Why Simulated and Theoretical Maximum Drawdowns Can Differ

Summary

The document raises a comparison between a theoretical expected maximum drawdown for Brownian motion and a Monte Carlo estimate produced by an R package. It reports that, for the stated drift, volatility, and horizon inputs, the package’s theoretical statistic exceeds the simulated average. The questioner says that increasing the simulation sample and changing scaling conventions did not resolve the gap, and asks what might explain it.

The post refers to a research paper on Brownian-motion drawdowns and package documentation, but it contains no accepted explanation or follow-up analysis. As a result, it does not identify whether the discrepancy comes from differing definitions, assumptions, time discretization, or an implementation detail. Its useful contribution is to highlight that a theoretical statistic and a simulated statistic must be compared using matching drawdown conventions and model assumptions. The reported figures are a single unresolved example, not evidence that either method is generally biased or that one calculation is correct.

Key ideas

• The post compares a theoretical expected maximum drawdown with a simulated average.
• The two reported estimates differ for the specified Brownian-motion inputs.
• The questioner reports that larger simulation samples and scaling adjustments did not close the gap.
• The document offers references but does not resolve the discrepancy.
• A valid comparison requires consistent assumptions and drawdown definitions.

Tags

# Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown


# Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown












I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2].

I do not understand how for some choices of parameters the value from sampling the maximum drawdown via rmaxdd and the corresponding statistic from maxddStats are so far apart.

```
require(fBasics)

maxddStats(mean = 0.01, sd = 0.0427, horizon = 135)
> 0.3142337
mean(rmaxdd(n = 100000, mean = 0.01, sd = 0.0427, horizon = 135))
> 0.2637941
```

It doesn't seem to be an issue of convergence (increasing small n in rmaxdd), nor of scaling (scaling mean and mu). Am I missing something basic here?

[1] https://www.cs.rpi.edu/~magdon/ps/journal/drawdown_journal.pdf [2] https://www.rdocumentation.org/packages/fPortfolio/versions/260.72/topics/DrawdownStatistics

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