{"id":7440,"date":"2012-06-04T10:19:58","date_gmt":"2012-06-04T09:19:58","guid":{"rendered":"https:\/\/www.portfolioprobe.com\/?p=7440"},"modified":"2012-07-04T08:35:53","modified_gmt":"2012-07-04T07:35:53","slug":"variability-in-maximum-drawdown","status":"publish","type":"post","link":"https:\/\/www.portfolioprobe.com\/2012\/06\/04\/variability-in-maximum-drawdown\/","title":{"rendered":"Variability in maximum drawdown"},"content":{"rendered":"

Maximum drawdown is blazingly variable.<\/p>\n

Psychology<\/h2>\n

Probably the most salient feature that an investor notices is the amount lost since the peak: that is, the maximum drawdown.<\/p>\n

Just because drawdown is noticeable doesn’t mean it is best to notice.<\/p>\n

Statistics<\/h2>\n

The paper “About the statistics of the maximum drawdown in financial time series”<\/a> explores drawdown analytically.\u00a0 The interesting part for me is Figures 14 onwards (which start on page 14).\u00a0 The pictures imply that the maximum drawdown could have been pretty much anything.<\/p>\n

In the post “A minimum variance portfolio in 2011”<\/a> we explored a particular portfolio. Figure 1 shows its value through 2011.\u00a0 We’ll use this as our example.<\/p>\n

Figure 1: The value of the portfolio throughout 2011. \"\"<\/a><\/p>\n

The maximum drawdown of this portfolio is 11.1%.<\/p>\n

It is almost true that we shouldn’t care what order the returns arrive in as long as we get the same set of returns over the year.\u00a0 In other words, we should be just as happy with each permutation of the order of the returns of our portfolio.<\/p>\n

Figure 2 shows the maximum drawdown distribution for random permutations of the returns of our portfolio.\u00a0 Remember that the return over the year is precisely the same for all the portfolios with these drawdowns.<\/p>\n

Figure 2: Distribution of maximum drawdown from permutations of the portfolio returns, with 95% confidence interval (gold lines). \"\"<\/a>We can easily get maximum drawdowns from 7% to 17% using the exact same returns<\/strong>.<\/p>\n

Given the big drop at the beginning of August, we might be led to think that the real maximum drawdown would be large relative to the possibilities over the permutations.\u00a0 Actually it is only slightly larger than the mean.<\/p>\n

What do worst-case and best-case profiles look like?\u00a0 Figure 3 shows a permutation that comes close to maximizing the drawdown and Figure 4 shows one that comes close to minimizing drawdown.<\/p>\n

Figure 3: Permutation of the portfolio returns that produces a large drawdown. \"\"<\/a><\/p>\n

Figure 4: Permutation of the portfolio returns that produces a small drawdown. \"\"<\/a><\/p>\n

Large drawdowns occur when the negative returns are all bunched together.\u00a0 Small drawdowns happen when the negative returns are evenly spread.<\/p>\n

Above I said we should be almost<\/strong> indifferent to different permutations of the returns.\u00a0 The catch is that we need to believe that there is neither momentum nor mean-reversion<\/a>.\u00a0 In Figure 3 we have selected for momentum, and we have selected for mean-reversion in Figure 4.<\/p>\n

You may be wondering what the equivalent pictures to Figures 3 and 4 are when the overall return is negative.\u00a0 Obviously the maximum drawdown has to be at least as big as the overall loss.\u00a0 To get big drawdowns the strategy remains the same — push all the negative returns together.<\/p>\n

Figure 5 shows an example of\u00a0 minimizing the maximum drawdown when there is an overall loss — create multiple drawdowns of the same size.\u00a0 That is the strategy with overall gains as well, but less obvious on first sight.<\/p>\n

Figure 5: Permutation of the negative portfolio returns that produces a small drawdown. \"\"<\/a><\/p>\n

 <\/p>\n

Summary<\/h2>\n

While maximum drawdown is emotionally compelling, it is not statistically compelling at all.\u00a0 My guess is that trying to ignore it might lead to better investment decisions.\u00a0 Other opinions?<\/p>\n

See also<\/h2>\n

It so happens that Timely Portfolio<\/strong> just had a post on drawdown<\/a> as well.<\/p>\n

Update (2012 July 04)<\/strong>: Slides from a talk on a theoretical analysis of drawdown are “An Analysis of the Maximum Drawdown Risk Measure”<\/a> by Malik Magdon-Ismail.<\/p>\n

Epilogue<\/h2>\n

I’m a poor wayfarin’ stranger
\nI have nothin’ left to lose
\nI have fallen down the mountain
\nWretched righteous, flesh for fools<\/p><\/blockquote>\n

from \u201cDown the Mountain\u201d by Robin E. Contreras
\n<\/object><\/p>\n

Appendix R<\/h2>\n

There were two R<\/a> functions written for this.\u00a0 The first computes the maximum drawdown given a wealth curve:<\/p>\n

> pp.maxdrawdown\r\nfunction (wealth)\r\n{\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 max(1 - wealth \/ cummax(wealth))\r\n}<\/pre>\n
If you have simple returns, then you can use the maxDrawdown<\/code> function from the PerformanceAnalytics<\/code> package.\u00a0 Note that the use of log returns is not supported — at least currently — in this function.<\/p>\n
The second function that was written collects the minimum and maximum of the maximum drawdowns over permutations of the returns:<\/p>\n
> pp.collectdd\r\nfunction (ret, trials=1e4)\r\n{\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 mind <- Inf\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 maxd <- -Inf\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 for(i in 1:trials) {\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 wealth <- exp(c(0, cumsum(sample(ret))))\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 td <- pp.maxdrawdown(wealth)\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if(td < mind) {\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 mind <- td\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 minw <- wealth\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 }\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 if(td > maxd) {\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 maxd <- td\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 maxw <- wealth\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 }\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 }\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 list(min.drawdown=mind, min.wealth=minw,\r\n                max.drawdown=maxd, max.wealth=maxw)\r\n}<\/pre>\n
There was not much sense in writing a function to get the distribution of maximum drawdowns from permutations — just a couple lines of code sufficed:<\/p>\n
> op2.mv.mddd <- numeric(1e4)\r\n> for(i in 1:1e4) op2.mv.mddd[i] <- pp.maxdrawdown(\r\n+     exp(cumsum(c(0, sample(op2.mv.logret)))))\r\n> plot(density(op2.mv.mddd * 100))<\/pre>\n
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Maximum drawdown is blazingly variable. Psychology Probably the most salient feature that an investor notices is the amount lost since the peak: that is, the maximum drawdown. Just because drawdown is noticeable doesn’t mean it is best to notice. Statistics The paper “About the statistics of the maximum drawdown in financial time series” explores drawdown … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,6],"tags":[248],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/posts\/7440"}],"collection":[{"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/comments?post=7440"}],"version-history":[{"count":0,"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/posts\/7440\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/media?parent=7440"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/categories?post=7440"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.portfolioprobe.com\/wp-json\/wp\/v2\/tags?post=7440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}