Given log returns for some stocks $A$ and $B$, which are the constituents of our hypothetical portfolio in equal weights, how does one actually come up with a distribution of the log returns of the portfolio?

I have read somewhere that log returns are not linear in the portfolio return calculation sense. So that if I have log returns for stock $A$ and stock $B$ at time $t$, then the portfolio log return is not $0.5$*log return of stock $A$ at $t$ + $0.5$*log return of stock $B$ at $t$, in such a case, it would be incorrect to do this calculation and then create a histogram. What is then a correct way to go about this?

EDIT: I can describe a method that I have in mind. But it would be computationally ineffective. So let's denote the log return as $r^*$ and simple relative return as $r$. We can establish the relationship between the two with: $e^{r^*}-1=r$. Therefore I can do the following, at each time $t$, convert the log returns into simple returns, apply the portfolio weights of each stock and then convert this new simple return into the log return (this is the correct log return). Now plot the histogram with these values. Actually, I may end up doing this...

Actually I would take a $\log(1+\text{simple portfolio return})$ and not just a log of simple portfolio return.

EDIT EDIT: I believe since I am looking at daily prices, I can apply the weights directly onto the log returns and I will not obtain much of a discrepancy, since log and simple returns are virtually the same for the small values.


The most correct way if you want to do it with log returns is the way you stated on your first edit, but indeed for daily data the approximation error is negligible.


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