The Efficient Frontier
Mean-variance optimisation is the mathematical procedure that constructs the efficient frontier. The inputs are expected returns for each asset, variances (standard deviations squared) for each asset, and covariances between every pair of assets. The output is the set of portfolios that minimise variance for each possible level of expected return — or equivalently, maximise expected return for each possible level of variance.
When plotted in risk-return space (standard deviation on the x-axis, expected return on the y-axis), the feasible set of all possible portfolios forms a roughly parabolic region. The efficient frontier is the upper-left boundary of this region: the portfolios you cannot improve upon without either accepting more risk or reducing your return target. Every portfolio below and to the right of the frontier is inefficient.
The leftmost point on the frontier is the global minimum variance portfolio — the combination of assets that produces the lowest possible volatility, regardless of expected return. A rational investor who is extremely risk-averse would hold this portfolio. More return-focused investors would hold portfolios further along the frontier, accepting more variance in exchange for higher expected return.
Adding the Risk-Free Asset
The efficient frontier changes shape once a risk-free asset is introduced — typically short-term government bills. An investor can now mix any risky portfolio with the risk-free asset, holding part of their wealth in risky assets and part in the risk-free asset. This creates a straight line in risk-return space from the risk-free rate through any chosen risky portfolio.
The optimal line is the one with the steepest slope — the Capital Market Line (CML). It touches the efficient frontier at exactly one point: the tangency portfolio. The tangency portfolio is the unique risky portfolio that maximises the Sharpe ratio (excess return per unit of risk). The CAPM's conclusion is that in equilibrium, this tangency portfolio must be the market portfolio — every asset, held in proportion to its market capitalisation.
The CML is a powerful result: all investors, regardless of their risk preference, should hold a combination of the risk-free asset and the same risky portfolio (the tangency portfolio). Risk preference determines only how much you allocate to each — a conservative investor holds more in the risk-free asset; an aggressive investor may even borrow at the risk-free rate to hold more than 100% in the risky portfolio.
The Inputs Problem
Mean-variance optimisation is mathematically elegant but practically fragile. It requires precise estimates of expected returns, variances, and all pairwise covariances. For a portfolio of N assets, that is N expected returns, N variances, and N(N−1)/2 covariances — 1,275 distinct parameters for a 50-asset portfolio.
All of these must be estimated from historical data, and all of them are noisy. Expected return estimates from historical data are particularly unreliable: the standard error of a mean return estimated from 20 years of monthly data is large enough to make most estimates statistically indistinguishable from each other. The optimisation procedure is highly sensitive to these inputs: small changes in expected return assumptions can flip the "optimal" portfolio from heavily overweighting one asset to heavily underweighting it.
This fragility — the tendency of mean-variance optimisation to produce extreme, concentrated, and practically unusable portfolios — is one of the central criticisms of MPT in practice. It motivates approaches like the Black-Litterman model, which starts from the market portfolio as a neutral baseline and adjusts only in proportion to the investor's conviction in specific views.