index (VIX) is based on consideration of all of the available market prices of the S&P 500 index
options. Such an approach facilitates the approximation of the expected aggregate volatility of the S&P
500 index during the subsequent 30 calendar-day period,
and indeed, this method has been used not
only as a measure of sentiment but also as an instrument for timing the market, particularly in the
aftermath of the subprime mortgage crisis.
Following Whaley (2000), who proposes the use of the VIX as an effective fear indicator, Giot (2005)
identifies a strongly negative correlation between contemporaneous changes and future market index
returns, along with a positive correlation between such future returns and current levels of the implied
volatility indices. Both Guo and Whitelaw (2006) and Banerjee, Doran, and Peterson (2007) report
similar findings. Zhu (2013) shows that VIX has the significant predictive power for extreme low and
high stock returns. Furthermore, Bekaert and Hoerova (2014) show that the variance risk premium (VRP)
from decomposing the squared VIX can predict stock returns. Overall, their results provide empirical
support for the predictive ability of the VIX-related variables on future portfolio returns.
Because the current version of the VIX is compiled for each maturity and incorporates almost all of
the available contracts, it should be more informative than the old version when used to investigate its
predictive ability on future equity returns. However, the VIX is dependent on maturity, with 30-day
maturity being the most frequently used. This dependence on maturity gives rise to the interesting and
important question of whether the VIX term structure contains any useful information for potential use
in the forecasting of returns.
Furthermore, because the literature clearly shows that volatility has some special stylized facts, such
as clustering and mean reversion, the relative positions of the VIX levels for different maturity periods
may imply the expectations of market participants on market volatility and, thus, on changes in the S&P
500 index due to the mean–variance relation in the conventional theory of the capital asset pricing model.
Therefore, this study contributes to the extant related literature by comprehensively investigating
whether the VIX term structure contains any useful information on future returns in the S&P 500 index.
Based on the stochastic volatility with correlated jumps (SVCJ) model proposed by Eraker (2004),
we refine the theoretical work of Banerjee et al. (2007) to derive a theoretical model that reveals a
positive relation between expected excess returns in the S&P 500 index and both the squared VIX
levels and the difference between forward and current squared VIX levels. Because the forward
squared VIX level can be computed from two squared VIX values with different horizons, it can be
regarded as a proxy for the squared VIX term structure. Hence, this model provides theoretical
fundamentals for the potential of the VIX term structure with regard to the prediction of excess returns
in the S&P 500 index.
We propose three alternative empirical methods to compile the variables representing the
information in both the squared VIX level and its term structure to investigate whether the information
content of the term structure contributes to the forecasting of future excess returns in the S&P 500
index. First, adhering closely to the theoretical model, we use the 30-day squared current VIX level and
The CBOE's revision of the methodology of the VIX formula was based largely on the results of Carr and Madan (1998)
and Demeterfi, Derman, Kamal, and Zou (1999) . Regarding the theoretical fundamentals, Britten-Jones and Neuberger
(2000) derive model-free implied volatility under the diffusion assumption, and Jiang and Tian (2005) subsequently extend
the model to the jump diffusion assumption.
When it was introduced in 1993, the VIX was compiled from the implied volatility of eight S&P 100 index options,
comprising near at-the-money and nearby and second nearby calls and puts; however, ever since 2003, the VIX has been
calculated from the prices of S&P 500 index options using a model-free formula with almost all of the available contracts
(i.e., with a wide range of strike prices).
See, for example, Bakshi et al. (2011), Feunou et al. (2014), Luo and Zhang (2017), and Andreou et al. (2017).