Historical Simulation vs. Monte Carlo: A Comparative Analysis for Stress Scenario Generation

At the heart of any stress-testing engine lies a scenario generation method. The two most dominant methodologies are Historical Simulation (HS) and Monte Carlo (MC) simulation. The choice between them is not merely a technical detail; it represents a fundamental trade-off between model risk and statistical robustness, between the realism of past events and the flexibility of simulated futures. Understanding the strengths and weaknesses of each is important for building a scenario analysis framework that is both relevant and reliable.

Historical Simulation: The Wisdom of the Past

Historical Simulation is conceptually simple and intuitively appealing. It makes no assumptions about the statistical distribution of risk factors. Instead, it assumes that the future will, in some sense, resemble the past. The methodology involves three steps:

1. Collect Data: Gather a history of daily (or weekly) changes for all relevant market risk factors (e.g., stock prices, interest rates, FX rates) over a specified look-back period (e.g., the last 5 years, yielding approximately 1260 daily scenarios).
2. Create Scenarios: Each day in the historical dataset is treated as a potential scenario for tomorrow. For example, the scenario for day t is the vector of actual market changes that occurred between day t-1 and day t.
3. Revalue Portfolio: The current portfolio is revalued under each of these historical scenarios. The resulting distribution of P&L provides an estimate of the portfolio's risk. The 1st percentile of this distribution is the 99% VaR.

The primary advantage of HS is its freedom from model risk. It does not require the estimation of a covariance matrix or any assumptions about normality or correlation. It naturally captures the fat tails and non-linear dependencies that were present in the historical data. If there was a flash crash or a correlation breakdown in the look-back period, it will be included in the set of scenarios.

However, HS has significant drawbacks. Its primary weakness is that it is entirely constrained by the past. If a particular type of event has not occurred in the look-back period, the model assumes it has a zero probability of occurring tomorrow. This makes it slow to adapt to new market conditions and blind to unprecedented events. Furthermore, for a typical 2-5 year look-back period, the number of scenarios is statistically small, leading to noisy and unstable risk estimates.

Monte Carlo Simulation: The Power of Randomness

Monte Carlo simulation takes the opposite approach. It begins by building an explicit statistical model of the risk factors. This typically involves:

1. Choosing Distributions: Specifying a marginal probability distribution for each risk factor (e.g., a Student's t-distribution to capture fat tails).
2. Modeling Dependence: Estimating a correlation matrix or, more sophisticatedly, a copula to model the dependence structure between the risk factors.
3. Simulating Paths: Drawing thousands (or millions) of random samples from this joint distribution. Each sample represents a possible state of the market in the next period. These simulated scenarios are then used to revalue the portfolio.

The great strength of MC is its flexibility. It can generate an essentially unlimited number of scenarios, leading to stable and precise risk estimates. It is not constrained by history; by design, it can generate events that are more extreme than anything seen in the past. By adjusting the parameters of the distribution (e.g., increasing the volatility or the tail index), a risk manager can easily create forward-looking stress scenarios that reflect a change in the market environment.

The glaring weakness of MC, however, is its complete dependence on the underlying statistical model. This introduces significant model risk. If the chosen distribution is wrong, or if the correlation structure is misspecified, the generated scenarios will be unrealistic and the risk estimates will be misleading. The model may fail to capture the complex, non-linear dependencies that characterize real market crises.

A Hybrid Approach: Filtered Historical Simulation

Given the complementary strengths and weaknesses of HS and MC, a natural solution is to combine them. Filtered Historical Simulation (FHS) is a popular hybrid method that attempts to do just that. FHS works as follows:

1. Fit a Volatility Model: For each risk factor, fit a volatility model like GARCH to the historical data. This captures the time-varying nature of volatility.
2. Extract Standardized Residuals: Divide the historical returns by their GARCH-predicted volatility. This creates a set of standardized residuals that are, in theory, independent and identically distributed (i.i.d.).
3. Simulate: To generate a scenario for tomorrow, first forecast the volatility for tomorrow using the GARCH model. Then, randomly draw one of the historical standardized residuals and multiply it by the forecasted volatility. This is the simulated return for that risk factor.

FHS combines the best of both worlds. Like MC, it is forward-looking, as it uses the current, forecasted level of volatility. This makes it much more responsive to changing market conditions than simple HS. Like HS, it uses the actual historical residuals, so it does not make strong assumptions about the shape of the distribution and naturally captures any fat tails or skewness that were present in the data. It is a semi-parametric method that lets the data speak for itself, while still allowing for a forward-looking adjustment based on current volatility.

Choosing the Right Tool for the Job

There is no single best method for all applications. The choice depends on the specific goal of the analysis.

• For a standard, daily VaR calculation where responsiveness to current market volatility is key, Filtered Historical Simulation is often the superior choice.
• For exploring unprecedented,