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I was asked yesterday by a colleague why we are doing asset allocation using optimizers which target, for a minimum expected return:

  • the portfolio with the minimum variance

or

  • the portfolio with the minimum expected shortfall

and why nobody uses drawdown optimizations.

I do know that the list above is not exhaustive, but I have never seen an allocation method which was aiming to minimize the drawdown for a given expected return.

The first thing that came to my mind is that the drawdown problem would not be convex and hence it would be difficult to find an optimal allocation using classic optimization model.

I also wanted to point out that drawdown measure are based on past measures and are not useful to "predict" the future -- drawdowns are history. So, an optimization based on drawdowns would probably result in data mining bias.

Do you see a point I missed? Is one of my points wrong?

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Maximum drawdown is extremely sensitive to minute changes in weights and to the specific time period examined. –  Tal Fishman Jul 6 '12 at 18:46
    
The convexity issue seems to be resolved here arxiv.org/abs/1404.7493 but I haven't tried this yet. –  John May 5 at 22:16
    
You may find "Investing for Retirement: The Defined Contribution Challenge" by Ben Inker of GMO interesting. He derives a dynamic investment strategy which minimizes the drawdown from an expected terminal wealth (he calls this expected shortfall and defines the objective as "minimize, in expectation, how much wealth falls short of what is needed") –  SpeedBoots May 19 at 13:04
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5 Answers 5

up vote 8 down vote accepted

I can think of three reasons.

First, and simplest, is that people care about variance.

Second, if you really do care about draw-downs, if returns are close to normally distributed, the distribution of draw-downs is just a function of the variance, so there's no need to include draw-downs explicitly in your portfolio construction objective. Minimizing variance is the same as minimizing expected draw-downs.

Third, (and here's the main point) if returns are very non-normal and you really do want to find portfolio weights that minimize expected draw-downs, you still wouldn't choose weights that minimize historical draw-down. Why? Because minimizing historical draw-down is effectively the same as taking all your returns that weren't part of a draw-down, and hiding them from your optimizer, which, as Tal mentioned, will lead to portfolio weights that are a lot less accurately estimated than if you let your optimizer see all the data you have. Instead, you might just include terms in your optimization objective that penalize negative skew and penalize positive kurtosis.

Or, if you wanted to get fancy, you could use all your historical data to fit your favorite fat-tailed distribution. As far as I know, none of those distributions have both finite variance and are closed under linear combinations, so your portfolio returns wouldn't have the same distribution. So to get optimal portfolio weights, you'd have to simulate some very large number of returns from each of these distributions, and then calculate (let's say) the 99th percentile portfolio draw-down for each set of portfolio weights, and use that function in your optimization. I would think that would give you more robust results. Though since you're trying to estimate a tail event, it would probably require a huge amount of simulated data and take a long long time to run. After you finish, you might fairly conclude that the entire exercise wasn't worth the effort, and just go back to mean variance optimization...

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Good answer. I disagree with your first point though, non-sophisticated investors do not care about variance, they care about drawdowns: "how much can I use if I give you my money". I guess it's a matter of point of view. My initial first answer was, any strategy with any kind of asset has a max drawdown of 100% anyway, so it's better to care about variance or, better, expected shortfall of fat-tailed distributions. –  SRKX Jul 9 '12 at 22:22
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Of all the professional investors I work with, not one of them uses simple mean-variance optimization. In fact, every one starts with a maximum drawdown parameter to bound the space of acceptable solutions (among other constraints), then applies some optimization within that space. As for assuming a parametric distribution of returns, I've never encountered anyone willing to make such an assumption. –  pteetor Oct 24 '12 at 0:13
    
@pteetor: Agreed; I don't know anyone who uses simple MV optimization. Just about everyone I've worked with uses heavily modified MV optimization, typically with regularization terms are added, along with hard constraints on the weights. –  Marc Shivers Oct 24 '12 at 12:30
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Actually, Ralph Vince's Leverage Space Trading Model does utilise draw down. A short introductory pdf is available here, and the R-forge package is here.

Briefly, a genetic algorithm is used to model the maximum expected portfolio return based on a joint probability distribution of the portfolio component returns, subject to an overall maximum draw down constraint.

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Technically, Ralph Vince uses the probability of drawdown as a constraint. The LSPM package does have a function for max drawdown too though. –  Joshua Ulrich Jul 6 '12 at 14:45
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With minimum variance, the covariance matrix does not change when you change the holdings. So all the optimizer needs to do is change the weights. This makes it easy to calculate the gradients.

To construct the drawdown statistic, you would need the distribution of returns in each period to your horizon. You would then need to calculate the path of profits given your holdings and then calculate the drawdown. You would not be able to write a function for the gradients, which means that you could potentially get into trouble during the optimization.

One alternative is to construct a mean-variance frontier and then using those portfolios calculate the drawdowns. You could then choose the optimal portfolio based on some measure that incorporates drawdown. This can also be done with expected shortfall.

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Just have a look at Chekhlov/Uryasev/Zabarankin paper on the subject. People do not use it because they do not understand it that s it.

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Can you provide a link to the article and maybe slightly develop the idea. Because trust me, clients do know very well what a drawdown is. –  SRKX Jul 1 '13 at 10:06
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Drawdown Measure in Portfolio Optimization June 25, 2003 papers.ssrn.com/sol3/papers.cfm?abstract_id=544742 Portfolio Optimization with Drawdown Constraints April 8, 2000 papers.ssrn.com/sol3/papers.cfm?abstract_id=223323 DRAWDOWN MEASURE IN PORTFOLIO OPTIMIZATION Accepted 25 May 2004 ise.ufl.edu/uryasev/files/2011/11/IJTAF_DrawDown_Paper.pdf PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS January 13, 2003 ise.ufl.edu/uryasev/files/2011/11/drawdown.pdf –  montyhall Jul 1 '13 at 19:42
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In response to the original question:

Drawdown optimization is a convex problem, see our recent article:

http://ssrn.com/abstract=2430918

We do not address the issue of choosing a "good" risk model to feed the optimizer. However, even when using the history, drawdown does capture something that volatility and expected shortfall do not account for, namely path dependency (serial correlation). An initial analysis of the path dependency of drawdown is in the above article. The value added (if any) by optimizing drawdown rather than variance or shortfall is a project we are currently working on.

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