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I know this might be a trivial question, however, I would be grateful for some clarification. I am working on weekly log-return data, doing volatility-foracasting using GARCH models and then using these forecasts I solve a mean-variance asset allocation problem and obtain weights for each of the considered assets.

Having done so, I want to obtain realized portfolio returns. And here is my question: can I simply multiply the weights by the corresponding log-return series? Thus I would obtain a series of realized portfolio log-return which I can later easily transform into simple returns. I was reading this paper and in formula $1.21$ the author compares the results of portfolio returns obtained from simple and log returns and in this comparison he multiplies the weights by corresponding log-returns. In this paper the author works on the log-returns and than in formula $6$ uses the exponential transformation for obtaining simple returns in the asset allocation problem.

So my question in general is: when modelling the volatility on log-returns using GARCH models and obtaining weights in the log-return framework can I use the log-returns of assets to obtain the realized portfolio return? Or should I solve the asset allocation problem on simple returns, as in the second reference I mentioned.

I know that for daily data the difference is negligible, so for weekly data it might be the same.

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