Average correlation of index/portfolio

We try to analyze the average correlation of a portfolio as it can be found here in section 2 b), the same formula which is also used by the CBOE to calculate implied correlations:

$$\rho_{av(2)} = \frac{\sigma^2 - \sum_{i=1}^N w_i^2\sigma_i^2}{2 \sum_{i=1}^N \sum_{j>i}^N w_i w_j \sigma_i \sigma_j}$$

EDIT:Assuming that $\sigma^2 = \sum_{i=1}^N \sum_{j=1}^N w_i w_j \sigma_i \sigma_j \rho_{i,j}$, where $\rho_{i,i}=1$, for $i=1,\ldots,N$, the above expression can be written as

$$\rho_{av(2)} = \frac{\sum_{i=1}^N \sum_{j>i}^N w_i w_j \sigma_i \sigma_j \rho_{i,j}}{\sum_{i=1}^N \sum_{j>i}^N w_i w_j \sigma_i \sigma_j}.$$

The following questions arise.

1. Assuming that $w_i \in \mathbb{R}$, i.e. long/short leverage is allowed, is it possible that $|\rho_{av(2)}|>1$ ? Note that we don't assume $\sum w_i=1$.


2. Does there already exist the notion of contribution to average correlation? Meaning that e.g. in a long/short portfolio, where average correlation should be close to zero, I can identify positions that drive the average correlation up (in absolute value).

Answer

I did some calculations in mathematica in the 3 asset case. Assume we have exposures $w_i,i=1,2,3$ and volatilities $\sigma_i,i=1,2,3$ and correlations $\rho_{1,2},\rho_{1,3},\rho_{2,3}$. Let's assume $\sigma_1=\sigma_2=\sigma_3=\sigma$ for some arbitrary positive $\sigma$. For the weights we assume $w_2=w_3 = 0.5$ and we have a short in asset 1 of $w_1 = -0.5$. Then the above formula becomes

$$\rho_{av(2)} = \rho_{1,2}+\rho_{1,3}-\rho_{2,3}.$$ Then the question is whether we can find valid (pos.definite correlation matrix) values for the correlations such that the above formula delivers a results out side of the unit interval. A possible choice is $\rho_{1,2}=0.95, \rho_{1,3}=0.95$ and $\rho_{2,3}=0.89$ with the result $1.01$!

The mathematica code is the following:

pfvar[w1_, w2_, w3_] := w1^2*[Sigma]1^2 + w2^2*[Sigma]2^2 + w3^2*[Sigma]3^2 + 2*([Sigma]1*[Sigma]2*[Rho]12*w1*w2 + [Sigma]1*[Sigma]3*[Rho]13*w1*w3 + [Sigma]3*[Sigma]2*[Rho]23*w3*w2)

impliedCorr[w1_, w2_, w3_] := (pfvar[w1, w2, w3] - (w1^2*\[Sigma]1^2 + w2^2*\[Sigma]2^2 
   + w3^2*\[Sigma]3^2))/(  2*(\[Sigma]1*\[Sigma]2*w1*w2 + \[Sigma]1*\[Sigma]3*w1*
       w3 + \[Sigma]3*\[Sigma]2*w3*w2)  )

impliedCorr[w1, w2, w3] /. w2 -> w3 /. [Sigma]2 -> [Sigma]3 /. [Sigma]3 -> [Sigma]1 /. w3 -> 0.5 /. w1 -> -0.5 // Simplify

[Rho]12 + [Rho]13 - [Rho]23 /. [Rho]12 -> 0.95 /. [Rho]13 -> 0.95 /. [Rho]23 -> 0.89

EDIT: Thanks to @John I found a mistake and corrected $\rho_{2,3}$ to $0.89$.