Question18 Which of the following statements is true? Risky assets on the efficient frontier… provide returns which are higher than or equal to the Global Minimum Variance Portfolio. have risk which is higher than or equal to the Global Minimum Variance Portfolio. are a subset of the Minimum Variance Frontier. represent the optimal allocations among risky assets at each risk level. All of the above statements are true. ResetMaximum marks: 3 Flag question undefined

Question

BlackTom AI · Accepted Answer

The efficient frontier is the upper branch of the minimum-variance parabola, with the Global Minimum Variance Portfolio (GMVP) as its leftmost (minimum-risk) point. This geometric and optimization-based view underpins the correct choice: All of the above statements are true.

1) Relevant knowledge points and concepts
- Minimum Variance Frontier (MVF): the set of portfolios that achieve the minimum variance for each possible expected return; it forms a parabola in risk-return space.
- Global Minimum Variance Portfolio (GMVP): the single portfolio on the MVF with the lowest possible variance (leftmost point).
- Efficient frontier (among risky assets): the upper portion of the MVF starting at the GMVP; for any given level of risk it gives the highest expected return (or equivalently, for any given expected return it gives the lowest risk).

2) Step-by-step reasoning
- By definition GMVP has the smallest variance of all risky portfolios, so any portfolio on the efficient frontier cannot have variance below GMVP’s variance. Hence risk of efficient portfolios ≥ risk of GMVP.
- The efficient frontier is the upper branch of the MVF that starts at the GMVP and extends to portfolios with higher variance; therefore it is a subset of the MVF.
- The efficient frontier, as the upper branch, is constructed by solving the mean-variance optimization that maximizes expected return for each risk level—so it represents optimal allocations at each risk level.
- Because the efficient frontier contains GMVP at its leftmost point and then portfolios with non-decreasing expected return as variance increases along the upper branch, efficient portfolios have expected returns greater than or equal to the GMVP’s return.

3) Intermediate “equation-style” statement
Let sigma_g^2 be GMVP variance and mu_g its return. For any efficient portfolio p: sigma_p^2 ≥ sigma_g^2 and mu_p ≥ mu_g. Efficient frontier ⊂ MVF by construction.

4) Why the correct answer is correct
Each option directly follows from definitions: GMVP minimizes variance, efficient frontier is the MVF upper branch (subset) and yields maximal return at each risk—so all four listed properties hold simultaneously.

5) Why the other options are not correct choices by themselves
Each individual option is true, so none are incorrect; selecting any single option would omit other true properties. The only fully correct choice is that all are true.

6) Conclusion
All of the above statements are true. This confirms the provided answer: "All of the above statements are true."

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