Variance of Multi-Period Returns and Return Autocorrelation

Summary

The document poses a variance question for a sum of four weakly stationary log returns. It asks how to express the four-period variance in terms of the variance of the first three returns and the return variance together with autocorrelations at several lags. The attempted derivation assumes that the variance of a sum can be split into the separate variances, which is generally valid only when the terms are uncorrelated.

The central statistical lesson is that covariance terms matter when returns are serially dependent. Weak stationarity supplies a common return variance and lag-based autocovariances, so the variance of a cumulative return must account for pairwise relationships among periods. The document provides no answer or proof, and its proposed equality is not established by the displayed steps. Resolving it requires expanding the variance of the sums using covariance identities and checking the specific relationship being claimed.

Key ideas

• The variance of a sum includes covariance terms between its components.
• Weak stationarity allows return covariances to be described by lag.
• The displayed derivation incorrectly treats return variances as additive without addressing dependence.
• A proof must expand the cumulative-return variance and account for pairwise autocovariances.

Tags

# Variance of Log Returns


# Variance of Log Returns












Consider an asset held for $n$ time periods with weakly stationary log-returns $r_t$, $1≤t≤n$.

Show that $var(r_1 +r_2 +r_3 +r_4)=var(r_1 +r_2 +r_3)+var(r_1)(1+2ρ_3 +2ρ_2 +2ρ_1)$, where $ρ_k$ is the autocorrelation of ${r_t}$ at lag $k$.

I assume there is something simple I am missing, I have got to..

$var(r_1+r_2+r_3+r_4)= var(r_1) + var(r_2) + var(r_3) + var(r_4)$

$ = var(r_1+r_2+r_3) + var(r_4)$

==> $var(r_4) = var(r_1)(1+2ρ_3+2ρ_2+2ρ_1) $

But I'm struggling to prove the final result, any help would be appreciated!

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