BF Link (Per-Group Carino)¶

1. Calculation Name¶

Brinson–Fachler Group Linking (Carino Method)

2. Description and Mathematical Formula¶

The BF Link plugin scales period-by-period allocation/selection/interaction (A/S/I) effects into a horizon total while preserving attribution across groups.

For each period \( t \):

Compute the raw active return \( a_t = r^P_t - r^B_t \)
Sum raw group contributions \( \sum_{g} E_{g,t}^{\text{raw}} = a_t \) after normalization
Carino scaling factor:

\[ C_t = \frac{1 + r^P_t}{1 + r^B_t} \prod_{k = t+1}^{T} \frac{1 + r^B_k}{1 + r^P_k} \]

Linked group effect:

\[ E^{\text{linked}}_{g} = \sum_{t=1}^{T} E_{g,t}^{\text{raw}} \times C_t \]

3. Input Sample Data¶

Month	Group	\( r^P_t \)	\( r^B_t \)	Allocation (raw)	Selection (raw)	Interaction (raw)
Jan	Rates	2.50%	2.20%	0.06%	0.10%	0.02%
Jan	Spread	2.50%	2.20%	0.04%	0.03%	0.05%
Feb	Rates	1.80%	1.50%	0.07%	0.11%	0.03%
Feb	Spread	1.80%	1.50%	0.05%	0.04%	0.00%
Mar	Rates	-0.50%	-0.80%	0.05%	0.12%	0.04%
Mar	Spread	-0.50%	-0.80%	0.03%	0.04%	0.02%

Each month’s raw contributions sum to the month’s active return (0.30%).

4. Mathematical Solution¶

Carino factors using the monthly returns:
\( C_{\text{Jan}} = 0.9969648 \), \( C_{\text{Feb}} = 0.9999317 \), \( C_{\text{Mar}} = 1.0030242 \)
Normalize raw contributions so they add to monthly active return.
Apply each Carino factor to the normalized group effects.
Sum across months to obtain linked group totals.

Linked results (basis points):

Rates: Allocation 18.00, Selection 33.01, Interaction 9.01
Spread: Allocation 11.99, Selection 11.00, Interaction 6.99
Horizon active (all groups) ≈ 90.00 bps

5. Sample Python and R Code¶

Python

import pandas as pd
import numpy as np

data = pd.DataFrame(
    {
        "month": ["Jan", "Jan", "Feb", "Feb", "Mar", "Mar"],
        "group": ["Rates", "Spread", "Rates", "Spread", "Rates", "Spread"],
        "rp": [0.025, 0.025, 0.018, 0.018, -0.005, -0.005],
        "rb": [0.022, 0.022, 0.015, 0.015, -0.008, -0.008],
        "alloc": [0.0006, 0.0004, 0.0007, 0.0005, 0.0005, 0.0003],
        "select": [0.0010, 0.0003, 0.0011, 0.0004, 0.0012, 0.0004],
        "inter": [0.0002, 0.0005, 0.0003, 0.0000, 0.0004, 0.0002],
    }
)

factors = []
months = data["month"].unique()
for month in months:
    row = data.loc[data["month"] == month].iloc[0]
    rp = row["rp"]
    rb = row["rb"]
    tail = 1.0
    later = data[data["month"].isin(months[list(months).index(month)+1:])]
    for _, later_row in later.drop_duplicates("month").iterrows():
        tail *= (1 + later_row["rb"]) / (1 + later_row["rp"])
    factor = ((1 + rp) / (1 + rb)) * tail
    factors.append((month, factor))
factor_map = dict(factors)

def normalize(group):
    active = group["rp"].iloc[0] - group["rb"].iloc[0]
    total_raw = (group["alloc"] + group["select"] + group["inter"]).sum()
    scale = active / total_raw if total_raw else 0
    return group.assign(
        alloc_norm=group["alloc"] * scale,
        select_norm=group["select"] * scale,
        inter_norm=group["inter"] * scale,
    )

normalized = data.groupby("month", group_keys=False).apply(normalize)
normalized["factor"] = normalized["month"].map(factor_map)

normalized["alloc_linked"] = normalized["alloc_norm"] * normalized["factor"]
normalized["select_linked"] = normalized["select_norm"] * normalized["factor"]
normalized["inter_linked"] = normalized["inter_norm"] * normalized["factor"]

totals = (
    normalized.groupby("group")[["alloc_linked", "select_linked", "inter_linked"]]
    .sum()
    .mul(10000)  # convert to bps
)
print(totals)

R

library(dplyr)

data <- tibble::tibble(
  month = c("Jan", "Jan", "Feb", "Feb", "Mar", "Mar"),
  group = c("Rates", "Spread", "Rates", "Spread", "Rates", "Spread"),
  rp = c(0.025, 0.025, 0.018, 0.018, -0.005, -0.005),
  rb = c(0.022, 0.022, 0.015, 0.015, -0.008, -0.008),
  alloc = c(0.0006, 0.0004, 0.0007, 0.0005, 0.0005, 0.0003),
  select = c(0.0010, 0.0003, 0.0011, 0.0004, 0.0012, 0.0004),
  inter = c(0.0002, 0.0005, 0.0003, 0.0000, 0.0004, 0.0002)
)

months <- unique(data$month)
factors <- purrr::map_dbl(seq_along(months), function(i) {
  rp <- data %>% filter(month == months[i]) %>% pull(rp) %>% first()
  rb <- data %>% filter(month == months[i]) %>% pull(rb) %>% first()
  tail_ratio <- prod((1 + (data %>% filter(month %in% months[(i+1):length(months)]) %>% distinct(month, rb, rp) %>% pull(rb))) /
                     (1 + (data %>% filter(month %in% months[(i+1):length(months)]) %>% distinct(month, rb, rp) %>% pull(rp))))
  ((1 + rp) / (1 + rb)) * ifelse(is.finite(tail_ratio), tail_ratio, 1)
})
factor_map <- setNames(factors, months)

normalized <- data %>%
  group_by(month) %>%
  mutate(
    active = first(rp) - first(rb),
    total_raw = sum(alloc + select + inter),
    scale = ifelse(total_raw == 0, 0, active / total_raw),
    alloc_norm = alloc * scale,
    select_norm = select * scale,
    inter_norm = inter * scale,
    factor = factor_map[month]
  ) %>%
  ungroup() %>%
  mutate(
    alloc_linked = alloc_norm * factor,
    select_linked = select_norm * factor,
    inter_linked = inter_norm * factor
  )

totals <- normalized %>%
  group_by(group) %>%
  summarise(
    alloc_bps = sum(alloc_linked) * 10000,
    select_bps = sum(select_linked) * 10000,
    inter_bps = sum(inter_linked) * 10000,
    .groups = "drop"
  )
totals

6. Output Table¶

Group	Allocation (bps)	Selection (bps)	Interaction (bps)	Total (bps)
Rates	18.00	33.01	9.01	60.01
Spread	11.99	11.00	6.99	29.99
Total	29.99	44.01	16.00	90.00

7. Conclusion¶

This template demonstrates how FinFacts scales A/S/I contributions across groups using Carino linking. It is ideal for validating horizon totals against desktop exports and for onboarding analysts to the group-aware attribution workflow.