# 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?

• Welcome! And nice markup. You know that there is the option of latex too? $N(\mu=1, \sigma=10)$? – Richard Sep 26 '18 at 8:32
• oook next time latex 😋 – elemolotiv Sep 27 '18 at 6:01

## 1 Answer

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.