The performance of a portfolio during a single period can be attributed to a set of factors, but in order to aggregate those daily factors and get a breakdown of the portfolio’s total performance during a multi-period (for example 1Y), we have to make use of an smoothing algorithm.

This is due to the fact that single-period factor returns are not additive; they are referred to different bases. To overcome this difficulty, we use attribution linking algorithms that distort single-period factors to make them sumable, avoiding unexplained residual terms.

In his paper Linking Single Period Attribution Results [2], Andrew Frongello states

“Despite the abundance of single period attribution methodologies, there continues to be no clear industry standard for linking these single period results”.

Indeed, there are a number of linking algorithms, some of the most famous are those proposed by Cariño or Menchero [2]. In “Linking Attribution Factors” , it is explained why linking smoothing algorithms are necessary for performance attribution problems, centering the discussion in the method of base-period adjustment of attribution contributions.

Starting point

The advantages of the base-adjustment linking method for attribution problems have been largely explained in the bibliography [1],[2]. So we will take this as a starting point. This is, if we have the daily returns presented in an arithmetic fashion in terms of the attribution factors:

$$
r_t = \sum_i f_{i,t}
$$

and want to express the total return of a multiple period in terms of those same factors:

$$
R_T = \sum_i F_{i,T}
$$

the correct expression for the multi-period contribution of the i-th factor, \(F_i\), is:

$$
F_{i,T}=\sum_{t=1}^T f_{i,t}(1+R_{t-1}),
$$

each contribution is base-adjusted by the cumulative-to-date portfolio return from the base date of the analysis.

Taking this as a departure point, we are going to explain the linking impact focusing in the case of divergence attribution, that is, when the matter is explaining the divergence between the performance of a certain portfolio and a reference.

Divergence attribution in currency hedging

In [3] Frongello investigates the performance of a portfolio relative to a benchmark over a reporting period, attributing its over/underperformance to allocation and selection. In the context of currency hedging, we would want to see how a hedged or unhedged class performed, relative to a reference, due to currency effects.

Imagine we are to compare the performance of an unhedged share class with the underlying local class. 

The daily performance of the unhedged share class is compounded by the returns of the underlying, the spot and the interaction term between them, i.e., the Asset Value Uncertainty (this factor is well explained in [5]):

$$
r_t^{SH} = r_t^{UND} + r_t^{SPOT} + r_t^{UND}r_t^{SPOT}
$$

hence, according to the above, the total return of this unhedged class through the total period is

$$
R_T^{SH}=\sum_{t=1}^T(r_t^{UND} + r_t^{SPOT} + r_t^{UND}r_t^{SPOT})(1+R_{t-1}^{SH}).
$$

While the underlying local class we are taking as reference does

$$
R_T^{UND}=\sum_{t=1}^Tr_t^{UND}(1+R_{t-1}^{UND}).
$$

Our analysis is about attributing the divergence of those two classes:

$$
\begin{align}
Div_T = R_T^{SH} – R_T^{UND}=\sum_{t=1}^T \left[ (r_t^{UND} + r_t^{SPOT} + r_t^{AVU} )(1+R_{t-1}^{SH}) – r_t^{UND}(1+R_{t-1}^{UND}) \right].
\end{align}
$$

We then arrive at:

$$
\begin{align}
Div_T = R_T^{SH} – R_T^{UND}=\sum_{t=1}^T \left[ ( r_t^{SPOT} + r_t^{AVU} )(1+R_{t-1}^{SH}) + r_t^{UND}(R_{t-1}^{SH}-R_{t-1}^{UND}) \right].
\end{align}
$$

The first term of this expression looks very similar to that of the performance attribution linking model, each factor is scaled by the cumulative total return earned by the portfolio through the prior period.

The key point here is that the underlying is just another source or factor when we talk about performance attribution (in the example of the unhedged share class, the sources of returns are the underlying, the spot, and the asset value uncertainty). However, it is not a source of divergence. Observe that for daily returns, the divergence is:

$$
Div_t = r_t^{SH} – r_t^{UND} = r_t^{UND} + r_t^{SPOT} + r_t^{AVU} – r_t^{UND} = r_t^{SPOT} + r_t^{AVU}.
$$

So the only sources of divergence are the spot and the asset value uncertainty. Hence, one would expect to see the total divergence through the period as the sum of the two (base-adjusted) factors

$$
R_T^{SPOT}=\sum_{t=1}^Tr_t^{SPOT}(1+R_{t-1}^{SH}); \quad R_T^{AVU}=\sum_{t=1}^Tr_t^{AVU}(1+R_{t-1}^{SH}).
$$

 However, we encounter and additional term:

$$
\sum_{t=1}^T r_t^{UND}(R_{t-1}^{SH}-R_{t-1}^{UND}).
$$

Interpreting the linking impact factor

This factor arises from the fact that the base adjustment of the underlying’s return is different when it is by itself and when it is embedded with other factors.

