Squared Returns as a Daily Volatility Proxy

Summary

The document asks how a sample standard-deviation estimate of return volatility relates to a daily measure based on squared returns. It contrasts estimating dispersion across a series of returns around their sample mean with using the square of an individual daily return, and questions why the latter is called volatility when it does not subtract the mean. The central topic is the intuition behind squared returns as a proxy for changing volatility.

No answer or derivation is included, so the relationship is left unresolved. The text provides no assumptions about the return process, no evidence comparing the estimators, and no discussion of whether the daily measure is intended as an instantaneous variance proxy or a volatility estimate. Readers should therefore treat this as a question framing a statistical issue, not as guidance establishing that squared returns equal volatility. It also writes the sample-volatility formula with a square root while labeling the result as variance, leaving a notation inconsistency that would need clarification in a full explanation.

Key ideas

• The document contrasts sample dispersion around average returns with a daily squared-return measure.
• It asks why squaring one return can serve as a proxy for volatility without subtracting the mean.
• The text provides no answer, derivation, or empirical support for the proposed relationship.
• Its sample-volatility expression labels a square-rooted quantity as variance, creating a notation inconsistency.

Tags

# Prove that $\sigma^2=R^2$


# Prove that $\sigma^2=R^2$












Suppose we have the following series: $R_1$, $R_2$, ...., $R_N,$ which are returns of a stock for $t=1,2,...,N$. If we want to estimate a sample volatility, then we can use $$\sigma^2=\sqrt{\frac{1}{N-1}\sum(R_i-\mu_R)^2}$$, where $\mu_R$ is the average of the series. However, if we have a daily data and we want to estimate daily volatility, in the literature it is used the following equation: $$\sigma_t^2=R_t^2$$. What is the connection of the first equation and the second? Why the latter is called a volatility? (Volatility is defined as deviation from them mean, but in the second estimation we do not have mean.) I do not understand the intuition of the second equation. Please explain. Thanks!

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