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Historical vs Monte Carlo Simulation for Risk
Both methods estimate how much a portfolio could lose, but they source their scenarios differently. Historical simulation replays returns that actually happened; Monte Carlo simulation generates thousands of returns from an assumed model. The choice shapes every risk number that follows.
Key Takeaways
- Historical simulation reuses the real, observed return series, so it captures actual correlations and fat tails without assuming any distribution.
- Monte Carlo simulation draws returns from a chosen statistical model, so it can generate far more scenarios but is only as good as that model.
- Both are common ways to compute value at risk (VaR); they often disagree precisely because one is bound to the past and the other to assumptions.
- Use historical simulation when you trust your data window; use Monte Carlo when you need many scenarios, forward-looking shocks, or instruments with no long history.
Key Takeaways
- Historical simulation reuses the real, observed return series, so it captures actual correlations and fat tails without assuming any distribution.
- Monte Carlo simulation draws returns from a chosen statistical model, so it can generate far more scenarios but is only as good as that model.
- Both are common ways to compute value at risk (VaR); they often disagree precisely because one is bound to the past and the other to assumptions.
- Use historical simulation when you trust your data window; use Monte Carlo when you need many scenarios, forward-looking shocks, or instruments with no long history.
What It Is
Historical simulation estimates risk by applying a portfolio's current holdings to a window of past market moves. If you have 500 days of returns, you get 500 hypothetical outcomes, then read the loss at the chosen percentile. No distribution is assumed; the data speaks for itself.
Monte Carlo simulation estimates risk by specifying a model for how returns behave, for example a normal distribution with a given mean and volatility, or something richer with fat tails and correlations. A computer then draws thousands or millions of random scenarios from that model and reads the same percentile loss.
The shared output is usually a value at risk figure: the loss that will not be exceeded with a stated confidence over a stated horizon.
The Intuition
Historical simulation is a recording. It says, in effect, "here is exactly what markets did, so here is what would have happened to today's portfolio." Its honesty is also its limit: it can only replay events inside the sample window.
Monte Carlo simulation is a generator. It says, "here is a rule for how markets behave, so let me manufacture as many plausible futures as I like." That freedom lets it explore combinations that never occurred, but every number it produces inherits the assumptions of the rule.
How It Works
Historical simulation follows four steps: collect a window of past returns for each risk factor, apply those moves to the current position, build the distribution of hypothetical profit and loss, then read the percentile. A 95% one-day VaR sits at the 5th percentile of that distribution.
Monte Carlo simulation follows five: estimate the model parameters (means, volatilities, correlations), choose a distribution, draw a large number of random scenarios, revalue the portfolio in each, then read the same percentile from the simulated distribution. Because the scenario count is a free choice, the tail estimate is smoother, but it centers on whatever distribution you picked.
Worked Example
A portfolio is worth $1,000,000. We want a 95% one-day VaR two ways.
Historical simulation. Use the last 20 daily returns, sorted worst to best. The 95% VaR sits at the 5th percentile, and 5% of 20 observations equals 1, so the single worst day sets the cutoff. That day was -4.0%.
Historical VaR = 4.0% x $1,000,000 = $40,000.
Monte Carlo simulation. Assume daily returns are normal with a mean of 0.05% and a volatility of 1.5%. The 5th-percentile return of a normal distribution is the mean minus 1.645 volatilities:
0.05% - (1.645 x 1.5%) = 0.05% - 2.4675% = -2.4175%.
Monte Carlo VaR = 2.4175% x $1,000,000 = $24,175, roughly $24,200.
Same portfolio, two answers: $40,000 versus about $24,200. The historical window contained one brutal -4.0% day that a well-behaved normal model treats as very unlikely, so the normal Monte Carlo understates the tail. Swap in a fat-tailed distribution and the Monte Carlo figure could easily exceed the historical one. That gap is the lesson: the number depends on where the scenarios come from.
Common Mistakes
- Treating either output as fact. Both are estimates. Historical VaR is hostage to its window; Monte Carlo VaR is hostage to its distribution.
- Using a calm-market window. Historical simulation drawn from a placid year will miss crashes entirely, since it can only replay what the window contains.
- Assuming normality by default. A normal Monte Carlo systematically understates extreme losses because real returns have fatter tails than the bell curve.
- Ignoring correlation shifts. Both methods assume the correlations in the model or window still hold; in a crisis, correlations often jump toward one.
- Confusing more scenarios with more accuracy. Running a million Monte Carlo paths tightens the estimate of the model, not of reality, if the model is wrong.
Frequently Asked Questions
Q: What is the core difference in historical vs monte carlo simulation? Historical simulation replays real past returns and assumes no distribution, while Monte Carlo simulation generates synthetic returns from an assumed statistical model. One is anchored to observed data; the other to your modeling choices.
Q: Which is more accurate for value at risk? Neither wins universally. Historical simulation is more faithful to what actually happened, and Monte Carlo simulation can explore scenarios outside the sample. Accuracy depends on whether your data window or your model better represents future risk.
Q: When should I prefer historical vs monte carlo simulation? Prefer historical simulation when you have a long, representative return history and want no distributional assumptions. Prefer Monte Carlo when history is short, you need many forward-looking scenarios, or you must price complex instruments.
Q: Does Monte Carlo simulation require a normal distribution? No. Normality is a common simplification, but Monte Carlo can draw from fat-tailed, skewed, or regime-switching models. The method is flexible; the normal assumption is just the easiest and often the least realistic choice.
Q: Can the two methods give very different VaR numbers? Yes, and they frequently do. If the historical window holds a shock that a normal Monte Carlo treats as improbable, the historical figure will be larger, as the worked example shows. Reconciling the gap is often more informative than either number alone.
Sources
- Investopedia. "Value at Risk (VaR)." https://www.investopedia.com/terms/v/var.asp
- Investopedia. "Monte Carlo Simulation." https://www.investopedia.com/terms/m/montecarlosimulation.asp
- Investopedia. "Historical Simulation Method." https://www.investopedia.com/terms/h/historical-method.asp
- Investopedia. "Stress Testing." https://www.investopedia.com/terms/s/stresstesting.asp
Disclaimer
This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.