Zero Covariance?

CFA Level I
#1

Other than when you have one riskless asset and another risky asset, how do you get a covariance of zero? Riskless asset has a fixed return, so there is no deviation from its expected return, which causes the covariance to equal zero, i.e. R1-E(R1) * (R2-E(R2) = 0, because R1=E(R1). If two variables are *independent* then their covariance=0, it is claimed. Isn’t it possible to have two variables that are independent, yet their covariance does not equal 0?

#2

Nope independence => covariance = 0 covariance = 0 does not => independence UNLESS they are normals and then normal => if covariance = 0 => indpendence if covariance = 0 => independence then => normal

#3

Joey, yes I agree as I stated above that if two variables are *independent* then their covariance=0, and not vice versa. That’s clear. What I would like to know is if it is possible to have two variables that are independent, yet their covariance does not equal 0?

#4

No if they are independent their cov = 0.

#5

I see. So, if I think that stock returns in Angola are independent of U.S. stock market returns, then I gather monthly returns of the past 20 years of each market, and I calculate the covariance, then I should expect its value to be zero, right? So, I cannot say they are independent unless the covariance = 0? Isn’t it awfully difficult to actually get zero? That is 0.00? Would any value close to zero do? If not, then it is terribly difficult to prove independence using covariance, right? Of course if I find two stock markets whose covariance=0, it does not mean they are necessarily independent.

#6

You should expect it to be 0 - doesn’t mean sample cov is 0.

#7

Gotcha… but If no sample will show a covariance=0, then it is impossible (almost) to show empirically a covariance of zero, or at least we should not expect a sample to show a covariance=0, unless it’s a trivial case, like risk-free asset with risky asset. Is that a fair conclusion?

#8

Right

#9

The expected value of covariance is 0, but because the data points are not infinite or very large, it is likely that the covariance is not 0 due to chance.