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.


  1. To quote the great Economist and Philosopher Mary Poppins: "Why make things more complicated than they really are?" In doing so, you miss the true answer to this so-called "dilemma" which is that RETURN COMPONENTS COMPOUND - they do not add. While we would like for returns to be "built up" from their components, this is not how it really works. The most familiar example of this is the calculation of real (that is, inflation adjusted) returns. We might simply subtract inflation from the nominal return to get a reasonable approximation, but the correct way to calculate a real return is to DISCOUNT it by inflation. So, we divide (1 + nominal return) by (1 + inflation) and then subtract 1 to get the real return. For example, a 9% nominal return adjusted for 3% inflation is (1.09)/(1.03)-1 or 5.83% rather than 9 - 3 = 6 percent. Key point to recognize is that the effect on return is greater than we might have expected when we discount as opposed to when we subtract.

    In your example, subtracting the fee would produce a "net" return of 26.82% - 1.5% = 25.32% instead of the true net return of (1.2682)/1.05) -1 = 24.95 percent. You'll see that the difference between the approximate and the true net returns is 37 bps, which is the difference between the true actual annual fee (187 bps) and the stated annual fee of 150 bps. So, once again it's not that "things don't add up" but rather that we have to state what "the things" (in this case the net returns) really are. It may appear that the fees are actually higher; this is NOT the case. In fact, it's the RETURNS that are actually lower, because the impact of DISCOUNTING by the fees is more powerful than we thought. Once again, this also shows that fees matter - perhaps more than we thought!


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