Why Rolling Averages May Not Reduce Price-Level Variance

Summary

The document asks why a rolling average of a compounded price series can appear to have the same variance as the underlying price levels, even as the averaging window grows. The question contrasts this observation with the familiar intuition that averaging reduces variance by a factor related to the number of observations. It sketches a derivation starting from prices formed by compounding normally distributed returns and a finite moving average, then considers using a power-series expansion.

No resolution or empirical dataset is provided, so the proposed convergence remains an observation rather than an established result. The distinction between price-level variance and return variance is central: price levels in a compounded series are dependent and potentially nonstationary, so the usual variance reduction intuition for independent observations does not directly apply. The document is useful as a prompt to examine dependence, sample length, and the assumptions behind variance calculations, but does not supply a completed analytic comparison.

Key ideas

• The question concerns variance of price levels, not variance of returns.
• A compounded price series creates dependent observations across time.
• The usual variance reduction intuition for averages relies on assumptions that may not hold for price levels.
• The document presents a conjectured large-sample behavior but does not establish it analytically.

Tags

# Moving average variance


# Moving average variance












I have generated a random series of returns drawn from a normal distribution and generated a random price series by compounding these returns (X) so $P_i = P_1(1+X)^i$. I want to show the analytic comparison between the variance of this price series to the variance of a k-period moving average of the price.

What I am finding is that the variance of the price series (not price returns) converges to the variance of k period average price for $all$ values of k (so the variance of moving average price = price variance regardless of the averaging period) for large data sets. This surprised me as I would have expected the variance of the moving average to be scaled down by k regardless of the period.

I have tried to derive this result mathematically but without success. Here is the starting point of my calculation: $$MA_k = 1/k . {\sum^k_{i=1}P_i} = $$ $$=1/k . (P_1 + P_1(1+x) + ... + P_1(1+x)^{k-1})$$ $$P_1/k.{\sum^{k-1}_{i=0}}(1+x)^i$$

From here I use a power series expansion, ignore higher order terms, and take the variance of the expression. I think I should find this expression $var(MA_k) = P_1^2.var(x)$ but that's a guess. I can see I am not incorporating the dependency on the size of the data because there appears to convergence in variances between the rolling average of the price and the price, but I am not sure how to do that.

Any assistance in filling in the missing steps would be very welcome here.

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.