Mean-variance efficient portfolios are optimal as Modern Portfolio Theory alleges, only if risk were foreseeable, that is under the hypothesis that price (co)variance is known with certainty. Admitting uncertainty changes the perception. If portfolios are presumed vulnerable to unforeseen price shocks as well, risk optimality is no longer obtained by minimising variance but also pertains to the diversification in the portfolio, for that provides protection against unforeseen events.
Generalising MPT in this respect leads to the double risk-objective to minimise variance and maximise diversification. We demonstrate that a series of portfolio construction techniques developed as an alternative to MPT, in fact address this double objective, under which Bayesian optimisation, entropy-based optimisation, risk parity and covariance shrinkage. We give an analytical demonstration and provide by that new theoretical backing for these techniques.