5.1 Modern Portfolio Theory & Mean–Variance

Basic Prerequisites: Linear Algebra & Matrices and Calculus & KKT Optimization. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Modern Portfolio Theory (Markowitz 1952) is the discipline that turned portfolio construction into an explicit quadratic program with a closed-form answer. Its core claim is that an asset’s value to a portfolio is not its own return and risk but how it covaries with every other holding - so the portfolio’s return $w^T\mu$ and variance $w^T\Sigma w$ are the entire decision surface. From that one quadratic object, everything in this folder follows: the efficient frontier, the tangency (Sharpe-optimal) portfolio, two-fund separation, the capital market line / CAPM, and the minimum-variance portfolio.

This page is the hub: it gives the fast formula lookup (job #1) and routes you to six sub-pages that walk from raw intuition through the derivation, the frontier, tangency/CAPM, constraints, failure modes, and extensions. Every formula below is transcribed from Merton (1972) ($A,B,C,D$ machinery, frontier parabola, min-variance, tangency, SML) and cross-checked against Markowitz (1952), Sharpe (1964), and Tobin (1958); the check-column numbers were re-executed and reproduced exactly from the verified corpus (§3).

The one-sentence essence. “Minimize portfolio variance subject to a target expected return - the solution is linear in ${\Sigma }^{-1}$; add a riskless asset and the Sharpe-optimal portfolio is the tangency portfolio, and a linear combination of the riskless asset with any two frontier funds spans the whole efficient set.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Worked on the folder’s worked universe - three assets with $\mu =\left[\begin{array}{c}0.08\\ 0.12\\ 0.16\end{array}\right]$, $\Sigma =\left[\begin{array}{ccc}0.100& 0.040& 0.016\\ 0.040& 0.180& 0.032\\ 0.016& 0.032& 0.250\end{array}\right]$, $r_f=0.04$. ($\Sigma$ from annual volatilities 0.10/0.18/0.25 with correlations 0.30, 0.10, 0.15.) All numbers below were re-run and reproduced exactly (see §3).

Merton scalars ($1$ the ones-vector, ${\Sigma }^{-1}$ the inverse covariance):

Worked values: $A=1.53150,B=0.18447,C=14.48325,D=0.32622$.

The ${\Sigma }^{-1}$-linearity is the whole story. Because the objective is quadratic, every optimizing weight is linear in ${\Sigma }^{-1}$. That makes it beautiful and simultaneously fragile: whatever estimation error lives in $\Sigma$ (and especially in $\mu$) is amplified by ${\Sigma }^{-1}$, whose small eigenvalues blow up - see 05 · Failure Modes.

3. Computational Implementation - the frontier engine

This runs on the standard library only (Gauss–Jordan matrix inversion + dot products). It reproduces every verified number above. (All results were cross-checked against numpy.linalg.solve/inv and match to machine precision.)

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. MVO is an “estimation-error maximizer.” Because $w\propto {\Sigma }^{-1}(\mu -r_{f}1)$, small errors in $\mu$ (the hardest inputs to estimate) are amplified by ${\Sigma }^{-1}$; Best & Grauer (1991) show a 0.08% mean change can drive the most-sensitive asset out of a 100-asset portfolio.
2. Covariance inversion instability. Near-collinear assets make $\Sigma$ ill-conditioned ($\kappa (\Sigma )={\lambda }_{\max }/{\lambda }_{\min }$ large); ${\Sigma }^{-1}$ then explodes tiny eigenvalue noise into extreme long/short weights.
3. Unconstrained solutions are not investable. The frontier assumes shorting is free; real long-only and turnover constraints push optimal weights to corner solutions - the bridge into 04 · Constraints and Transaction Costs.

5. References

• Merton, Robert C.: An Analytic Derivation of the Efficient Portfolio Frontier, JFQA 7(4):1851–1872 (1972)
• Markowitz, Harry: Portfolio Selection, Journal of Finance 7(1):77–91 (1952)
• Tobin, James: Liquidity Preference as Behavior Toward Risk, Review of Economic Studies 25(2):65–86 (1958)
• Sharpe, William F.: Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance 19(3):425–442 (1964)
• Best & Grauer: On the Sensitivity of Mean–Variance-Efficient Portfolios to Changes in Asset Means, Review of Financial Studies 4(2):315–342 (1991)

6. Connected Graph Bridges

• Foundational base: Linear Algebra · Calculus & KKT Optimization
• Sub-pages (in-folder): 01 From Zero · 02 Efficient Frontier · 03 Tangency & CAPM · 04 Min-Variance & Constraints · 05 Failure Modes · 06 Advanced Extensions
• Sibling topics: Covariance Shrinkage & RMT Denoising · Black–Litterman · Risk Parity · Transaction Costs

Beginner: start at 01 · Practitioner: start at 05