Sources of Portfolio Returns

Simple Attribution

Return attribution approaches are a way to break down the returns of a portfolio and determine where they came from. They can be highly informative for all stakeholders from portfolio managers to clients.

The simplest form of return attribution is an arithmetic approach. In this version, the difference between the return on the portfolio and the return on the benchmark is calculated, which is known as the excess return.

$$
\text{Portfolio Return} – \text{Benchmark Return} = \text{Excess Return} $$

The advantage of this approach is its simplicity and is often used when explaining returns to clients. The disadvantage of this approach is that over multiple periods, the effects of compounding are not captured, and thus the approach values must be ‘smoothed’. This is where the geometric approach comes into play. The geometric approach also calculates excess return but accounts for compounding:

$$
G = \left[\frac { (1+R) }{ (1+B) }\right] – 1 $$

Where:

\(G\) = Geometric return (excess).

\(R\) = Return on portfolio.

\(B\) = Return on benchmark.

Example

In this example, the hypothetical portfolio outperformed its benchmark by 2.0%. This represents the excess return. A further breakdown of allocation and selection effects is given. While it may be obvious that the portfolio outperformed the benchmark by 2.0%, the allocation effect of +3.0% shows that this was the reason why. It was not selection of securities that improved performance, but rather the manager's tilt toward specific asset classes which performed well, and the avoidance of asset classes which performed poorly. The value of return attribution is apparent from this example. The reading now switches to more advanced attribution techniques.

$$ \textbf{Arithmetic Return Attribution Approach} \\
\begin{array}{c|c|c|c|c}
\textbf{Portfolio} & \textbf{Benchmark} & \textbf{Excess} & \textbf{Allocation} & \textbf{Selection} \\
\textbf{Return} & \textbf{Return} & \textbf{Return} & \textbf{Effect} & \textbf{Effect} \\ \hline
7.44\% & 5.44\% & 2.00\% & 3.0\% & -1.0\% \end{array} $$

Brinson Model

The Brinson Model is another method of decomposing portfolio returns. It states that the total portfolio and benchmark returns are calculated by summing the weights and returns of the sectors within the portfolio and the benchmark, as such:

$$
R_i = \sum w_i \times R_i $$

and,

$$
B_i = \sum W_i \times B_i $$

Where:

\(w_i\) = Weight of the i^th sector in the portfolio.

\(R_i\) = Return of the portfolio assets in the i^th sector.

\(W_i\) = Weight of the i^th sector in the benchmark.

\(B_i\) = Return of the benchmark in the i^th sector.

\(n\) = Number of sectors or securities.

Example

$$ \begin{array}{c|c|c|c|c}
\textbf{Sector} & \textbf{Port. Weight} & \textbf{Benchmark} & \textbf{Portfolio} & \textbf{Benchmark} \\
& & \textbf{Weight} & \textbf{Return} & \textbf{Return} \\ \hline
\textbf{Industrials} & 60\% & 50\% & 18\% & 10\% \\ \hline
\textbf{Technology} & 40\% & 50\% & 10\% & 12\%
\end{array} $$

$$ 60\% \times 18\% + 40\% \times 10\% = 10.8\% + 4\% = 14.8\% $$

$$ 40\% \times 10\% + 50\% \times 12\% = 4\% + 6\% = 10\% $$

Thus, excess return is \((14.8\% – 10\%) = 4.8\%\)

Candidates also need to calculate the basic attribution effects using the Brinson model, including the:

• Allocation effect.
• Security selection effect.
• The interaction effect.

The Allocation Effect

To calculate the allocation effect, first, calculate the contribution to allocation \((A_i)\) for each sector. The contribution to allocation in the ith sector is equal to the portfolio's sector weight minus the benchmark's sector weight, times the benchmark sector return:

$$ A_i = (w_i – W_i)B_i $$

Using the data from the table, calculate individual sector allocation effects as follows:

Industrials: \((60\% – 50\%) \times 10\% = 1\% \)

Technology: \((40\% – 50\%) \times 12\% = –1.2\%\)

To find the total portfolio allocation effect, A, we sum the individual sector contributions to allocation:

Total allocation effect \(= 1\% – 1.2\% = –0.2\%.\)

The Selection Effect

Security selection is the value the portfolio manager adds by holding individual securities or instruments within the sector in different-from-benchmark weights.

