# Uncovering a positive risk-return relation: the role of implied volatility index

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## Abstract

We report empirical evidence suggesting a strong and positive risk-return relation for the daily S&P 100 market index if the implied volatility index is included as an exogenous variable in the conditional variance equation. This result holds for alternative GARCH specifications and conditional distributions. Monte Carlo evidence suggests that if implied volatility is not included, whilst is should be, the risk-return relation is more likely to be negative or weak.

## Keywords

S&P 100 Implied volatility index GARCH-M Risk-return relation## JEL Classification

G12 C22## 1 Introduction

This paper unveils the importance of the CBOE implied volatility index in uncovering a strong and positive risk-return relation for the daily S&P 100 market index within the framework of GARCH-in-Mean (GARCH-M) models. On one hand, empirical evidence on the risk-return relation is conflicting, with recent studies finding a weak or negative relation (Goyal and Santa-Clara 2003; Lettau and Ludvigson 2003).^{1} On the other hand, there is evidence supporting the ability of implied volatility to predict future volatility, highlighting the information content of implied volatility (Guo and Whitelaw 2006; Day and Lewis 1992). This paper is an attempt to bridge these two pieces of literature: by adding the implied volatility index (*IVI*) as an exogenous variable in conditional variance, it explores whether the incremental information content of *IVI* is useful in uncovering a strong and positive risk-return relation.

We consider 3 alternative GARCH specifications which have been employed by previous studies (Guo and Neely 2008; Lundblad 2007), including the standard GARCH, the Exponential GARCH (Nelson 1991) (EGARCH), and the Component GARCH (CGARCH). Following the findings of Baillie and DeGennaro (1990) that the risk-return relation is dependent on the distribution of returns, we consider 3 alternative distributions including the Normal, the *t*-student, and the Generalized Error Distribution (GED). We find that without *IVI* in conditional variance, the risk-return relation is weak. After allowing for *IVI*, all specifications yield a strong relation regardless of the distribution considered. Monte Carlo evidence suggests that if *IVI* is not included, the risk-return relation is more likely to be negative or weak. Our results are consistent with the contention that, as conditional variance is unobservable, the risk-return parameter is poorly measured thereby often yielding a weak relation (Guo and Whitelaw 2006).^{2} As implied volatility is a measure of market expectations of future risk, it embodies important forward-looking information which, when allowed for in the conditional variance equation, significantly improves upon the measurement of conditional variance and the risk-return parameter, thereby uncovering a strong risk-return relation.

## 2 Econometric specifications and data

*r*

_{ t }is the excess market returns,

*h*

_{ t }is the conditional variance, λ is the risk-return parameter (GARCH-in-Mean parameter). Equation (1) is the conditional mean equation. To examine the robustness of our estimation results to alternative distributional assumptions (Baillie and DeGennaro 1990), we consider that

*z*

_{ t }follows a

*t*-distribution, the GED, and the Normal.

*IVI*is being included as an exogenous variable:

In the above conditional variance specifications, *h* _{ t } denotes the conditional variance measure, and ε_{ t }, and *z* _{ t } as defined in (1). The GARCH model in (2) is the parsimonious conditional variance equation, with (α + β) measuring volatility persistence. The EGARCH specification captures the leverage (asymmetry) effect of ‘good’ and ‘bad’ news to stock returns, reflected in parameter θ. A significant body of literature (Hamilton 1994, page 672) supports the use of EGARCH over competing models (e.g. the GJR-GARCH of Glosten et al. 1993), as one of the features of EGARCH is the log form of conditional variance which ensures its non-negativity. In CGARCH, *q* _{ t } and (*h* _{ t } − *q* _{ t }) refer to the ‘long-run’ (trend) and the ‘short-run’ components respectively. As a special case, the CGARCH(1,1) specification reduces to the parsimonious GARCH(1,1) if α = β = 0. In all specifications, parameter *g* captures the statistical significance of *IVI* in the conditional variance equation.

*VIX*) from 2 January 1986 to 31 December 2008.

