Empirical Reverse Engineering of the Pricing Kernel

This paper proposes an econometric procedure that allows the estimation of the pricing kernel without either any assumptions about the investors preferences or the use of the consumption data. We propose a model of equity price dynamics that allows for (i) simultaneous consideration of multiple stock prices, (ii) analytical formulas for derivatives such as futures, options and bonds, and (iii) a realistic description of all of these assets. The analytical specification of the model allows us to infer the dynamics of the pricing kernel. The model, calibrated to a comprehensive dataset including the S&P 500 index, individual equities, T-bills and gold futures, yields the conditional filter of the unobservable pricing kernel. As a result we obtain the estimate of the kernel that is positive almost surely (i.e. precludes arbitrage), consistent with the equity risk premium, the risk-free discounting, and with the observed asset prices by construction. The pricing kernel estimate involves a highly nonlinear function of the contemporaneous and lagged returns on the S&P 500 index. This contradicts typical implementations of CAPM that use a linear function of the market proxy return as the pricing kernel. Hence, the S&P 500 index does not have to coincide with the market portfolio if it is used in conjunction with nonlinear asset pricing models. We also find that our best estimate of the pricing kernel is not consistent with the standard time-separable utilities, but potentially could be cast into the stochastic habit formation framework of Campbell and Cochrane (J. Political Economy 107 (1999) 205).