Estimating Maximum Drawdown from Normally Distributed Returns

Summary

The document asks how maximum drawdown varies with the mean and standard deviation of daily returns and the length of the return series. It frames the problem with simulated paths: draw daily observations from a normal distribution, calculate each path’s maximum drawdown, and compare the resulting values. The question is how to generalize that simulated distribution across different inputs.

The response points to a related result for Brownian motion and notes that an analytical expression is available for the mean maximum drawdown under a log-return formulation. It does not provide that expression, derive a full probability distribution, or show simulation results. The suggested connection is therefore a starting point rather than a complete solution. Applying Brownian-motion results requires care about how returns or portfolio value are modeled, and knowing the mean alone does not describe the spread or shape of maximum drawdown outcomes.

Key ideas

• Simulating return paths gives a sample of maximum drawdowns for fixed model parameters and horizon.
• The question seeks to understand how drawdown outcomes change with drift, volatility, and path length.
• A related Brownian-motion treatment may provide an analytical mean for maximum drawdown under log returns.
• The response does not supply a full distribution or establish how well the continuous-time model fits a given return process.

Tags

# What is the Probability Distribution of Max-Drawdown?


# What is the Probability Distribution of Max-Drawdown?












How to obtain the probability distribution of Maximum Drawdown, starting from the probability distribution of Daily Returns? Here the details:

Suppose I have a time serie of N=1000 daily returns.

Each daily return is normally distributed, like 𝒩[μ=1\$,σ=10\$]

Suppose I run 100 simulations of that time series. At every simulation I create a new realisation of the time series, by I pulling 1000 random values from that normal distribution 𝒩[μ=1\$,σ=10\$].

At every simulation I calculate the Maximum Drawdown (MDD).

Obviously, I get a different MDD every time, because each of the 100 realisations of the time series is different, although they all originate from the same normal distribution.

I want to generalise these results and understand how MDD varies as a function on μ, σ, N days. How can I do that?

## Answer by numerairX (score 3)

https://quant.stackexchange.com/a/41913

I think the answer you're looking for is very similar to this question Expectation of maximum draw down in the Brownian motion case.

just like your assumption that return is normally distributed with mu and sig, say price/portfolio value follows Brownian motion with same property, and if you're using log return, I found this article that provides an analytical formula for the mean of distribution of max drawdown.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.