To calculate selection, first calculate the contribution to selection \((S_i)\) for each sector. Excess return multiplied by the weight afforded to that security.

$$ S_i = W_i(R_i – B_i) $$

$$ \begin{array}{c|c|c|c|c}
\textbf{Sector} & \textbf{Port. Weight} & \textbf{Benchmark} & \textbf{Portfolio} & \textbf{Benchmark} \\
& & \textbf{Weight} & \textbf{Return} & \textbf{Return} \\ \hline
\textbf{Industrials} & 60\% & 50\% & 18\% & 10\% \\
\textbf{Technology} & 40\% & 50\% & 10\% & 12\% \\ \hline
\textbf{Total} & 100\% & 100\% & 14.8\% & 11\%
\end{array} $$

Calculate individual sector selection effects as follows:

Industrials: \(60\% \times (18\% – 10\%) = 4.8\%\)

Technology: \(40\% \times (10\% – 12.0\%) = –0.8\%\)

To find the total portfolio selection effect, S, sum the individual sector contributions to selection:

Total selection effect \(= 4.8\% + –0.8\% = 4.0\%\)

Interaction Effect

The Brinson model will not explain all of the arithmetic differences between the portfolio return and the benchmark return. To explain this remaining difference in the excess return, the Brinson model uses a third attribution effect, called “interaction.” This effect is generated by the interaction between allocation and selection decisions combined. To calculate interaction, first, calculate the contribution to interaction for each sector.

The contribution to interaction in the ith sector is equal to the portfolio sector weight minus the benchmark sector weight, times the portfolio sector return minus the benchmark sector return:

$$
I_i = (w_i – W_i)(R_i – B_i) $$

Using the data from the table, calculate individual sector selection effects as follows:

Industrials: \((60\% – 50\%) \times (18\% – 10\%) = 0.8\%\)

Technology: \((40\% – 50\%) \times (10\% – 12.0\%) = 0.2\%\)

To find the total portfolio interaction effect, we sum the individual sector contributions to interaction:

Total interaction effect \(= 0.8\% + + 0.2\% = 1.0\%\)

Brinson-Fachler Model

The Brinson-Fachler Model can be considered an extension of the Brinson Model shown before, with one notable difference:

Individual sector allocation effects are calculated as such:

$$ A_i = (w_i – W_i)(B_i – B) $$

Where:

\(A_i\) = Sector allocation effect.

\(w_i\) = Weight of the i^th sector in the portfolio.

\(R_i\) = Return of the portfolio assets in the i^th sector.

\(W_i\) = Weight of the i^th sector in the benchmark.

\(B_i\) = Return of the benchmark in the i^th sector.

\(B\) = Total benchmark return.

This is as compared to the traditional Brinson Model sector allocation effect:

$$ A_i = (w_i – W_i)B_i $$

This traditional calculation looks at the relative weightings of the portfolio versus the benchmark, and then scales the difference by the return of the benchmark sector. This allows for proper allocation effect calculation, regardless of whether or not the sector return is less than the overall benchmark return.

Question

Which of the following is a true statement about simple arithmetic returns as compared to geometric returns?

1. For all positive series of returns with positive standard deviation, arithmetic returns \(\le\) geometric returns.
2. Accounts for the geometric nature of returns.
3. Arithmetic returns are easier to calculate.

Solution

The correct answer is B.

The advantage of using simple arithmetic averages is its simplicity and is often used to account for the geometric nature of returns.

A is incorrect. Arithmetic returns are not always less than or equal to geometric returns. Geometric returns tend to be less than arithmetic returns for a series of returns with positive standard deviation because they account for compounding and the volatility of returns.

C is incorrect. Geometric returns are typically more challenging to calculate than arithmetic returns, as they involve compounding and require additional mathematical steps compared to simple arithmetic returns. 

Performance Measurement: Learning Module 1: Portfolio Performance Evaluation; Los 1(e) Interpret the sources of portfolio returns using a specified attribution approach