Questions: Efficient Frontier Construction and Mean-Variance Analysis

Question 1

A portfolio manager is considering adding a new stock that has higher individual volatility than any existing holding. Under mean-variance analysis, she should:

• A. Reject it immediately — adding a higher-volatility asset always increases portfolio risk
• B. Add it if its correlation with the existing portfolio is sufficiently low, even if standalone volatility is high
• C. Accept it only if its expected return exceeds that of every existing holding
• D. Add it only if it becomes the smallest position by weight, to limit its impact

Answer: (B)

An asset's contribution to portfolio risk is determined by how it covaries with the rest of the portfolio, not by its standalone volatility. A highly volatile asset with low correlation can actually reduce overall portfolio risk by providing diversification. This is the central insight of mean-variance analysis: σ²_p = w'Σw depends on the full covariance structure, not on individual variances. Option A is the classic misconception — it conflates standalone volatility with portfolio risk contribution.

Question 2

In mean-standard deviation space, the set of all minimum-variance portfolios (before restricting to the efficient upper portion) traces out:

• A. A straight line from zero-risk to maximum-return
• B. A downward-sloping curve showing the risk-return tradeoff
• C. A hyperbola, with the minimum-variance portfolio at the leftmost point
• D. A horizontal line at the minimum achievable variance level

Answer: (C)

The full set of minimum-variance portfolios forms a parabola in mean-variance space, which appears as a hyperbola in mean-standard deviation space. The leftmost point is the global minimum-variance portfolio — the lowest risk achievable with the available assets. The efficient frontier is the upper portion of this curve: for any portfolio on the lower portion, there exists another portfolio with the same risk and higher expected return, so rational investors would never choose the lower portion.

Question 3

Small errors in expected return estimates can produce large swings in optimal portfolio weights, making mean-variance optimization sensitive to input quality.

• T. True
• F. False

Answer: True

Mean-variance optimization is often called 'error-maximizing' with respect to expected return inputs. Small perturbations in estimated expected returns cause the optimizer to dramatically shift weights toward assets with slightly higher estimated returns. This makes real-world efficient frontiers fragile: the theoretically optimal portfolio depends heavily on expected return estimates that are themselves noisy, leading practitioners to use shrinkage, factor models, or explicit constraints to produce stable and robust portfolios.

Question 4

A portfolio that lies below the minimum-variance portfolio on the efficient frontier offers lower risk for the same expected return as portfolios on the efficient upper portion.

• T. True
• F. False

Answer: False

The minimum-variance portfolio is the leftmost point — the lowest-risk portfolio achievable. Portfolios below it on the curve (the lower, 'inefficient' portion) offer lower expected return at the same or higher risk compared to portfolios on the efficient upper portion. They are dominated: you can always find an efficient portfolio with the same risk but higher return, or the same return but lower risk. No rational mean-variance investor would hold a portfolio on the lower portion of the frontier.

Question 5

Explain why adding a highly volatile asset to a portfolio can sometimes reduce the portfolio's overall risk.

Model answer: Portfolio variance is σ²_p = w'Σw — a function of all pairwise covariances, not just individual variances. When a new asset has low or negative correlation with existing holdings, its movements partially offset theirs, smoothing out the portfolio's overall swings. Even if the asset is individually volatile, its diversification benefit (reducing covariance contributions) can exceed its variance contribution, resulting in a net decrease in portfolio risk. This is the formal generalization of the two-asset intuition: correlation below 1 creates diversification benefit, and sufficiently low correlation can dominate even high standalone volatility.

The key is the distinction between standalone variance and the covariance contribution. The misconception (option A in MC1) is to judge an asset by σ_i alone, ignoring that what matters to the portfolio is σ_{ip} — the covariance with the existing portfolio.