^{3}

*VIX*is used to measure

*IVI*. We use the S&P 100 index return (defined as the first logarithmic difference of the index level) and the 1 month risk free rate to construct excess returns (return on S&P 100 minus the risk free rate). Figure 1 plots the two series. The first graph plots the returns series and the second graph plots the level of the

*VIX*series over the sample period. Descriptive statistics, reported in Table 1, suggest that both series depart from normality on the basis of the Jarque–Bera normality test. Further, using the DF-GLS unit root test of Elliott et al. (1996), we find a test statistic of −2.70 (with the 5 % critical value being −1.9411) and thus, we conclude that the excess returns series is stationary.

Descriptive statistics

Excess returns | VIX | |
---|---|---|

Mean | −0.017062 | 21.00069 |

Std. Dev. | 0.014257 | 9.190209 |

Skewness | −0.709872 | 3.052879 |

Kurtosis | 22.73182 | 24.67120 |

JB | 94,611.38* | 122,548.2* |

Observations | 5,802 | 5,802 |

## 3 Empirical results

*IVI*in the conditional variance equation are reported in Table 2. This Table reports the estimated parameters of both the conditional mean and the conditional variance equations under the alternative GARCH specifications discussed above. Based on likelihood ratio (LR) tests of alternative lags, a lag length of 3 is chosen for the conditional mean equation. Statistical inference is based on robust standard errors (Bollerslev and Wooldridge 1992). As shown in this Table, in 1 case (EGARCH-M with the

*t*-distribution), parameter λ is negative and statistically significant at the 5 % level, reflecting a negative risk-return relation In all other 8 cases, λ takes values ranging from −0.002 (for the EGARCH-M with GED) to 1.91 (for the CGARCH-M with the Normal distribution), and is statistically insignificant. Thus, λ is not positive and statistically significant in any of the 9 models estimated. These results indicate that the risk-return relation for the S&P 100 is either negative or weak, echoeing the findings of several previous studies (Goyal and Santa-Clara 2003; Lettau and Ludvigson 2003). An interpretation of these findings is that, failure to allow for

*IVI*in the conditional variance equation produces an inability of GARCH-M models to reveal a positive and strong risk-return relation. The conditional variance parameters provide evidence of variance persistence and asymmetry, and diagnostics based on the Ljung–Box test of order 36 [LB(36)] on squared standardized residuals show that there are no remaining nonlinearities.

Models without the implied volatility index in the conditional variance equation

GARCH-M | EGARCH-M | CGARCH-M | |||||||
---|---|---|---|---|---|---|---|---|---|

Normal | t-distribution | GED | Normal | t-distribution | GED | Normal | t-distribution | GED | |

Conditional mean equation parameters | |||||||||

| −0.006** (−17.60) | −0.006** (−18.52) | −0.006** (−19.16) | −0.01** (−19.74) | −0.01** (−7.11) | −0.01** (−19.65) | −0.01** (−17.98) | −0.01** (−18.19) | −0.01** (−18.77) |

| 0.21** (15.40) | 0.21** (15.58) | 0.20** (15.38) | 0.22** (16.32) | 0.20** (15.60) | 0.20** (15.58) | 0.21** (16.19) | 0.21** (15.38) | 0.21** (15.93) |

| 0.22** (16.86) | 0.22** (16.80) | 0.22** (16.98) | 0.23** (18.37) | 0.22** (16.93) | 0.22** (17.34) | 0.22** (17.16) | 0.22** (16.58) | 0.22** (16.77) |

| 0.23** (16.32) | 0.23** (16.90) | 0.22** (16.99) | 0.23** (18.01) | 0.22** (17.17) | 0.22** (17.29) | 0.22** (16.77) | 0.22** (16.62) | 0.22** (16.70) |

λ | 1.72 (1.29) | 0.39 (0.29) | 0.51 (0.38) | 0.27 (0.20) | −0.001** (−4.14) | −0.002 (−0.001) | 1.91 (1.57) | 0.27 (0.19) | 0.42 (0.32) |

