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Why Portfolio and Index Volatility Can Be Compared Directly

Article Quant Q&A · Author: AltTabsen
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Summary

The post asks whether the volatility of a stock portfolio can be compared with the volatility of the S&P 500 when one is calculated from constituent volatilities and correlations and the other from observed index returns. The accepted answer says the direct return series of an index already reflects the combined behavior of its constituents, including their correlations and diversification effects. Therefore, the two volatility measures can be compared when they represent comparable return periods and conventions.

A second answer describes an index as a portfolio and gives the portfolio variance relation: for asset weights and returns, the variance of the weighted return equals the weights applied to the return covariance matrix. This provides the look-through interpretation of index variance. The post also flags changing constituents and time-varying weights as practical details. It does not specify annualization, sampling frequency, rebalancing assumptions, or other choices that must be aligned for a rigorous empirical comparison.

Key ideas

  • Volatility computed directly from index returns already includes constituent co-movement and diversification effects.
  • An index can be treated as a portfolio whose variance depends on asset weights and return covariances.
  • Portfolio volatility and index volatility can be compared when their measurement conventions are aligned.
  • Changing constituents and weights over time can affect a look-through analysis.
  • The post does not prescribe sampling, annualization, or rebalancing conventions.

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


# 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?

Thanks in advance!

## Answer by QuantK (score 1, accepted)

https://quant.stackexchange.com/a/16225

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 by Richi Wa (score 1)

https://quant.stackexchange.com/a/16226

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.