Portfolios that lie on the Efficient Frontier dominate all other portfolios because they maximize standard deviation for a specific level of expected return.

Question

BlackTom AI · Accepted Answer

Efficient frontier portfolios provide the highest expected return for a given level of risk or, equivalently, the lowest risk for a given expected return; they do not maximize risk (standard deviation). This is a core result of mean-variance (Markowitz) portfolio theory.

1. Relevant knowledge points and concepts
- Expected return of a portfolio: E[R_p] = w'μ (weights w, asset expected returns μ).
- Portfolio variance (risk): σ_p^2 = w'Σw (covariance matrix Σ).
- Efficient frontier: set of portfolios that solve the mean-variance optimization problem — either maximize E[R_p] subject to a fixed σ_p or minimize σ_p for a fixed E[R_p].
- Dominance: a portfolio is dominated if another portfolio has both higher expected return and lower (or equal) risk.

2. Step-by-step reasoning
- Formulate the constrained optimization: minimize w'Σw subject to w'μ = μ_target and w'1 = 1. This produces the minimum-variance portfolio for that target return.
- Alternatively, maximize w'μ subject to w'Σw ≤ σ_target^2 and w'1 = 1; this yields the maximum-return portfolio for the given risk.
- The efficient frontier is obtained by varying μ_target (or σ_target) and collecting the optimal portfolios.

3. Intermediate steps (mathematical form)
- Lagrangian: L = w'Σw − λ1(w'μ − μ_target) − λ2(w'1 − 1). Solving first-order conditions gives the weights that minimize variance for the target return.
- The solution shows that the frontier portfolios are those with minimal variance for their return level.

4. Why the correct answer is correct
- The given statement claims efficient frontier portfolios "maximize standard deviation for a specific level of expected return." That is the opposite of the truth: they minimize standard deviation (risk) for a given expected return (or maximize expected return for given risk). Investors prefer portfolios on the frontier because they are not dominated.

5. Why the other option is incorrect
- Interpreting the statement as true would imply investors choose portfolios with the highest risk for a given return, which contradicts rational preference for lower risk and the optimization formulation above. Therefore "True" is incorrect.

6. Conclusion
- The statement is False. This matches the provided answer: False.

Portfolios that lie on the Efficient Frontier dominate all other portfolios because they maximize standard deviation for a specific level of expected return.True/False

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