## Tuesday, April 22, 2014

### Why geometric excess return? Yes, WHY?

When I'm presented with the same question in relatively short period of time, I suspect it's something I should opine about, even if I've done so before.

I was teaching our Fundamentals of Investment Performance Measurement class for an asset owner (large pension fund) recently, and was asked by a couple folks there what the benefits are of geometric excess return; it seems they have a UK manager who insists on presenting them their excess returns geometrically. They've attempted to understand why it's supposed to be better, but haven't had any luck.

Well, join the crowd!

This week I received an email from the head of performance for a large institution, who also inquired into this subject.

Some, though not many, believe it's a better way to represent the manager's outperformance. The math for both methods is pretty simple:
If, for example, the portfolio's return is 7.00% and the benchmark's 5.00%, the arithmetic excess return is 2.00%, while the geometric's is 1.90 percent.

We can also look at the differences by using money. If the portfolio began with \$1 million, for example, the 7% return would add \$70,000. Had the money been invested in the benchmark, it would have returned a profit of \$50,000. If we subtract \$50,000 from \$70,000, we get \$20,000; if we then divide this by what we began with (\$1 million), we'd get our 2.00% excess return. Geometric, however, would have us divide the \$20,000 by where we would have been, had we been in the benchmark (i.e., \$1,050,000), which yields our 1.90% geometric excess return.

To me, the one legitimate advantage of geometric is that it's proportionate. For example, if one manager beats his benchmark 50% to 49%, while another outperforms her benchmark 2% to 1%, arithmetic yields a 1.00% return for both. Geometric, however, would give us a 0.67% return for the first manager, and a 0.99% for the second. To me, this is a legitimate advantage; however, it isn't sufficient to overcome the challenges in understanding it, thus the almost universal preference for arithmetic.

The UK is the bastion for geometric excess return. Ironically, our research has shown that while the portfolio managers there prefer geometric (by roughly a two-to-one margin), their clients prefer arithmetic (by the same two-to-one margin).

Geometric attribution is different from arithmetic because it reconciles to a geometric excess return. It's more complicated to execute and more difficult to discern and explain. And while I wouldn't argue for something simply because it's easy to explain, given that after many attempts, I've yet to see the true benefit of geometric, I remain solidly in the arithmetic camp.