Market Risk Measurement - VaR and Beyond › Historical Simulation VaR
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Historical simulation sidesteps distributional assumptions by using the actual empirical distribution of past returns. The method revalues the portfolio using historical return scenarios and directly reads VaR from the resulting distribution of portfolio values.
Historical simulation avoids theoretical models by using actual data. Whatever patterns exist in historical returns, including fat tails, skewness, and complex dependencies, are automatically captured when we use actual historical scenarios as the basis for risk measurement.
The Historical Simulation Approach
The procedure begins by assembling a database of historical returns for all assets in the portfolio. Each historical day provides one complete scenario: a vector of returns for every asset that occurred together in the market. This preserves the natural co-movement patterns that existed on each date, including any unusual correlation structures during market stress periods.
The steps proceed systematically:
- Collect historical returns for all portfolio assets over a lookback window (typically 250-500 days)
- Apply each historical return scenario to the current portfolio
- Calculate the portfolio profit or loss for each scenario
- Sort the P&L outcomes and select the appropriate quantile as VaR
For 500 historical observations and 99% confidence, VaR is the 5th worst loss (since 1% of 500 is 5).
The key conceptual shift from parametric VaR is that we no longer assume any particular distributional form. Instead, we treat the historical return vectors as equally likely future scenarios. Each past day contributes one potential outcome, and the empirical distribution of these outcomes becomes our estimate of the true return distribution. VaR is then read directly from this empirical distribution as the appropriate order statistic.
# Generate historical scenarios for 3 assets using Cholesky decomposition
np.random.seed(42)
cholesky = np.linalg.cholesky(cov_matrix)
random_normals = np.random.randn(n_obs, n_assets)
asset_returns = random_normals @ cholesky.T
# Calculate portfolio returns for each historical scenario
portfolio_returns = asset_returns @ weights
# Calculate portfolio P&L for each scenario
portfolio_pnl = portfolio_value * portfolio_returns
# Sort P&L from worst to best
sorted_pnl = np.sort(portfolio_pnl)
# Historical simulation VaR
# For 95% confidence with 500 observations, use the 25th worst outcome (5% * 500)
idx_95 = int(np.floor(n_obs * (1 - alpha_95)))
idx_99 = int(np.floor(n_obs * (1 - alpha_99)))
var_95_hist = -sorted_pnl[idx_95]
var_99_hist = -sorted_pnl[idx_99]Historical Simulation VaR ======================================== Number of scenarios: 504 Index for 95% VaR: 25 (worst outcome #26) Index for 99% VaR: 5 (worst outcome #6) 95% VaR: $233,226 99% VaR: $303,960
The historical simulation results are subject to sampling noise given the limited history (504 days). With 99% confidence, we are looking at the 5th or 6th worst outcome, making the estimate sensitive to individual extreme points in the dataset.
Visualizing the P&L Distribution
The histogram shows that losses beyond the 99% VaR threshold are rare but do occur. Historical simulation captures the actual shape of the return distribution, including any fat tails present in the data.
Advantages and Limitations of Historical Simulation
Historical simulation requires no distributional assumptions and captures non-normal and nonlinear behavior present in historical data. It handles options and other nonlinear instruments naturally since you revalue the actual portfolio under each scenario.
The method has significant drawbacks, however. It assumes the past is representative of the future: if the lookback window contains only calm markets, VaR will underestimate risk. The method is also sensitive to the choice of window length. Too short a window produces noisy estimates; too long a window includes stale data that may no longer reflect current market dynamics. Additionally, rare extreme events in the lookback period can cause VaR to jump discontinuously as those observations enter or exit the window.