  Portfolio optimization methods suffer from a catalogue of known problems,
mainly due to the facts that pair correlations of asset returns are unstable,
and that extremal risk measures such as maximum drawdown are difficult to
predict due to the non-Gaussianity of portfolio returns. \\ In order to look at
optimal portfolios for arbitrary risk penalty functions, we construct portfolio
shapes where the penalty is proportional to a moment of the returns of
arbitrary order $p>2$. \\ The resulting component weight in the portfolio
scales sub-linearly with its return, with the power-law $w \propto
\mu^{1/(p-1)}$. This leads to significantly improved diversification when
compared to Kelly portfolios, due to the dilution of the winner-takes-all
effect.\\ In the limit of penalty order $p\rightarrow\infty$, we recover the
simple trading heuristic whereby assets are allocated a fixed positive weight
when their return exceeds the hurdle rate, and zero otherwise. Infinite order
power-law portfolios thus fall into the class of perfectly diversified
portfolios.
