With regards to the portfolio variance, variance of the portfolio = (1/n)(avg. individual var) + [(n-1)/n][Avg Cov] Is this purely a “theoretical concept”? My natural instinct suggests that it is rather easy to find a very large sample n while maintaining an average covariance of 0, and then this equation will give a very small portfolio variance, but it will be inaccurate. For example, i will 1) long apple 2) short google 3) long temperature in new york 4) short temp in new jersey 5) long the weight growth of my dog 6) short the weight growth of my other twin dog A portfolio like that will probably give me an avg cov of close to zero (I could be wrong here?). And if i use n=500 then the formula will give a VERY small portfolio variance, but will this small variance be accurate? Thanks all in advance. Jeffrey
and what are you doing in the CFA program?
going long an apple? what happened to Orange? and why are dogs and temperature in your portfolio?
You can always find a portfolio with zero variance. A trivial case would be long AAPL and short AAPL, which obviously has zero volatility. But why would you do that? Generally, you are trying to minimize volatility given some positive return target, not minimize volatility only.
In addition, there are some things that you can’t really invest in. For instance, there is no futures market on the weight of your dog, so that would be a constraint against your ideal portfolio. In the real world, a stock you want to short might be hard to borrow (GRPN), there might not be a good tracking security (VIX), or you might have some other trading constraint.
What ohai said. It’s not unrealistic or some made up concept. It’s statistics. If you long/short two things that are extremely similar, you’re taking a position in their difference, which as you seem to understand is very small. Taking a position in something that is, in general, very small implies that your position’s volatility is also very small.
And the reason it gets even closer to zero as you increase n is that some differences will be positive and others will be negative, so on net they will tend to cancel each other out symmetrically.
The main point of that formula (also note that it only works for equallyweighted portfolio) is that as n gets larger, the covariance effect dominates at the expense of variance effect.