Carino Linking

1. Calculation Name

Carino Multi-Period Active Return Linking

2. Description and Mathematical Formula

The desktop Linking (Carino) plugin expects a single effect series at a time (for example, run it once for Allocation, again for Selection, etc.). For each period \( t \) with portfolio return \( r^P_t \), benchmark return \( r^B_t \), and raw effect \( E_t \) (in decimal form):

1. Compute the Carino factor
   [ k^{\text{Carino}}_t = \frac{\ln(1 + r^P_t) - \ln(1 + r^B_t)}{rP_t - r^B_t} ]
2. Scale the raw effect: \( \tilde{E}_t = E_t \times k^{\text{Carino}}_t \)
3. Find the cumulative active return
   \( R^{\ast} = \prod_{t}(1 + r^P_t) / \prod_{t}(1 + r^B_t) - 1 \)
4. Apply a horizon scaling factor
   \( \lambda = R^{\ast} / \sum_t \tilde{E}_t \)
5. Linked effect: \( E^{\text{linked}}_t = \tilde{E}_t \times \lambda \)

Summing \( E^{\text{linked}}_t \) matches the geometric excess return (subject to rounding).

3. Input Sample Data

Month	Portfolio Return	Benchmark Return	Effect (bps)
Jan	2.50%	2.20%	30
Feb	1.80%	1.50%	30
Mar	-0.50%	-0.80%	30

Effect values are shown in basis points (30 bps = 0.003 in decimal form). Run separate exports if you need Allocation, Selection, and Interaction contributions.

4. Mathematical Solution

1. Carino factors:
   \( k_{\text{Jan}} = 0.977040 \)
   \( k_{\text{Feb}} = 0.983769 \)
   \( k_{\text{Mar}} = 1.006543 \)
2. Scaled (log) effects:
   Jan \( = 30 \times 0.977040 = 29.31 \) bps
   Feb \( = 30 \times 0.983769 = 29.51 \) bps
   Mar \( = 30 \times 1.006543 = 30.20 \) bps
3. Cumulative excess return \( R^{\ast} = 8.9418\% \) (894.18 bps)
   Scaling factor \( \lambda = 1.004464 \)
4. Linked effects:
   Jan \( = 29.31 \times \lambda = 29.44 \) bps
   Feb \( = 29.51 \times \lambda = 29.64 \) bps
   Mar \( = 30.20 \times \lambda = 30.33 \) bps
   Total linked active \( = 89.42 \) bps (matches \( R^{\ast} \) to rounding).

5. Sample Python and R Code

Python

from math import log

portfolio = [0.025, 0.018, -0.005]
benchmark = [0.022, 0.015, -0.008]
effect = [0.003, 0.003, 0.003]  # raw effect in decimal form

factors = []
scaled = []
for rp, rb, eff in zip(portfolio, benchmark, effect):
    active = rp - rb
    numerator = log(1 + rp) - log(1 + rb)
    k = numerator / active if abs(active) > 1e-12 else 1.0
    factors.append(k)
    scaled.append(eff * k)

cum_port = 1.0
cum_bench = 1.0
for rp, rb in zip(portfolio, benchmark):
    cum_port *= 1 + rp
    cum_bench *= 1 + rb
cum_active = cum_port / cum_bench - 1
link_scale = cum_active / sum(scaled)
linked = [s * link_scale for s in scaled]

print("k:", [round(k, 6) for k in factors])
print("scaled_bps:", [round(s * 10000, 2) for s in scaled])
print("linked_bps:", [round(l * 10000, 2) for l in linked])
print("total_active_bps:", round(sum(linked) * 10000, 2))

R

portfolio <- c(0.025, 0.018, -0.005)
benchmark <- c(0.022, 0.015, -0.008)
effect <- c(0.003, 0.003, 0.003)

k <- mapply(function(rp, rb) {
  active <- rp - rb
  num <- log(1 + rp) - log(1 + rb)
  if (abs(active) < 1e-12) 1 else num / active
}, portfolio, benchmark)

scaled <- effect * k
cum_port <- prod(1 + portfolio)
cum_bench <- prod(1 + benchmark)
cum_active <- cum_port / cum_bench - 1
link_scale <- cum_active / sum(scaled)
linked <- scaled * link_scale

round(k, 6)
round(scaled * 10000, 2)
round(linked * 10000, 2)
round(sum(linked) * 10000, 2)

6. Output Table

Month	\( k^{\text{Carino}}_t \)	Scaled (log bps)	Linked (bps)
Jan	0.9770	29.31	29.44
Feb	0.9838	29.51	29.64
Mar	1.0065	30.20	30.33
Total	—	89.02	89.42

Totals differ slightly due to rounding when converting between decimal values and basis points.

7. Conclusion

Run the Carino plugin once per effect stream to obtain linked contributions that reconcile to the geometric excess return. The steps above mirror the desktop app’s calculations, making it easy to validate outputs or generate documentation-ready tables.