Asset Allocation

Choose weights w to minimize variance wᵀΣw subject to a target expected return wᵀμ = μ* and Σwᵢ = 1. Trade-off between expected return and variance defines the efficient frontier.

The set of portfolios that maximize expected return for each level of variance (or minimize variance for each level of expected return). In (σ, μ) space it forms an upward-sloping hyperbola; only the upper branch is efficient.

Without a risk-free asset, every mean–variance efficient portfolio is a linear combination of any two distinct efficient portfolios. Investors differ only in the mix, not in the underlying risky funds held.

With a risk-free asset, every efficient portfolio is a mix of the risk-free asset and one risky tangency portfolio (max Sharpe). All investors hold the same risky portfolio, scaled by risk preference.

E[Rₚ] = Rf + ((E[Rₘ] − Rf)/σₘ) · σₚ. The CML connects the risk-free asset to the market (tangency) portfolio; its slope is the market Sharpe ratio.

Investors are mean–variance optimizers with homogeneous expectations, can lend/borrow at a single risk-free rate, face no taxes or transaction costs, and all assets are tradable in a frictionless market. In equilibrium the market portfolio is the tangency portfolio.