  Modern portfolio theory(MPT) addresses the problem of determining the optimum
allocation of investment resources among a set of candidate assets. In the
original mean-variance approach of Markowitz, volatility is taken as a proxy
for risk, conflating uncertainty with risk. There have been many subsequent
attempts to alleviate that weakness which, typically, combine utility and risk.
We present here a modification of MPT based on the inclusion of separate risk
and utility criteria. We define risk as the probability of failure to meet a
pre-established investment goal. We define utility as the expectation of a
utility function with positive and decreasing marginal value as a function of
yield. The emphasis throughout is on long investment horizons for which
risk-free assets do not exist. Analytic results are presented for a Gaussian
probability distribution. Risk-utility relations are explored via empirical
stock-price data, and an illustrative portfolio is optimized using the
empirical data.

Statistical Mechanics;Portfolio Management