Performance Perspectives Blog

How come the math doesn’t work out?

by | Mar 3, 2010

We recently received an inquiry from a client regarding netting of advisory fees:

If an annual fee schedule is 60bps after you link the returns for 1 year it looks like returns were reduced by 88bps. I know is because we are linking and not adding or subtracting but for some reason is not making any sense to others.

GOF Return for 1 year ending 01/31/2010= 67.52%
NOF return for 1 year ending 01/31/2010= 66.64%
Annual fee schedule: 60bps

This doesn’t make sense, right? And so, let’s walk through the math. First, I want to find the monthly equivalent of the annual GOF return: I simply add one to the annual return (1.6752) and raise it to the 1/12 power, and then subtract one. My answer: 4.39% (there are some extra decimal places which you’ll no doubt retain in your spreadsheet). I geometrically link these and tie to the 67.52% annual return as provided.

Next, I divide the annual fee (0.60%) by 12 (0.05%) and enter this for each month; I geometrically link these and the result is my 60 bps return. (Actually, there are some trailing decimal places: 6016527…). It is important to point out that our arithmetically derived monthly return won’t geometrically link to the starting annual return; they’re only this close when the numbers are quite small, as in this case. We’ll discuss this further.

For each month, I derive the net-of-fee return by simply subtracting the 5 bp fee from the monthly gross-of-fee return: 4.34 percent. I then geometrically link these and obtain an annual return of 66.56%; note that this doesn’t match the NOF return our client has, but this can be for a variety of reasons, one being that their returns varied from month to month, while I kept them equal for simplicity purposes. Our results:

(NOTE: click on the above figure to see the entire spreadsheet)

We expect that our GOF minus NOF annual returns should equal our annual fee of 0.60%; however, it doesn’t! It’s 0.96%. How come?

It has to do with compounding. Think of this problem as if we were dealing with excess returns, where our fee is the benchmark and the NOF return represents our excess return. We know, from numerous articles, that arithmetically derived excess returns don’t link: this is why we have such tools as those developed by Menchero, Carino, and Frongello for multi-period attribution.

Our monthly net-of-fee returns will not geometrically link at the same rate as our gross-of-fee returns, because there’s a size difference: compounding builds upon prior periods, and the larger the prior period’s value, the greater the rate of compounding. I hope this explanation makes sense. The results are correct; they just don’t tie out as we’d like them to or believe they should.

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