You've compounded long and short returns over a period, and you notice they don't add up to total returns. But why?

If you've grappled with this question, you'll need to understand something about "smoothing." The video below provides an introductory example.

Smoothing is a technique whereby long and short returns for each period are smoothed to ensure additivity is preserved. The Novus Platform allows you to access both un-smoothed and smoothed returns.

Note that the concept applies to any partition of your portfolio, not just longs and shorts. For example sectors, market cap, geography, or custom partitions. In the following though, we'll continue to use longs and shorts as an example.

For reporting purposes, investors prefer smoothed returns, as this is the only way to ensure that the sum of contributions of longs and shorts add up to the total portfolio contribution. However, when analyzing how a specific position contributed over time, the unsmoothed contribution is often more useful, as position-level analysis can be (and should be) analyzed independently from overall contribution.

Let's work out an example together using Excel. Click on the video below if you prefer watching it play out. Or, skip the video and keep reading.

You can download the Excel we reference by clicking on the below.

As an example, assume that in January and February, long, short and total contributions add up to the values as in the table below. For sake of clarity, green cells indicate input numbers, while a white cell indicates that the number has been derived from others via formulas. Orange cells indicate numbers upon which the reader’s attention is drawn at that particular time.

For each month, the long and short contributions add up to the total contribution. However, after compounding each set of returns independently, we arrive at a Long Contribution of 2,091.00, a Short Contribution of 712.00, and a Total Contribution of 2,876.00.

You’ll notice that 2,091.00 + 712.00 does not equal 2,876.00, hence the need for smoothing.

In simple terms, smoothing is a technique whereby one adjusts contributions (for each point in time where they are available) by multiplying them by some coefficient in order to preserve additivity over time.

The smoothing coefficient is the same for all the possible portfolio partitions (whether you’re breaking down into Longs and Shorts, Sectors, Market Cap bands, or else), and is derived from unsmoothed contributions through the following process.

In this example, we will focus on long and short contributions. Note that there are many possible ways to smooth returns. At Novus, we chose to use the log-smoothing approach.

The steps are as follows:

In our example, we start with USD 100 mn of AUM in January. Further, assume the securities we buy and sell generate a monthly P&L of USD 13 and 3 mn respectively. That is, we end the month with an AUM of USD 116 mn, and a total contribution of 1,600 bps (or 16%).

Further, assume that the long and short book generate a P&L of USD 8.12 and 4.64 mn respectively during February. The total contribution for the month (total P&L of USD 12.76 mn divided by beginning of month AUM of USD 116 mn) is 1,100 bps (or 11%).

First, we need to calculate the portfolio's total contribution over the period by compounding the Total Contribution for January and February.
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Next, we divide this by 10,000 to convert from bps, yielding our total returns (RT).

Now we have everything we need to solve for the Total Linking Coefficient.

Plugging in our value for RT, we get 0.88.

Note that the Total Linking Coefficient depends on the start and end date of our analysis. Therefore, it must be recalculated every time the time interval changes.

The total contribution for each period, as one can deduce from the table below, is 16% for January, and 11% for February.

We have everything to calculate Individual Smoothing Coefficients for each month.

We can now use each month's RT and the already calculated Total Linking Coefficient to find the smoothing coefficients, as follows:

Finally, we calculate the Smoothed Contribution of each partition of the portfolio by taking the unsmoothed contribution and multiplying it by its Individual Smoothing Coefficient.

For January, the unsmoothed contributions are as follows:

Plugging these into the formula results in:

For February, the unsmoothed contributions are as follows:

Plugging these into the formula results in:

Let’s now verify that we’ve solved the problem we started with. The total (smoothed) long and short contributions for the period are 2,127.61 and 748.39 bps respectively. Their total adds up to 2,876 bps, which is what we wanted.

Note that, in order to calculate the total smoothed returns for the long book over the period, we added (as opposed to compounded) the smoothed returns for each period. While this may sound counter-intuitive, as one was always taught to compound (as opposed to sum) over time, it is the correct decision.

There are multiple ways to achieve smoothing, not just the one described above. Of the various approaches available, Novus uses the log-smoothing approach developed by Carino (Carino, D. (1999), "Combining attribution effects over time". The Journal of Performance measurement Summer, 5-14). Those that are interested in the mathematical manipulations underlying the approach can refer to the video below.

If you are interested in a review of all approaches available (including the log-smoothing methodology presented here), please refer to Bacon (Bacon, C. (2008), "Practical portfolio performance measurement and attribution". Chapter 8: Multi-period attribution).

In accessing contribution fields on the Novus Platform, you can apply smoothing by selecting Contribution, and then in the settings for Contribution, click the checkbox to apply smoothing.

If you want to see a 'live' example (where we also verify manually that the Novus platform is doing its job correctly), click on the video below. If you prefer reading about it, go on.

After having toggled 'Enable Smoothing', group by both "Top Level" and "Exposure Type" as in the table below. At this point, smoothing is now being applied to each position, having been adjusted using the cumulative gross returns, such that:

Grouping by "Exposure Type" will also ensure that any positions labeled “Other” are included as well, allowing the Total Gross to be accurately represented by Long, Short and Other. 

As discussed, there is no unique way of smoothing returns so that additivity is preserved. Of the other numerous approaches available in the literature, one deserves particular attention (let's denote it as 'simple smoothing').

In this approach,

In other words, every security's contribution (or every portfolio's partition contribution) is smoothed by multiplying its unsmoothed returns by the compounded portfolio returns up until the preceding period. Note that, in fact, the simple Smoothing Coefficient above only depends on portfolio returns up until t-1.

Like for log-smoothing, smoothed contributions can be added over time (as opposed to compounded) and over securities (or portfolio partitions) to add up to portfolio returns. However, there are pros and cons. Please refer to the video below for a rigorous comparison.

Practitioners have been arguing about one method's superiority versus another one for a few years now. In truth, the insights derived from smoothed contribution analyses do not depend on one methodology versus another one.

At Novus, we chose to adopt log smoothing because of the inherently desirable properties of the log operator when applies to returns. For more details on that, please refer to the video above.

Remember, it is better not to smooth if you are looking at individual positions or partitions of the portfolio in isolation.

Also bear in mind that the effect of smoothing is exaggerated when returns are high and/or for long periods. This is exactly what happened in our example above. In fact, you can see how the smoothed long and short contributions over the period (2,127.61 and 748.39 bps) are higher than their respective un-smoothed version (2,091.00 and 712.00 bps) by quite the amount.

Let’s briefly examine why this happens. Note that ln (1+r) < r for all r. Hence, total linking coefficients are always <1. The smaller r is though, the more ln (1+r) tends toward r. In other words, while the upper boundary for total linking coefficients is one, the lower the r the closer the total linking coefficients get to that boundary (without even reaching it).

Now consider the individual smoothing coefficient to be applied to contributions in a given period. When spelling the formula entirely and substituting the total linking coefficient with its definition, we have

In other words, you can see the individual smoothing coefficient as the ratio of two identical equations. The one pertaining to the individual period appears as the numerator, and the other pertaining to the overall period appears as the denominator.

Typically, and especially if returns were positive during the period as a result of multiple periods with positive Rt, one has that RT>Rt. In this case, because of the observations above on how total linking coefficients evolve with r, we have that the smoothing coefficient is >1.