# Reference request for research on the maximum drawdown **ratio** (NOT value)

Let's suppose the asset price process follows a Geometric Brownian motion $$S_t \sim GBM(\mu, \sigma),\,t\ge 0$$, and define the two process: \begin{align} \text{MSF}_t &:= \max_{\tau\in[0,t]} S_\tau\\ \text{MDD_Ratio}_t &: = \max_{\tau\in[0,t]} ((\text{MSF}_\tau- S_\tau) / \text{MSF}_\tau) \end{align} where MSF means "maximum so far" and MDD means maximum drawdown. As $$S_t$$ is forever positive, we see that MDD_Ratio is always well defined.

Is there any research done on the distribution, or at least the mean value of MDD_Ratio?

For what I saw, most research literature on DD seems to focus on the max drawdown value, i.e., $$\max_{0\le t\le T}(\text{MSF}_t - S_t)$$ (and most of the time the asset price process is assumed to follow a BM instead of a GBM), instead of maximum drawdown ratio whch IMHO is much commoner in practice.

• Do you mean $S_{\tau}$ in your second equation? – Gordon Jan 17 '19 at 14:30
• @Gordon yes. All should be $\tau$ instead. – Vim Jan 17 '19 at 15:42
• @vim please allow me to ask "why you ask?". From an investor's perspective the series [+1,+5,-1] and [+1,-1,+5] have different mdd ratios but are equally risky. Aren't they, in your opinion? – elemolotiv Jan 18 '19 at 8:51
• @elemolotiv I assume you're talking about the percentages of return. Then your examples have the same mdd ratio (-1%). – Vim Jan 18 '19 at 9:07
• @elemolotiv MDD ratios are very important both to investors and sometimes also to regulators. Let's consider a risky fund: if at some point the MDD ratio reaches an intolerable point for investors (say 10%), they will withdraw; if the MDD ratio reaches some regulatory threshold (say 20%), the fund may be forced to close down. – Vim Jan 18 '19 at 9:09