Unit 12 of 14
Portfolio mathematics and allocation
- 8 lessons
- Videos 89 to 96
Portfolio choice turns expected returns, covariances and constraints into weights. This unit develops mean-variance optimization and methods for estimation error, risk budgets, growth and liabilities.
Mean-variance optimization is a quadratic program whose solutions trace the efficient frontier. Factor models, shrinkage and principal components structure covariance estimates, and Black-Litterman updates a prior with uncertain views by Gaussian conditioning. Worst-case optimization over an uncertainty set acts as a penalty for estimation error. Euler allocation splits risk into contributions, leading to risk parity. Expected log utility gives the Kelly fraction, the Hamilton-Jacobi-Bellman equation gives the Merton allocation, and immunization matches asset value and duration to liabilities.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Mean-variance optimization and covariance estimation
Mean variance optimization and efficient frontiers
How fixed weights give a portfolio mean and variance, how the budget and target-mean constraints define the minimum-variance quadratic program, why its solutions trace the efficient frontier, and what an active constraint changes.
Video 89
Factor models and covariance estimation
How a factor model splits covariance into loadings, factor covariance and residual covariance, why sample and shrunk covariance matrices are positive semidefinite, and how principal components maximize the Rayleigh quotient.
Video 90
Views, estimation risk and risk parity
Black Litterman and Bayesian portfolio views
How the Black-Litterman model combines an equilibrium prior with uncertain views through a view matrix, how view confidence shapes the posterior mean, and why the predictive covariance adds mean uncertainty to return risk.
Video 91
Robust optimization and estimation risk
How a mean uncertainty set turns estimation error into a worst-case objective, why its support function adds a norm or weighted absolute-value penalty, and when that regularization sets a position exactly to zero.
Video 92
Risk contributions and risk parity
How marginal risk and component risk split portfolio volatility exactly by Euler allocation, how a risk budget fixes each holding's share, and how a convex problem builds the equal risk contribution portfolio of risk parity.
Video 93
Growth, dynamic allocation and liabilities
Expected utility and Kelly growth
How expected utility and log utility lead to the Kelly criterion, why the positive-wealth constraint limits the stake, and how the Kelly fraction of a binary wager maximizes expected log growth.
Video 94
Dynamic allocation and stochastic control
How the Bellman principle leads to the Hamilton-Jacobi-Bellman equation for dynamic allocation, how CRRA utility gives the Merton allocation and its value, and what a verification argument proves about admissible controls.
Video 95
Liability hedging and immunization
How immunization protects an asset-liability surplus by matching value and duration, what normalized convexity adds, why a nonparallel curve shift can break the hedge, and how a key rate hedge matches several exposures.
Video 96