Quantitative model · Portfolio construction
Efficient-frontier geometry under uncertain inputs
Expected returns, volatility, correlations, and constraints decide which portfolios look efficient. Every input here is illustrative, not a forecast. Illustrative inputs
Decision summary
An efficient portfolio is only as stable as its inputs
Interpretation
The efficient frontier depends on expected returns and covariances that are not known in advance. A precise optimizer can amplify small estimation errors into concentrated allocations.
Decision context
Choose an allocation method according to confidence in the inputs. In the lab’s locked test, risk-based portfolios reduced out-of-sample volatility and drawdown. They did not reliably deliver higher returns.
Intended analytical use
Asset allocators, investment committees, model validators, and analysts use it to compare optimization with risk budgeting.
Principal limitation
This explorer uses illustrative capital-market assumptions. The empirical result comes from one frozen 2015–2026 ETF sample and one sealed evaluation window.
Universe
Parameters
Highlighted portfolios
| Portfolio | E[r] | σ | Sharpe |
|---|---|---|---|
| Min variance | — | — | — |
| Max Sharpe | — | — | — |
| Equal weight | — | — | — |
Efficient frontier risk-return plot
Highlighted portfolio weights
View portfolio weight data
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, 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. Mean-variance optimization is maximally sensitive to them.
Research question
How does portfolio construction change when the objective moves 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 the comparison. The solution is highly sensitive to estimated expected returns. Small input changes can produce materially 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 removing it from the objective. That implicitly assumes risk premia per unit of risk are roughly comparable across assets. Reaching equity-like returns then typically requires leverage on low-volatility assets.
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 trades in-sample optimality for greater stability under covariance-estimation error.
MVO, risk parity, and HRP encode different assumptions about expected returns and covariance estimates. Their allocations therefore differ in concentration, sensitivity to estimation error, leverage requirements, and dependence on the realized market regime.