For the unhedged share class, the smoothing or linking coefficient that scales the return of the underlying is \((1+R_{t-1}^{SH})\), since the cumulative total return earned by the class through the prior period is precisely \(R_{t-1}^{SH}\). While for the underlying standing alone, the smoothing coefficient is just its own return during the prior period; \((1+R_{t-1}^{UND})\).

This gives place to another source of divergence between the share class and the reference, when we do multiple-period attribution.

Lets see this with a numerical example

Imagine both classes had started with an initial value of $100.

• Isolated local class: During period 1, the reference class would have earned $11, so that for period 2, the base value would have been $111. Therefore this class earns $9.99 during period 2. At the end of the total period, this class would have a value of $120.99. Hence the overall return, that can be calculated as the sum of the base-adjusted daily returns, is 20.99%.
• Local class as underlying of the unhedged share class: Again, during period 1, the reference class would have earned $11, but in this case there are other sources of performance. The dollar return of the unhedged share class during period 1 is $17.66, so the base value for period 2 would have been $117.66. During period 2, the underlying/reference class has a return of 9% over a base of $117.66, therefore earning a dollar return of $10.59.  At the end of the total period, this class would have earned $10.59+$11=$21.59. Hence the total return of the underlying when it is within this share class is 21.59% . 

There is a 0.6% difference in return between the underlying by itself and the underlying together with the other factors. This is precisely what is captured by the linking impact factor:

$$
LI_T= \sum_{t=1}^T r_t^{UND}(R_{t-1}^{SH}-R_{t-1}^{UND})\Rightarrow \\
LI_2= r_2^{UND}(R_1^{SH}-R_1^{UND})=9\%(17.66\%-11\%)=0.6\%.
$$

Second Adjustment

One could argue that this is difficult to understand as a proper financial source of divergence and that it is more of a “mathematical” effect. Currency managers could be uncomfortable with having an additional factor in the multi-period divergence attribution that is not present in the single-period attribution, or they could find it difficult to explain the source of this term to their clients.

An approach to this problem is relocating this “linking impact” among the other factors, so we can get the desired divergence attribution

$$
Div_T = R_T^{SH}-R_T^{UND} = \widetilde{R}_T^{SPOT} + \widetilde{R}_T^{AVU}.
$$

The new factors \(\widetilde{R}_{i,T}\) absorb the linking impact factor, presenting therefore an extra distortion in addition to the base-period adjustment. This is what Frongello names Second Adjustment in [3].

This relocation can be done by expressing the overall return of the share class through the prior period as sum of its factors; following with the 2 period example:

$$
Div_1 = r_1^{SPOT}+r_1^{AVU}
$$
\(\require{cancel}\)
\begin{align}
Div_2 &= (r_2^{SPOT}+r_2^{AVU})(1+R_1^{SH})-r_2^{UND}(R_1^{SH}-R_1^{UND})\\&= r_2^{SPOT}(1+R_1^{SH})+r_2^{AVU}(1+R_1^{SH})+r_2^{UND}(\cancel{r_1^{UND}}+ r_1^{SPOT}+r_1^{AVU}-\cancel{r_1^{UND}})\\&= r_2^{SPOT}(1+R_1^{SH})+r_2^{UND}r_1^{SPOT}+r_2^{AVU}(1+R_1^{SH})+r_2^{UND}r_1^{AVU}.
\end{align}

The sumable daily factors are more distorted than with the 1st base adjustment; for period 2 an additional term arises:

$$
\begin{align}
\widetilde{r}_1^{SPOT}&= r_1^{SPOT}; \quad &\widetilde{r}_2^{SPOT} = r_2^{SPOT}(1+R_1^{SH})+r_2^{UND}r_1^{SPOT} \\
\widetilde{r}_1^{AVU}&= r_1^{AVU};\quad &\widetilde{r}_2^{AVU} = r_2^{AVU}(1+R_1^{SH})+r_2^{UND}r_1^{AVU}.
\end{align}
$$

Thus, the contribution of each factor through the total period would be:

$$
\widetilde{R}_{i,2}=\widetilde{r}_{i,1} + \widetilde{r}_{i,2} = r_{i,1} + r_{i,2}(1+R_1^{SH})+r_2^{UND}r_{i,1}.
$$

Here we have illustrated the simplest case of two periods. For the general case, the math is a little more elaborated, but it can be shown (by induction, for example) that the expression for the multi-period contribution of a factor is

$$
\widetilde{R}_{i,T}=\sum_{t=1}^T \widetilde{r}_{i,t} = \sum_{t=1}^T \left( r_{i,t}(1+R_{t-1}^{SH})+r_t^{UND}\sum_{j=1}^{t-1} \widetilde{r}_{i,j}\right).
$$

Considerations about the linking impact factor

Although mathematically correct, this Second adjustment somehow obscures the meaning of the attribution factors. It introduces a very strong dependence between the return of the factor through the period and return of the reference. Intuitively, the linking distortion of the factors should depend on the overall performance of the share class during the previous period, not on the reference’s performance during the current period.

Instead, by understanding the meaning and source of this linking impact factor, one can take advantage of the information it gives when it appears separately in the divergence attribution model.

Thanks for reading!

References