A multi-currency return puzzle for you to solve (part 1)

A Spaulding Group client posed the following situation to me, which I hope you find of interest:

They calculated the return on a portfolio in US Dollars (USD). They next converted the return to Pound Sterling (GBP). Everything worked correctly; i.e., the returns make sense and are accurate.

They next converted the portfolio's assets from USD to GBP. They measured the return on the GBP-based portfolio and got a different return than they got when they simply converted the USD portfolio's return using an FX conversion formula, which I'll show you shortly. And, as expected, when they converted the GBP return they just calculated to USD, it didn't match the earlier return. The following graphic summarizes what occurred and our expectations:

Here are the details behind what is occurring, which only makes this problem a bit more challenging.

As you can see from this table, we began with $23 million, took out $4 million on the first, another $1 million on the 28th, and ended the month with $18 million. Simple arithmetic reveals that we made nothing and lost nothing. Therefore, our return of 0.00% is accurate. This activity occurred in March 2012 (a 31-day month), and the weighting reflects it. Modified Dietz was used to derive the return.

We next want to convert the return to GBP. This can be done by using a fairly simple formula:

With this in hand, along with the following FX Rates:

We find that the GBP return equals the FX return for the period (-0.0188%). This makes sense, since the USD return is zero, the only thing that would affect our return would be the conversion from USD to GBP, which is reflected in the FX rate change, from the beginning to the end of the month.

Now, we want to convert our USD values (both the starting and ending values, as well as our cash flows) to GBP. We use the FX rates that correspond to the date of each value, and get the following:

The table shows the return we get when we use Modified Dietz. And it's obviously not what we got earlier: it's materially different.

At this point it's probably not a surprise for you to learn that when we convert the GBP return back to USD (using our FX conversion formula) we get something other than 0.00%; we get 0.0626 percent, which gain is a materially different value.

What's wrong? Does it not make sense that we should have a closed system? That going from USD to GBP, then from GBP to USD, we should have returns that all tie out?

I invite you to check the math: you'll see that it's all correct. And so, what is happening? If  you can figure it out, send me an email. I will reveal the solution tomorrow (at least the one I have!).

p.s., I thank Sarah Ringle of Alliance Bernstein for introducing me to the FX conversion formula more than a decade ago, when I was doing a consulting assignment for Alliance Capital.