Comparing Portfolio Volatility with Index Volatility seems a wrong method?

Question

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?

Thanks in advance!

Answer 1

The volatility of the index already incorporates the correlation (diversification benefits), even if you calculate it directly as you stated.

So yes, you can compare them.

Answer 2

There is no real difference between an index and a portfolio - at least an index usually can be seen as a portfolio. There are things to consider if index constituents change. One thing that has to be taken into account is that the weights of the constituants change over time due to chaging market prices.

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.