When a portfolio brief has no feasible answer

The brief combined two requirements: an 8% annual return threshold and less than a 20% probability of falling below it. The team used five years of weekly observations for CMG, ORLY, MNST and TSLA to estimate a mean–variance frontier and the probability of missing that threshold.

Minimum variance is a different objective

The minimum-variance portfolio had less volatility, but also a lower expected return. The best portfolio in the report’s underperformance analysis accepted more volatility in exchange for a higher expected return.

The best reported result still missed the probability constraint by 3.07 percentage points. An expected return above 8% did not make the probability of falling below 8% small enough.

Describe the boundary of the answer

Under the study’s normal-return assumption, the probability is Φ((8% − μ)/σ). Meeting the brief requires μ > 8% + 0.8416σ. Both expected return and volatility matter; reducing either in isolation does not settle the question.

The study found no qualifying portfolio in its reported analysis. That is not a proof about every possible allocation or asset universe. Historical means and covariances are estimates, normality is an assumption, and a discrete search can miss solutions. The stated allowance for short selling also needs to be reconciled with the weights used in the search.

The next step would be to reproduce those constraints and test estimation sensitivity before changing the universe or the brief. I would present the unmet constraint explicitly, so a decision-maker can see what must change rather than mistaking the best available result for a feasible one.