## Friday, April 27, 2012

### When compounding of returns doesn't make sense

For many, it only seems logical that when you deduct fees monthly or quarterly, that your annual net-of-fee and gross-of-fee return difference should equal the annual fee. How can it not?

It's quite simple to derive the monthly amount to deduct: just take your annual fee (e.g., 1.50%), add one, raise this value to the 1/12th power, and subtract one (in the case of a 1.50% annual fee we get 0.12414877 percent. And, we can validate that this is the right number by linking it for 12 months (I'll let you do the validation). Okay, so we know we can link the monthly fee and get our annual fee; so why, when we link our monthly returns net this fee don't we get the annual fee as the difference between annual gross and net?

Alone, the monthly fee, when linked, does yield the annual fee. However, when combined with a return, it will compound at a very different rate. Take a look at the following table to see how this can happen:

"But it doesn't make any sense!," you might explain. And yet, it does! Let's consider a situation where we begin with \$100,000 and are able to have a consistent 2% return for each month of the year. Our annual fee is 1.50 percent. Let's calculate our gross-of-fee (GOF) and net-of-fee (NOF) returns, by (a) linking the monthly returns and (b) comparing our ending market value with our beginning.

What do we see? We get the same GOF (26.82%) and NOF (24.98%) returns. In addition, our difference is not 1.50 percent. Like it or not, this  is the way compounding works.

p.s., I thank my colleague, Jed Schneider, for suggesting that I demonstrate this using dollars.