Module 02 · Portfolio Theory
Efficient Frontier Explorer
Twenty thousand long-only portfolios over six asset classes, sampled from a Dirichlet distribution and mapped into risk-return space. The frontier emerges as the upper edge of the cloud; the assumptions that drew it are the real subject. Illustrative inputs
Universe
Parameters
Highlighted portfolios
| Portfolio | E[r] | σ | Sharpe |
|---|---|---|---|
| Min variance | — | — | — |
| Max Sharpe | — | — | — |
| Equal weight | — | — | — |
Long-only random portfolios colored by Sharpe ratio. Diamonds are the individual assets; the line from the risk-free rate through the max-Sharpe portfolio is the capital market line. Extremes are located by sampling, so they are close to — not exactly on — the true frontier.
Weights of the highlighted portfolios. Note how concentrated the max-Sharpe solution tends to be relative to minimum variance — optimizers amplify whatever the inputs whisper.
Inputs
Illustrative capital market assumptions
The explorer uses the annualized expected returns, volatilities, and correlations below. They are illustrative — plausible in magnitude and ordering, chosen for pedagogy, and not estimated from any particular sample. That is precisely the point: change them by amounts well within estimation error and the "optimal" portfolio can change dramatically. Expected returns are the least estimable quantity in finance, and mean-variance optimization is maximally sensitive to them.
Research question
How does portfolio construction change when we move from classical return optimization to uncertainty-aware risk allocation?
Three answers to the same problem
Mean-variance optimization (Markowitz, 1952) solves \( \min_w \; w^\top \Sigma w \) subject to a target return \( w^\top \mu = \bar\mu \). It is the intellectual foundation of everything else, and in raw form it is an error-maximizing machine: it takes the noisiest inputs — expected returns — and loads onto whichever asset's estimation error happens to look most attractive. Small input perturbations produce violently different weights.
Risk parity abandons return forecasts entirely and equalizes each asset's contribution to portfolio risk, \( w_i (\Sigma w)_i = w_j (\Sigma w)_j \). It answers uncertainty about \(\mu\) by refusing to estimate it — an epistemically humble move that implicitly bets risk premia per unit of risk are roughly comparable across assets, and that typically requires leverage on low-vol assets to reach equity-like returns.
Hierarchical risk parity (López de Prado, 2016) attacks a different weakness: inverting an estimated covariance matrix. HRP clusters assets by correlation distance, then allocates down the resulting tree by inverse variance — no matrix inversion, no return forecasts. It sacrifices in-sample optimality for out-of-sample robustness, and in many horse races that trade is a good one.
The progression MVO → risk parity → HRP is a progression in what you admit you do not know. Each method encodes a different level of confidence in the inputs, and the portfolios differ accordingly: more diversified, less leveraged on estimation error, less spectacular in backtests, and more likely to survive contact with the future.