3 Diversification and Modern Portfolio Theory
3.1 The portfolio as the unit of analysis
Harry Markowitz’s central insight is that asset selection should account for interactions among assets. The variance of a portfolio includes cross terms: what matters is not simply how volatile the holdings are, but how their returns move together.
A basic mean-variance problem is
\[\min_{\mathbf w}\;\mathbf w^\mathsf T\boldsymbol\Sigma\mathbf w\]
subject to \(\mathbf 1^\mathsf T\mathbf w=1\) and, for a long-only portfolio, \(w_i\ge 0\). Additional constraints may limit holdings, turnover, industries, or individual positions.
3.2 Efficient does not mean optimal for everyone
An efficient frontier displays portfolios with minimum estimated risk for a given estimated expected return, or maximum return at a stated risk. It does not choose an investor’s objective or estimate the future correctly.
When covariance is estimated from a finite sample, tightly optimized weights can react strongly to small input changes. Position limits, regularization, and scenarios are pragmatic ways to explore robustness.
3.3 From MPT to broader portfolio engineering
Classical variance is one useful risk measure, but it does not distinguish upside from downside. Extensions may introduce loss-tail measures, constraints across time, explicit turnover costs, liabilities, and scenario-dependent decisions. Those extensions should recover the classical model when its original assumptions are restored.
Think about it: What changes if the portfolio must sell no more than 10% of its holdings each month?
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.