Constructing Portfolio Log Returns from Constituent Returns
Summary
The document explains how to build a time series of portfolio log returns from constituent stock returns when portfolio weights are specified. Because log returns do not combine linearly across assets in the same way as simple returns, directly taking a weighted average of constituent log returns is not the exact portfolio calculation. The proposed procedure is to convert each asset’s log return to a simple return, combine those simple returns using the portfolio weights, and then convert the resulting portfolio simple return back to a log return before making a histogram.
The accepted response agrees with this calculation and notes that, for daily data, the approximation error from weighting log returns directly is negligible. That observation is limited to the small return magnitudes typical of the stated daily-data setting; it is not a general identity. The document does not discuss rebalancing, changing weights, transaction costs, or how the resulting histogram should be interpreted.
Key ideas
- Log returns do not combine linearly across portfolio constituents for an exact return calculation.
- Convert constituent log returns to simple returns before applying portfolio weights.
- Convert the weighted portfolio simple return back to a log return to form the portfolio series.
- A histogram should use the correctly calculated portfolio log returns for each observation time.
- For daily data, directly weighting log returns may be a close approximation, but it is not exact.
Tags
Full text
# Creating the histogram for the distribution of the portfolio returns
# Creating the histogram for the distribution of the portfolio returns
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.
## Answer by phdstudent (score 1, accepted)
https://quant.stackexchange.com/a/22880
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.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.