## Thursday, June 30, 2011

### Compounding problems with hurdles

Some folks in a Linkedin group I'm a member of have been discussing the challenge of dealing with hurdle rates when calculating returns. As I showed in a December 2010 newsletter, you cannot compound these rates, as their rate of compounding is proportional with the monthly returns they're added to. In the examples I provided, where the annual hurdle rate is 1.00%, compounding can result in the actual annual hurdle rate ranging from 0.53% to 1.36 percent. So much for the agreed to 100 basis point hurdle.

And so, what is one to do if you want your compounded returns with the hurdle rates to tie out to what you've contracted to, or to track your hurdles on a monthly basis? The "simple" solution:
1. Take your annual hurdle and divide by 12 to arrive at the monthly rate
2. For each month that's being linked, add a multiple of the monthly hurdle rate to the linked return value.
That is, we link the returns first, then add the hurdle. Let's take an example. We'll begin with the way most firms do it, which we can call the "geometric" approach, since we are compounding the fee, along with the return. As you'll see, with this approach we add the hurdle, then link.

Suppose that we have an annual hurdle rate of 3.00 percent. If we intend to include the hurdle with our returns as they're compounding (i.e., to add the monthly hurdle first, then compound) we would take the hurdle rate (3.00% or 0.03), add 1 to it, raise it to the 1/12th power, and then subtract one, which gives us 0.2466 percent. This is our monthly hurdle rate. We add this amount to each monthly return, and geometrically link these values. The following table provides our data:

I've labeled the columns to make the explanation hopefully clearer:
• (A) is our monthly return (just numbers I made up for this exercise)
• (B) shows the inclusion of the hurdle rates, on a monthly basis. Since we're doing this geometrically, we add the hurdle to each monthly return, and then link them. I'm showing the cumulative effect of this linking, so that February is the two-month linked return, March the three-month, and so on.
• (C) is the cumulative hurdle rate (since it's for the geometric approach, I linked the monthly hurdles). You can see that by themselves they link to the agreed to 3.00 percent.
• (D) shows the cumulative returns, based on the values in (A) (i.e., without the hurdle rate)
• (E) is the difference between (B) (the linking of the returns with the inclusion of the hurdle rates) and (C) (the linked hurdle rates)
• (F) is the difference between (B) and (D) (the cumulative returns without the hurdles included).
First, we notice that our return for the year with the hurdle is 21.3556%; when we compare this to our annual return without the hurdle (17.87%), we see that our hurdle isn't the 3.00% we would have expected, but significantly higher (3.49%). The additional 49 basis points makes success more challenging for the manager than what they had agreed to, yes?

I included columns (E) and (F) to show that on a monthly basis the numbers don't tie out as we might expect. Column (E)'s values should equal the cumulative returns, but we see that they don't match what's in column (D). Column (F)'s values should equal the cumulative hurdles (C), but here, too, the numbers fail to agree.

The problem with this method is that the resulting annual return, with the hurdle compounded along with it, will not agree with what your contract or client calls for (i.e., what you've agreed to), and will either be lower (and therefore easier to obtain) or higher (therefore more of a challenge), but in neither case correct. Who would agree to a hurdle that will vary, depending on market conditions? If I'm expected to deliver a return 300 basis points above the LIBOR rate, for example, isn't it reasonable to expect that at the end of the year, the compounded benchmark would be exactly 300 basis points higher than the compounded LIBOR rate?

Now let's consider the arithmetic approach. With this method, we will compound or link the monthly returns first, and then add the hurdle. To derive our monthly hurdle rate we take 1/12th of the 3.00% annual rate, which gives us 0.2500 percent. The following table provides us with the data for the full application of the method:

The column's meanings are consistent with the geometric table version. First, notice that in this case our annual return with the hurdles (20.8682%) is 3.00% higher than our linked return (17.87%; the return without the hurdle), meaning we reconcile to the agreed upon annual hurdle rate. In addition, on a monthly basis, our column (E) matches the cumulative returns in column (D), and column (F)'s values match the cumulative hurdles shown in (C).

While most firms no doubt utilize the first method (the "geometric" approach), I would argue that it's flawed, since the annual hurdle will only equal what the agreed upon value is, if the return for the year is 0.00 percent; otherwise, it will be higher or lower than the hurdle, which to me justifies a switch in methods to the recommended approach (arithmetic), where we tie out exactly on both an annual as well as a monthly cumulative basis. Certain numbers aren't supposed to compound, and you can include in this group returns with hurdles; compound the returns, then add the hurdles to them.

If you'd like a copy of the spreadsheets, send me a note.

1. Hi Dave, nice posting. I'm fully with you on this one. One little comment from my side: "Certain numbers aren't supposed to compound" - they do compound, they just don't chain-link. The underlying issues are the same as when compounding contributions and attribution effects.

2. Andreas, thanks for your comment.

3. Dave, First of all I thank you for the effort. I am also with Andreas, that it is a very nice posting and all numbers need to be compounded. If we are geometrically linking target rate, we have to geometrically link hurdle rate also.

4. Alakh, thanks for your note. Ir respectfully disagree: there is no need or obligation to compound the hurdle rates w/the returns; as is clearly obvious, to do so generates errors. In my view it's plainly wrong. Certain things (e.g., returns) compound, others (e.g., excess returns, hurdles) do not.

5. David,
The method you propose here is exactly the same as the first method that I proposed in the LinkedIn discussion about Reporting Performance for Custom Benchmarks that you reference.
Thus, it exhibits exactly the anomalies that Daragh Pollard pointed out there.
This can be seen in your example if you simply change the December return from 1.7% to 17.0%. For this new case, calculate the new single-month return for December by taking the new YTD return factor for December and dividing it by the YTD return factor for November and then subtract one = 1.356006/1.186479 -1 = 16.8167. This new December monthly return (including hurdle) is less than the 17% new return for December without the hurdle. Very strange!
The question that was then discussed in the blog is whether this is OK or not.
I am inclined to agree with what Jeroen Geenen suggested on the LinkedIn discussion, that this kind of strange result is a necessary evil.
But it is important to recognize that it can occur.

6. Andre, first, sorry that I didn't recognize that we agreed on a methodology that would eliminate the problem. Second, to correct your wording, "it is important to recognize that it WILL occur," and will ALWAYS occur when the benchmark's return is anything but 0.00 percent. Thus the problem: the agreed upon hurdle (3.00% on an annual basis) will never be what the manager is being tested against; it will always be lower or higher, and clearly invalid. It makes no sense to compound when it produces unpredictable results.