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.

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.

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.

E[Rᵢ] = Rf + βᵢ · (E[Rₘ] − Rf), with βᵢ = Cov(Rᵢ, Rₘ)/Var(Rₘ). Plots expected return vs systematic risk β; in equilibrium every asset lies on the SML.

Assumes returns follow a linear K-factor model Rᵢ = αᵢ + Σₖ βᵢₖFₖ + εᵢ. No-arbitrage implies E[Rᵢ] = Rf + Σₖ βᵢₖλₖ, where λₖ are factor risk premia. No utility or market-portfolio assumption needed.

E[Rᵢ] − Rf = βᵢ,MKT · (Rₘ − Rf) + βᵢ,SMB · SMB + βᵢ,HML · HML. SMB = small minus big (size), HML = high minus low book-to-market (value).