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Uncorrelated Does NOT Imply Independent - Adam Drake
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<meta name=twitter:description content="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.">
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<meta property="og:description" content="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.">
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<h1 class=page-title>Uncorrelated Does NOT Imply Independent</h1>
<h2 class=content-date>June 21, 2011</h2>
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<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>
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<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>
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