I’m reviewing a client’s performance measurement system and came across an interesting example of how negative market values can wreak havoc with your reporting. What I’ll be showing aren’t exactly the numbers, but the problem is the same.

We begin the month with $20 and end with $10; we have no cash flows. And so what’s our return? Hopefully you’ll respond -50 percent. And what will their system show? Minus 180.46 percent! But how can this be?

The client uses a daily return approach and use Modified Dietz as their formula. They take the absolute value of the denominator in all cases. I want to pause here and say that while I’ve seen others employ this approach, it isn’t clear that it’s always a good idea, but with no evidence to refute its use, we’ll conclude for now that it’s okay.

The account in question is an interesting one in that it has cash and one security; the cash balance is negative for the entire time, while the security’s price fluctuates greatly, causing it to go low enough on one day that the overall portfolio becomes negative on that day, meaning it swings from positive to negative and then back to positive.

On the day it begins negative and ends positive, it starts at -$3.00 and rises to +9.85. On that day, if we use the more standard version of Modified Dietz (or, what might be called the “true daily” formula: EMV/BMV), we get a return of -428.33 percent; if we use the absolute value in the denominator, we get +428.33. Now which makes more sense? If we go from a minus to a positive, wouldn’t that seem to be a positive return? In this case only, I can see using the absolute value in the denominator as it provides (to me and many of my colleagues) a more reasonable and intuitive result).

Okay, so let’s quickly look at our daily values and returns, as well as the return as derived by linking these daily numbers:

On June 23 we see the portfolio swing from a negative to a positive market value. If we link our daily returns we will get the -180.46% return shown at the top left; but if we calculate the return directly from the month’s ending and beginning market values, we get a return of -50.00 percent. Also, if we don’t take the absolute value of the denominator on 23 June (which results in a -428.33% return that day), then we WILL get a -50% return.

It looks like we have two choices: have a less intuitive daily return but a more accurate monthly, or a more intuitive daily but inaccurate monthly. But there IS a solution that resolves our difficulty.

The problem lies in how we *link* returns that come from short positions; we can arguably characterize our June 23rd starting position as being short, so we can employ it here. Traditional linking (i.e., (1+r1)*((1+r2)*…*(1+rn)-1) doesn’t work with shorts. We have to __subtract__ the return from 1 on the date with the short position. In our case, it will change the value from 5.2833 to -3.2833. And, when we link it to the other daily values we achieve the -50% return we would expect.

I often say that the move to daily returns actually creates more challenges to accurate returns; this is just one example. I will comment on this in greater detail in next month’s newsletter.