Portfolio Construction: Theory

Portfolio construction determines how much capital to allocate to each stock. Even the best stock picks can produce poor portfolio results if position sizing is wrong. This article covers the three optimization approaches available in QuanterLab and the theoretical foundations behind them.

The Position Sizing Problem

After screening identifies 10 candidate stocks, you need to decide: how much in each? Equal weight (10% each) is simple but ignores risk. Concentration in your highest-conviction pick maximizes potential return but also maximizes risk. Portfolio optimization finds the mathematically optimal balance.

Mean-Variance Optimization (MVO)

Developed by Harry Markowitz in 1952. The foundational framework of modern portfolio theory.

The Efficient Frontier

For any set of assets, there exists a curve of portfolios that offer the maximum possible return for each level of risk. Portfolios on this curve are "efficient" — no other portfolio offers higher return at the same risk level.

Maximize: E[Rp] - λ × σ²p Where: E[Rp] = expected portfolio return = Σ(wi × E[Ri]) σ²p = portfolio variance = w^T × Σ × w λ = risk aversion parameter Σ = covariance matrix of asset returns

Two Key Portfolios

Maximum Sharpe Ratio — The portfolio with the highest risk-adjusted return: (Return - Risk-Free Rate) / Volatility. This is the theoretically optimal portfolio for a rational investor
Minimum Variance — The portfolio with the lowest possible volatility. Ignores expected returns entirely; purely a risk-minimization approach

Return Estimation Methods

MVO requires expected return estimates — notoriously difficult to forecast accurately. Three approaches:

Historical Mean — Uses past average returns as the forecast. Simple but assumes the past repeats
CAPM — Estimates returns from each stock's beta and the market risk premium. More theoretically grounded but depends on beta stability
Black-Litterman — Starts from market-implied equilibrium returns (derived from market-cap weights) and blends in investor views. Most sophisticated; reduces extreme positions common in basic MVO

Covariance Estimation

The covariance matrix captures how assets move together. Three estimation methods:

Sample Covariance — Direct calculation from historical return data. Accurate with enough data but noisy with limited history
Ledoit-Wolf Shrinkage — Shrinks the sample covariance toward a structured target (constant correlation model). Reduces estimation noise. Recommended when the number of assets approaches the number of time periods
Exponential Weighted — Gives more weight to recent returns. Captures changing market dynamics but may overreact to short-term events

MVO Sensitivity

MVO is sensitive to its inputs. Small changes in expected returns can produce drastically different optimal weights. This is why covariance shrinkage and the Black-Litterman model exist — they reduce this sensitivity. If you see very concentrated results (90%+ in one stock), consider switching to Ledoit-Wolf covariance or using HRP instead.

Hierarchical Risk Parity (HRP)

Developed by Marcos López de Prado (2016). A fundamentally different approach that does not require expected return estimates.

How It Works

Correlation clustering — Assets are organized into a tree structure (dendrogram) based on their correlation patterns. Highly correlated assets cluster together
Quasi-diagonalization — The correlation matrix is reordered so similar assets are adjacent, forming a block-diagonal pattern
Recursive bisection — The algorithm splits the tree in half repeatedly, allocating more weight to the less risky branch at each split

Distance: d(i,j) = √((1 - ρ(i,j)) / 2) Where ρ(i,j) is the correlation between assets i and j Weight allocation per split: α = 1 - (Var_cluster1 / (Var_cluster1 + Var_cluster2))

Advantages Over MVO

No return estimates needed — Eliminates the biggest source of estimation error in MVO
Naturally diversified — The clustering algorithm ensures each group of similar assets gets proportional allocation
More stable — Small changes in correlations produce small changes in weights, unlike MVO where small input changes can flip the entire portfolio

Inverse Volatility (IVO)

The simplest approach: allocate more capital to less volatile assets.

w_i = (1 / σ_i) / Σ(1 / σ_j) Where σ_i is the annualized volatility of asset i

No expected returns needed. No covariance matrix needed. Just individual asset volatilities. The intuition: less volatile assets are more predictable, so they deserve larger positions. More volatile assets are riskier, so position sizes should be smaller.

When to Use Each

Choosing an Optimizer

MVO — When you have strong views on expected returns and want to maximize risk-adjusted return. Best with Black-Litterman or CAPM return estimates and Ledoit-Wolf covariance
HRP — When you want diversification without making return predictions. Best default choice for most investors. Produces stable, well-diversified portfolios
IVO — When you want the simplest possible approach. Good for portfolios where all stocks have similar expected returns (e.g., index components after screening)