In the business of performance measurement, a recurrent task is the breakdown of a stream of returns into meaningful contributions from different factors, in order to identify the driving financial forces or sources of risk.

Eventually, these daily contributions have to be aggregated to explain the complete period performance or the divergence between two different streams of returns. This step is known as attribution linking, and it requires a return smoothing algorithm. The smoothing algorithm will slightly distort single-period contribution returns so that they acquire an additive property that geometric returns lack.

Today we are going to explain one of many smoothing procedures;

the base-period adjustment of attribution

contributions.

Together with theoretical considerations regarding the distortion, we will show its implications with a numerical example. The first two blocks of our post will cover the fundamentals of base-adjusting returns, a third one will explain the difference between geometric and arithmetic attribution models, and finally the fourth one will sum up the whole with the numerical example.

Percentage Returns And Mark-To-Markets

We start our way by recalling a silly yet subtle property of daily returns: they are referred to the previous base.

Let’s take the example of two portfolios, Win-Lose (WL) and Lose-Win (LW). Both of them see the same stream of returns, \(\{10\%, -10\%\}\), albeit in different order.

As we can see in the graph and table, the same percentage figures correspond to different amounts. Realizing this is of vital importance to understand why, when the returns are smoothed to become addable, the return value changes.

The order in which returns take place is important too. Despite the fact that from a blunt algebraic point of view, both paths are equivalent,

$$
(1 + 10\%)(1 – 10\%) = (1 – 10\%)(1 + 10\%),
$$

from a money point of view they are not. In absolute terms, the first path displaced a larger amount of money (21) than the second one (19). This would be relevant, for instance, if we had an ongoing fee based in the Assets Under Management (AUM).

Geometric vs. Arithmetic Breakdowns

According to the kind of valuation problem we have at hand, from our perspective two types of breakdowns can arise for the daily return:

• Arithmetic: \(r_t = \sum_i f_{t,i}\)
• Geometric: \(r_t = \prod_i (1+ f_{t,i})\)

On the one hand, arithmetic breakdowns arise naturally for valuation problems where two or more financial obligations sit down next to each other and evolve independently: stock long-only portfolios, hedged portfolios, etc. (Hedged positions are usually updated according to the overall size of the portfolio, but if the hedges are placed and their size is not updated, they evolve independently according to the evolution of the derivative used to hedge).

TnA = Asset 1 + Asset 2 + … + Futures MtM + Options MtM + …

On the other hand, geometric breakdowns could be typical for valuation problems such as service fees, taxes, or the inclusion of currency effects,

$$
1 + r_t^{BASE} = (1 + r_t^{LOC})(1 + r_t^{FX}),
$$

basically, any scenario where the contribution amount depends on the overall amount of the portfolio.

Bacon [1] argues that geometric attribution models are preferred over arithmetic ones since they have better properties in terms of compounding attribution effects across time. Although I agree with him from a mathematical point of view, I doubt if it is always possible to obtain such a breakdown.

How could we build a geometric attribution model for the daily return of a long-only portfolio based on the sector exposures? If we take a look at the geometric excess return model in [2], it falls back to an arithmetic one if the benchmark is set to zero. That is, the natural breakdown of a long-only portfolio is

$$
r_t = r_t^\text{Energy} + r_t^\text{Healthcare} + r_t^\text{Utilities} + \ldots,
$$

rather than

$$
r_t = (1+r_t^\text{Energy})(1+ r_t^\text{Healthcare})(1 + r_t^\text{Utilities})(\ldots)-1.
$$

Base-Adjusting Contributions

If we depart from an arithmetic breakdown of a stream of returns, \(r_t = \sum_i f_{t,i}\), we would like to aggregate each contribution in isolation, so that the total period return is explained by the sum individual contribution magnitudes, without the need to include any interaction effect:

$$
\begin{align}
R_T &= \prod_{t}^{T} (1+r_t) – 1\\
R_T &= \sum_i F_{i}
\end{align}
$$

Unfortunately, due to the compounding nature of geometric returns, we cannot naively add the single-period contributions,

$$
F_i \neq \sum_t f_{t,i}.
$$

Methodology

To obtain \(F_i\) for each contribution as a sum, we need to base-adjust each single-period contribution [3]. If we depart from the fact that the difference in prices within a period can be expressed as the telescopic sum of the daily differences in prices,

$$
P_T \, – P_0 = (P_{T} \,- P_{T-1}) + \ldots + (P_{2} – P_{1}) + (P_{1} – P_{0}),
$$

by factoring out the previous price \(P_{t-1}\) from each summand and dividing both sides by \(P_0\), we obtain an expression that relates the period total return with the sum of the scaled daily returns,

$$
\begin{align}
\frac{P_{T}}{P_{0}} -1
&=
\left(\frac{P_{T}}{P_{T-1}} -1 \right)\left(\frac{P_{T-1}}{P_{0}}\right)
+
\ldots
+
\left(\frac{P_{2}}{P_{1}} -1 \right)\left(\frac{P_{1}}{P_{0}}\right)
+
\left(\frac{P_{1}}{P_{0}} -1 \right)
\\
\\
R_T &= r_T \cdot (1+R_{T-1}) + \ldots + r_2 \cdot (1+R_1) + r_1
\end{align}
$$

