Markowitz's mean-variance optimization takes a universe of assets with expected returns, variances, and pairwise correlations as inputs, and solves for the set of portfolios that achieve maximum return for each level of risk (or minimum risk for each level of return). Plotting these optimal portfolios produces the Efficient Frontier — a curved boundary in risk/return space where no further improvement is possible without either accepting more risk or sacrificing return. All portfolios below the frontier are suboptimal (dominated by frontier portfolios with better risk/return profiles).

The Capital Market Line (CML) extends the efficient frontier concept by adding a risk-free asset. Investors can combine the risk-free asset (zero volatility, risk-free rate return) with the optimal risky portfolio (the Tangency Portfolio, where the CML is tangent to the Efficient Frontier). Combinations of the risk-free asset and the Tangency Portfolio dominate all other portfolios in risk/return space. This result — the Separation Theorem — means all investors should hold the same risky portfolio (the Tangency Portfolio) and adjust their risk level by varying the allocation to cash, not by changing which risky assets they hold.

MPT requires estimates of expected returns, variances, and all pairwise correlations. For a 100-asset universe, this requires 100 expected returns, 100 variances, and 4,950 pairwise correlations — 5,150 inputs, all subject to estimation error. Small errors in expected return estimates produce dramatically different optimal portfolios, because the optimizer is highly sensitive to inputs. The 'error maximization' problem: optimization concentrates weight in assets with the highest estimated returns, which are also the assets where estimation error is most likely to have inflated the estimate.

The instability of MPT portfolios in practice is well-documented. Optimal weights derived from historical data frequently reverse wildly when recalculated with updated data — suggesting the portfolios reflect overfitting to historical noise rather than genuine optimization. Practitioners address this through Bayesian approaches (shrinking estimates toward priors like the market portfolio), robust optimization (explicitly modeling estimation uncertainty), risk-parity (bypassing expected return estimates entirely and focusing only on risk inputs), or simple heuristics (equal weight, factor-based allocation).

Despite practical implementation difficulties, MPT's conceptual legacy is enormous. It established that diversification is mathematically quantifiable (not just intuitive wisdom), that portfolio risk depends on correlations not just individual asset risks, and that the relevant risk measure for a security in a portfolio context is its marginal contribution to portfolio variance (covariance with the portfolio) rather than its standalone volatility. These insights directly spawned CAPM, factor investing, risk parity, and the entire field of quantitative portfolio management.

MPT's assumptions that fail in practice: (1) returns are normally distributed (fat tails exist); (2) correlations are stable over time (they spike in crises); (3) expected returns can be estimated with reasonable precision from historical data (they cannot); (4) investors are fully rational and purely mean-variance optimizing (behavioral biases are pervasive). The theory is most useful as a conceptual framework for thinking about diversification and risk decomposition — less useful as a direct optimization tool with empirical inputs.