Over the past 25 years the relation between implied equity volatility (VIX) and market returns has been non-linear. When VIX was low there was no meaningful relation. However, when volatility increased above average higher equity and lower bond returns followed. This is evidence for “flight to quality“, where investors pay a rising premium for safe and liquid assets as volatility increases. Fear of redemptions, liquidity constraints, and deteriorating market intermediation are plausible causes of this effect.

Adrian, Tobias, Richard Crump, and Erik Vogt (2015), “Nonlinearity and Flight to Safety in the Risk-Return Trade-Off for Stocks and Bonds”, Federal Reserve Bank of New York Staff Reports, No. 723, April 2015. http://www.newyorkfed.org/research/staff_reports/sr723.pdf

On the general information value of VIX view post here.

The below are excerpts from the paper. Headings and cursive text have been added.

Evidence of non-linear relation between volatility and returns

“We document a highly significant, strongly nonlinear dependence of stock and bond returns on past equity-market volatility as measured by the VIX…Expected returns increase for stocks when volatility increases from moderate to high levels, while they decline for Treasuries.”

N.B. VIX is the Chicago Board Options Exchange Volatility Index. It measures 30-day forward volatility implied in S&P 500 index call and put options.

“There are three notable regions that characterize the nature of the nonlinear risk-return tradeoff, defined by the VIX median of 18 and the VIX 99.3rd-percentile of 50. When the VIX is below its median of 18, both stocks and bonds exhibit a risk return tradeoff that is relatively insensitive to changes in the VIX. In the intermediate 18-50 percent range of the VIX, the nonlinearity is very pronounced: as the VIX increases above its unconditional median, expected Treasury returns tend to fall, while expected stock returns rise. This finding is consistent with a flight-to-safety from stocks to bonds, raising expected returns to stocks and compressing expected returns to bonds.”


“For levels of the VIX above 50, which has only occurred in the aftermath of the Lehman failure, this logic reverses, and a further increase in the VIX is associated with lower stock and higher bond returns. The latter finding for very high values of the VIX likely reflects the fact that severe financial crises are followed by abysmal stock returns and aggressive interest rate cuts, due to a collapse in real activity.”

N.B. The experience of the great financial illustrates that earning high volatility premia comes with a tail risk and requires confidence that the underlying uncertainty or crisis is ultimately not escalatory.

“The nonlinear relationship is highly significant for the 1990-2007 sample which excludes the 2008-09 financial crisis. Importantly, the shape of the nonlinearity in the 1990-2007 and the 1990-2014 sample resemble each other closely, even though the tail events in those samples are distinct.”

Non-linear forecasting

“Linear regression using the VIX does not forecast stock or bond returns significantly at any horizon. Nonlinear regressions, on the other hand, do forecast stock and bond returns with very high statistical significance.”

“The [preferred forecasting function] is a nonlinear function of volatility and is the best common predictor for the whole cross section of stock and bond returns…The VIX strongly forecasts stock and bond returns up to 24 months into the future when the nonlinearity is accounted for, in sharp contrast to the insignificant linear relationship.>

The causes of non-linear volatility effects

“The dynamic asset pricing results indicate that the pricing of risk over time is related to the level of volatility in a nonlinear fashion. A number of alternative theories are compatible with such a finding, including 1) flight-to-safety theories due to redemption constraints on asset managers, 2) macro-finance models with financial intermediaries, and 3) representative agent models with habit formation.”

“Our findings are particularly in line with the theory of Vayanos (2004), where asset managers are subject to funding constraints that (endogenously) depend on the level of market volatility… When volatility is low, managers are not concerned with withdrawals and hence the component of the risk premium that corresponds to withdrawals is very small and almost insensitive to volatility…When volatility increases, the likelihood of redemptions rises, leading to a decline in the risk appetite of the asset managers. Increases in volatility generate flight-to-safety as managers attempt to mitigate the impact of higher volatility on redemption risk by allocating more to relatively safe assets…Assets’ illiquidity arises as trading is subject to fixed transactions costs. Fund managers are subject to withdrawals when fund performance is poor, generating a preference for liquidity that is a time varying function of volatility… In equilibrium, those transaction costs are also related to the level of volatility, and to the withdrawal intensity of mutual fund investors.”

“When the aggregate quantity of liquidity is limited…agents grow concerned with liquidity shortages and they therefore sell risky financial claims in favor of safe and uncontingent claims, i.e. there is flight to safety. Even though the flight to safety seems prudent from individuals’ point of views, it is collectively costly for the macroeconomy.”

On the basics of market liquidity risk view post here.

“Our findings are also closely linked to intermediary asset pricing theories…Intermediaries are subject to value at risk (VaR) constraints that directly link intermediaries’ risk taking ability to the level of volatility. Prices of risk are a nonlinear function of intermediary leverage, which has a one-to-one relationship to the level of volatility.”