• Introduction
• Notation
• Brinson-Fachler attribution
• Brinson-Hood-Beebower attribution
• Normalised stock selection
• Bottom-up attribution
• Off-benchmark holdings
• Worked examples and documentation
• References

Introduction

The classical Brinson model decomposes active return into return from asset allocation decisions, and return from stock selection decisions.

This page provides a concise, mathematical description of the equity attribution models available in FIA 2.4.2 upwards. The models may be classified as

• standard Brinson
• Brinsin with normalised stock selection
• Bottom-up equity attribution

Depending on the model selected, additional sources of return may also be available from interaction and price return. These terms are defined below.

Notation

In the following presentation, lower-case variables refer to the portfolio, and upper-case variables to the benchmark.

At the security level, $r_i$ denotes the return of security $i$ in a portfolio, while $R_i$ is the return of the same security in a benchmark. We do not necessarily assume that these quantities are the same; they may come from different sources, or the portfolio return may include transactions.

Similarly, $w_i$ is the weight of security $i$ in a portfolio, while $W_i$ is the return of the same security in a benchmark.

In the same way, $r_S$ and $R_S$ are the returns of sector $S$ in portfolio and benchmark, and $w_S$ and $W_S$ are the weights of the sectors. These are calculated as

Lastly, $r$ and $R$ (without subscripts) denote the return of portfolio and benchmark respectively, given by

and

Brinson-Fachler attribution

Consider a portfolio where the manager makes a single asset allocation decision in terms of allocations to sectors. The active return of the portfolio against its benchmark is given by $r-R$.

Asset allocation returns $r_S^{AA}$ are given by

Note that asset allocation return is only described at the sector level.

Stock selection returns $r_i^{SS}$are given by

Interaction returns $r_i^I$ are given by

Lastly, pricing returns $r_i^P$ are given by

The sum of the four sources of return over all securities and sectors equals the overall active return of the portfolio against its benchmark:

Brinson-Hood-Beebower attribution

The Brinson-Hood-Beebower (BHB) model is identical to the Brinson-Fachler model, except that the asset allocation term becomes

From the user's perspective, the only difference is that the sector-level asset allocation returns are different. However, the overall return due to asset allocation decisions remains the same.

• Equity attribution always requires a benchmark. Unlike fixed income attribution, where it makes sense to decompose a portfolio's return by source of risk, equity attribution is always measured by comparing a portfolio against a benchmark.
• The presence of an interaction term is a shortcoming in the Brinson model. It is often combined with the stock selection return to give $r_i^{SS} = W_i \times (r_S - R_S)$.
• In the Brinson model, asset allocation is only shown at the sector level, while all other sources of return are shown at the security level.
• In the case that $r_i=R_i$, price return for security $i$ is zero.
• Brinson-Fachler and Brinson-Hood-Beebower models are collectively referred to as 'Brinson'.

Normalised stock selection

Using the same notation as above, the normalised stock selection approach calculates stock selection as follows:

In this model, returns due to asset allocation and price are the same as for the Brinson approach. Interaction return does not arise using this model.

Bottom-up attribution

Using bottom-up attribution, asset allocation return is given by

and stock selection return by

• Bottom-up attribution is the only model in which it makes sense to show asset allocation returns at the security level.
• Stock selection in the bottom-up model is the same as price return in the previous two models.
• Interaction return and price return does not arise in this model.

Off-benchmark holdings

A security that lies in the portfolio but not in the benchmark is referred to as an 'off-benchmark' holding. FIA allows the return contribution from such holdings to be directed to asset allocation, stock selection, or both. Click here for information on how to do this.

Worked examples and documentation

References

'Mastering Attribution in Finance', chapter 3 ('Equity Attribution'), Andrew Colin, FT Press, 2015