Estimating Drawdown Duration and Depth Under Portfolio Dynamics

Summary

The document asks how to estimate the distributions of drawdown duration and depth under specified portfolio dynamics. It notes that the arcsine law for Brownian motion suggests extended periods below a prior high, and asks how the analysis might change for a geometric Brownian motion with given drift and variance. The answer points to research references on drawdowns but does not summarize their results or derive a distribution.

For a portfolio whose returns are available historically, its observed dynamics can inform an estimate; alternatively, a model can be specified and simulated. The answer recommends focusing on the portfolio’s own dynamics, cautioning that standard distributions may not closely represent a diversified portfolio. Under a geometric Brownian motion assumption, Monte Carlo simulation is offered as a practical way to estimate drawdown behavior. The document gives no simulation setup, calibration method, or quantitative findings, so results would depend on the chosen dynamics and assumptions.

Key ideas

• Drawdown analysis can examine both the time spent below a previous high and the depth of the decline.
• The arcsine law for Brownian motion motivates the question but does not answer it for other portfolio models.
• Historical returns or an explicit model can provide portfolio dynamics for analysis.
• Monte Carlo simulation can estimate drawdowns under an assumed geometric Brownian motion.
• Results depend on how well the assumed dynamics represent the actual portfolio.

Tags

# Expected length and depth of drawdown


# Expected length and depth of drawdown












Does anyone know of any model to estimate the distribution of drawdown length and depth assuming a certain portfolio dynamics? The arcsine law seems to suggest that a portfolio can spend a large portion of time under water if it follows a Brownian motion. I am wondering whether there are research papers that generalize similar concepts under different assumptions of the dynamics. For example, what would be the expected time spent under water if a portfolio follows a GBM with mean mu and variance sigma?

## Answer by Matt Wolf (score 0, accepted)

https://quant.stackexchange.com/a/4451

Here you go,

http://arxiv.org/pdf/cond-mat/9808295.pdf

http://www.cs.rpi.edu/~magdon/talks/mdd_NYU04.pdf

http://www.intelligenthedgefundinvesting.com/pubs/rb-kwcmlr.pdf

However, you mentioned you make an assumption of the portfolio dynamics. That means you either have historical data available about your portfolio returns and standard deviation or you must be able to formulate a model of your specific portfolio under question that allows you to construct portfolio dynamics to be used in a monte carlo simulation. Either case, I strongly recommend you to look as closely as possible at the specific portfolio dynamics assumptions and go from there, because most likely you will never be able to even come close to mimic diversified asset portfolio dynamics with standard distributions.

You mentioned in the end an example. If you assume your portfolio valuations follow a GBM then you can most easily construct a mc simulation. Hope this helps.

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