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Are there any good papers/ references on the statistical distribution of Max Drawdown over a specified amount of time given a specified Sharpe? Assuming returns are iid normally distributed

I’ve been running some Monte Carlo simulations but wondering if there is any theory

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    $\begingroup$ Magdon-Ismail et al, Rej et al, among others. $\endgroup$
    – steveo'america
    Jan 25, 2021 at 17:48
  • $\begingroup$ Given the nature of your resonses to the already provided answer I think it would be helpful if you provided more details in the question about what you you are looking for (mathematically speaking). In the absense of a trading strategy, are you simply looking for the statistical distribution of the minimum of a sum of normal iid variables? $\endgroup$
    – Attack68 ♦
    Feb 4, 2021 at 7:49

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The distribution of drawdown is highly sensitive to the trading strategy you are running. The distributions of drawdown for a Black Swan strategy and carry strategy are very different. Only you know what you are doing. If you are good, drawdown will have thin upper tail. If you are not good, it may have fat upper tail (or its distribution may fail in other ways). No paper will be able to encompass all the possibilities.

You are on the right track simulating drawdown via Monte Carlo and via backtesting on the historical data.

UPDATE: not only is the assumption of iid Gaussian returns unrealistic; your question is ill-posed for the following reason. The distribution of drawdown can take many shapes depending on the notional rules. The way you

  1. choose the notional of a new trade,
  2. readjust the notionals of the open trades

on any given day affects the distribution of the portfolio mark-to-market. Do you ever double down? Do you ever scale down? Are you saying: "If I have 10 trades open already, the 11-th one is not allowed no matter what"? Or are you assuming unlimited balance sheet?

As naive as is, the iid Gaussian framework can be distantly related to returns on a particular asset or static portfolio. However, the framework is completely inapplicable to a portfolio which is rebalanced. Everything depends on how you rebalance. Everything depends on your strategy.

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  • $\begingroup$ Question was assuming returns are normally distributed. That should remove the dependence on type of trading strategy $\endgroup$
    – Michael
    Jan 27, 2021 at 17:14
  • $\begingroup$ No, it shouldn't because a lot depends on the correlation of returns over time. Also, the normality assumption is quite unrealistic. Where are the fat tails?... Instead, sample from the historical distribution. $\endgroup$
    – stans
    Jan 27, 2021 at 18:51
  • $\begingroup$ Not asking whether it's realistic or not. I specifically stated in the question, what is max drawdown distribution for normal iid returns, e.g. brownian motion. $\endgroup$
    – Michael
    Feb 3, 2021 at 15:07
  • $\begingroup$ See my detailed update in the answer above. $\endgroup$
    – stans
    Feb 4, 2021 at 2:59
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    $\begingroup$ :) "Drawdown" is defined only for a trading strategy. It is the maximum continuous loss. Loss on what? On trading. If you are trading, then you have rules on how to trade. Your rules. Your strategy. Clear?... Normal iid returns can only be discussed if you are looking at a static (non-changing) portfolio, and even then it is an abstract "detour"... It is very peculiar on your part to say: I would like to discuss P&L but there is no strategy. $\endgroup$
    – stans
    Feb 4, 2021 at 4:14
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I assume Michael meant distribution of maximum negative returns of iid Normal distribution. I guess the word drowdowns is meant to represent sequence of max negative returns, not the drowdowns of any specific strategy

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