If we introduce in the above the arithmetic breakdown for each daily return, we see that the correct expression for the multi-period contribution \(F_i\) is

$$
\begin{align}
F_i &= \sum_t f_{t,i}\left(1 + R_{t-1}\right), \quad R_0 = 0, \\
F_i &= \sum_t f_{t,i} +\sum_t R_{t-1} f_{t,i}.
\end{align}
$$

By inspection, we understand the origin of the name of the smoothing methodology. Each contribution is scaled by the cumulative-to-date portfolio return from the base date of the analysis. An alternative interpretation is that the period contribution equals the “naïve” sum of the contributions plus the sum of the contributions scaled by the preceding portfolio return. A very good presentation of this property is given by Frongello in [8] in terms of monetary amounts. On a final note, we refer to the term \(\left(1 + R_{t-1}\right)\) as the smoothing or linking factor.

Overall, this procedure has several advantages:

• Easy-to-understand mathematics.
• Forward-looking scaling: future market movements do not affect past contributions.
• Order matters: each % value is scaled to its actual size in amount.
• The scaling is “contained” within the attribution model: no further assumptions or optimizations [6] were required; simple algebraic operations were enough.
• No residuals or interaction terms are introduced due to the aggregation.

Surprisingly, this method [3] is not present in the extensive comparison made by Yindeng and Sáenz [9], but when one looks at the Frongello method [7], we realize base-adjusting is at its core [8].

Alternative Linking Methodologies

Alternative linking methods are available, such as the renown Cariño [4], [5], and Menchero [6] methods. However accurate from a mathematical point of view, their methodology is ill-posed from a business perspective.

They approach the problem with the distribution of the second summand we saw in our attribution equation, \(\sum_t R_{t-1} f_{t,i}\) among all contributions. Due to their methodology, the outcome is that if the analysis period is enlarged, the smoothing factor changes retrospectively, that is, August attribution results are affected by the performance in December! How could we explain this meaningfully to a client?

Practical Applications

Now that we have a feeling for the effects of our smoothing algorithm, we are going to quantify them with an actual implementation.

Win-Lose vs. Lose-Win Portfolios

Between our two toy portfolios, we see how, when base-adjusted, the losses incurred by the first one become larger and the profits made by the second one become smaller.

This change in the % values is correctly reflecting the value of the MtM amounts with respect to the beginning amount of the portfolio.

Currency Unhedged Portfolio

A more elaborate example, yet simple, would be the performance attribution of an unhedged portfolio.

Due to the interaction between spot and underlying movements, the return of the unhedged position is not exclusively the sum of the spot and underlying returns,

$$
r_t^{BASE} = r_t^{LOC} + r_t^{FX} + r_t^{LOC} \cdot r_t^{FX},
$$

with a third term, coined Asset Value Uncertainty, showing up. For the yearly period from April 2020 to April 2021, how would the attribution aggregation look like? We will compare our base-adjusted aggregation with the alternative geometric attribution,

$$
R_t^{BASE} = R_t^{LOC} + R_t^{FX} + R_t^{LOC} \cdot R_t^{FX},
$$

which makes use of the fact that this breakdown is originally geometric. The comparison, shown in the following table is interesting, we can see that the base-adjusted results are not the same.

The AVU value for the base-adjusted model is smaller than the one estimated by the geometric model. Additionally, the Local and Spot total returns are not the same as if we had only invested in one of those two positions. Despite the initial strangeness of these results, we are keen to say that the base-adjusted ones are correct.

The reason for them to differ is mainly due to the order-dependency clause, but this again is correct. The geometric model implies that all wealth was generated in one leap, whereas the other one considers the daily evolution. As Bacon states in [1], although daily analysis of performance is unfair, daily valuation is desired for accurate return calculations.

Thanks for reading, and see you next time!

References

[1] Carl R. Bacon, “Practical Portfolio Performance Measurement and Attribution”.
[2] Carl R. Bacon and Marc A. Wright, “Reading 5: Return Attribution”.
[3] Mark R. David, “Sector-Level Attribution Effects with Compounded Notional Portfolios”.
[4] David Cariño, “Combining Attribution Effects Over Time”.
[5] David Cariño, “Refinements in Multi-Period Attribution”.
[6] Jose Menchero, “An Optimized Approach To Linking Attribution Effects Over Time”.
[7] Andrew Frongello, “Linking Single Period Attribution Results”.
[8] Andrew Frongello, “Attribution Linking: Proofed and Clarified”.
[9] Jiang Yindeng and Joseph Sáenz, “Comparing Performance Attribution Linking Methods: An Empirical Study”.