Portfolio Risk Analytics - VaR and ES Estimation

Project Overview

This notebook implements a comprehensive risk analytics framework featuring 17 distinct Value-at-Risk (VaR) and Expected Shortfall (ES) methodologies for portfolio risk measurement. The analysis provides a comparative study of traditional, advanced, and modern risk estimation techniques.

Objective

Evaluate and compare multiple VaR/ES estimation methodologies to identify the most accurate and robust approaches for portfolio risk measurement, with validation through rigorous backtesting.

Analysis Specifications

Configuration Parameters

Analysis Period: January 1, 2019 - December 22, 2025
Confidence Level: 99% (α = 0.01)
Rolling Window: 250 days (~1 trading year)
Portfolio: 11 stocks (ADBE, BALL, DASH, DUK, IFF, SO, TJX, UNH, AXON, JPM, MAR)
Random Seed: 3407 (reproducibility)

Risk Parameters

EWMA Lambda: 0.94 (exponential decay)
Bootstrap Samples: 1,000 iterations
Bootstrap Block Size: 5
Volatility Window: 20 days
Age-Weighted Decay: 0.98
Correlation Window: 60 days

Methodology: 17 VaR/ES Methods

1. Parametric Methods (6 methods)

1.1 Normal Distribution VaR/ES

Assumption: Gaussian return distribution
Formula: \(VaR = -(\mu + z_\alpha \sigma)\)
ES Formula: \(ES = -(\mu - \sigma \frac{\phi(z_\alpha)}{\alpha})\)
Advantage: Simple, closed-form solution
Limitation: Underestimates tail risk

1.2 Student’s t-Distribution VaR/ES

Assumption: Heavy-tailed distribution
Method: Maximum likelihood estimation of degrees of freedom
Formula: \(VaR = -(loc + t_\alpha(df) \times scale)\)
Advantage: Better captures fat-tail events
Application: More realistic for financial returns

1.3 EWMA (Exponentially Weighted Moving Average) VaR

Decay Factor: λ = 0.94
Variance Formula: \(\sigma_t^2 = \lambda \sigma_{t-1}^2 + (1-\lambda) r_t^2\)
Advantage: Adapts to changing volatility
Use Case: Trending markets

1.4 Cornish-Fisher VaR

Method: Moment expansion accounting for skewness and kurtosis
Adjustment: \(z_{CF} = z + \frac{(z^2-1)S}{6} + \frac{(z^3-3z)K}{24} - \frac{(2z^3-5z)S^2}{36}\)
Advantage: Non-normal distribution accommodation
Parameters: Incorporates 3rd and 4th moments

2. Non-Parametric Methods (4 methods)

2.1 Historical Simulation VaR/ES

Method: Empirical quantile estimation
VaR: \(VaR = -\text{percentile}(returns, \alpha \times 100)\)
ES: Mean of tail losses beyond VaR
Advantage: No distributional assumptions
Limitation: Limited by historical data availability

2.2 Kernel Density Estimation (KDE) VaR/ES

Method: Smoothed continuous density estimation
Bandwidth: Scott’s rule
Grid Size: 10,000 points
Advantage: Smooth distribution estimate
Application: Better tail interpolation than raw historical

3. Advanced Historical Simulation Methods (10 methods)

3.1 Bootstrapped Historical Simulation (BHS) VaR/ES

Technique: Block bootstrap resampling
Parameters:
- 1,000 bootstrap samples
- Block size: 5 observations
- Median of bootstrap estimates
Advantage: Confidence intervals for VaR
Method: Preserves temporal dependencies

3.2 Age-Weighted Historical Simulation (AWHS) VaR/ES

Weight Formula: \(w_i = \lambda^{n-i}\) (exponential decay)
Decay Factor: λ = 0.98
Normalization: Weights sum to 1
Advantage: Recent observations more influential
Application: Adapts to regime changes

3.3 Volatility-Weighted Historical Simulation (VWHS) VaR/ES

Method: Scale historical returns by current volatility regime
Formula: \(r_{scaled} = r_{hist} \times \frac{\sigma_{current}}{\sigma_{hist}}\)
Volatility Window: 20 days
Advantage: Adjusts for volatility clustering
Use Case: Changing market conditions

3.4 Correlation-Weighted Historical Simulation (CWHS) VaR/ES

Method: Weight scenarios by correlation similarity to current regime
Distance Metric: Frobenius norm of correlation matrices
Weight Formula: \(w_i = \exp(-\frac{d_i}{2})\)
Correlation Window: 60 days
Advantage: Captures regime-dependent correlations
Application: Multi-asset portfolios

3.5 Filtered Historical Simulation (FHS) VaR/ES

Method: Volatility-adjusted historical scenarios
Volatility Estimation: Optional GARCH(1,1) or rolling window
Filter Window: 30 days
Scaling: Current vol / historical vol
Advantage: Combines parametric volatility with non-parametric shocks
Fallback: Simple volatility scaling if GARCH unavailable

Technical Implementation

Core Libraries

NumPy/Pandas: Numerical computation and data handling
yfinance: Historical stock price data
SciPy: Statistical distributions and optimization
- norm, t, chi2: Distribution functions
- gaussian_kde: Kernel density estimation
Matplotlib: Visualization
arch: GARCH modeling (optional for FHS)

Key Functions

Data Processing

download_price_data() - Yahoo Finance data retrieval with robust error handling
construct_portfolio() - Build portfolio from notional amounts
- Calculate shares from notional and latest prices
- Compute position values, P&L, and returns
- Handle zero prices gracefully

VaR/ES Calculation Functions

Parametric:
- parametric_var_normal() / parametric_es_normal()
- parametric_var_student_t() / parametric_es_student_t()
- ewma_var() with rolling EWMA variance calculation
- cornish_fisher_var() with moment adjustments
Non-Parametric:
- historical_var() / historical_es()
- kde_historical_var() / kde_historical_es()
Advanced Historical:
- bootstrapped_historical_var() / bootstrapped_historical_es()
- age_weighted_historical_var() / age_weighted_historical_es()
- volatility_weighted_historical_var() / volatility_weighted_historical_es()
- correlation_weighted_historical_var() / correlation_weighted_historical_es()
- filtered_historical_var_enhanced() / filtered_historical_es_enhanced()

Rolling Computation

compute_comprehensive_rolling_var() - Master function computing all 17 methods
- Handles stochastic methods with seed control
- Robust error handling for each method
- Progress tracking (17 steps)
- Multi-asset correlation-weighted calculations