Log Returns and Portfolio Return Aggregation

Summary

The document concerns portfolio construction with monthly returns for several assets. It asks whether asset log returns should be averaged arithmetically and whether multiplying asset returns by portfolio weights gives the portfolio return when the optimization uses a variance-covariance matrix and seeks Sharpe ratio maximization.

The brief answer highlights a key distinction: log returns add across time for a single asset, but they do not add linearly across assets in a portfolio. Thus, a weighted sum of component log returns is not generally the portfolio’s realized log return. The source answer does not provide a full calculation procedure, discuss the arithmetic-versus-geometric return choice in detail, or explain implications for estimating expected returns and portfolio variance. Those questions require care about whether returns are simple or logarithmic and about the portfolio’s rebalancing convention.

Key ideas

• Log returns are additive across successive time periods for an individual asset.
• Log returns are not generally additive across assets within a portfolio.
• A weighted sum of component log returns should not automatically be treated as the realized portfolio log return.
• Return aggregation depends on the return definition and the portfolio rebalancing convention.

Tags

# Log returns of individual assets and calculating portfolio returns


# Log returns of individual assets and calculating portfolio returns












I am researching optimal asset allocations and am wondering if I am making mistake(s) in calculating the portfolio return. I have three assets, of which I have monthly return data. I have calculated the returns by log(pt/pt-1) (because I was told to do this). Next, I calculate the variance-covariance matrix, from which I get the portfolio variance etc. Then I multiply the weights by the realized returns to get the portfolio return. I use an algorithm to get the Sharpe-maximizing weights.

- Is it even correct to calculate the return of the assets as the arithmetic mean of the log returns? How should it be done?

- How should the portfolio return be calculated? Is it wrong to multiply weights by expected returns because the log returns aren't cross asset additive?

## Answer by CFW (score 1)

https://quant.stackexchange.com/a/42903

As you already mention in point 2, log returns are not linearly additive across portfolio components. However, they are additive over time (point 1).

See this technical answer on SE Economics.

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.