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# I sense a need to clarify the differences between arithmetic and geometric excess returns

by | Sep 24, 2014

## Don’t focus on arithmetic vs. geometric attribution

Of late there have been several comments regarding arithmetic and geometric attribution; but that’s not what’s important. What matters is

## The difference between arithmetic and geometric excess returns!

And today’s post will attempt to shed some light on this subject. But first,

# What’s the point of attribution?

Well, a simple explanation is that it provides insights into what the return or excess return is attributable to (thus the word “attribution”). We attempt to reconcile to the portfolio’s return (in the case of “absolute attribution” or contribution) or the excess return (in the case of relative attribution). This post is limited to relative attribution, which involves models such as Brinson, Fachler and Brinson, Hood, Beebower for equity attribution, and Campisi and van Breukelen for fixed income attribution.

The choice of whether to use arithmetic or geometric attribution rests solely on the way we calculate excess return.

## What geometric excess returns isn’t!

Many investment professionals associate the word “geometric” with the way we compound rates of return across time, but that’s not what it’s about when we discuss geometric excess returns (or geometric attribution, for that matter). Let’s clarify …

## The math behind the differences

Arithmetic excess returns are calculated using a rather simple formula:

Geometric excess return is derived as follows:

Let’s say that you begin with \$100,000 and end with \$110,000. The return’s easy to derive, right? It’s 10 percent.

If we’re told the benchmark had a return of 6%, then we could consider what our portfolio would have looked like had it been invested in the benchmark: it would have risen to \$106,000.

And so, our relative gain or benefit in dollars is the difference, or \$4,000.

# What are our excess returns?

Well, our arithmetic excess return is simply the difference, or 4.00 percent.

And, our geometric excess return is 3.77 percent.

We can arrive at these results by using our dollar results, too.

For arithmetic, take our relative gain (\$4,000) and divide it by our portfolio’s beginning value (\$100,000): we get 4.00 percent.

For geometric, take our relative gain and divide it by where we would have been, had we been invested in the benchmark (\$106,000): we again get 3.77 percent. To paraphrase the A-Team’s Col. John “Hannibal” Smith,

# It’s nice when the numbers work out!

We can see that math for arithmetic excess returns compares the gain with the initial amount invested, while for geometric excess returns, it compares it to our initial amount, adjusted for what has happened with the benchmark.

## The survey says …

Our research has shown a consistent preference for the arithmetic approach. The only place we know of where there’s a preference for geometric is the United Kingdom; and even in the UK, when reporting to clients, arithmetic dominates.

Which is better? It’s up to you to decide. Most individuals and firms clearly prefer arithmetic. Arithmetic is typically the way we derive differences (e.g., Team A beats Team B 21 to 7, their margin of victory was 14 points). I prefer arithmetic because it’s clearer, easier to justify, and what most folks want to see. I’ve yet to hear an argument for geometric that I find compelling enough to switch, but I’ll try to keep an open mind.

## And so …

We shouldn’t be discussing a preference for attribution, but rather a preference for excess return:

• If you prefer to see arithmetic, then you’ll want an arithmetic attribution model, so that the results reconcile to the arithmetic excess returns.
• If you like geometric better, then you’ll want a geometric attribution model since it will reconcile to the geometric excess returns. It’s really that simple.