Value-at-Risk (VaR) Modeling

This project implements Value-at-Risk (VaR) using two standard approaches:

1. Historical Simulation (non-parametric)
2. Parametric Variance–Covariance (assuming normal returns)

The objective is to estimate potential portfolio losses at 95% and 99% confidence levels, and to compare how assumptions about return distributions affect risk estimates.

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1. Project Structure

• main.ipynb — Jupyter notebook with full implementation
• plots/ — saved figures (histograms, VaR cutoffs, comparisons)
• requirements.txt — dependencies
• README.md — project documentation

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2. Methods

• Data: Daily adjusted close prices from Yahoo Finance (yfinance) for selected tickers (AAPL, MSFT, GOOGL, AMZN, META, JPM, XOM, SPY).
• Returns: Daily log returns.
• Historical Simulation VaR: Empirical quantiles of the return distribution.
• Parametric VaR: Normal distribution assumption, formula:
  [ \text{VaR}\alpha = - \big( \mu + z\alpha \sigma \big) ]
• Portfolio: Equal-weight portfolio of all tickers.

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3. Results & Interpretation (1-Day Horizon)

Sample window: 2020-01-01 → Current
Portfolio: Equal-weight, daily log returns

Historical Simulation VaR

• Asset AAPL: VaR_95 = 3.17 %, VaR_99 = 5.22 %
• Portfolio: VaR_95 = 2.3 %, VaR_99 = 4.47 %

Parametric VaR

• Asset AAPL: VaR_95 = 3.25 % , VaR_99 = 4.63 %
• Portfolio: VaR_95 = 2.49 % , VaR_99 = 3.55 %

Comparison (portfolio).

• Historical VaR is typically more conservative, capturing empirical fat tails.
• Parametric VaR assumes normality, which can underestimate tail risk.
• Diversification reduced risk at the portfolio level vs single assets.

Figures:

• hist_var_portfolio.png — histogram of portfolio returns with VaR cutoffs
• var_comparison_portfolio.png — overlay of historical vs parametric VaR

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4. Reflection

This project highlighted the importance of distributional assumptions in risk modeling. Historical VaR captured heavier tails compared to normal-based estimates, underscoring how parametric methods can underestimate risk.

Diversification effects were visible in the portfolio VaR, though tail sensitivity remains.

Next steps:

• Extend to multi-day horizons (10-day VaR via square-root-of-time rule)
• Implement Conditional VaR (Expected Shortfall)
• Evaluate robustness with rolling windows and alternative weighting schemes

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5. Requirements

Install dependencies with:

pip install -r requirements.txt