Questions: Mean-Variance Optimization (Markowitz Framework)

Question 1

The efficient frontier is best described as the set of portfolios that...

• A. Maximize expected return without any constraint on risk
• B. Minimize portfolio variance for every possible level of expected return
• C. Offer the maximum expected return for each level of variance, forming the upper portion of the minimum-variance frontier
• D. Have zero correlation with every other available asset

Answer: (C)

The minimum-variance frontier contains the lowest-variance portfolio for each target expected return — including the bottom half where higher variance comes with *lower* expected return. The efficient frontier is only the upper portion: portfolios where no other portfolio offers strictly higher expected return for the same variance. Option B describes the full minimum-variance frontier, not the efficient subset. An investor would never rationally choose an inefficient portfolio.

Question 2

Adding a new asset to the investment universe can only keep the efficient frontier the same or move it outward (toward better risk-return combinations) — it can never make it worse.

• T. True
• F. False

Answer: True

Expanding the opportunity set is weakly beneficial. Any portfolio achievable before is still achievable after adding the new asset (by setting its weight to zero). If the new asset has any correlation less than +1 with the existing portfolio, it creates diversification opportunities that shift the frontier outward. In the degenerate case of perfect positive correlation with an existing asset, the frontier is unchanged.

Question 3

Why is mean-variance optimization sometimes described as an 'error maximizer' in practice?

Model answer: Small errors in expected return estimates cause the optimizer to concentrate large portfolio weights in the slightly-overestimated assets. Because the objective function is very sensitive to expected return inputs, estimation noise gets amplified into extreme and unstable weight allocations, rather than being averaged out.

This is a well-known critique of Markowitz optimization in implementation. Expected returns are notoriously difficult to estimate reliably from historical data. Practitioners use techniques like shrinkage estimators, Black-Litterman views, or constraints on maximum weights to reduce this sensitivity and produce more robust portfolios.