# Portfolio Performance Attribution Using Carino Smoothing

I'm trying to conduct portfolio performance attribution using Carino smoothing, but it seems that the active returns do not match and I don't know why. Here is the example I use:

$$\begin{array} {|r|r|r|r|r|r|r|r|r|r|} \hline portfolio &R_{t1} &R_{t2}&R_{t3}&R_{t4}&w_{t1}&w_{t2}&w_{t3}&w_{t4}\\ \hline sector1 &-0.06& 0.04& 0.04& 0.04 &0.30& 0.40& 0.20 &0.20\\ \hline sector2 &0.02 & -0.12 &-0.02 &0.06 & 0.10 & 0.40 &0.10 & 0.20 \\ \hline sector3 &-0.12 &0.04 & 0.21 & 0.06 & 0.60 &0.20 & 0.70 & 0.60 \\ \hline \end{array}$$

$$\begin{array} {|r|r|r|r|r|r|r|r|r|r|} \hline benchmark &R_{t1} &R_{t2}&R_{t3}&R_{t4}&w_{t1}&w_{t2}&w_{t3}&w_{t4}\\ \hline sector1 &0.00 & 0.03 & -0.06 &0.08 &0.10 & 0.10 &0.30 &0.40 \\ \hline sector2 &0.04 & 0.00 & -0.04& 0.06 & 0.20 & 0.40 & 0.20 & 0.20 \\ \hline sector3 &0.14 &0.00 & -0.10& 0.00 & 0.70 &0.50 &0.50 & 0.40 \\ \hline \end{array}$$

So I calculate the portfolio return 15.06% and benchmark return 10.84%, and thus the active return is 4.21%. Then, by using Carino smoothing, I get those coefficients: $$k = 0.8854, k_1 = 0.9996, k_2 = 1.0054, k_3 = 1.0004, k_4 = 0.8696$$. I also calculate the asset allocation, security selection and interaction term for single period and use those coefficients to adjust them. For example: $$Asset Allocation_{t, adjusted} = Asset Allocation_t * k_t / k$$

After the adjustment, I sum up all those effects for 4 periods and find that the portfolio return minus the benchmark return becomes nearly 2% (I'm quite confused about this). I would expect the active return should be the same as the previous one, but it's far less than 4.21%. I guess there are some problems with how to use adjusted returns. Could someone explain how to use Carino attribution when I want to calculate the asset allocation, security selection, interaction for multiperiod for this example?

• Excuse me but, when I compute the benchmark returns I get [0.106 0.003 -0.076 0.044] which add up to 0.077 and compound to 0.07011. It does not match your 10.84 pct. Did I make a stupid error or is there a typo somewhere or something else. – noob2 Sep 4 '20 at 9:27
• Ah, I guess that's where the problem lies! When I imported benchmark data into excel, I made a mistake that the columns did not match. Thank you so much! I thought there were some problems with my adjustment previously. hahaha that's sad. – March Sep 4 '20 at 9:40