Conditional variance equation parameters | |||||||||

ω | 0.001** (7.73) | 0.001** (4.11) | 0.001** (4.67) | −0.32** (−13.91) | −0.19** (−8.03) | −0.23** (−8.94) | 0.01** (3.77) | 0.02* (2.02) | 0.02* (1.95) |

α | 0.10** (28.22) | 0.07** (11.03) | 0.08** (14.04) | 0.20** (26.35) | 0.15** (12.90) | 0.17** (17.21) | 0.10** (28.50) | 0.06** (6.52) | 0.08** (8.13) |

β | 0.90** (192.80) | 0.92** (134.88) | 0.91** (136.27) | 0.98** (438.10) | 0.99** (449.57) | 0.99** (415.62) | 0.89** (54.12) | 0.89** (49.69) | 0.88** (53.00) |

θ | −0.06** (−12.83) | −0.04** (−6.20) | −0.04** (−6.10) | ||||||

ρ | 0.99** (368.11) | 0.99** (1,077.20) | 0.99** 1,247.60) | ||||||

ϕ | 0.02** (3.70) | 0.02** (3.75) | 0.02** (3.77) | ||||||

Additional estimation output and diagnostics | |||||||||

Distribution parameter | 8.78** (12.60) | 1.50** (59.80) | 8.94** (11.72) | 1.49** (60.19) | 8.90** (12.26) | 1.52** (56.36) | |||

Likelihood | 17,993.1 | 18,103.90 | 18,068.01 | 18,029.82 | 18,129.84 | 18,072.03 | 17,998.20 | 18,118.31 | 18,082.50 |

LBSQ(36) | 33.90 [0.54] | 34.60 [0.54] | 34.00 [0.54] | 32.90 [0.56] | 33.10 [0.55] | 32.9 [0.55] | 34.00 [0.55] | 33.4 [0.55] | 33.00 [0.55] |

*IVI*in the conditional variance equation are reported in Table 3. This Table reports all the estimated parameters of both the conditional mean and the conditional variance equations under the alternative GARCH specifications discussed above after the inclusion of

*IVI*. Thus, the difference between Tables 2 and 3 is that the latter incorporates the additional

*IVI*in the conditional variance. In sharp contrast to Table 2, λ is now positive in all cases, taking values ranging from 3.20 (for the EGARCH-M with the Normal and GED) to 5.12 (for the standard GARCH-M with the Normal distribution). Importantly, λ is statistically significant in all models, with it being significant at the 1 % level in 8 out of 9 models. In the conditional variance equations, parameter

*g*(the parameter of

*IVI*in the variance equation) is strongly statistically significant in all GARCH specifications. This result suggests that

*IVI*is important in the correct specification of the (unobservable) conditional variance, indicating that

*IVI*possesses information content in measuring conditional variance. Our result differs from Day and Lewis (1992), who used weekly data on implied volatility (but not the index) to assess its information content. Although implied volatility is found to have information content for future volatility, they find that the risk-return parameter is either insignificant or negative using GARCH and EGARCH specifications.

^{4}

Models with the implied volatility index in the conditional variance equation

GARCH-M | EGARCH-M | CGARCH-M | |||||||
---|---|---|---|---|---|---|---|---|---|

Normal | t-distribution | GED | Normal | t-distribution | GED | Normal | t-distribution | GED | |

Conditional mean equation parameters | |||||||||

| −0.006** (−19.80) | −0.006** (−22.68) | −0.006** (−21.87) | −0.006** (−22.80) | −0.006** (−22.02) | −0.006** (−22.25) | −0.01** (−19.35) | −0.01** (−21.50) | −0.01** (−22.01) |

| 0.22** (14.78) | 0.21** (15.90) | 0.21** (15.53) | 0.23** (17.60) | 0.22** (16.85) | 0.22** (17.02) | 0.22** (14.90) | 0.21** (15.68) | 0.21** (15.45) |

| 0.22** (17.46) | 0.22** (17.46) | 0.22** (17.25) | 0.21** (17.52) | 0.22** (17.99) | 0.22** (17.85) | 0.23** (16.10) | 0.22** (17.17) | 0.22** (17.07) |

| 0.23** (16.90) | 0.23** (18.11) | 0.23** (17.98) | 0.22** (17.45) | 0.23** (18.33) | 0.23** (18.19) | 0.23** (16.20) | 0.22** (17.78) | 0.22** (17.80) |

