Brinson–Fachler Attribution (A/S/I)¶

1. Calculation Name¶

Brinson–Fachler Allocation, Selection, Interaction Attribution

2. Description and Mathematical Formula¶

Using Brinson–Fachler arithmetic attribution, performance is decomposed into:

Allocation effect: \( A_i = (w^P_i - w^B_i) \times (r^B_i - R^B) \)
Selection effect: \( S_i = w^B_i \times (r^P_i - r^B_i) \)
Interaction effect: \( I_i = (w^P_i - w^B_i) \times (r^P_i - r^B_i) \)

Where:

\( w^P_i, r^P_i \) are portfolio weights and returns
\( w^B_i, r^B_i \) are benchmark weights and returns
\( R^B = \sum_i w^B_i r^B_i \) is the total benchmark return
The active return \( R^P - R^B = \sum_i (A_i + S_i + I_i) \)

3. Input Sample Data¶

Segment	\( w^P_i \)	\( r^P_i \)	\( w^B_i \)	\( r^B_i \)
Government	35%	2.10%	40%	1.80%
Credit	30%	4.50%	25%	3.80%
Mortgages	15%	3.20%	20%	3.00%
High Yield	10%	6.50%	5%	5.00%
Cash	10%	0.50%	10%	0.40%

Benchmark return \( R^B = 2.56\% \).

4. Mathematical Solution¶

Allocation effects (bps):
Government \(= (0.35 - 0.40) \times (1.80\% - 2.56\%) = +3.8\)
Credit \(= (0.30 - 0.25) \times (3.80\% - 2.56\%) = +6.2\)
Mortgages \(= (0.15 - 0.20) \times (3.00\% - 2.56\%) = -2.2\)
High Yield \(= (0.10 - 0.05) \times (5.00\% - 2.56\%) = +12.2\)
Cash \(≈ 0.0\)
Selection effects (bps):
Government \(= 0.40 \times (2.10\% - 1.80\%) = +12.0\)
Credit \(= 0.25 \times (4.50\% - 3.80\%) = +17.5\)
Mortgages \(= 0.20 \times (3.20\% - 3.00\%) = +4.0\)
High Yield \(= 0.05 \times (6.50\% - 5.00\%) = +7.5\)
Cash \(= 0.10 \times (0.50\% - 0.40\%) = +1.0\)
Interaction effects (bps):
Government \(= (0.35 - 0.40) \times (2.10\% - 1.80\%) = -1.5\)
Credit \(= (0.30 - 0.25) \times (4.50\% - 3.80\%) = +3.5\)
Mortgages \(= (0.15 - 0.20) \times (3.20\% - 3.00\%) = -1.0\)
High Yield \(= (0.10 - 0.05) \times (6.50\% - 5.00\%) = +7.5\)
Cash \(≈ 0.0\)
Totals:
Allocation = 20.0 bps
Selection = 42.0 bps
Interaction = 8.5 bps
Active return \(= 0.705\%\) (70.5 bps)

5. Sample Python and R Code¶

Python

import pandas as pd

data = pd.DataFrame(
    {
        "segment": ["Government", "Credit", "Mortgages", "High Yield", "Cash"],
        "w_p": [0.35, 0.30, 0.15, 0.10, 0.10],
        "r_p": [0.021, 0.045, 0.032, 0.065, 0.005],
        "w_b": [0.40, 0.25, 0.20, 0.05, 0.10],
        "r_b": [0.018, 0.038, 0.030, 0.050, 0.004],
    }
)

R_b = (data["w_b"] * data["r_b"]).sum()
data["allocation"] = (data["w_p"] - data["w_b"]) * (data["r_b"] - R_b)
data["selection"] = data["w_b"] * (data["r_p"] - data["r_b"])
data["interaction"] = (data["w_p"] - data["w_b"]) * (data["r_p"] - data["r_b"])

totals = data[["allocation", "selection", "interaction"]].sum()
active = totals.sum()
print(data)
print(totals, active)

R

library(dplyr)

data <- tibble::tibble(
  segment = c("Government", "Credit", "Mortgages", "High Yield", "Cash"),
  w_p = c(0.35, 0.30, 0.15, 0.10, 0.10),
  r_p = c(0.021, 0.045, 0.032, 0.065, 0.005),
  w_b = c(0.40, 0.25, 0.20, 0.05, 0.10),
  r_b = c(0.018, 0.038, 0.030, 0.050, 0.004)
)

R_b <- sum(data$w_b * data$r_b)

data <- data %>%
  mutate(
    allocation = (w_p - w_b) * (r_b - R_b),
    selection = w_b * (r_p - r_b),
    interaction = (w_p - w_b) * (r_p - r_b)
  )

totals <- summarise(data,
  allocation = sum(allocation),
  selection = sum(selection),
  interaction = sum(interaction)
)
active <- sum(totals)
data
totals
active

6. Output Table¶

Segment	Allocation (bps)	Selection (bps)	Interaction (bps)
Government	+3.8	+12.0	-1.5
Credit	+6.2	+17.5	+3.5
Mortgages	-2.2	+4.0	-1.0
High Yield	+12.2	+7.5	+7.5
Cash	+0.0	+1.0	+0.0
Total	+20.0	+42.0	+8.5

7. Conclusion¶

Brinson–Fachler attribution highlights whether excess performance comes from allocation tilts, security selection, or their interaction. This template aligns with FinFacts’ reporting views and is a reliable reference for QA, demos, and documentation.