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Stock Market Risk-Return Inference. An Unconditional, Non-Parametric Approach.

We propose an unconditional non-parametric approach to the simultaneous estimation of volatility and expected return. By means of a detailed analysis of the returns of the Standard & Poors 500 (S&P 500) composite stock index over the last fifty years we show how theoretical results and methodological recommendations from the statistical theory of non-parametric curve inference allow one to consistently estimate expected return and volatility. In this approach we do not postulate an a priori relationship risk-return nor do we specify the evolution of the first two moments through covariates. Our analysis gives statistical evidence that the expected return of the S&P 500 index as well as the market price of risk (the ratio expected return minus risk free interest rate over volatility) vary through time both in size and sign. In particular, the periods of negative (positive) expected return and market price of risk coincide with the bear (bull) markets of the index. A complex relationship between risk and expected return emerges which is far from the common assumption of a positive linear time-invariant relation.