λ | 5.12** (3.21) | 4.00** (2.56) | 3.80* 2.56) | 3.20** (2.61) | 3.40* (2.42) | 3.20** (2.60) | 4.60** (3.40) | 4.40** (2.86) | 4.60** (3.03) |

Conditional variance equation parameters | |||||||||

ω | −0.001** (−9.40) | 0.001** (7.23) | −0.001** (−8.02) | −0.22** (−18.38) | −0.20** (−11.95) | −0.20** (−15.58) | −0.001* (−24.85) | −0.001* (−21.42) | −0.001* (−23.82) |

α | 0.11** (20.27) | 0.06** (4.48) | 0.08** (7.96) | 0.06** (4.02) | 0.28** (2.64) | 0.28** (2.60) | 0.070 (1.08) | 0.07 (1.08) | 0.08 (1.24) |

β | 0.40** (8.50) | 0.13 (1.19) | 0.21* (2.50) | 0.02 (1.80) | 0.01 (0.33) | 0.02 (0.43) | 0.55** (7.00) | 0.68** (7.72) | 0.60** (7.12) |

θ | −0.40** (−5.30) | −0.29** (−2.64) | −0.32** (−3.62) | ||||||

ρ | 0.50** (13.68) | 0.67** (7.66) | 0.54** (5.32) | ||||||

ϕ | 0.04 (0.49) | 0.02 (0.75) | 0.024 (0.98) | ||||||

| 0.10** (10.04) | 0.10** (7.29) | 0.10** (8.20) | 0.31** (17.51) | 0.29** (11.80) | 0.30** (15.60) | 0.10** (12.10) | 0.10** (12.87) | 0.10** (12.90) |

Additional estimation output and diagnostics | |||||||||

Distribution parameter | 9.63** (12.66) | 1.55** (58.62) | 11.12** (11.98) | 1.60** (90.50) | 9.82** (12.50) | 1.55** (59.35) | |||

Likelihood | 18,071.20 | 18,174.95 | 18,134.0 | 18,160.0 | 18,241.70 | 18,203.30 | 18,080.10 | 18,181.91 | 18,141.21 |

LBSQ(36) | 31.11 [0.56] | 30.60 [0.57] | 31.90 [0.56] | 32.7 [0.55] | 33.00 [0.55] | 32.87 [0.55] | 32.81 [0.55] | 33.02 [0.55] | 33.05 [0.55] |

In the conditional variance equations, the estimates of the other volatility parameters reveal that the degree of volatility persistence is now reduced. Importantly, many of the coefficients in the conditional variance equations are no longer significant when adding the *IVI* as exogenous parameters. This result is by-and-large compatible with previous evidence which found that inclusion of an important variable in the conditional variance equation (such as the *IVI* variable in the present context) entails reduction of volatility persistence. Studies by Lamoureux and Lastrapes (1990), Kalev et al. (2004) have reached this conclusion by including trading volume in the variance equation. The insignificance of these coefficients and the reduction of persistence in conditional variance may be attributed to the fact that *IVI* is now capturing more persistence than (α + β). A further explanation of this finding refers to the effects of information arrival upon conditional variance. In other words, the information conveyed by the important variable may reduce the effects from the conditional variance at (t − 1) upon the conditional variance at (t), thereby reducing persistence.

We next turn to examining whether the statistical significance of *g* is linked with the statistical significance of λ. In other words, given that *IVI* contains important information for the conditional variance (and thus, given the statistical significance of *g*), we explore whether the inclusion of this information in the conditional variance specification helps to uncover a strong and positive risk-return relation (i.e. statistical significance of λ). Furthermore, we seek to establish whether failure to capture this information, by not including *IVI* in the conditional variance, entails insignificance of λ and thus reduction of the ability of GARCH-M models in uncovering such a relation. Monte Carlo simulations are used to address these issues.

