpha$ and $\gamma$. Note that the upper bounds of $\alpha$ and $\gamma$ using the hedge ratio test change drastically. This difference is primarily driven by the squared information ratio of the futures return.

Figure 2. Insignificant $\alpha$ bounds of safety-first hedges. This figure depicts the VaR-based hedges that are statistically equivalent to the MV hedge based on various significance tests. The $5\%$ significance level lines indicate that the lower parts of the sample hedged boundary below the line represent the statistical equivalence between the MV and VaR-based hedges. (A) reports the small differences of the $5\%$ significance lines ranging from $\alpha ^{*}=47.464\%$ to $\alpha ^{***}=49.782\%$. In (B), the return magnitude has been transformed to $10^{-2}$. This yields $\alpha$ bounds ranging between $\alpha ^{*}=9.022\%$ and $\alpha ^{***}=42.331\%$. Note that the difference is primarily driven by the squared information ratio of the futures return.

In sum, we have two observations. First, the hedge ratio test yields the narrowest region in which the safety-first hedge is statistically equivalent to the MV hedge among all the cases. Then, the hedged variance test gives a wider region and the hedged return test produces the widest insignificant region of $\alpha$. Second, we observe that the upper bounds of insignificant collapse probabilities depend critically on the squared information ratio of the futures return. All the tests indicate that the safety-first hedge with a broad range of collapse probabilities could be insignificantly different from the MV hedge.