# Constrained Optimization for performance attribution

I am trying to perform constrained opmitization for portfolio performance attribution analysis. Specifically, I am trying to determine the impact of sectors performance on the S&P 500 index.

Min y-(b1x1+b2x2+...+bnxn+bn+1xn+1+....bpxp)

subject to b1+b2+...+bp = 1,

0 <= bi <= 1 for i=1,2,...,p

yis the time-series return of S&P500 and xi is the time-series return of a sector for all i = 1, 2, ..., p.

The idea of the betas is that the higher the beta, the greater the impact of the variable (sector) is on the portfolio.

The problem with this optimization is that, sometimes, the sector and market are negatively correlated so beta is negative. But the constraint forces it to be between 0 and 1, so beta becomes 0, which suggests that the sector had no impact on the S&P 500 index (even though it DID have an impact, negatively).

What's the better way of solving this?

Thank you very much in advance!

• What is $n$? Are you minimizing $\sum_i \epsilon_i^2$ where $\epsilon_i = y - b_1x_1 \ldots$? – Matthew Gunn Jul 11 '18 at 13:57
• @MatthewGunn sorry there was a typo. n is just some index. p is the total number of sectors. And yes, I am trying to minimize the sum of errors squared. – Jun Jang Jul 11 '18 at 14:03
• Are you using daily data? Monthly data? How long a sample? – Matthew Gunn Jul 11 '18 at 14:50
• @MatthewGunn I have daily data, which I convert to weekly. The data sample is from 1/2/2003 to 6/10/2018. Pretty long. – Jun Jang Jul 11 '18 at 14:51