Defining Maximum Drawdown as a Proportional Loss

Summary

The document frames a research question about the maximum drawdown ratio under geometric Brownian motion. It defines the running maximum price and measures drawdown as the largest proportional decline from a prior peak over the observation period. This is distinct from maximum drawdown expressed as an absolute price difference. The distinction matters because a ratio expresses loss relative to the asset’s own peak, whereas an absolute drawdown depends on the price scale.

The author asks whether a distribution or expected value is known for this ratio and observes that the literature they encountered tends to study absolute drawdown, often under Brownian motion assumptions. The document does not provide a derivation, citation, estimate, or empirical result answering the question. It is therefore useful mainly for clarifying the quantity of interest and identifying a gap to investigate. Any result would depend on the process assumptions and time horizon; the document does not specify parameters or establish how the ratio behaves under other price models.

Key ideas

• Maximum drawdown ratio measures the largest decline from a running peak as a fraction of that peak.
• A proportional drawdown differs from an absolute drawdown measured in price units.
• The document poses a distribution and expectation question under a geometric Brownian motion model but does not answer it.
• Any drawdown result would need to be interpreted in light of the assumed price process and observation horizon.

Tags

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


# 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.

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