The Cariño Linking Algorithm

Summing monthly attribution effects gives a number that doesn't match the fund's actual compounded excess return, and that mismatch is more than an academic nuisance — client statements, GIPS reports, and manager review meetings all need a multi-period allocation and selection number that ties out exactly to the return the client actually experienced. David Cariño's 1999 paper solved this with a specific formula that's now the industry default, precisely because it makes the reconciliation exact rather than approximate.

The scaling factor

For each period $t$, define a linking coefficient:

$$k_t = \frac{\ln(1 + R_{p,t}) - \ln(1 + R_{b,t})}{R_{p,t} - R_{b,t}}$$

In words: $k_t$ compares how much the log of the portfolio's growth factor differs from the log of the benchmark's growth factor, relative to how much the raw returns differ. Logs turn compounding into addition, which is exactly the property needed to make period-by-period effects linkable. Each month's allocation, selection, and interaction effects get multiplied by $k_t$ before being summed, and a second, portfolio-level coefficient $K$ (built the same way from the total multi-period portfolio and benchmark returns) rescales the summed total back into ordinary percentage terms:

$$\text{Linked Effect} = \frac{1}{K} \sum_t k_t \times \text{Effect}_t$$

The reason this works is a property of logarithms: $\ln(1+R_p) - \ln(1+R_b)$ summed across periods equals $\ln(1+R_{p,total}) - \ln(1+R_{b,total})$ exactly, since logs of products become sums. Cariño's coefficients essentially convert each period's arithmetic active return into its log-equivalent contribution, sum those (which is now valid), then convert back.

Worked example

Two months. Month 1: portfolio return 3%, benchmark return 2%, allocation effect +0.5%. Month 2: portfolio return -1%, benchmark return -2%, allocation effect +0.3%.

1. Month 1 coefficient. $k_1 = \dfrac{\ln(1.03) - \ln(1.02)}{0.03 - 0.02} = \dfrac{0.02956 - 0.01980}{0.01} = \dfrac{0.00976}{0.01} = 0.976$.
2. Month 2 coefficient. $k_2 = \dfrac{\ln(0.99) - \ln(0.98)}{-0.01 - (-0.02)} = \dfrac{-0.01005 - (-0.02020)}{0.01} = \dfrac{0.01015}{0.01} = 1.015$.
3. Total-period coefficient $K$. Using compounded portfolio return $1.03 \times 0.99 - 1 = 1.9\%$ and compounded benchmark return $1.02 \times 0.98 - 1 = -0.04\%$: $K = \dfrac{\ln(1.019) - \ln(0.9996)}{0.019 - (-0.0004)} = \dfrac{0.01882 - (-0.0004)}{0.0194} \approx 0.991$.
4. Linked allocation total. $\dfrac{1}{0.991}\big(0.976 \times 0.5\% + 1.015 \times 0.3\%\big) = \dfrac{0.488\% + 0.3045\%}{0.991} \approx 0.80\%$.

That 0.80% is the correctly-linked two-month allocation effect — it will sum with the equivalently-linked selection and interaction totals to exactly the fund's true two-month active return, unlike a naive sum of +0.5% and +0.3%.

What this means in practice

Performance systems compute Cariño coefficients automatically; an analyst rarely does this by hand, but understanding the mechanism explains why linked attribution numbers for the same period can look slightly different from the raw monthly effects you'd expect from simple addition. GIPS-compliant firms must disclose their linking methodology, since Cariño is common but not universal — some shops use alternatives like the Menchero or Frongello methods, which distribute the residual differently.