Nonlinearity and flight to safety in the risk-return trade-off for stocks and bonds

We document a highly significant, strongly nonlinear dependence of stock and bond returns on past equity-market volatility as measured by the VIX. We propose a new estimator for the shape of the nonlinear forecasting relationship that exploits additional variation in the cross section of returns. The nonlinearities are mirror images for stocks and bonds, revealing flight to safety: Expected returns increase for stocks when volatility increases from moderate to high levels, while they decline for Treasuries. We further demonstrate that these findings are evidence of dynamic asset pricing theories where the time variation of the price of risk is a function of the level of the VIX.