Is It Expected Volatility or Expected Precision?

Abstract

There is little evidence in support of a normal distribution for most financial assets, including the VIX. This paper concludes that the lambda parameter, in the one-parameter Box & Cox (1964) family, appropriate for VIX to be normal, is minus one (expected precision), which is very far from values of one (no transformation) and zero (logarithm), typically assumed by the financial literature. We provide an economic meaning to the Gaussian transformation of VIX and call this variable "expected precision'' (EP). The EP measures the level of consensus of expectations in the market about future returns, regardless of the expectations accuracy. We show the pdf/acf for VIX from the EP statistical moments and report evidence in favor of a higher leverage effect when the EP is low (less consensus of expectations). From comparing realized and expected precisions, we define the "precision risk premium" (PRP) that informs about the premium the investors are willing to pay to avoid the uncertainty related to the lack of consensus of expectations, regardless of their level of accuracy. The PRP is Gaussian and relates to the VRP and market returns dynamics. Our result suggests that not only the accuracy but the precision of expectations (more or less consensus) of future returns affect the process of price formation in the market.

Keywords: volatility index, time series analysis, forecasting, leverage effect, expected precision, risk premium

JEL Classification: C32, G13, G15