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I heard a professor say: "We know the return of the risk free asset must be less than the expected return of the global minimum variance portfolio, otherwise there would be arbitrage opportunities", but he did not elaborate on it.

Is this result true? What is the argument? I could not find anything proving this.

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The capital market line used in the context of the efficient frontier represents the allocation of capital between the riskless asset and an optimal portfolio (say the tangent portfolio of highest Sharpe). If we move along the capital market line and go beyond tangent portfolio, we are applying leverage by borrowing at the riskless rate and further investing in the tangent portfolio.

Using this logic, if the riskless rate is higher than the ER of the global minimum variable portfolio, we can draw a negatively sloped capital market line from the riskless asset to the global minimum variance portfolio.

Using this negatively sloped capital market line, we can short the global minimum variance portfolio to invest further in the riskless asset (your portfolio is >100% riskless asset and <0% global minimum variance portfolio, both adding up to 100%), increasing returns at no risk (which is the arbitrage I guess your Professor is talking about).

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    $\begingroup$ Perfect intuition. Thank you @KaiSqDist. We could short the GMV portfolio and increase the weight on the riskless asset. The net effect of this increase would a lower variance and higher return. $\endgroup$
    – Gustavo Amarante
    Commented Apr 27 at 11:54

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