Why Logarithmic Portfolio Returns Complicate Variance Weighting

Summary

The document raises a question about calculating portfolio variance when portfolio performance is expressed as a logarithmic return. It gives a portfolio log-return expression formed by taking the logarithm of the weighted sum of assets’ exponentiated returns. The central issue is that the familiar portfolio variance formula, which combines asset variances and covariances using portfolio weights, is derived for linear simple returns.

The document does not supply a solution or compare alternative variance calculations. It highlights a modeling distinction: the logarithm of a weighted sum is generally not the same as a weighted sum of log returns, so applying the standard linear-return variance formula directly requires justification. Readers would need to specify the return horizon and portfolio rebalancing convention, then derive or approximate the distribution of the portfolio’s log return under those assumptions. No data, empirical evidence, or proposed method is included.

Key ideas

• The stated portfolio log return is the logarithm of a weighted sum of exponentiated asset returns.
• The standard portfolio variance formula is formulated for linear simple returns.
• A logarithm applied after summing weighted returns does not generally preserve linear weighting.
• The document poses the variance question but does not provide a derivation or answer.
• Any calculation needs clear assumptions about horizon and portfolio rebalancing.

Tags

# Applying portfolio variance weight based on logarithmic returns?


# Applying portfolio variance weight based on logarithmic returns?












The expected logarithmic return of a portfolio is calculated as :

$$𝐸_p = \log\left(\sum_i w_i e^{R_i}\right)$$

Therefore, I was wondering that how can I apply weight to use with the variance based on logarithmic returns in order to compute the portfolio variance? This is because the modern portfolio theory using simple weight multiply with variance is based on linear returns.

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