Performance attribution

A PMS beat its benchmark by 2.84%. Good — but why? Did the manager hold more of the asset classes that did well? Pick better securities within each asset class? Or both at once?

Performance attribution answers that. It is, in the workbook's words, the second phase of performance evaluation: first measure the return, then dissect the difference from the benchmark into its sources, to establish whether it came from the manager's skill or from random factors (luck).

The framework the workbook calls dominant is the one propounded by Fama.

The workbook's decomposition (section 20.6.3) uses four inputs for each segment i: the fund's weight Wai and return Rai, and the benchmark's weight Wpi and return Rpi.

• Allocation effect — excess earned only because weights differ: Σ Rpi × (Wai − Wpi).
• Selection effect — excess earned only because returns within segments differ, at benchmark weights: Σ Wpi × (Rai − Rpi).
• Interaction effect — the residual from doing both: Σ (Wai − Wpi) × (Rai − Rpi).

The three add up exactly to the fund's excess return, Σ (Wai·Rai − Wpi·Rpi).

The workbook's example gives an excess return of 2.84% = allocation 1.25% + selection 1.19% + interaction 0.40%.

Excess return  = Σ (Wai × Rai) − Σ (Wpi × Rpi)

Allocation     = Σ Rpi × (Wai − Wpi)
Selection      = Σ Wpi × (Rai − Rpi)
Interaction    = Σ (Wai − Wpi) × (Rai − Rpi)

Allocation + Selection + Interaction = Excess return

The workbook's example, applied to an illustrative ₹10 crore balanced PMS portfolio.

Fund return      = 0.75×8 + 0.15×2 + 0.10×0.5 = 6.35%
Benchmark return = 0.50×6 + 0.30×1.5 + 0.20×0.3 = 3.51%
Excess           = 2.84%

Allocation  = 6×0.25 + 1.5×(−0.15) + 0.3×(−0.10) = 1.245%
Selection   = 0.50×2 + 0.30×0.5 + 0.20×0.2      = 1.190%
Interaction = 0.25×2 + (−0.15)×0.5 + (−0.10)×0.2 = 0.405%

The manager earned roughly equal amounts from being overweight equity and from picking better securities. A client paying for stock selection now knows that less than half of the outperformance came from it.

Chapter 20 (Performance Measurement and Evaluation of Portfolio Managers), section 20.6, covers allocation, selection and the full decomposition with the three-asset example above. Chapter 18 mentions attribution analysis as one of the uses of factor models and, in a footnote, lists sector selection, stock selection and timing as the sources of excess return. Expect a computation of one effect, or of all three.

• The workbook gives two different selection formulas. Section 20.6.2 weights the return difference by the fund's weights (Wai); section 20.6.3 uses the benchmark's weights (Wpi) and adds a separate interaction term. The workbook explicitly advises not to compare the formulas across the two approaches. For a question asking for allocation, selection and interaction, use the 20.6.3 set.
• Section 20.6.1 also prints an allocation formula using (Rpi − Rp) — segment return relative to the total benchmark — before offering the Rpi × (Wai − Wpi) version. Use whichever the question's wording matches.
• Allocation uses benchmark returns; selection uses benchmark weights. Swapping them is the most common error.
• The three effects must sum to the excess return. If they do not, recheck the signs on underweight segments.
• Interaction can be negative — for example, overweighting a segment where your selection underperformed.