Funded by The Danish National Research Foundation

MPS-RR 2004-22

October 2004

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 as defined in the literature. A complex
relationship between risk and expected return
emerges which is far from the common assumption
of a positive linear time-invariant relation.

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