Mean-variance optimisation across eight international equity indices in USD, with and without a short-sale constraint, then rolled forward five years at a time.
Daily log returns across eight indices converted to a common USD base. Before any optimisation, the distributional check: skewness is negative nearly everywhere, TSX at -0.941 and the S&P 500 at -0.638. Excess kurtosis runs from about 10.5 to 21.4 against a Gaussian benchmark of 3, with TSX highest at 21.40.
Lag-1 autocorrelation is small, around -0.155 for the S&P 500, so the independence assumption is not badly violated. The Gaussian one is. Fat tails and negative skew mean mean-variance understates how bad the bad days are, and that caveat belongs on the output rather than in a footnote.
The correlation structure explains most of what follows. The developed indices are tightly linked, up to 0.978 between the closest pairs. NIKKEI225 sits apart at 0.226 to 0.422 against the others, and that separation is the only real diversification in the set.
With short selling allowed the tangency portfolio reaches a Sharpe ratio of 1.098. It gets there with weights of +5.47 in DAX, +4.25 in CAC and -9.11 in Eurostoxx50.
Those three holdings are almost the same asset. The optimiser has found tiny differences in estimated means between near-identical indices and levered them enormously. That is not a portfolio, it is an estimation-error amplifier, and the Sharpe ratio attached to it is not something anyone would realise.
Imposing non-negative weights collapses the portfolio to 89.4% S&P 500 and 10.6% NIKKEI225, with every other index at zero. Sharpe falls to 0.348.
The concentration is not a failure of the constraint. It is the optimiser telling the truth: once you cannot short, the only diversification available in this universe is the Japanese index, because everything else is a near-duplicate of the S&P 500.
A third of the paper Sharpe survives contact with a tradeable constraint. I would rather present 0.348 that can be implemented than 1.098 that cannot.
Re-estimating on five-year windows from 2010-14 through 2017-21 and comparing against market-cap weights is the test of whether any of this is stable. Unconstrained weights swing violently window to window, which is the estimation-error problem showing up as instability rather than as a number.
The constrained weights are far steadier, and the comparison against market-cap weights is the useful benchmark: if the optimised portfolio cannot beat capitalisation weighting out of sample, the optimisation is decoration.