Uncorrelated Does NOT Imply Independent

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 Risk Fundamentals (link removed due to being broken as of 2018-06-13) website commits this error fairly egregiously:

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.

Again, uncorrelated does not in any way imply independence (also called statistical independence). Here is a proof in case you don’t want to look it up on Wikipedia. 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 $Y$:

• Are uncorrelated
• Have the same normal distribution
• Are NOT independent

Proof that $ X $ and $ Y $ are uncorrelated:

To show this we need only demonstrate that their covariance is 0. $$ cov(X,Y) = E[XY]-E[X]E[Y] = E[XY] - 0 = E[E[XY \| W]] $$ Now $ W $ is very simple, so this conditional expectation can be expressed easily. $E[E[XY| W]]= E[X^2]Pr(W=1) + E[-X^2]Pr(W=-1) = 1\\\times\\\frac{1}{2} + -1\\\times\\\frac{1}{2} = 0$ Since their covariance is 0, they are uncorrelated. QED

Proof that $X$ and $Y$ have the same normal distribution:

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 $Pr(Y \le x) = Pr(X \le x)$. QED.

Proof that $X$ and $Y$ are NOT independent:

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. QED.

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.