Portfolio performance attribution is a crucial tool for assessing how investment decisions influence returns relative to a benchmark. Among the widely used methods are arithmetic attribution and geometric attribution, which analyze and explain excess returns—key to evaluating the effectiveness of fund management.
This article explores the fundamentals of arithmetic and geometric attribution, with a focus on their applications, advantages, and use cases, alongside an introduction to the Brinson model, a standard framework for equity return attribution.
Arithmetic vs. Geometric Attribution
Arithmetic Attribution
- Definition: Arithmetic attribution calculates excess return as the difference between the portfolio return (R) and the benchmark return (B). It provides a straightforward, intuitive way to evaluate how well a portfolio performed relative to its benchmark.
- Formula:
- Use Case: Best suited for non-technical audiences, such as retail investors or clients, due to its simplicity. It is commonly used in marketing reports and presentations.
- Limitations: Arithmetic attribution requires smoothing adjustments to account for effects over multiple time periods, which can complicate long-term evaluations.
Geometric Attribution
- Definition: Geometric attribution improves upon the arithmetic approach by normalizing excess return over the benchmark wealth ratio, making it more accurate for compounding effects over multiple periods.
- Formula:
- Use Case: Preferred by industry professionals and institutional investors for its precision and ability to reflect compounding effects over time. Unlike arithmetic attribution, geometric methods do not require smoothing adjustments for multi-period evaluations.
- Advantages: Geometric attribution is ideal for detailed performance analysis, ensuring accuracy in long-term evaluations and consistency across periods.
The Brinson Model: An Equity Attribution Framework
The Brinson model, introduced by Brinson and Fachler (1985) and expanded by Brinson, Hood, and Beebower (1986), is a cornerstone of return attribution analysis. It dissects portfolio performance into three effects:
- Allocation Effect: Measures the impact of sector weight deviations from the benchmark.
- Selection Effect: Assesses the impact of stock selection within each sector.
- Interaction Effect: Captures the residual effects of combining allocation and selection decisions.
Key Equations in the Brinson Model
Practical Example Using the Brinson Model
Portfolio and Benchmark Data
| Sector | Portfolio Weight (wiw_iwi) | Benchmark Weight (WiW_iWi) | Portfolio Return (RiR_iRi) | Benchmark Return (BiB_iBi) |
|---|---|---|---|---|
| Financial | 0.20 | 0.50 | 6% | 4% |
| Technology | 0.80 | 0.50 | –12% | 2% |
1. Allocation Effect
Quantifies the impact of over/underweighting sectors relative to the benchmark.
2. Selection Effect
Measures the manager’s ability to choose securities within sectors that outperform the benchmark.
3. Interaction Effect
Captures the combined effects of allocation and selection decisions.
Summary of Attribution Effects
| Effect | Contribution |
|---|---|
| Allocation | -0.6% |
| Selection | -6.0% |
| Interaction | -4.8% |
| Total Excess | -11.4% |
Insights from the Example
- The portfolio manager’s decision to underweight financials detracted value, despite the sector outperforming the benchmark.
- Overweighting technology further reduced returns, as the sector significantly underperformed.
- Combined allocation, selection, and interaction effects resulted in a total excess return of -11.4%.
Conclusion
The Brinson model, coupled with arithmetic and geometric attribution methods, provides a robust framework for dissecting portfolio performance. While arithmetic attribution offers simplicity for non-technical audiences, geometric attribution ensures precision over time. For institutional and professional investors, the Brinson model’s detailed decomposition of allocation, selection, and interaction effects delivers actionable insights into a portfolio manager’s strategy and effectiveness.
By leveraging these tools, stakeholders can better assess the value added—or detracted—by active management, paving the way for more informed investment decisions and client transparency.
