# Comparing Portfolio Volatility with Index Volatility seems a wrong method?

thanks for looking into this question.

I am comparing an investment strategy against the S&P 500 for a financial article I'm writing.

I compute volatility of the Portfolio in this way, as the square root from the variance computed like this: [This is for 4 stocks, but it can of course be extended to an amount of i stocks]
So for that I take volatility of all individual stocks and correlation between these stocks, that constitute the portfolio, into account.

However, when I calculate the volatility of the index, I just compute the standard deviation from the logreturns on the index. So no correlation of constituents is taken into account for the index.

What I am wondering: is comparison of the volatility of a portfolio against the volatility of an index valid? Or should you also treat the portfolio as an index when you want to perform this comparison?

But putting these details aside: If the return $r_p$ is calculated by $$r_p = \sum_{i=1}^n w_i r_i = w r,$$ with a vector of weights $w$ and a vector of returns $r$ and $n$ is the number of assets, then $$VAR(r_p) = VAR(w r) = w^T \Sigma w.$$ Thus the variance of the index (left hand side) without look-through equals the variance of the portfolio if you look through.