  Utility and risk are two often competing measurements on the investment
success. We show that efficient trade-off between these two measurements for
investment portfolios happens, in general, on a convex curve in the two
dimensional space of utility and risk. This is a rather general pattern. The
modern portfolio theory of Markowitz [H. Markowitz, Portfolio Selection, 1959]
and its natural generalization, the capital market pricing model, [W. F.
Sharpe, Mutual fund performance , 1966] are special cases of our general
framework when the risk measure is taken to be the standard deviation and the
utility function is the identity mapping. Using our general framework, we also
recover the results in [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Master
funds in portfolio analysis with general deviation measures, 2006] that extends
the capital market pricing model to allow for the use of more general deviation
measures. This generalized capital asset pricing model also applies to e.g.
when an approximation of the maximum drawdown is considered as a risk measure.
Furthermore, the consideration of a general utility function allows to go
beyond the "additive" performance measure to a "multiplicative" one of
cumulative returns by using the log utility. As a result, the growth optimal
portfolio theory [J. Lintner, The valuation of risk assets and the selection of
risky investments in stock portfolios and capital budgets, 1965] and the
leverage space portfolio theory [R. Vince, The Leverage Space Trading Model,
2009] can also be understood under our general framework. Thus, this general
framework allows a unification of several important existing portfolio theories
and goes much beyond.