## 4 Monte Carlo simulations

The Monte Carlo experiments are designed as follows. Assuming that the true conditional variance equation includes *IVI*, we simulate the distributions of λ if *IVI* is *not* included (λ_{ NO_IVI }) and of λ if *IVI* *is* included (λ_{ IVI }) in the variance equation. Simulated excess returns, \( \tilde{r}_{t} \), are constructed from equation \( \tilde{r}_{t} = \hat{c} + \lambda h_{t - 1} + h_{t - 1} \varepsilon_{t} \), where ε_{ t } is drawn from a N(0,1), and parameter λ is fixed at 5.12 (equal to its estimated value). The conditional variance is assumed to evolve according to the following calibrated GARCH(1,1) model with the *IVI* included: \( h_{t} = 0.0001 + 0.10\varepsilon_{t - 1}^{2} + 0.50h_{t - 1} + 0.01IVI_{t - 1} \), where \( \varepsilon_{t} = h_{t - 1} \varepsilon_{t} \). Taking a simulated sample of *T* = 5,800 observations, a GARCH(1,1)-M model without *IVI* in the conditional variance equation (i.e. imposing *g* = 0) is estimated 5,000 times, generating a distribution for \( \hat{\lambda }_{NO\_IVI} \). Similarly, a GARCH(1,1)-M with *IVI* in the conditional variance equation is estimated 5,000 times, generating a distribution for \( \hat{\lambda }_{IVI} \).

*IVI*in the conditional variance equation when it should be included dilutes an existing strong and positive risk-return relation. Being a measure of market expectations of future risk, implied volatility incorporates important forward-looking risk-related information. If this information is allowed for in the conditional variance equation, it significantly improves upon the measurement of conditional variance which, in turn, is reflected upon the risk-return parameter (λ) thereby uncovering a strong risk-return relation.

## 5 Conclusion

This paper has shown that, if the implied volatility index is included as an exogenous variable in the conditional variance equation, the information content of the index is important in uncovering a strong risk-return relation for the daily S&P 100 series. This result holds for various GARCH specifications and distributions. These results are further corroborated by Monte Carlo simulations which indicate that if the implied volatility index is not included whilst it should, the estimated risk-return parameter is more likely to be either negative or close to 0. Thus, uncovering a strong risk-return relation depends on correctly specifying the conditional variance equation by allowing for the information content of the implied volatility index.

## Footnotes

- 1.
- 2.
Guo and Whitelaw (2006) found that omitting the ‘hedge component’ of stock returns from the estimation methodology of the risk-return relation causes a large downward bias in the estimate of relative risk aversion. These authors estimated the hedge component using a linear function of a vector of state variables including the dividend yield, the yield spread between Baa-rated and Aaa-rated bonds, and the yield spread between the 6-month commercial paper and the 3-month Treasury Bill rate. Using a GMM estimation approach and employing past realized volatility or implied volatility as instruments for conditional volatility, these authors found that incorporating the previously defined hedge component reveals a positive risk-return relation. The present study differs from Guo and Whitelaw (2006) in several ways. Firstly, we adopt a GARCH-in-Mean estimation framework, with the conditional variance being estimated on the basis of a GARCH-type specification and not being approximated using instruments. Secondly, in the present study,

*IVI*is not used as an instrument variable for conditional variance but as an important variable for the specification of the conditional variance equation. Thirdly, we illustrate that it is the*IVI*, and not the hedge component, the important variable which is capable of revealing a positive risk-return relation. - 3.
The CBOE implied volatility index is also known as VIX. There exist an old VIX series (distributed under the new ticker VXO) which is highly correlated with the former VIX series (Banerjee et al. 2007).

- 4.
Day and Lewis (1992, Tables 2 and 3, pages 277, 280).

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