# Return correlations

Assume an equity fund sample shows returns negatively correlated with the S&P 500.

Are we more inclined to say that a) these funds are invested outside the S&P 500, perhaps non-US stocks; b) these funds have selected stocks belonging to the S&P 500, but substantially uncorrelated with the index?

Update

I give a quantitative context for the problem above.

Let $R_{it}$, $M_t$ be resp. the $i$-th fund, and the index return in $t$. Let $s_i$ be the total wealth of the $i$-th fund invested in stocks belonging to the index. Therefore, $\bar{s}_i=1- s_i$ identifies portfolio assets not belonging to the index.

Consider the linear model:

$$R_{it} = \alpha_i + \beta_iM_t + e_{it}$$

For a large sample of funds, if $\beta_i$ are consistently and significantly negative, can we say that $\bar{s}_i$ is large, that is, on average funds' wealth is not invested in the S&P index?

## 1 Answer

When you say "an equity fund sample shows returns negatively correlated with the S&P 500". So if I assume that the benchmark (S & P 500 in your case) to measure the performance of the returns is correct and it is negatively correlated, then it may be because the fund has the inverse positions as that of S & P 500. For example it might be possible that the 10 stocks which are there in the fund as long positions are there in the S & P 500 as short positions which will depict a negative correlation.

Now if talk about your two given options:-

a) "these funds are invested outside the S&P 500, perhaps non-US stocks" - If this is the case then the obviously the choice of benchmark is entirely wrong i.e. it is obvious if the fund has invested in non US stocks, then it will show a negative corelation when compared to S & P 500.

b) "these funds have selected stocks belonging to the S&P 500, but substantially uncorrelated with the index?" - We can say this option as I have given the explanation at the start of my answer.

Hope this helps!!

• You are adding an extra hypothesis! I did not claim that the index used is the "benchmark" of the sample funds, nor did I say that all funds in the sample share the same benchmark. BTW, if the sample exhibits significant negative correlation with the S&P, how can one affirm that this index is the funds' benchmark? Also you are not considering that managers might not go short, but still implement an active management focused on a subset of the S&P stocks. – antonio Jan 10 '17 at 2:45
• Then why are you measuring the correlation with S & P 500? Tell me simply when does a Fund exhibit a negative correlation with any index? What here index refers to? A fund shows a negative correlation only when it is having a inverse positions as compared to index or if the fund is having most of the stocks outside of the index or entire fund is having different stocks as compared to index. It can be any of the case and you can only explain them by assuming or taking some hypotheses. – Manish Jan 10 '17 at 17:16
• Right, you got the problem. That's why the question says "are we more inclined too...", that is, given the lack of info, which is the more likely case (if there is a more likely one). Further reasonable hypotheses in the answers are welcomed, but they should be clearly labeled as new assumptions. Specifically, taking the S&P as the benchmark of each sample fund, somewhat clashes with the initial negative-correlation assumption. – antonio Jan 10 '17 at 20:26