Yesterday, I introduced a case of going from a USD to a GBP return (via an FX rate conversion), then taking the underlying USD assets and converting them to GBP, deriving the return that way, and converting (via the FX converter) to USD [quite a mouthful]. Recall that the GBP and USD results were different, depending on how they were derived. Thus, the problem: what's going on?
Simple answer: Modified Dietz.
Recall that Modified Dietz (MD) is an approximation formula; that is, its result approximates the true, exact TWRR (time-weighted rate of return). We began with a USD portfolio where there was neither a gain nor a loss for the month; even though there were cash flows, our MD return equaled zero, which is what we would have gotten with the exact method. Therefore, the FX conversion worked perfectly fine.
When we converted the USD values (beginning and ending value, along with the cash flows) to GBP, however, we had a problem. This was because we were now dealing with a non-zero (approximate) return, and MD is sensitive to the size, timing, and frequency of cash flows. Plus, you may have noticed that one of the flows was more than 17% (i.e., "large"), which should result in a revaluation, at least on that date.
For our client I recalculated the GBP return, but this time revalued the portfolio daily. This graphic summarizes all the details (you can click on it to make it larger, for easier viewing):
The spreadsheet is repeated here:
And as you can see, by calculating our GBP return on GBP values, and revaluing when flows arise, we tie out to the earlier derived return of -0.0188 percent.
Well, that was fun for me ... I always enjoy solving puzzles. Hope you found it of interest. Feel free to chime in with your thoughts.
p.s., I mentioned yesterday that Sarah Ringle introduced the FX return conversion formula to me. She cautioned that this is a quick conversion methodology, and that the standard approach is to convert the underlying valuations to the alternate currency, in order to capture the correct unrealized and realized gain/loss including the derivative estimated amounts.