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<!doctype html><html lang=en><head><meta charset=utf-8><meta name=mobile-web-app-capable content="yes"><meta name=viewport content="width=device-width,initial-scale=1"><title>Uncorrelated Does NOT Imply Independent - Adam Drake
</title><meta name=description content="Adam Drake is an advisor to scale-up tech companies. He writes about ML/AI/crypto/data, leadership, and building tech teams."><link rel="shortcut icon" href=https://adamdrake.com/static/favicon.ico><link rel=authorization_endpoint href=https://indieauth.com/auth><link rel=token_endpoint href=https://tokens.indieauth.com/token><link rel=me href=https://github.com/adamdrake><link rel=stylesheet href=https://adamdrake.com/css/style.min.css crossorigin=anonymous media=screen><meta property="og:url" content="https://adamdrake.com/"><meta property="og:title" content="Adam Drake"><meta property="og:site_name" content="Adam Drake"><meta property="og:type" content="website"><meta property="og:description" content="Adam Drake is an advisor to scale-up tech companies. He writes about ML/AI/crypto/data, leadership, and building tech teams."><meta property="og:image" content="/static/images/twitter-card.jpg"><meta name=twitter:title content="Adam Drake"><meta name=twitter:description content="Adam Drake is an advisor to scale-up tech companies. He writes about ML/AI/crypto/data, leadership, and building tech teams."><meta name=twitter:card content="summary_large_image"><meta name=twitter:image content="/static/images/twitter-card.jpg"></head><body><header><section><div class="header flex row"><div class="header__item flex row"><a id=site__name href=https://adamdrake.com/>Adam Drake</a></div><div class="flex row"><nav aria-label="page menu" class="flex row"><ul role=menubar class="flex row"><li role=none><a class=sidebar-nav-itemmenu__item href=/ title>Latest</a></li><li role=none><a class=sidebar-nav-itemmenu__item href=/about.html title>About</a></li><li role=none><a class=sidebar-nav-itemmenu__item href=/cases.html title>Case Studies</a></li><li role=none><a class=sidebar-nav-itemmenu__item href=/contact.html title>Contact</a></li><li role=none><a class="sidebar-nav-item activemenu__item" href=/posts.html title=Posts>Posts</a></li><li role=none><a class=sidebar-nav-itemmenu__item href=/press.html title>Press</a></li><li><button class="subscribe subscribe-btn">
<a href=https://www.digitalmaneuver.com/#/portal>Subscribe to newsletter</a></button></li></ul></nav></div></div></section></header><main aria-role=main><section><ul id=feed__ul><li class="feed__li h-entry"><div class=feed__content><time class="hidden dt-published">2011-06-21 00:00:00 +0000 UTC</time><div class="flex properties__row"><div rel=author class="flex left p-author h-card hidden"><img class=u-photo src=https://adamdrake.com/static/images/adam_drake_240.jpg alt="Adam Drake" id=author-img><div><p rel=me class=p-name id=author-name>Adam Drake</p><p class=properties>Jun 21, 2011</p></div></div><div class="flex right properties"></div></div><article class="md p-summary e-content"><h1 class=p-name>Uncorrelated Does NOT Imply Independent</h1><p>This is just an aside to describe a misconception that we have seen some
money managers make when describing their strategies or portfolios. When
you are discussing the correlation of your portfolio to another
portfolio or the market in general, the fact that your portfolio may be
fairly uncorrelated does not have anything to do with the independence
of your portfolio from a reference portfolio or the market in general.
In fact, even if the portfolio you have developed is completely
uncorrelated, it still probably isn’t independent. The <a href>Risk
Fundamentals</a> (link removed due to being broken as of 2018-06-13) website commits this error fairly egregiously:</p><blockquote><p>However, if the performance of the two funds were uncorrelated
statistically independent the standard deviation of a portfolio
comprised of the two funds would decline to 7.1% compared with 10%
for each of the individual funds.</p></blockquote><p>Again, uncorrelated does not in any way imply independence (also called
statistical independence). Here is a proof in case you don’t want to
<a href=https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent>look it up on Wikipedia</a>. In the symmetric case, suppose you have two
random variables, both normally distributed and perfectly uncorrelated.
Let $X$ be a normally distributed random variable with mean
0 and standard deviation 1. Let $W$ be a simple random
variable that is -1 with probability 1/2 and 1 with probability 1/2. Now
let $Y = WX$. Then $X$ and $Y$:</p><ul><li>Are uncorrelated</li><li>Have the same normal distribution</li><li>Are NOT independent</li></ul><h3 id=proof-that--x--and--y--are-uncorrelated class=anchor-link><a href=#proof-that--x--and--y--are-uncorrelated>Proof that <code>$ X $</code> and <code>$ Y $</code> are uncorrelated:</a></h3><p>To show this we need only demonstrate that their covariance is 0.
<code>$$ cov(X,Y) = E[XY]-E[X]E[Y] = E[XY] - 0 = E[E[XY \| W]] $$</code> Now
<code>$ W $</code> is very simple, so this conditional expectation can be
expressed easily. <code>$E[E[XY| W]]= E[X^2]Pr(W=1) + E[-X^2]Pr(W=-1) = 1\\\times\\\frac{1}{2} + -1\\\times\\\frac{1}{2} = 0$</code> Since their covariance is 0, they are uncorrelated. <strong>QED</strong></p><h3 id=proof-that-x-and-y-have-the-same-normal-distribution class=anchor-link><a href=#proof-that-x-and-y-have-the-same-normal-distribution>Proof that $X$ and $Y$ have the same normal distribution:</a></h3><p>To do this we can show that both random variables have
the same cumulative probability density function. $Pr(Y \le x) = E[Pr(Y \le x | W)] = Pr(X \le x)Pr(W=1) + Pr(-X \le
x)Pr(W=-1)$ Now since $X$ and $-X$ have
the same normal distribution, this just becomes $Pr(X \le
x)\times\frac{1}{2} + Pr(X \le x)\times\frac{1}{2} = Pr(X \le
x)$ Therefore $Pr(Y \le x) = Pr(X \le x)$. <strong>QED</strong>.</p><h3 id=proof-that-x-and-y-are-not-independent class=anchor-link><a href=#proof-that-x-and-y-are-not-independent>Proof that $X$ and $Y$ are NOT independent:</a></h3><p>To see this consider the fact that $Pr(Y > 1 | X =
\frac{1}{2}) = 0$. In other words, $Y$ being
greater than 1 is dependent on $X$ NOT being
$\frac{1}{2}$ and therefore, they are NOT independent. <strong>QED</strong>.</p><p>As you can see, the fact that two things are uncorrelated does
not mean they are independent. If you are fund manager, investment
advisor, or any individual writing materials to be distributed to
investors or potential investors you should realize that every time you
say your portfolio is uncorrelated with and therefore independent of the
market, you’re probably lying.</p><a class=hidden href=https://brid.gy/publish/mastodon></a><a class=hidden href=https://brid.gy/publish/twitter></a><a class=hidden href=https://fed.brid.gy/></a><data class=p-bridgy-omit-link value=false></data></article></div><div id=webmentions></div></li></ul></section></main><hr><footer class="flex col"><section class="footer-bio content"><p><strong>Adam Drake</strong> leads technical business transformations in global and multi-cultural environments. He has a passion for helping companies become more productive by improving internal leadership capabilities, and accelerating product development through technology and data architecture guidance. Adam has served as a White House Presidential Innovation Fellow and is an IEEE Senior Member.</p></section><button class="subscribe subscribe-btn">
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