Markowitz Portfolio Optimizer

Live demo: portfolio-optimizer.marcpartensky.com

Mean-variance portfolio optimization with efficient frontier, Monte Carlo simulation, and Value at Risk — built with Python and Streamlit.

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Theory

Markowitz mean-variance optimization

Harry Markowitz (1952) showed that for a given level of expected return, there exists a portfolio that minimizes risk (volatility). The set of all such portfolios forms the efficient frontier.

For a portfolio with weights w, expected returns μ and covariance matrix Σ:

Return    : r = w · μ · 252
Volatility: σ = √(wᵀ Σ w) · √252
Sharpe    : S = (r - rf) / σ

Two special portfolios are identified:

• Minimum Variance — lowest volatility regardless of return
• Maximum Sharpe — best risk-adjusted return given a risk-free rate

Both are found via constrained optimization (SLSQP) with the constraint Σwᵢ = 1 and optionally wᵢ ∈ [0, 1] (long-only). Enabling leverage removes the upper bound, allowing short positions (wᵢ < 0) and leverage (wᵢ > 1).

Capital Market Line

The CML connects the risk-free rate to the Maximum Sharpe portfolio. Any portfolio on this line dominates portfolios on the efficient frontier for the same level of risk.

CML: r(σ) = rf + (r_ms - rf) / σ_ms · σ

Monte Carlo cloud

5000 random portfolios are generated via Dirichlet sampling and plotted in risk-return space. This visualizes the feasible set and shows how the efficient frontier bounds it.

Value at Risk

VaR at confidence level α over horizon h is the loss not exceeded with probability α. Three methods are implemented:

Parametric (Normal) — assumes returns follow a normal distribution. Fast but underestimates tail risk.

VaR_1d = -(μ + z_α · σ)
VaR_hd = VaR_1d · √h          (square-root-of-time scaling)
CVaR   = σ · φ(z_α) / (1-α) - μ

Historical — uses the empirical percentile of past returns. No distributional assumption, limited by history length.

VaR_1d = -percentile(returns, (1-α) · 100)

Monte Carlo GBM — simulates future asset paths via Geometric Brownian Motion with Cholesky decomposition to preserve inter-asset correlations.

dS = μ·S·dt + σ·S·dW

The three methods give different results by design — each reflects different assumptions about the return distribution.

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Project structure

portfolio-optimizer/
├── app.py          # Streamlit UI — routing only, no business logic
├── optimizer.py    # Markowitz core: efficient frontier, min variance, max sharpe
├── var.py          # VaR: parametric, historical, Monte Carlo GBM
├── data.py         # yfinance download + Euribor fetch with caching
├── charts.py       # Plotly figures
├── settings.py     # YAML config loader
├── settings.yml    # All parameters (tickers, windows, VaR defaults)
└── Justfile        # Task runner

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Quickstart

Requirements: Python 3.11+, uv

git clone https://github.com/marcpartensky/portfolio-optimizer
cd portfolio-optimizer
uv run streamlit run app.py

Open http://localhost:8501.

Available commands

just app     # run the app
just lint    # ruff linter
just clean   # remove venv and cache

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Configuration

All parameters are in settings.yml:

data:
  default_tickers: "AAPL, MSFT, GOOGL, AMZN, JPM, GS, XOM, JNJ"
  cache_ttl: 3600

portfolio:
  trading_days: 252
  default_rf_pct: 4.5
  frontier_points: 80
  mc_portfolios: 5000

var:
  default_confidence: 0.99
  default_horizon: 1
  mc_simulations: 10000
  default_notional: 1000000

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Stack

• Streamlit — UI
• scipy — SLSQP optimizer
• yfinance — market data
• Plotly — charts
• uv — package manager