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Why the Risk-Free Rate Must Sit Below the Minimum-Variance Return

Article Quant Q&A · Author: Gustavo Amarante
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Summary

The document explains an arbitrage argument linking the risk-free rate to the expected return of the global minimum-variance portfolio. If the risk-free rate exceeds that portfolio’s expected return, the line connecting their risk-return points slopes downward. An investor could then short the minimum-variance portfolio and put the proceeds into the risk-free asset, creating a portfolio with a higher return and no greater variance under the stated setup.

This is an intuitive capital-market-line argument rather than a full formal proof. It relies on being able to borrow or lend at the risk-free rate, short the portfolio, and treat that asset as genuinely riskless. The note does not discuss transaction costs, short-sale constraints, or other market frictions, which can limit whether the theoretical arbitrage can be implemented.

Key ideas

  • If the risk-free rate exceeds the minimum-variance portfolio’s expected return, the line between them slopes downward.
  • Shorting the minimum-variance portfolio and investing the proceeds at the risk-free rate can raise return without adding variance.
  • The arbitrage conclusion assumes frictionless borrowing, lending, and short selling.

Tags

ArbitragePortfolio constructionRisk management
Full text
# Risk free rate must be lower than expected return of global minimum variance portfolio


# Risk free rate must be lower than expected return of global minimum variance portfolio












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.

## Answer by KaiSqDist (score 2, accepted)

https://quant.stackexchange.com/a/79144

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.