Mean-variance optimization selects portfolios by minimizing variance and maximizing expected return jointly. Markowitz (1952) proved portfolio variance equals V = Σ σ_ij X_i X_j in "Portfolio Selection." Maximizing expected return alone never implies diversification. The E-V rule implies diversification for a wide range of μ_i, σ_ij combinations.
What the Study Found
Markowitz (1952) proves maximizing expected return always concentrates all weight in a single security. The E-V rule implies diversification for a wide range of μ_i, σ_ij. Two equal-variance portfolios, when combined, produce a compound portfolio with lower variance than either original — except when returns are perfectly correlated. Sixty railway securities are less diversified than an equal-size cross-industry portfolio. Within-industry covariances exceed cross-industry covariances.
Methodology
Markowitz (1952) presents a theoretical model, not an empirical study — no dataset or sample period applies. Inputs are μ_i (expected return) and σ_ij (covariance) per security. Portfolio weights X_i ≥ 0 sum to 1, with no short sales permitted. The 3- and 4-security cases are solved geometrically; the N-security result is stated analytically.
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| Portfolio variance | V = Σ_i Σ_j σ_ij X_i X_j | Double-weighted covariance sum across all N securities |
| Portfolio expected return | E = Σ_i X_i μ_i | Weighted sum of individual security expected returns |
| Two-portfolio diversification | Variance of compound < variance of either original | Holds unless returns are perfectly correlated |
| Inadequate diversification example | Sixty railway securities | Less diversified than equal-size cross-industry portfolio |
| Efficient set structure | Series of connected line segments | Applies to 3-, 4-, and N-security cases |
Why This Matters
The E-V framework gave portfolio construction a mathematical foundation separate from stock-picking heuristics. Markowitz's proof that cross-industry diversification reduces variance in ways within-industry diversification cannot remains the basis for institutional asset allocation and risk budgeting. The efficient frontier gives practitioners a precise vocabulary for the risk-return trade-off.