Why Time Aggregation and Asset Diversification Affect Variance Differently

Summary

The document asks why two variance claims seem to conflict. One concerns aggregating returns across time: serial dependence can change the variance of a multi-period return relative to the variance expected under independent increments. The other concerns combining asset returns: portfolio variance depends on each asset’s variance and on their covariance, so positive covariance raises portfolio risk relative to an otherwise comparable less-correlated mix.

These are different operations. Time aggregation combines observations of a return process, where serial covariance across dates matters; portfolio construction combines returns across assets at a given horizon, where cross-asset covariance matters. The first claim’s conclusion depends on its setup and is not generally true as written: identical consecutive returns, for example, do not make the two-period sum less variable than twice the one-period variance. The document gives no numerical example or formal resolution, so the relationship needs explicit assumptions about whether variance refers to sums, averages, or scaled returns.

Key ideas

• Time aggregation depends on covariance between returns observed at different dates.
• Portfolio variance depends on covariances between asset returns over the same horizon.
• Positive cross-asset covariance increases portfolio variance relative to a lower-correlation mix.
• The claim that correlated consecutive returns necessarily lower multi-period variance is not generally correct.
• Clarify whether the comparison concerns summed returns, averages, or scaled returns.

Tags

# Portfolio variance over time vs. Portfolio variance from mix of two assets


# Portfolio variance over time vs. Portfolio variance from mix of two assets












Here are two facts about finance:

1. If stock returns are not independent, for example tomorrow's return equals today's return, then the variance over two days will be less than the variance of 1day*2. A test of market efficiency is that the variance over time does not decrease but is stable, meaning the returns are independent. Observation #1:When adding highly correlated observations, the combined will have a lower variance over the period.

2. When combining returns, the variance of the sum is equal to the sum of the variances and covariances Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). Therefore a portfolio of highly correlated observations will have the highest variance and one with negative correlation will have the lowest variance. Observation#2:When adding highly correlated observations, the combined will have a higher variance.

What is wrong in my reasoning that Observation #1 and #2 are exactly opposite statements? Is Observation #1 incorrect somehow?

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